near-infrared optical absorption loss characteristics of

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NEAR-INFRARED OPTICAL ABSORPTION LOSS CHARACTERISTICS OF NONLINEAR OPTICAL POLYMERS FOR ELECTRO-OPTIC WAVEGUIDE APPLICATIONS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Richard Ronald Barto, Jr. August 2004

© Copyright by Richard Ronald Barto, Jr. 2004 All Rights Reserved

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

_________________________________ Curtis W. Frank, Principal Advisor

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

_________________________________ William D. Nix

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

_________________________________ Reinhold H. Dauskardt

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

_________________________________ Michael D. McGehee

Approved for the University Committee on Graduate Studies.

Abstract

Photonics technology has enabled remarkable growth in the rate and efficiency of data transfer over long distances, and is continually advancing to keep pace with data traffic for land, airborne, and space-based communication networks.

For space

applications, photonic links based on electro-optic (E-O) modulators offer greater architectural simplicity compared with conventional RF communications, with greatly reduced on-satellite transmission loss, major size and weight savings, reduced power consumption, improved phase performance, and immunity to electromagnetic interference and crosstalk. The most prevalent E-O modulators are based on lithium niobate single crystals, with bandwidths of 10 GHz in commercial-off-the-shelf products. The cost and bandwidth of this technology have impeded wide implementation of photonics communications in space at present.

Significant advances in E-O device

performance have been made in the past several years using organic polymer-based nonlinear optical (NLO) materials as the electroactive component. In these materials, a nonlinear optical (NLO) chromophore, comprised of an electron donor and acceptor moiety bridged by a long p-conjugated segment, and exhibiting a high microscopic firstorder nonlinear susceptibility b and strong induced dipole m, is incorporated into the polymer. Upon heavy doping followed by alignment of the chromophore under an applied field, films of these materials display large linear electro-optic coefficients, r33. Great progress has been recently reported by Dalton and Jen in synthesizing organic NLO dyes of the merocyanine class with tricyano acceptors, demonstrating electro-optic coefficients greater than 100 pm/V at wavelengths of 1300 nm in thin films of polycarbonate, compared to a maximum reported value of 35 pm/V for lithium niobate. Devices based on these materials are expected to possess significantly higher sensitivity and bandwidth than lithium niobate-based devices, making them attractive for potential application for satellite RF devices extending beyond 20 GHz. Moreover, compared to lithium niobate, polymer-based devices are more well suited for hybridization into low cost thin film manufacturing technology, possess small ionizing

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radiation cross-sections, and have development potential for devices with RF gain and low noise figure, low power operation, and multiple functions such as frequency conversion and frequency generation.

While E-O device sensitivity and bandwidth

performance enhancements are expected to ensue from the recent improvements in chromophore activity, the RF gain and noise figure, critical for space applications, also depend on the device total fiber-to-fiber optical insertion loss at standard near-infrared telecommunication frequencies (1310 nm and 1550 nm). At present, this property is higher and less predictable for NLO polymers than for lithium niobate devices. A fundamental component of the device optical insertion loss is that due to absorption of the propagating waveguide mode by the active material.

The physical mechanisms

controlling absorption loss at telecommunication wavelengths are not well understood, and there have been very few systematic studies of dye-polymer structure - property behavior in this regard. In this study, the optical absorption behavior of organic nonlinear optical dyes combined with polymers for potential electro-optic waveguide applications are studied for structure-property relationships between absorption spectral features and polymer and dye molecular structure.

An esoteric measurement technique, called photothermal

deflection spectroscopy, in conjunction with conventional UV-Vis transmission spectroscopy, is used for spectral characterization. Of central interest are the roles of polymer and dye structure and dielectric properties in the shape of the absorption spectrum in the near infrared region, at optical frequencies important to telecommunication applications. A highly active thiophene-bridged dye, LMCO-46M, closely related to the well characterized dye FTC-2 from Larry Dalton and coworkers, is investigated as a homologous series of di-alkyl side-chain spacer group lengths as a guest-host system with polycarbonate, as well as a single di-alkyl spacer group length in a series of polycarbonates and a series of rigid amorphous aromatic polymers. The roles of spacer groups and polymer structure on dye-dye and dye-polymer interactions, and associated effects on inhomogeneous broadening and solvatochromism of the electronic absorption peak are assessed for these materials.

Strong relationships between the

features of the dye main electronic absorption peak and near-IR loss are established. Predictive models of near-IR absorption loss based dye geometry and dipole properties

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and polymer dielectric properties are developed. The structure-property relationships will be used to aid in the design of organic nonlinear optical materials with minimized optical loss. Finally, a study of azo dyes incorporated into various linear aliphatic polymers is presented, showing effects of polymer backbone chemistry, azo dye substitution, and covalent incorporation of the dye into the polymer on complex loss vs. concentration dependence and spectral features, which in turn are used to deduce physical mechanisms responsible for loss for each of the disparate polymer backbone structures.

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Acknowledgements

First and foremost, I want to express my deepest appreciation and gratitude for the unending love and support of my wife Kathryn McKellar and daughter Jacqueline during the vast span of time spent away while I earned my PhD. Kathryn’s and Jackie’s encouragement and patience truly made it possible for me to complete my dissertation, and for this, I am forever indebted to them. My advisor, Prof. Curtis Frank in the Department of Chemical Engineering provided enduring support, encouragement, expert guidance, and constructive criticism over the course of my PhD program. His discernment, circumspection, and foresight were invaluable to the conception and scientific significance of the study, and his inscrutable standards and candor significantly improved the quality of the work and the richness of the experience. I owe my parents, Letty and Dick Barto a debt of gratitude for imbuing the value of education and for creating a loving and supportive environment for learning and achievement throughout my childhood. The encouragement and moral support of my parents, as well as my sister Valerie and brother Andrew, were of immeasurable value throughout the entirety of my PhD program. I would like to acknowledge and thank the professors and staff of the Department of Materials Science and Engineering and the Stanford University community, whom have helped and encouraged me as I worked towards my PhD. I have made many friends during my long attendance at Stanford, and consider myself fortunate to have had the opportunity to associate with world-class scholars in an engaging and fertile intellectual environment. I want to thank Professor William Nix for persuading me to apply to the Materials Science and Engineering graduate program in the fist place, and for his many words of encouragement.

I’d like to thank Professor Bruce Clemens for his vote of

confidence and for inviting me to enter the PhD program in Materials Science and Engineering.

Professor David Barnett has given me tremendous advice and

encouragement over the years, both as an academic advisor and as a friend, which have been of profound value in my PhD pursuits.

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I received extraordinary support from my management and colleagues at Lockheed Martin over the course of my PhD program.

I want to express deep

appreciation to my manager, Dr. Susan Ermer for creating the opportunity for me to join her research team at the Advanced Technology Center (ATC) in Palo Alto that made it possible to do my dissertation as a continuation of my career at Lockheed Martin. Dr. Ermer has provided me with financial and technical resources, entrée to many key academic and industry collaborations, access to her extensive scientific network in the nonlinear optical materials community, and the facilities and flexibility to conduct academic research in an industrial setting, frequently providing a buffer between business pressures and research needs.

I’m forever grateful for her encouragement, stalwart

support, and extreme patience over the long time span of my dissertation. I wish to thank Dr. Willy Anderson, the co-principal investigator at the time I joined the research team at Lockheed Martin ATC, for the initial opportunity to join his research program, and for his aegis and sage advice on many key aspects of the dissertation study. My current supervisor, Dr. Rebecca Taylor has provided critical scientific review and strategic guidance for my thesis work and its relevance and integration with the polymer materials research and electro-optic device development efforts at Lockheed Martin.

Dr. Taylor has given me substantial guidance in

spectroscopic measurement principles and interpretations, has been a staunch advocate of my research, has secured funding for a large portion of my dissertation activities, and has helped me solve many vexing technical and logistical problems along the way. I would like to thank my colleagues at Lockheed Martin ATC, Dr. Peter Bedworth, Dr. Larry Dries, Dr. Wendell Eades, Gil Mendenilla, Angie Moss, Joseph Epstein, Dr. Anthony Cooper, and Dr. Steven Lovejoy for their significant contributions during my dissertation studies. Dr. Bedworth has taught me almost every aspect of the chemistry and nonlinear optical behavior of the materials of this study, provided critical scientific review of experimental approaches and results, provided synthesis of nonlinear optical dyes, and guidance on synthetic functionalization, purification, and recovery of dyes for this study. Dr. Dries provided expert guidance in optical test bed development and detection electronics, as well as overall consulting on measurement principles and error sources. Dr. Eades taught me nearly all aspects of clean room procedure and

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waveguide film processing, and provided extensive help and guidance in developing experimental techniques for depositing nonlinear optical polymer films of optical quality for spectral characterization. Mr. Mendenilla and Ms. Moss provided excellent support in providing substrate and wafer preparation processing in the clean room for the film samples. Ms. Moss also performed film preparation for two dye/polymer series. Mr. Epstein provided synthesis of several of the nonlinear optical dyes of this study. Dr. Cooper provided polymer molecular weight determinations by gel permeation chromatography.

Dr. Lovejoy provided measurements of polymer glass transition

temperatures by differential scanning calorimetry. My colleague, Dr. Dexter Girton deserves special recognition for his monumental contribution to the development of the research photothermal spectroscopy test bed for this study.

Dr. Girton worked tirelessly over the span of three years, in close

coordination with me during the optical test bed development, to design a working, reliable, LabVIEW instrument control and data acquisition program that allowed the test bed to be operated in an automated fashion. The control program design had to be completely reconfigured and re-optimized as the test bed evolved through radically different states, in three major phases of development.

Dr. Girton devised several

innovative instrument control processes in the course of developing the program, most significantly the implementation of a machine state table capturing the operation decision logic amassed over three years. Dr. Girton also provided significant technical guidance in troubleshooting several key aspects of the optical test bed, most notably the electronic and ambient noise characteristics. His pleasant demeanor and calm, rationale approach make him a joy to work with, and an indispensable asset to his peers. Appreciation and gratitude are extended to my former supervisor at Lockheed Martin, Dr. Scott Selover for his advocacy of my acceptance into the Lockheed Martin – Stanford Honors Co-op program, and for his active support, flexibility, and encouragement as my supervisor during my first several years in the program. I would also like to thank my former managers at Lockheed Martin who have supported and encouraged my participation in this program over the years: Arnold Hultquist (deceased), Dr. James Litton, Joyce Steakley, Jean Riley, Bernie Mulroy, and Claudia Kennon. A special thanks goes to Dr. James Ryder, Director of Science and Technology at the ATC,

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who encouraged and facilitated my relocation into the Materials and Structures department at the ATC to complete my dissertation studies. My most sincere gratitude goes to Lockheed Martin Space Systems Company for their generous and enduring support through financial coverage of my participation in the Stanford Honors – Co-op program, and in particular the ATC for funding of laboratory equipment, facilities, and materials, labor budget for a significant portion of my research, and flexible and accommodating work policies supporting academic pursuits. I want to specifically express my appreciation to the management of the ATC for fostering a collegial environment that actively supported my dissertation research. Sincere thanks and appreciation is expressed to Dr. Andrew Skumanich for his expert help and guidance in the initial design and construction of the photothermal deflection spectroscopy (PDS) test bed, as well as teaching me the fine points of the art of this measurement technique. I would also like to thank Dr. Carleton Seager from Sandia National Laboratories for his willingness to share with me his expertise in PDS, for providing PDS measurements of NLO polymer materials for this study, and for guidance and several extremely enlightening discussions on signal detection, noise mitigation, and data reduction. I’d like to thank Prof. Chris Moylan in the Department of Chemistry for critical scientific review and guidance in interpretation of spectral data and trends for this study. I wish to thank my collaborators, Prof. Alex Jen from the Department of Materials Science and Engineering at the University of Washington and his students Hong Ma, Jingdong Luo, and Hong-Zhi Tang, who provided azodye functionalization and polymer synthesis; Prof. Dennis Smith from the Department of Chemistry at Clemson University, and his students Suresh Iyer and Shengrong Chen, who provided polymer synthesis and dye-doped specimen preparation; Dr. Albert Ren, formerly from the Department of Chemistry at University of Southern California and Dr. Michael Lee of APIC Corporation, each of whom provided azodye polymer synthesis. This talented group of chemists provided expert materials synthesis and guidance on structure-property experimental approaches, and was a pleasure to work with. I would also like to thank Dr. James Cella and Chris Kapusta from General Electric Global Research for providing experimental polycarbonate polymers for theses studies.

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I would like to acknowledge and thank Prof. David A. B. Miller from the Department of Electrical Engineering for graciously serving as chair on my defense committee. Finally, I wish to thank each of the members of my reading committee, who have graciously agreed to review and critique this document before the final publication: Prof. William Nix, Prof. Reinhold Dauskardt, and Prof. Michael McGehee, all from the Department of Materials Science and Engineering.

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Table of Contents

Abstract iv Acknowledgements vii Table of Contents xii List of Tables xiv List of Figures xv Abbreviations xvi Chapter 1 Introduction 1 1.1 Technological relevance and motivation 1 1.2 Objectives and outline 12 1.3 References 14 Chapter 2 Background 22 2.0 Outline 22 2.1 General review of optical loss mechanisms in NLO polymers 23 2.2 Dalton theory and measurements of E-O attenuation 28 2.3 Bond-length alternation theory of nonlinear optical properties 30 2.4 Voight profile treatment of Vis-NIR absorption spectra 34 2.5 Overtone spectra and bonding contributions 35 2.6 Introduction to inhomogeneous broadening 38 2.7 Introduction to solvatochromism 40 2.8 Theories of inhomogeneous broadening and solvatochromism 45 2.8.1 Marcus formalism for polar contributions to electronic state free energies 45 2.8.2 Matyushov and Schmid treatment of inhomogeneous broadening and solvatochromism 47 2.8.3 Kador lattice irregularity theory of inhomogeneous broadening 51 2.8.4 Obato, Machida, and Horie treatment of inhomogeneous broadening and solvatochromism 53 2.8.5 Loring linearized solvation model of inhomogeneous broadening 55 2.8.6 Sevian and Skinner microscopic theory of inhomogeneous broadening 57 2.8.7 Gonzalez – Liebsch treatment of inhomogeneous broadening in cermet topology 59 2.9 Solvent polarity-induced band shift relationships 61 2.10 Theoretical and experimental treatments of correlated and anti-correlated broadening distributions 68 2.11 Quantum mechanical treatments of order and disorder effects on inhomogeneous broadening 74 2.12 Optical band shapes in different spectral ranges in amorphous inorganics 75

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2.13

2.14

2.15 2.16

2.17

2.12.1 High absorption region A 2.12.2 Exponential absorption region B 2.12.3 Weak absorption tail region C Studies of disorder and defect states in band tail regions of amorphous inorganics 2.13.1 Robertson theoretical treatment of defects and disorder in amorphous carbon Optical bandshapes in different spectral ranges in organic materials 2.14.1 Ulstrup bandshape formalism 2.14.1.1 Bandshape asymmetry in the spectral wings 2.14.1.2 Weak electronic-vibrational coupling limit 2.14.2 Experimental Bandshape Measurements for N-pyridinium Phenolates 2.14.3 Urbach tail behavior in organic systems Theory and measurements of disorder and π − localization in linear organic polymers Organic polymer and chromophore property prediction methods 2.16.1 Bicerano hydrogen-suppressed graph model QSPR methods for polymer properties 2.16.2 Seitz method of predicting polymer elastic modulus 2.16.3 ZINDO semi-empirical quantum mechanical model for chromophore molecular properties References

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78 81 88 90 104 108 108 114 115 116 117 120 126 126 131 137 143

List of Tables

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List of Figures

xv

Abbreviations

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Chapter 1

Introduction 1.1 Technological relevance and motivation Photonics technology harnesses the generation and propagation of light and its interactions with electrons to sense and encode electrical signals onto optical signals by either phase or amplitude modulation over long distances. This technology has had an explosive impact on the growth of the Internet, by dramatically increasing the rate and efficiency of long distance data transfer, and is poised to have an ever-increasing influence on the information society of the early 21st century. The rate of growth of photonics technology capacity is outpacing Moore’s Law, and is being driven by optical network applications1. Sales of optoelectronic equipment was projected in year 2000 to be $34 billion in the year 2006, due to continued growth of the Internet, deregulation of the telephone industry, and expanding use of video and teleconferencing1.

Optical

networks are expected to reach data transmission rates of terabits per seconds, which is equivalent to the total Internet capacity in the year 2000 per second. These advances are anticipated to enable achievement of such lofty goals as instant access to complete information libraries, three-dimensional multi-sensory displays, downlinking of massive amounts of imagery, metrometry, and spectroscopic data from space payloads, viewerspecific real-time high definition video, or sensors capable of determining identities, densities and distributions of drugs and pathogens in living tissue1. The projections for these enormous improvements in the data handling capacity of optical networks stems from the relationship between the bandwidth of a communications channel, given by the carrier frequency, and the data transfer rate, as defined by Shannon’s theorem2: 2 B = 2∆f Log 2 1 + ( AS AN )   

1

(1.1)

Chapter 1 where B is the data transmission rate in units of number of binary units per second (b/s),

∆f is the bandwidth of the signal, and AS and AN are the peak signal and r.m.s. noise of the receiver, respectively. The ratio AS AN defines the signal dynamic range, and B is referred to as the channel capacity. Since the carrier frequency in optical fibers at near infrared wavelengths is of the order 100 THz, optical communication networks have the potential for much greater capacity than present systems limited by microwave bandwidths on the order of 1 – 10 GHz3, 4. The advent of fiber optics with losses of 0.2 dB/km in 19795 allowed the replacement of traditional twisted wire pairs for long haul telephone networks. The first wavelength to be adopted by industry was 0.85 µm , almost exclusively in multimode optical fiber, which required signal repeaters every few kilometers due to signal degradation by modal dispersion. This fiber is still in use today for local-area networks operating at a few hundred Mb/s. Single mode optical fiber has become the most widely used transmission medium today due to its extremely low loss and lack of modal dispersion. Standard single-mode fiber has minimum chromatic dispersion at 1.3 µm and minimum loss at 1.55 µm , at which it is compatible with erbium-doped fiber amplifiers (EDFAs)6. The implementation of fibers operating at 1.3 µm allowed the

distance between repeaters to be extended to 40 km, limited almost entirely by the optical loss of the fiber, and operated at a few hundred Mb/s. The lower loss of 1.55 µm fiber allowed the repeater distance to be extended even further. The dot com boom of the late 1990’s and early 2000’s saw extensive development of devices aimed at 1.55 µm , such as single longitudinal mode (SLM) distributed feedback (DFB) lasers exhibiting a narrow spectral bandwidth.

This greatly reduced smearing of signal pulses by chromatic

dispersion at 1.55 µm , increasing transmission rates to above 1 Gb/s.

The

implementation of EDFAs in place of repeaters has led to deployment of high-capacity wavelength division multiplexing (WDM) systems operating with 8 – 32 wavelengths per fiber, at data rates of 2.5 Gb/s. Conventional time-division multiplexing (TDM) is driving data transmission rates in these systems to 10 Gb/s and beyond. The applications described above are focused primarily on optical systems to route digital signals for telephone and computer networking. However, applications that

1

Chapter 1

involve point-to-point distribution of analog signals have also taken advantage of the superb signal carrying properties of optical fiber, through the use of RF-photonic links. For applications such as RF antenna remoting, and cellular communications and cable television signal distribution networks, it is practical and economical to use RF as the carrier through space, but for on-board processing and re-routing to distribution hubs, transmission over optical fiber represents several key benefits. The benefits of RFphotonic links for aircraft and space communications have been recognized for many years, including the light weight and small size of the optical fiber, its immunity to electromagnetic interference and electromagnetic pulses, low crosstalk, low on-board transmission loss, reduced power consumption, and greatly improved phase stability over a wide range of temperatures4. The performance requirements for RF-photonic links for remoting RF signals in phased-array radar antenna systems and satellite RF systems are particularly strict. However, major performance improvements have been made for these systems, as well as for cellular and cable television systems, over the past ten years6. An RF-photonic link has the generic form of Figure 1.1. The modulation of an RF signal onto an optical carrier can be achieved by either directly modulating the current of a semiconductor laser or light emitting diode or by external modulation of a continuous wave (CW) laser. Direct modulation has the advantages of being simple and inexpensive since a single device serves as both the optical source and the RF/optical modulator. The major disadvantage of direct modulation is the presence of appreciable chirp in the modulated pulses, causing variation of the transmitted carrier frequency with time, leading to broadening of the transmitted spectrum. Chirp can be mitigated by raising the power of a zero bit such that the laser remains above its threshold, but this reduces the on/off extinction ratio, degrading system performance5. In addition, the coupling loss between the semiconductor laser chip and a single-mode fiber in direct modulation is a significant limitation to the device slope efficiency sm 7 (expressed in W/A, see below), which is the efficiency of conversion of RF modulation current to modulated optical power6. Further, maintaining a high degree of linearity is difficult with direct modulation, and high speed current modulation in semiconductor lasers yields wavelength modulation, which is deleterious to WDM systems2. The highest reported bandwidth for external modulation is nearly a factor of 2 that reported for direct 2

Chapter 1

modulation, and the highest reported bandwidth-slope efficiency product for external modulation is a factor of 14 that reported for direct modulation6. External modulation minimizes the undesirable effects seen in direct modulation, including chirp, coupling loss-limited slope efficiency, poor linearity, and wavelength modulation. A Mach-Zehnder modulator (MZM), a schematic of which is shown in Figure 1.2, is an example of a commonly used external electro-optic (E-O) modulator. In these devices, phase modulation is imposed on the optical carrier in one of two parallel waveguides by an incoming RF (signal) field, which changes the refractive index of the waveguide core material. Amplitude modulation is achieved through interference of the modulated light with that of the unmodulated, parallel waveguide. A phase change in the receiving arm of π radians yields complete destructive interference at the device output, corresponding to the device “half-wave” voltage, Vπ . A schematic representation of an MZM in operation is shown in Figure 1.3. The figures of merit for an RF-photonic link are gain, slope efficiency, noise figure, and intermodulation-free dynamic range6. The primary figure of merit for an MZM is the link gain g , RLOAD RM

g = sm2 rd2

(1.2)

where RM is the modulator electrode resistance, rd is the photodetector responsivity and sm , the modulator slope efficiency at the quadrature point of sinusoidal transfer function,

is given by sm =

πηFF PI RM 2Vπ

3

(1.3)

Chapter 1

Figure 1.1 Schematic representation of a generic intensity modulation direct-detection (IMDD) RF-photonic link (from reference 6)

a)

b)

Figure 1.2 Schematic representation of a Mach-Zehnder modulator (MZM) external electro-optic (E-O) modulator: a) top-view layout showing a separate electrode (rf drive and dc bias) located over each arm of the modulator; b) schematic showing principle of operation: with no applied drive voltage, refractive index of NLO polymer in channel waveguide is matched in both arms, yielding no phase change; with applied drive voltage, refractive index varies with applied voltage in one arm, leading to voltagedependent light phase and amplitude modulation; complete interference (“off” state) occurs at half-wave voltage Vπ .

4

Chapter 1

Figure 1.3 Schematic representation of MZM (in cross-section, poled in the standard manner) in operation.

where and η FF is the total modulator optical efficiency between the input- and outputfibers and PI is the total input laser power. The modulator optical efficiency is inversely proportional to the total modulator fiber-to-fiber optical insertion loss TFF . From Eqs. (1.2) and (1.3), it is seen that the link gain has quadratic dependence on the optical input power PI , and inverse quadratic dependence on the total insertion loss TFF and halfwave voltage Vπ .

Thus, to maximize device performance, it is necessary to

simultaneously minimize each of Vπ and TFF while maximizing the modulator power handling capacity PI . The link noise figure NF is given by6 N  NF ≡ 10 Log  out   kTg 

(1.4)

where N out is the total output spectral noise power density, k is Boltzmann’s constant, and T is the absolute temperature. The main components of N out are the photodetector relative intensity noise (RIN) and shot noise, and the thermal noise from the ohmic impedances of the RF source, modulation device, photodetector, and source-to-modulator and output load-to-detector interface circuits. For an external modulator operating with a solid-state CW laser, shot noise is usually the dominant component, given by6

5

Chapter 1 N shot = 2eI D RLOAD

(1.5)

where I D is the output-detected photocurrent. Thus, from Eq. (1.4), the noise figure is seen to exhibit a linear-log relationship with the inverse of the link gain, so that minimizing Vπ and TFF while maximizing PI to maximize gain also serves to minimize the noise figure of the external link. The intermodulation-free dynamic range (IMDR) is a measure of the degree to which nonlinearities lead to signal distortion in the link, which is of vital importance in RF remoting applications6. The IMDR of standard MZMs biased to the quadrature point is very similar to that of direct modulation. A key performance goal for large-scale implementation of E–O modulators is a Vπ well below 1 V 8. The switching voltage Vπ is a function of several device and material properties, but the principal degree of freedom available for improvement is the electro-optic coefficient rIJK of the core material, which is a measure of the material’s macroscopic nonlinear optical activity. Since link performance improves ad infinitum with lower Vπ (see equations 1.2 – 1.4 above), the focus in recent years has been on developing materials with higher electro-optic coefficients. The electro-optic effect is rigorously defined as the change in impermeability ( ε −1 ) of the material, where ε is the bulk permittivity, in response to an applied field:

( )

δ ε −1 = rIJK EK + pIJKL EK EL +

(1.6)

where I , J , K = X , Y , Z are in the laboratory reference frame, the tensor r refers to the first-order, linear electro-optic coefficient, as a consequence of second-order optical nonlinearity, and the tensor p refers to the quadratic electro-optic effect, also known as the Kerr effect9, which is a consequence of third-order optical nonlinearity. For materials

with no inversion symmetry (noncentrosymmetric), the linear E-O effect dominates, and for small changes of diagonal components of r , the linear electro-optic effect can be expressed

6

Chapter 1 1 2

δ nI = − nI3rIJK EK

(1.7)

where nI is the bulk material refractive index in the X − coordinate direction. In a condensed notation, IJ = ZZ = 3 and K = Z = 3 , so that the macroscopic electro-optic coefficient r33 corresponds to TM polarization, i.e. the optical field is polarized parallel to modulating field vector9. The electro-optic tensor component r13 corresponds to TE polarization, i.e. the optical field is polarized perpendicular to the applied field vector, which is typically 1/3 the value of r33 in NLO polymer films. Lithium niobate is the most widely used E-O material in commercial high-speed external modulators6, 10. The wide availability of LiNbO3 ferroelectric crystals is driven by the large market for surface acoustic-wave filters for mobile communication receivers7. LiNbO3 modulators have optical propagation losses as low as 120 pm/V for CF3-modified FTC chromophore analogs54 possessing bulky inert moieties to hinder close dipole approach. Marks and co-workers have reported fabrication of thick (> 1 µm ) films of organic NLO materials using layer-by-layer self-assembly of Langmuir-Blodgett films that exploit hydrogen bonding interactions to direct the assembly56. Further, it may be possibly to fabricate films of waveguide thicknesses (> 1.5 µm ) by macromolecular assembly of dendrimers incorporated with high- β chromophores. Experimental work using second-harmonic generation57-63, dielectric relaxation64, 65

, and direct measurements of r33 40, 66 or Vπ

67

has been done to elucidate the potential

for long-term thermal temporal stability of poling order of NLO polymers, showing that the relaxation rates follow a Kohlrausch-Williams-Watts stretched exponential with time constants that depend on polymer segmental motion and which are strongly correlated to the glass transition temperature. However, as shown in Eqs. (1.2) – (1.4) above, high activity (and low Vπ ), while important, is only one of several parameters, which must be considered for optimal link performance. Low fiber-to-fiber optical insertion loss is also crucial, which leads to stringent requirements on propagation loss in the EO material. The ability to achieve high activity, high thermal-temporal stability, and low propagation loss concurrently in a single material has been hindered by a lack of understanding of mechanisms of optical loss in the near-IR (i.e. at standard telecommunication frequencies 1.06 µm , 1.3 µm , 1.55 µm ).

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Chapter 1 1.2 Objectives and outline

This work attempts to address the above limitation by systematically characterizing optical loss in these materials and developing structure-property relationships that will lead to a better understanding of optical loss mechanisms and strategies for minimizing loss without sacrificing nonlinear optical activity. We will develop structure-optical property relations for guest-host materials due to their process simplicity and reproducibility, and in the case of dye structure studies, the ease of synthesis of a homologous structural series, obviating complications such as polymer polydispersity varying from sample to sample.

Many of these materials may find

practical utility in device applications to meet near-term technological development goals, such as improvements in device gain and thermal-temporal stability. We explore more qualitatively the effects of various dye incorporation strategies, such as pendant covalent attachment to or incorporation into the polymer main chain and ester protection of highly interacting dye hydroxyl moieties, for azo-dyes in aliphatic polymers, compared to simple guest-host mixtures of unmodified dyes. We hope to glean an understanding of the loss-concentration behavior from these materials representing systems with the potential for strong-binding or specific interactions. The overall goal is to apply the lessons learned through this work to some of the more complex material systems that should eventually aid in the development or more versatile, highly active, processible materials for large-scale device implementation. While a number of previous studies performed on NLO polymers have provided considerable insights into the loss behavior of aromatic-based polymer optical fibers, there is a great deal of uncertainty as to the controlling features for absorption losses in NLO guest-host amorphous polymers. The telecommunication wavelengths at 1.3 µm and 1.55 µm lying between vibrational peaks frequently correspond to acceptable absorption losses for undoped polymers or lower activity guest-host polymers. However, to achieve higher E-O activity, the trend has been toward longer conjugation length of the dye molecule, pushing the dye absorption maximum out to longer wavelengths, with tails that can extend into the telecommunication wavelengths.

Dye-polymer or dye-dye

interactions can give rise to inhomogeneous broadening of the electronic absorption peak, and defect states arising from processing or local domains can create “sub-band”

12

Chapter 1 structures that exhibit an exponential distribution of states in the absorption peak red tail68, 69. Absorption band shift behavior based on polymer host polarity is frequently observed, and linewidth contraction has been associated with this solvatochromic behavior70. Although the position, width, and shape of the dye absorption peak can influence the net optical loss in the near-IR, the influence of these spectral properties on near-IR absorption has not been investigated in detail. Moreover, the dye concentration dependence of near-IR loss is seldom reported. We report here on three systematic, quantitative spectral studies of optical absorption loss in high-activity guest-host materials. In Chapter 4, we explore materials doped with the same high-activity dye, LMCO-4E6m (2-(3-cyano-4-{2-[5-(2-{4-[ethyl(2-methoxyethyl)amino]phenyl}vinyl)-3,4-diethylthiophen-2-yl]vinyl}-5,5-dimethyl-5Hfuran-2-ylidene)malononitrile), closely related to the well-characterized dye FTC-2 from Dalton and co-workers51, incorporated into a series of four related Bisphenol A polycarbonate polymers, to assess the effects of polymer host polarity on near-IR loss. This work had been published in Journal of Physical Chemistry B71. In Chapter 5, we investigate the same dye incorporated as a guest in a series of four rigid, aromatic, amorphous polymers, to evaluate the effects of polymer backbone rigidity on near-IR loss. In Chapter 6, we characterize a homologous series of dyes related to LMCO-4E6m, in which the length of dialkyl spacer groups attached to the mid-section of the π conjugated bridge is varied, each incorporated into the polycarbonate copolymer, to determine the effects of alkane spacer lengths on near-IR loss. For each of these studies, we make use of the linear loss vs. dye concentration slope as a fundamental near-IR material property, and evaluate the influence of the main dye electronic absorption peak features on this near-IR property. To our knowledge, this is the first near-IR structureproperty study to systematically investigate loss-concentration behavior and the relationships between near-IR loss and the main dye absorption peak. The results of Chapters 5 and 6 are in preparation for submission to Journal of Physical Chemistry B. Finally, in Chapter 7 we report on a qualitative study of Disperse Red azo dyes incorporated into three unrelated aliphatic polymers, under three dye incorporation schemes: unmodified guest-host systems, esterified dye guest-host systems, and covalently attached dye-polymer systems.

13

As above, we systematically vary dye

Chapter 1 concentration under each of these schemes, and qualitatively assess the effects of polymer structure and dye incorporation scheme on molecular interaction mechanisms that influence the low energy tail of the main dye absorption peak and concomitant effects on near-IR loss. The results of Chapter 7 are in preparation for submission to Chemistry of Materials. We make use of an esoteric, highly sensitive near-IR spectroscopy technique, called photothermal deflection spectroscopy, to resolve and quantify the near-IR spectral fine structure for each of these studies, in conjunction with conventional UV-Vis transmission spectroscopy, to resolve the main dye absorption peak. 1.3 References

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21

Chapter 2

Background

2.0 Outline

In this chapter, we provide a review of previous work, experimental and theoretical treatments of optical absorption spectra for both inorganic and organic systems, and property prediction methods for organic polymers and NLO chromophores, each of which are relevant to, drawn upon, or extended in the studies of Chapters 4 – 7. In Section 2.1, a general review of previous studies of NLO polymer optical loss mechanisms is presented. In Section 2.2, a review of theory and measurements of E-O attenuation by Dalton and co-workers is given, which will be shown in Chapters 4 and 6 to have analogies to observed optical absorption loss behavior and mechanisms in these systems. In Section 2.3, we provide an account of Bond Length Alternation theory of nonlinear optical properties, with relevance for solvatochromic behavior and near-IR loss. In Section 2.3, a Voight profile treatment of experimental absorption spectra for extrapolating near-IR absorption behavior is reviewed, with a critique of its adequacy for the NLO materials of the studies of Chapters 4 – 7. Section 2.5 gives a description of near-IR overtone spectra as they relate to the chemical bonding contributions of materials. Section 2.6 introduces the concept of inhomogeneous broadening, and how it can relate to the chromophore – polymer systems of these studies. Section 2.7 introduces the concept of solvatochromism, implications for chromophore structure and electronic properties, empirical scales, and relevance to optical absorption spectra. Section 2.8 gives a description of a various related theories of inhomogeneous broadening and solvatochromism that will be important in interpreting the results of Chapters 4 – 6. Section 2.9 provides a description of generalized solvent polarity – band shift 22

Chapter 2 relationships that will be exploited in Chapter 4 to correlate solvatochromic behavior and concentration-dependent near-IR loss with host polymer dielectric properties. Section 2.10 reviews theoretical and experimental treatments of absorption peak broadening distributions, that will be useful in understanding effects of dye structure variation on near-IR loss in Chapter 6. Section 2.11 briefly describes quantum mechanical treatments of order and disorder effects on inhomogeneous broadening in organic systems, introducing the concepts of diagonal and nondiagonal disorder. Section 2.12 discusses optical band shape behavior that has been reported extensively in the literature for amorphous inorganic materials, from which analogies may be drawn for amorphous inorganic materials. Likewise, Section 2.13 describes a cross-section of the wealth of experimental and theoretical studies reported in the literature for disorder and defect states in the band tail regions of amorphous inorganics, which may be extended to organic systems exhibiting short- and medium-range order, as shown in Chapter 5. Section 2.14 gives a an account of theoretical treatments and experimental measurements of optical band shape behavior in organic materials, which has been reported to a more limited extent compared to amorphous inorganics. Section 2.15 reviews theories and measurements of effects of disorder and π − localization defects on optical band tails in organic materials, which will be useful in interpreting the results of Chapter 5. Finally, Section 2.16 describes empirical and semi-empirical techniques and models for predicting the properties of polymers and organic NLO chromophores that are critical to correlating and understanding the optical absorption spectra of these materials, as shown in Chapters 4 – 6. 2.1 General review of optical loss mechanisms in NLO polymers

In order to develop dye-polymer structure – optical loss relationships, we must examine the various sources and mechanisms of optical loss. The two major components of loss in a Mach-Zehnder modulator (MZM) are fiber input and output coupling losses, due to optical mode size mismatches and scattering, and propagation loss within the device. The studies of this thesis will address only contributions to propagation loss in detail. Propagation losses are due to a combination of: scattering from defects and rough

23

Chapter 2 interfaces within the waveguide; poor mode confinement due to waveguide geometry and insufficient refractive index contrast; absorption losses due to optical mode interaction with metal electrodes; losses induced by abrupt directional changes such as S-bends; and losses within the electro-optically active material.

Waveguide processing such as

photobleaching and electric field poling can contribute to any of the loss mechanism listed above. A low level of propagation loss in E-O waveguides is presently considered to be less than 1 dB/cm at the near-IR transmission wavelength for most external modulator applications. This level of loss is below detection limits for most optical spectroscopy techniques at waveguide films thicknesses (~2 – 4 µm), which is one of the key reasons for the scarcity of detailed near-IR spectroscopy studied reported for NLO polymers in the literature1. The two most general types of optical losses are those due to absorption and those due to scattering. These loss sources can act additively, and appear to be given roughly equal credence in describing optical losses in polymer waveguides for the majority of the research reported in the literature. Losses due to absorption are typically regarded as characteristic of the chemical structure and morphology of the E-O material, while scattering is more commonly associated with waveguide or fiber processing effects2, although cases to the contrary for both loss types have been suggested3-6. Three intrinsic loss mechanisms in amorphous polymers, which should describe the lower limit of loss in a polymer/chromophore system, have been defined as those arising from absorption associated with the dye π → π * electronic transition7, 8, those from absorption due to harmonics of molecular vibration7-11, and those from Rayleigh scattering8. Absorption losses induced by molecular vibrations are generally assumed to be larger than those from electronic transitions in the range of frequencies used for optical communications10. Molecular vibration losses are ascribed to overtones of the fundamental stretching vibration ν v and a bending vibration δ v 8 in the near-IR region, due primarily to hydrogen stretches in moieties such as C–H , N–H, O–H, and Si–H. These will be discussed in greater detail in Section 2.5. Rayleigh scattering loss is induced by thermally induced fluctuations of density, refractive index and dielectric constant within the polymer/chromophore system4, 7, 8, and is considered to be independent of the scattering angle. Localized nematic ordering

24

Chapter 2 processes in rigid aromatic polymers are thought to produce birefringence and refractive index fluctuations, and hence, Rayleigh scattering12. Further, the thermally induced fluctuation theory for structureless liquids predicts an increase in intrinsic scattering loss with increasing refractive index of the material system4. This is of particular interest for design optimization of electro-optic polymer materials, as the addition of high- β dyes causes a significant increase in the refractive index of the material, and the index change generally scales with β . The propagation losses in planar waveguides have been shown to be influenced by the bake process used to remove spin solvent from the film, such as extent and rate of solvent removal13, heating rate12, bake time and temperature13. Overly long bake times, or bake temperatures approaching the chromophore sublimation or decomposition temperature, may introduce inhomogeneity defects that can act as scattering sites, or degradation byproducts that are more absorbing than either the polymer or chromophore46

. The chromophore may be similarly vulnerable to thermochemical degradation and

concomitant loss contributions as a result of the poling process4; the morphology of the polymer matrix itself can also greatly influence the thermochemical stability of the chromophore during poling and subsequent thermal exposures5. Voids or pinholes caused by evaporation of complexed or trapped solvents or water can give rise to scattering centers3,

12

.

Evaporation of solvent or water may also give rise to high volume

contraction, which can induce strain heterogeneities or rough surfaces, both of which produce scattering losses3, 4, 6, 14, 15. It has been reported, on the basis of light scattering measurements and gas chromatography/mass spectrometry (GC/MS), that polymerization of residual, unreacted monomer causes volume contraction and void formation in PMMA4,

14

, from which

scattering arises. These authors also reported that residual monomer can, upon slight polymerization, form aggregates, and that unreacted initiator can aggregate and leave voids, both of which form scattering centers. The morphology of the host polymer can have pronounced effects on scattering losses. Size heterogeneities of 100’s of angstroms (10’s of nm) have been attributed to scattering losses of ~ 300 dB/km in PMMA, which was reduced to ~ 13 dB/km upon heat treatment above the Tg 4, eliminating the heterogeneous structure. Isotropic scattering 25

Chapter 2 loss induced by size heterogeneity has been shown to result from thermally induced dielectric fluctuations4, 7. A two-phase morphology would similarly be expected to introduce high scattering losses3. Heterogeneity in NLO polymers has been inferred from the angular dependence of polarized light scattering4, 7, 14. Aggregation of chromophores in polymers has been observed in solution16-22, in bulk solids or solid films17-20, 23-28, in monolayer and assembled multi-layer LangmuirBlodgett films29-31, and at film-substrate interfaces25, 26. Evaluation of aggregation states was carried out predominantly by photophysical techniques such as fluorescence and UV-Vis spectroscopies. Propagation losses in polymer waveguide films due to scattering from aggregated chromophore has been proposed2, 28, and aggregation has been shown to shift the position of the absorption maximum17, 29. The formation of charge transfer complexes has been suggested to produce absorption loss contributions12,

32, 33

. Intramolecular and intermolecular charge transfer

can occur in polyimides between amine acceptor and imide donor groups, or between the central phenyl group of the diimide and imide groups of neighboring chains34, that can be induced by ordering processes in rigid chain structures12, particularly near surfaces3. Inadequate separation between chromophores might also contribute to charge transfer between a donor group of one molecule and the acceptor group of an adjacent molecule. Dipole-dipole interactions between adjacent poled chromophores may also play a role in absorption loss behavior. Significant attenuation losses due to water absorption have been observed in polymer optical fibers and waveguides35, 36. For example, loss increases from 0.5 dB/cm to 2.0 dB/cm in polymer waveguides upon increasing the atmospheric relative humidity from 13 to 98% RH have been reported35. A variety of extrinsic causes of optical losses, that are associated to a greater extent with film processing and device fabrication rather than the polymer-chromophore material characteristics, have been documented. Photobleaching is a process frequently used for photodelineating a channel waveguide, because it affords the ability to precisely adjust the effective refractive index in a single step. Photobleaching has been implicated in increasing the surface roughness of the film due to chromophore photodecay and evaporation during annealing above the polymer Tg , giving rise to scattering15,

26

37

.

Chapter 2 Exposure to UV light during photodelineation, which is possible either by scattering from the exposed regions or by diffraction through the mask, has been observed to cause absorption losses that could not be removed by thermal annealing33; it has been proposed that the UV exposed regions next to the masked waveguide region will incur excess absorption that have deleterious effects on the evanescent tails of the optical waveguide modes that occupy the exposed regions. Exposure to visible light (or elevated temperatures) can induce a cis-trans isomerization of azo chromophores, which can increase absorption losses if the isomerized state has higher absorption than the initial state33; these losses are typically reversible. Process induced fluctuations in waveguide dimensions, such as rough interfaces to the cladding layers or rough channel sidewalls, produce scattering3,

33

. Impurities

imparted to the waveguide or cladding layers by the casting or spinning solvents, e.g. due to aged spin solvent, can introduce absorption losses3. Dust particles introduced into the film as contaminants during processing will behave as scattering sites8, 33. Losses associated with poling have been explained in terms of inhomogeneities in the orientation of the chromophore after poling, which was found to be strongly dependent on the uniformity of the waveguide film and cladding layers as well as the magnitude of the poling field38.

High propagation losses have been ascribed to

incomplete isolation of the electromagnetic wave from the silicon substrate, due to an insufficient thickness of the wafer silica buffer layer (1400 nm). For a waveguide built up as a modulator, losses can accrue from corners, and incoupling and outcoupling sites3, 33

.

Poling losses induced by metal electrodes have also been reported39, in which

significant changes in the optical field distribution were caused by small thickness variations in a 30 nm thick electrode. A number of authors have observed large effects on optical losses (both positive and negative) imparted by thermal annealing above the Tg . In most cases, annealing was associated with reduced losses by relieving stresses or smoothing out rough surfaces through polymer relaxations4,

15, 33, 37

. However, in more rigid amorphous aromatic

polymer where localized nematic ordering can occur, such as with HFDA-ODA polyimide, annealing can actually increase losses through scattering by ordered domains and charge transfer absorption losses within the domains12.

27

Chapter 2

2.2 Dalton theory and measurements of E-O attenuation

Measurements of electro-optic coefficient r33 by Dalton and co-workers for poled guest-host materials, comprised of chromophores with and without alkane groups attached to the thiophene unit near the center of the π − conjugated bridge, showed that a maximum in r33 occurs as a function of the number density of dye, and that this maximum occurs for a lower dye loading level at lower r33 for linear dyes containing no spacer groups (Figure 2.1)40.

Assuming no interactions between chromophores, the

expression in Equation 1.10 indicates that r33 should increase linearly with dye concentration without limit. However, statistical mechanical calculations by Dalton and co-workers showed that an electrostatic interaction field due to chromophore – chromophore dipole interactions competes with the poling field, such that above a particular dye concentration, dye molecules take on average separation distances below a critical value at which centric aggregation increases, i.e. clusters are formed which have reduced net noncentrosymmetric order, and hence reduced activity. This effect is manifested as an attenuation of the order factor expressed in Equations 1.11 – 1.12, which in turn attenuates r33 .

cos3 θ , as This internal

electrostatic interaction field is predicted to yield a maximum in r33 at high chromophore number densities at high r33 assuming a spherically shaped dye molecule (Figure 2.1). When the chromophore is more realistically treated as a prolate ellipsoid, the maximum in r33 is predicted to occur at a lower chromophore number density with a lower r33 , and shows strong agreement with measured values (Figure 2.1). Moreover, Dalton and coworkers have shown both theoretically and experimentally that this attenuation effect is stronger for chromophores with shorter spacer groups and with higher dipole moments, and propose that the ideal dye shape is spherical41. Measurements of both waveguide loss at a wavelength of 1.3 µm and r33 at a wavelength of 1.06 µm for the chromophore TCI doped in PMMA (Figure 2.2) showed that loss increases precipitously at a dye concentration corresponding to r33 attenuation from linear, which the Dalton group attributed to the onset of non-transient centric chromophore aggregation (Figure 2.2)42. 28

Chapter 2 They also found that the maximum in r33 increased by factors of two to three by changing the length of the alkane spacer group by 1-2Å. The combined experimental and theoretical work by Dalton’s group showed that both the dipolar and geometry properties of the chromophore control electro-optic attenuation. These same properties are known to influence the features of the optical absorption spectrum of the dye in solution43, 44, so it is asserted for this study that electro-optic attenuation and optical absorption behavior may share common origins.

N

CLD-2 O CN NC

NC

N

CLD-3 O CN NC

NC

1 Figure 2.1 E-O coefficients, r33 , vs. dye number density in poled guest-host materials as measured vs. calculated by Dalton et al. Circles designate measured values for the chromophore CLD-2 doped in poly(methylmethacrylate) (PMMA). Diamonds designate measured values for the chromophore CLD-3 doped in PMMA. Straight line is for predicted values assuming no chromophore interactions. Dotted line corresponds to calculated values using a chromophore intermolecular interaction model treating chromophores as spheres. Curved solid line corresponds to calculated values using a chromophore intermolecular interaction model treating chromophores as prolate ellipsoids (from reference 40).

29

Chapter 2

TCI

O H3CCO

CN

NC N

S CN

H3CCO

NC

O

Onset of non-transient, centric aggregation

Figure 2.2 Measured waveguide loss at 1.3 µm and r33 at 1.06 µm for the chromophore TCI doped in PMMA (from reference 42).

2.3 Bond-length alternation theory of nonlinear optical properties

Marder, Perry, Gorman, Brédas and coworkers45-50 have done extensive theoretical and experimental work in relating the linear and nonlinear optical properties of chromophores to their structure through the degree of bond-length alternation (BLA) due to distortion of the π − conjugation length from the equilibrium configuration by substituent and/or solvation effects.

A merocyanine dye may take on a range of

mesomeric structures of varying degrees of Zwitterionic character depending on the

π − conjugation length, the strength of electron donating and electron accepting character of the end groups, and the strength of either an applied external field or an effective internal field associated with the polarity of the surrounding matrix. This effect is depicted in Figure 2.3 (next page). The extent of geometric distortion is described as BLA, defined as the average difference between single and double bond lengths along the π − conjugated bridge. A closely related property to BLA defined by Marder et al. is bond-order alternation (BOA).

Calculations by Förster based on valence-bond theory showed that the

intermediate cyanine-like state (II), with equal contributions from the polyene-like (I) and polymethine-like (III) canonical resonance structures, exhibits the lowest energy

30

Chapter 2 transition51.

The degree of mixing between these two limiting resonance states

determines BOA, which Marder et al. have shown can be correlated to the polarizabilities

α , β , and γ 45. The polymethine (charge-separated) structure, of higher BOA, is more energetically stable for greater donor-acceptor strength (dye structure-dependent), for increased aromatic stabilization energy upon charge separation (dye structure-dependent), and for solvent-mediated polarization, i.e. a more polar solvent stabilizes charge separation (matrix structure-dependent).

The net effect of these three forces is an

effective internal field acting across the π − conjugated bridge that induces charge separation, which influences both the ground state structure and the response of the structure to a small perturbing field, i.e., the properties α , β , and γ .

Hypsochromic shift

Bathochromic shift R N R

n

O

I. Polyene-like

R δ+ N

n

R

R

δ− O

II. Cyanine limit

+ N

R

n

O

III. Polymethine-like

Increasing solvent polarity

Figure 2.3 Polyene-to-Zwitterion merocyanine solvent polarity scale

Quantum mechanical calculations by the Marder group with INDO (the intermediate neglect of differential overlap Hamiltonian), using configuration interactions to compute electronic states, revealed characteristic structure-property relationships for BOA vs. internal reaction field F , and for the properties µ g (ground state dipole moment), α , β , and γ vs. BOA; and further, that each higher-order polarizability is the first derivative of the next-lower polarizability with respect to F . Most significantly, these relationships showed that a maximum in β occurs for a BOA between structures I and II, and that γ (the third-order polarizability) is zero when β is optimized. These

31

Chapter 2 results were confirmed experimentally through electric field-induced second harmonic generation (EFISH) and by third harmonic generation. An extension of these analyses conducted by Marks, Ratner, and Albert52, involving INDO calculations of linear and nonlinear optical properties, transition dipole moments, and transition energies vs. applied and internal fields for three merocyanine chromophore of widely varying ground state resonance structures, found additional relationships between structure and optical properties. First, a maximum in linear polarizability occurs at intermediate field strengths (as shown by Marder et al.), and coincides with the variation in transition dipole moment and to the inverse of the transition energy to the first excited singlet state. The maxima in polarizability and transition dipole moment and minimum in transition energy occur at a field corresponding to the “cyanine limit,” i.e. structure II in Figure 2.3, at which all the

π − bridge bonds are of the same length. The reduction in transition energy (red-shifting to longer λmax ) with increasing field at small fields is called positive electrochromism, while the increase in transition energy at higher fields is called negative electrochromism, and was observed for all three dyes. This minimum in transition energy at the cyanine limit is consistent with the calculations by Förster discussed above. The maximum in transition dipole moment at the cyanine limit is consistent with a simple two-site model of the dipole oscillator strength f , which predicts that f is optimized by smoother charge distributions. The first hyperpolarizability β was predicted by Marks et al. increases to a maximum at low fields, decays sharply through zero to an equivalent negative maximum at higher fields, then increases monotonically to zero at still higher fields, and the secondorder hyperpolarizability γ was shown negatively peak for values of β near zero, again consistent with the predictions of Marder et al. Notably, the zero point in β and the negative maximum in γ were shown to occur at fields smaller than those of the cyanine limit for all three dyes. Of more immediate consequence for the present study, Marks et al. performed analogous calculations for the effects of the internal field induced by the polarity of the surrounding medium on optical and electronic properties.

32

They employed a self-

Chapter 2 consistent reaction field (SCRF) model53 to account for the solvent polarity-driven internal field. In the Onsager continuum dielectric model, the dye solute comprises a spherical cavity bathed by a continuum matrix of a given dielectric constant.

The

surrounding medium is polarized by the permanent dipole moment µ g of the dye, inducing a field in the polarized matrix that in turn acts back on the solute to stabilize charge separation in the ground and excited states. This polarized matrix field is referred to as the reaction field. Below the cyanine limit, the excited state stabilization has the effect of reducing the electronic transition energy between ground and excited states, yielding a red shift of the peak wavelength.

This red shift is called positive

solvatochromism, also known as bathochromism. The reaction field R is given by R = g (ε ) µ g

(2.1)

where g ( ε ) is the Onsager reaction factor, expressed as g (ε ) =

2 ( ε − 1)

a03 ( 2ε + 1)

(2.2)

The predicted values of transitions energies vs. SCRF showed positive solvatochromism for the three dyes, and a change to negative solvatochromism at higher SCRF for a dye exhibiting aromatic stabilization upon charge separation, which is generally the case for the class of NLO dyes considered in the present study. For the dyes absent of the aromatic stabilization, the calculated SCRFs for the highest realizable values of solvent polarities were too weak to achieve predicted turndowns in the polarizabilities α , β , and γ .

However, the dye possessing aromatic stabilization

showed SCRFs large enough to drive each property through maxima and minima to slightly beyond the cyanine limit. These results show that increasing the solvent polarity can improve the nonlinear optical activity (in particular, β ) up to a point, but physically accessible solvent polarities may be reached such that β is degraded or even driven to near-zero values. For the vast majority of polymer hosts considered for this study, it is

33

Chapter 2 expected that the solvent polarity scale be well within the regime of β rising with polarity. 2.4 Voight profile treatment of Vis-NIR absorption spectra

Stegeman and co-workers54 conducted a study of the effect of chromophore structure and polymer environment on the log slope of the infrared tail of the absorption peak, by treating the optical spectrum (on a wavelength scale) as a Voight profile. In this study, agreement was shown between predicted absorption loss by extrapolation of the log-linear Voight slope to 775 nm and planar waveguide propagation loss measurements at 775 nm, for azo dye guests in a PMMA host for absorption losses of at least 1db/cm, and a correlation was found between the log-linear Voight profile slope and glass transition temperature ( Tg ) through azo guest dyes in six different polymer hosts, in which higher projected loss at 775 nm was found for increasing polymer host Tg . The higher projected loss for increased Tg was attributed to greater inhomogeneous broadening of the dye absorption peak, interpreted as more rigid matrices imposing a greater range of interactions or conformations onto the chromophore. The magnitude change in Voight profile slope over the polymer host Tg range reported by Stegeman for two azo dyes in various polymer hosts was approximately 2.7 x 10-3 nm-1. While this treatment was found to be one of the more detailed, systematic approaches reported in the literature, a number of issues associated with the Voight profile extrapolation must be taken into account. First, the tail of the UV-Vis spectrum measured by conventional transmission techniques may be confounded by reflection, refraction, and scattering effects, introducing uncertainty into the pure optical absorption slope at lower energies. Similarly, the planar waveguide loss measurements can be significantly influenced by fabrication defects, introducing scattering. No structure – property relations were reported for the chromophores of the Stegeman et al. study, and while the argument for the effects of polymer Tg and associated chain stiffness on inhomogeneous broadening is sensible, the polymer structures of their study were not closely related, and no experimental confirmation of inhomogeneous broadening was provided. The uncertainties in measured absorption associated with reflection, refraction,

34

Chapter 2 and scattering in the tail of the main absorption peak discussed above make it very unlikely that extrapolation to the lower energy wavelengths of interest for telecommunications (1300 nm and 1550 nm) will be accurate or reproducible. Moreover, in the study by Stegeman and coworkers, solvatochromism was reported to be minimal. However, any polymer-induced solvatochromism would be expected to change the projected absorption loss values quite dramatically even if the Voight slopes were identical between polymer hosts. For these reasons, the Voight profile fitting approach was abandoned for developing a physical-structural model of near-IR loss behavior for the guest-host materials of the current studies. A more detailed analysis of the solvatochromic and inhomogeneous broadening behavior of the dye absorption peak is considered in order to more precisely describe the polymer structural attributes responsible for spectral absorption loss behavior.

2.5 Overtone spectra and bonding contributions

The individual chemical bonds of large molecules may be treated as independent oscillators, which give rise to fundamental vibrational transitions in the mid-infrared. The anharmonicity of these fundamental vibrations results in a series of higher order overtone transitions at higher energies, of successively weaker intensity. The near-IR spectral fine structure (800 – 2200 nm) of a wide range of materials has been shown to be dominated by the presence of stretching and combination vibrational overtones of hydrogen-containing bonds, such as C—H, O—H, N—H, and Si—H55-58. The positions of these overtones provide the physical basis for the choice of optical transmission frequencies, and are several orders of magnitude weaker in intensity than electronic transitions. These overtones may be expressed by a quadratic Morse potential of quantum energy levels En as55 En hc = ( n + 1 2 )ν h − ( n + 1 2 ) χν h 2

35

(2.3)

Chapter 2 where k is the force constant, n is the vibrational quantum number, νh is the frequency of the conjugate harmonic vibration [νh = (2 πc)-1 (k/ µ)1/2, µ is the reduced mass, and c is the velocity of light], and χ is the anharmonicity coefficient. The frequency of the stretching vibration ν0,n (n = 1, 2, 3, 4,...) can be expressed55

ν 0,n n = ν h (1 − χ ) − ν h χ n

(2.4)

It is commonly asserted that the optical loss of waveguide polymers can be minimized by reducing the number of hydrogen-containing bonds present in the material, which has led to strategies involving replacement of hydrogen in C—H bonds with heavier atoms such as deuterium or fluorine58-60, shifting the position of the overtone peaks to longer wavelengths. For instance, C—F overtones lie in the range 2024 – 8000 nm, with higher order (shorter wavelength) intensities decaying rapidly as a result of the lower anharmonicity constant χ relative to C—H bonds58. Pitois, Hult, and Weismann58 performed a detailed study of extrinsic and intrinsic near-IR optical loss in low-loss polymer waveguide materials based on crosslinked copolymers of pentafluorostyrene (PFS) and glycidyl methacrylate (GMA) monomers, using photothermal deflection spectroscopy (PDS) to measure the attenuation in the polymer due purely to absorption, and using the prism coupling method to measure the total optical attenuation in polymer slab waveguides.

Given in Figure 2.4 are PDS

spectra for a series of PFS-GMA copolymers measured by Pitois et al., showing spectral locations of the C—H stretching overtones and C—H stretching-plus-bending combination overtones, which occur at very similar positions for a wide range of materials. The overtone doublets at around 1710 nm, 1200 nm, and 900 nm were shown to correspond to v0,2 , v0,3 , and v0,4 C—H stretching overtones, respectively, using experimental fits to Eq. (2.4). A weak overtone at approximately 750 nm was assigned to the v0,5 C—H stretching overtone. Further, they assigned a broad overtone peak centered at 1400 nm to the C—H stretching-plus-bending combination overtone (v0,2 + δ1), and a shoulder at 1440 nm to the first overtone of the O—H stretching mode (v2); peaks at 980 nm and 755 nm were assigned to second and third O—H overtones (not shown).

36

Chapter 2

Figure 2.4 PDS spectra measured by Pitois et al. for a series of low-loss copolymers, illustrating near-IR spectral positions of C—H stretch ν0,n and C—H combination stretching-plus-bending (v0,n + δn-1) overtones (from reference 58).

Pitois et al. compared PDS-measured absolute absorption with prism-coupling measurements of waveguide loss at 1550 nm, providing for discrimination of the absorption and scattering components of total optical loss. Figure 2.5 shows optical loss measured at 1550 nm by both techniques as a function of calculated C—H bond concentration as the fluorine-containing comonomer (PFS) mole fraction was varied. While the absorption loss is seen to decrease as the C—H bond concentration in the polymer decreases, it remains appreciable even at the lowest C—H content. Further, the intrinsic scattering losses, over and above extrinsic losses due to particle contamination and roughness, was attributed to structural inhomogeneities caused by shrinkage during crosslinking.

37

Chapter 2

Figure 2.5 Optical loss measured by Pitois et al. on crosslinked PFS-GMA copolymers versus C—H composition (bond %) at 1550 nm, revealing absorption and scattering contributions (from reference 58).

Three important consequences ensue from this rigorous study by Pitois et al. First, only a modest reduction in absorption loss was affected by large changes in the C— H content, which runs counter to the conventional view. Note that the intensity of the C—H overtone peaks should follow the Beer-Lambert law and scale proportionally with C—H number density. Secondly, the intrinsic optical loss was shown to increase more substantially (relative to absorption) with increasing crosslink density. It is commonly held that the thermal-temporal stability of NLO polymers may be improved by crosslinking the polymer matrix. These results show that the crosslinking strategy comes with an intrinsic optical loss penalty. Lastly, PDS was demonstrated as an effective technique for identifying the lowest possible optical loss attainable by an organic material at given target wavelengths, allowing for screening of candidate materials in advance of more expensive characterization of device characteristics. 2.6 Introduction to inhomogeneous broadening

The breadth and frequency shifting behavior of the absorption peak can contain a great deal of information about the interactions between the molecules of the system. For liquids and glassy amorphous solids, the peak width is typically greater than 100 cm-1, which is considerably broader than those of crystals (~ 1 cm-1). The spectral broadening is brought about by statistical fluctuations of local chromophore environments, both in 38

Chapter 2 terms of variations in chromophore configurations with respect to the surrounding solvent molecules61,

62

, and different chromophore conformations63. These local environment

fluctuations give rise to a distribution of electronic energy gaps and an ensembleaveraged spectrum that is inhomogeneously broadened, as reflected in the standard deviation about the peak absorption frequency.

For dye/polymer systems, the

configurational relaxation in the excited state is well described by an inhomogeneous broadening model, in which the dye molecules in a glassy solid polymer solution are treated as an ensemble of centers of 0-0 transitions of varying frequencies62. A portion of the solvent vibrational, rotational, and translational motions can modulate the chromophore electronic transition61, and these dynamic processes can dominate the broadening behavior (as opposed to equilibrium solvation configurations). The transition frequency shift behavior in response to the solvent (“solvent shift”) is described as electronic dephasing, so that chromophores exhibiting larger dipole moments in the excite state than in the ground state experience stronger dephasing in the excited state, yielding greater spectral broadening64. Skinner and Moerner65 have pointed out that although the solvent shift and peak width provide details about the glass structure and chromophore-solvent interactions, systematic experimental studies of inhomogeneous broadening in amorphous glasses have not been widely reported. Kador66 suggested that since the interaction by each individual chromophore with its local surroundings determines its specific transition energy, the inhomogeneously broadened widths are dominated by the degree of amorphous matrix disorder. This can largely explain the vast differences in spectral broadening between crystals and glasses. Kjaer and Ulstrup67 showed that a single Franck-Condon envelop of Gaussian sub-peaks could be convolved to reproduce the asymmetric, broad, solvatochromic absorption peak of two merocyanine dyes (Betaine26 and Betaine-29) for a wide range of aprotic solvents. Theoretical and experimental studies of optical absorption band shape of solutesolvent systems has been extensively reported for both liquid and condensed phase systems61-63, 67-73. It should be pointed out that although glasses are in a non-equilibrium, quenched configuration, the relevant configurations may be treated as those of a lower temperature equilibrium liquid and use liquid equilibrium models to determine band

39

Chapter 2 shapes, allowing application of liquid solution theories based on statistical mechanics65. Loring69 based a computation of the inhomogeneously broadening absorption band of a chromophore in solution on the equilibrium theory of liquids, in conjunction with linear solvation theory, to predict a Gaussian absorption band shape. Meskers, et al.70 showed that for aggregates of dye molecules, changes in band shape are indicative of strong intermolecular interactions in the excited state. A detailed analysis by Liebsch and Gonzalez74 based on a coherent potential approximation revealed that increased nonuniformity of particle sizes of small, polarizable, randomly distributed Drude particles in a dielectric medium (cermet topology) leads to reduced red-shifting and increased inhomogeneous broadening of the absorption band. Experimental studies by Kador, Moerner, and Horne75 suggested that slowly decaying wings of the inhomogeneous peaks, extending much farther into the near-IR than predicted by a Gaussian bandshape, may be due to chromophores in close proximity to highly strained major lattice imperfections, displaced appreciably from their equilibrium positions. Kelley et al.76 reported that at high concentrations of chromophores in polymer matrices, a large inhomogeneous broadening and red or blue shift at any frequency influence the optical nonlinearity. 2.7 Introduction to solvatochromism

The spectral features of optical absorption and emission bands for intramolecular charge-transfer transitions reveal, through application of a suitable solute molecular bandshape theory and given a linear solute-solvent coupling relationship, the nuclear motions corresponding to the solvent polarization modes.

A solvent dielectric

permittivity function for induction by the solute field within homogeneous, isotropic media involves the entire solvent vibrational frequency spectrum. Within the framework of a statistical mechanical relationship between the dielectric permittivity dispersion and solvent molecular modes, the solute charge transfer transition band provides a powerful probe of the solute-solvent interactions and solvent molecular environment, upon exposure of the solute ground and excited state molecular wavefunctions to the solvent67, 77-80

.

40

Chapter 2 These strong solvent effects control not only the frequency position and shape of optical absorption bands, but also the rates and equilibrium positions of chemical reactions81. For example, rate acceleration on the order of 1011 with increasing solvent polarity is seen in the unimolecular heterolysis rate constants of 2-chloro-2methylpropane obtained in benzene and in water82, 83. The solvent effects on optical peak positions have facilitated development of empirical scales of solvent polarity through UV/Vis/near-IR spectroscopic measurements of positively and negatively solvatochromic compounds in solution.

Upon changing solvent from diphenyl ether to water, the

UV/Vis/near-IR absorption band of the solvatochromic indicator 2,6,-diphenyl(2,4,6triphenyl-1-pyridinio)phenolate

betaine

dye 36

λmax = 453 nm ( ∆λ = 357 nm, ∆ν = 9730 cm-1 ),

shifts

from

λmax = 810 nm

to

corresponding to a solvatochromic

transition energy change of 28 kcal/mol84. These solvent-mediated changes are caused by differential solvation between reactants and products or activated complexes (in the cases of equilibria or reaction rates, respectively) or between ground and excited molecular states (in the case of optical transitions), the extent of which depends on the strength of the solute-solvent intermolecular interactions. These can include specific interactions such as hydrogen bonding and electron-pair donor – electron-pair acceptor interactions, as well as nonspecific interactions such as electrostatic forces between charged ions or dipolar molecules and polarization forces from dipole moments induced in adjacent molecules8487

. In the case of optical transitions, the molecular structure and physical properties

of the solute (dye) and the solvent determine the solute-solvent interaction strength in the equilibrium ground state and the first excited state, which in turn control the extent and direction of solvatochromism.

The Franck-Condon principle states that nuclei of

absorbing chromophores in solution, within a solvation shell, undergo a negligible change in position during electronic transition, so that the solvation configuration of the first excited state is nearly identical to that of the equilibrium ground state. Positive solvatochromism (bathochromism) ensues if the solute molecule is more stabilized by solvation in the Franck-Condon excited state relative to its ground state as solvent polarity increases. This condition is generally accompanied by an increase in the solute 41

Chapter 2 permanent electric dipole moment upon excitation ( µ g < µe ). Moreover, the strength of the solvatochromism generally scales with the change in dipole moment ( ∆µ = µe − µ g ) from ground to excited state84. The converse of these conditions also generally holds, i.e., negative solvatochromism (hypsochromism) occurs if the solute molecule is more stabilized by solvation its ground state relative to its excited state as solvent polarity increases, which is usually exhibited when the solute permanent dipole moment decreases upon excitation ( µ e < µ g ). The magnitude of the solvent shift has been related to changes in the solute (dye) ground state electronic structure and dipole moment on electronic transition84, and can be directly related to the dielectric properties of the solvent medium through idealized electrostatic solvation models, such as the Onsager reaction field model88, depicting the solvent as a nonstructured, isotropic continuum, incorporating functions of solvent static dielectric constant ( ε r ), permanent dipole moment ( µ ), refractive index ( n ), as macroscopic parameters. An early formulation of such a model was put forth by Kundt in 187889, stating that positive solvatochromic shifts of the solute absorption occur for increased refractive index of the solvent. Reichardt84 has suggested that for a dipolar solute surrounded by a nonpolar solvent, dipole-induced dipole and van der Waals dispersion forces largely influence the solvent shift, so that the Franck-Condon excited state is more readily solvated by dipolesolvent polarization. For the case of a more polar solvent, stabilization of the solute ground state is modeled by orientation of a solvent “cage” around the solute; the excited state then forms in a cage of already oriented solvent dipoles, yielding better stabilization of the excited state with increased solvent polarity. These concepts are expected to apply equally for the case of a nonlinear optical dye dispersed in an amorphous, glassy polymer matrix, as is the case for the guest-host materials of this study. Several empirical parameters of polarity have been developed in response to the scarcity of accurate theories to predict solvent effects on chemical reactivity84,

85, 90-95

.

The implementation of well chosen, well understood chemical reactions or spectral absorptions displaying strong solvent dependence as reference processes have been used to develop various solvent polarity scales for practical use in the aid of selection of an

42

Chapter 2 appropriate solvent in chemical synthesis or optical absorption studies. One of the original empirical solvent polarity scales was based on solvatochromic dyes as color indicators of polarity, established by Brooker et al. in 195196. A variety of UV/Vis/nearIR spectroscopy derived polarity scales have since emerged, using dyes structurally tuned to respond to a full complement of intermolecular interactions, which has facilitated practical empirical measurements of the relative polarity of wide ranging condensed phase materials such as chromatographic media97-99, solvents and solvent mixtures100, polymers101-104, supercritical fluids105, 106, electrolyte liquids107, and microheterogeneous solutions108-112. Some of the most well known of these include the Kosower Z scale113,

114

, the

Brooker χ R and χ B scale98, the Dähne RPM scale115, the Armand ELMCT scale116, the Kamlet, Abboud, and Taft π * scale117, the Middleton et al. Ps scale (spectral polarity index)118, the Walther EK scale119, the Dubois φ scale120, the Zelinskii S scale121, the Dong and Winnik Py (pyrene) scale122, 123, and the Reichardt ET ( 30 ) scale84, 124. Each of these derives its utility from the ease of UV/Vis/near-IR spectroscopic measurements, and rely on the use of solvatochromic indicator compounds that represent the solvent response of a broad range of other processes or compounds. Many of the indicator dyes proposed for these polarity scales are soluble or stable in only a limited range of solvents, imposing severe restrictions their applicability. By far one of the most versatile and well publicized polarity scales is the ET ( 30 ) scale, first publicized in 1963124, which has been used to measure the relative polarity of several hundreds of solvents. The ET ( 30 ) scale is based on the molar electronic transition energy (in kcal/mol) of a well-established negatively solvatochromic indicator molecule, 2,6,-diphenyl(2,4,6triphenyl-1-pyridinio)phenolate betaine dye 36 (known generically as pyridinium Nphenolate betaine dye 36), at room temperature and pressure, in which the value ET ( 30 ) is expressed81

(

)

(

)

ET ( 30 )( kcal/mol ) = hcν max N A = 2.8591× 10−3 ν max cm-1 = 28591 λmax ( nm ) (2.5)

43

Chapter 2 where ν max is the transition frequency and λmax is the wavelength at the peak of the lowest energy, intramolecular charge-transfer π − π * absorption band of dye 36 in the solvent of interest. (In Reichardt’s first publication, the betaine dye 36 was assigned the formula 30.) In this empirical polarity scheme, the solvent polarity scales with the value of ET ( 30 ) . Consistent with the above discussion, stabilization of the highly dipolar, Zwitterionic ground state and a less dipolar first Franck-Condon excited state with increasing solvent polarity give rise to the unusually strong negative solvatochromism of this indicator dye. It exhibits a strong response to dipole-dipole and dipole-induced dipole interactions as a result of its large permanent ground state dipole moment, and due to its extended highly polarizable 42- π electron conjugated bridge, it is quite sensitive to dispersion interactions in the excited state.

This method of characterizing solvent

polarity registers the molecular microscopic solute-solvent interactions within a discontinuum of solvent molecules, as opposed to the electrostatic continuum dielectric models of solvent polarity, which describe the macroscopic solvent properties79, 81. Reichardt81 has measured ET ( 30 ) solvent shifts for betaine dye

36 in a

homologous series of n-alcohols (plus water), showing a monotonic decrease of polarity with increasing solvent chain length n (Figure 2.6).

Figure 2.6 Dependence of the ET ( 30 ) values of homologous 1-alkanols, H3C(CH2)nOH, on the chain length n of these alcohols [from n = 1 (methanol) to n = 11 (l-dodecanol)] with the inclusion of water (from reference 81). 44

Chapter 2

2.8 Theories of inhomogeneous broadening and solvatochromism

In this section we describe theoretical treatments of inhomogeneous broadening and solvatochromism in solute – solvent systems given by each of Marcus (Section 2.8.1); Matyushov and Schmid (Section 2.8.2); Kador (Section 2.8.3); Obato, Machida, and Horie (Section 2.8.4); Loring (Section 2.8.5); Sevian and Skinner (Section 2.8.6); and Gonzalez and Liebsch (Section 2.8.7). With the exception of the treatment by Gonzalez and Liebsch for a cermet topology of Drude-like particles in a dielectric matrix, each of the treatments of inhomogeneous broadening and solvatochromism will be seen to be consistent within the framework of Marcus’ theory of polar contributions to electronic state free energies for a polar solute in solution. 2.8.1 Marcus formalism for polar contributions to electronic state free energies In the seminal treatment by Marcus125, an electronic transition of a polar solute induces a change in solute dipole moment. The thermodynamic average energy difference between ground and excited states was estimated accounting for permanent dipoleinduced dipole moments for the polar part of the interactions. Marcus deduced that coupling between dispersion and polar forces arise in higher-order terms of a secondorder perturbation theory, and that contributions of London dispersions and polar interactions to the spectral shift are additive. Fluctuations in the polar term arise from rotational motions of the solvent molecules, while fluctuations in the nonpolar term are ascribed in part to translational fluctuations of solvent molecules.

Additional

contributions to the total potential energy U of the system come from an intermolecular nonpolar term, and solvent and solute intramolecular terms for isolated configurations. Within this theoretical framework, Marcus obtained the following expression for the polar contribution to the spectral shift of absorption

∆hν a = Fe − Fg + FeOP − g − Fe − g

45

(2.6)

Chapter 2 where Fe and Fg are polar components of the Helmholtz free energy at excited and ground electronic states, respectively, Fe− g is that of a virtual solute state of permanent dipole moment equal to excited state minus ground state dipole moments, and FeOP − g is that of Fe− g associated strictly with optical polarization. Each of these F ’s correspond to a polar contribution to the Gibbs’ solvation free energy, ∆Gsolv . The polar component of the Gibbs free energy of solvation is

(

Es = G (T , p,Ve ) − G 0 T , p,V (0)

)

(2.7)

where Ve is the volume of a charged system at equilibrium, V (0) the equilibrium volume of the uncharged system, and G (T , p, V ) = A (T , V ) − pV

(2.8)

( A (T , V ) is the Helmholtz free energy). Since each F in the above expressions is defined as A (T ,V0 ) − A(0) (T ,V0 ) , (V0 the ground state volume), they are also the polar components of the Gibbs’ solvation free energy G (T , p ) − G (0) (T , p ) , given by   U  F = − kT  ∫ exp  −  dτ  kT  

 U ( 0 )    dτ  kT  

∫ exp  −

(2.9)

Eq. (2.6) above can be written alternatively as hν m = Es + ∆F0

(2.10)

Es = ∆hν a = Fe − Fg + FeOP − g − Fe − g

(2.11)

∆F0 = ∆hν a0

(2.12)

with

46

Chapter 2 where

ν m = peak transition frequency ∆F0 = free energy gap between ground and excited states E s = solvent reorganization free energy, where reorganization is defined as changes in the nuclear equilibrium geometry among all solute vibrational modes plus collective solvation coordination. The reorganization energy has both intramolecular and solvent contributions, and for chromophore optical transitions in nonpolar solvents, nonzero reorganization energy is rooted in density fluctuations of induced dipoles126. Marcus also showed that the inhomogeneous broadening width σ s for a dielectrically unsaturated system could be written as

(

σ s2 = 2kT FeOP − g − Fe − g

)

(2.13)

which can be related to the solvent reorganization energy Es through Eq. (2.11) as

σ s = 2 Es kT

(2.14)

and when Fe− g is an even function of the charge distribution, the relationship between the broadening σ s and peak shift ∆ν a can be expressed

σ s2 = kTh∆ν a + A

(2.15)

Γ 2s = ( 8ln 2kTh ) ∆ν a + B

(2.16)

or

where A and B are constants. 2.8.2

Matyushov

and

Schmid

treatment

solvatochromism

47

of

inhomogeneous

broadening

and

Chapter 2 Expanding on Marcus’ treatment, Matyushov and Schmid126 showed that the chemical potential of solvation can be described as first and second order energy shifts, where a solvatochromic shift is a first order shift, and inhomogeneous broadening is a second order shift. Considering the chromophore a dipolar polarizable hard sphere and the solvent a polarizable hard sphere, the first order shift is shown to be dominated by dispersions in the case of large chromophore solutes. Es has a polar activation component induced by the solvent permanent dipoles, and a nonpolar activation part due to induction and dispersion interactions, and the total Es can be represented as ½ the Stokes shift127. For a positive change in dipole movement on excitation, the dispersion and induction components of the first order shift act constructively to cause a red (bathochromic) shift. Both the dispersion and induction contributions cause the solvent reorganization energy to rapidly diminish with increased chromophore size. The peak transition frequency is expressed as a sum of the various interactions as

ν a = ν ao + ∆ν rep + ∆ν disp + ∆ν p + ∆ν ss

(2.17)

where ∆ν rep is a solute-solvent repulsion term, ∆ν disp is a radially dependent dispersion attraction term, ∆ν p is an angular dependent dipolar interaction, and ∆ν ss is the solventsolvent interaction.

The solvent reorganization energy is expressed in the classic

continuum approximation as Esc =

∆µ 2  ε − 1 n 2 − 1  −   R03  2ε + 1 2n 2 + 1 

(2.18)

consistent with experimentally observed correlation of solvatochromic shifts of ν max in n2 − 1 n2 − 1 and , interpreted in the Onsager nonpolar fluids with functions such as 2 n +2 2n 2 + 1

reaction field model as the influence of solute-solvent dispersion interactions, which are stronger contributors in more highly polar solvents. Matyushov and Schmid expressed the ground and excited state energies Ei ( i = e, g ) of the inhomogeneously broadened transition in terms of a chemical potential of

48

Chapter 2 solvation ∆µ ( µi , γ i ) of a virtual solute of complex dipole moment µ~i and a complex solvent-solute dispersion interaction parameter γ~i as − ∆µ ( µ ,γ ) Ei = e ( i

i

kT )

(2.19)

with 1/ 2

ihν 2   µi =  µi2 + µ g − µe2  kT  

(

)

(

)

ihν   γ i = γ i + γ g −γe  kT  

(2.20)

(2.21)

allowing for a perturbation expansion of the solvation chemical potential into solutesolvent induction and dispersion terms, given as (1) 4 (2) 2 (2 ) −∆µ ( µ , γ ) = −∆µ 0 + µ 2 µ (1) p + γµ d + µ µ p + γ µ d

(2.22)

where ∆µ 0 is the chemical potential of cavity formation (repulsive), and µ (p1,d) and µ p( 2,d) are the first and second order terms, respectively, of the attractive part of the chemical potential of solvation. The first order terms correspond to the solvatochromic shift, ∆ν a , and second order terms correspond to the inhomogeneous width Γs . The subscript d refers to dispersion interactions while p refers to dipole-dipole interactions. The second order (inhomogeneous) terms can be represented as a product of pair distribution functions, arising from triplet solute-solvent-solute distributions within an effective liquid. The first order terms can be used to determine the solvent shift, expressed as

(

)

∆hν = ∆γµ d(1) + µe2 − µ g2 µ (1) p

(2.23)

where ∆γ = γ e − γ g . The solvent shift induced strictly by dispersion interactions can be expressed as:

49

Chapter 2

∆hν ( d )

3 n 2 − 1 3 ω0ω s α s   Rs  qd   = 2  +  r0  n + 2 2 ω0 + ω s Rs3   Rs + σ   

(2.24)

where r0 =

η=

qd =

1

(1 − η )2 Al =

Rs

σ

+

1 2

(2.25)

πσ 3 ρ

(2.26)

6

( A0 f0 + A1 f1 + A2 f 2 )

2

∑ H m j lm , j = e2π i 3

(2.28)

m =0

3 H 0 = − η + r0 (1 + 2η ) 2

H1 = −

(2.27)

(2.29)

 2   η2 2 1 η 4 η 4 η 2 η 1 1 η η 1 2 η + + − + + + − − + x r x r ( ) ( )    _ 0 0  +  2 4η f 2 + 1 8   

(

1

)

(

)

(2.30)  2   η2 2 H2 = − − η (1 + 2η ) r0    x+ 1 + η + 4η − 4η ( 2η + 1) r0 + x_ 1 + η −  2 4η f 2 + 1 8    1

(

)

(2.31)

(

x± = f ±

f 2 +1 8

)

(3 + 3η − η ) f =

1/ 3

(2.32)

2

4η 2

50

(2.33)

Chapter 2

2.8.3 Kador lattice irregularity theory of inhomogeneous broadening

Kador66 unambiguously derived the inhomogeneously broadened line shape for an ensemble of absorption or fluorescence lines in a disordered matrix, by performing an exact calculation of the distribution of lattice irregularities in a solid described by a stochastic theory based on Markoff statistics, and examining separately each of dipoledipole, van der Waals and modified Lennard-Jones intermolecular potentials. A precise definition of the inhomogeneous band shape was uncovered by defining a function J(x) J ( x) = ρ

− iν ( R ) x ) dR ∫ g ( R ) (1 − e

(2.34)

(V )

where ν~ (R ) denotes the interaction potential between dopant and matrix molecules,

ρ=

N = no. density of matrix molecules, R is the position of matrix units with respect V

to dopant molecules, and

{

g (R ) =

where

Rc

1, R ≥ Rc 0, R < Rc

(2.35)

is the dye solute cavity radius.

Making use of the identity

lim (1 − A N ) = e − A for very large values of N in the inhomogeneous distribution N

N →∞

1 I (ν ) = 2π



∫ −∞

N

1   iν x 1 − J ( x )  e dx  N 

(2.36)

yields 1 I (ν ) = 2π



e ∫ −∞

51

iν x − J ( x )

e

dx

(2.37)

Chapter 2 indicating the inhomogeneous distribution is the Fourier transform of the exponential of – J(x). The Fourier integral was solved by considering the case of the van der Waals interaction potential, appropriate for the solvent shift by apolar or weakly polar matrix materials

ν VDW ( R ) = −4ε u (σ R )

6

(2.38)

where

ε u = depth of the interaction potential well σ = diameter of matrix units. Defining the scaling parameters F = ρσ 3 ( Rc σ ) G = 4ε (σ Rc )

3

6

(2.39) (2.40)

and taking only the lowest order terms of a Taylor series expansion of the integrands of the real and imaginary parts of the Fourier integral, valid for large values of F, the inhomogeneous band shape reduces to I G (ν ) = ( 2πσ s )



1 2

 − (ν − ν )2  s  exp  2  2σ s   

(2.41)

where ν s , the “solvent shift,” is the displacement of ν max from the vacuum absorption frequency, and σ s is related to the inhomogeneous peak width (full-width at halfmaximum, FWHM) through

σ s = Γ s 2 2 ln 2 where Γs = FWHM . These are related back to the scaling terms through

52

(2.42)

Chapter 2

( 4π 9 ) FG 2

(2.43)

ν s = − ( 4π 3) FG

(2.44)

σs = and

This form of the inhomogeneous band shape is seen to be Gaussian, and the use of the lowest order terms in the series expansion of the Fourier integrands (valid for large values of F) is said to be the Gaussian approximation. The scaling parameters form the basis for solvent vs. peak width behavior statistically. When F is larger, a greater number of matrix units surround a dye molecule, which give a larger frequency shift due to the sum of their contributions, while the relative fluctuations in the shift are small owing to the interaction of each dye molecule with a larger number of matrix molecules, consistent with noise statistics. 2.8.4 Obato, Machida, and Horie treatment of inhomogeneous broadening and solvatochromism Obato, Machida, and Horie128 showed that the solvent shift ν s and broadening Γ s terms could be expressed as

ν s = ρ ∫ g ( R )ν ( R ) dR

(

(2.45)

Γ s ( FWHM ) = 2 2 ln 2 ρ ∫ g ( R ) ν ( R )  dR 2

)

1/ 2

(2.46)

and followed Kador’s analysis to show that these terms can be related to van der Waals

(W) and dipole-dipole (D) interaction terms, for a polar dye molecule in a nonpolar or weakly polar polymer, through

νs = −

4π ρ Rc3W 3

and

53

(2.47)

Chapter 2

Γs =

32π ln 2 ρ Rc3 W 2 + D 2 9

(

)

(2.48)

where

σ  W = 4ε    Rc  D=

6

2∆µ ⋅ µ M 4πε 0 hcRc3

(2.49)

(2.50)

and

µ M = matrix dipole moment. Eq. (2.50) for dipole broadening D implies that dipole-dipole interactions cause absorption lines to broaden symmetrically, while Eq. (2.49) for the solvent shift ν s (W ) implies that red shifts of the transition energy are controlled by the Van Der Waals interactions. The Γs (W , D ) expression indicates that both Van Der Waals and dipoledipole interactions contribute to Γs . The expression for inhomogeneous broadening Γ s (Eq. (2.48)) can be evaluated using the cohesive energy density, CED.

CED = δ t2 =

∆Ev Vm

(2.51)

where δ t is the total solubility parameter, ∆Ev is the energy of vaporization, and Vm is the molar volume.

In the solution theories of Hildebrand and Scott129 and of

Scatchard130, the interaction potential between dye molecules and polymer matrix units is the geometric mean of the CED of the dye and the polymer: 2 2 δ t2 ≅ δ dye δ polymer = δ dyeδ polymer

54

(2.52)

Chapter 2 This expression suggests that dye-polymer interactions are preceded by dissociation of dye-dye and polymer-polymer interactions. The expression for Γs then predicts a linear relationship between Γ s and δ dyeδ polymer .

2.8.5 Loring linearized solvation model of inhomogeneous broadening Loring69 derived a general relation between an equilibrium free energy of solvation and the inhomogeneously broadened solute absorption band at infinite dilution in a solvent for broadening dominated by dipole interactions. The solute-solvent system was modeled as an electronic two-level system with coupling to solvent nuclear coordinates of motion, and applied the linearized mean-spherical approximation (MSA) solvation theory to determine a low-power steady-state band shape. Solvent molecules are treated as non-interacting with the optical radiation, and a solute of ground state g , excited state e , and transition frequency Ω is coupled to the solvent nuclear coordinates through an electronically adiabatic Hamiltonian which sums the kinetic energy operator and Born-Oppenheimer many-body potential energy ( Vg , Ve ). In the inhomogeneous limit, the absorption band is broadened by the static distribution of solute environments, given by a line shape function I (ω )

(

I (ω ) = δ ω − Ω − Ve − Vg 

)

(2.53)

(where δ ( q ) is a canonical distribution function of adiabatic potential Vg ), which can be Fourier transformed with a transition dipole autocorrelation function C ( t ) as

I (ω ) = ( 2π )

−1





−∞

dt exp  −i (ω − Ω ) t C ( t )

(

C ( t ) = exp i Ve − Vg 

55

)

 

(2.54)

(2.55)

Chapter 2 The autocorrelation function is written as a function of an excess Helmholtz free

(

)

energy a ( λ ) of a condensed phase of potential Vg + λ Ve − Vg (relative to the ideal gas) as

{

C ( t ) = exp −  a ( −itkT

) − a ( 0 )

kT

}

(2.56)

For a polar solute experiencing a change in permanent dipole moment

∆µ = µe − µ g upon electronic transition and exhibiting a dipolar solvation free energy

α ( µ ) , a dipolar hard sphere model of solute of diameter D and solvent of diameter d yields an expression for C ( t ) as

{(

C ( t ) = exp α µ g − ( itkT

}

)  µe − µ g  ) − α ( µ g )

(2.57)

where α ( µ ) ⋅ kT is taken as the reversible work to change the solute dipole moment from 0 to µ when in solution. In the MSA linear solvation theory within the hard-sphere approximation,

(

)

α ( µ ) = µ 2 β D3 α s

(2.58)

where α s is a parameter relating to the solute diameter and thermodynamic state of the pure solvent.

C ( t ) is seen to be a Gaussian function of t due to the quadratic

dependence of α ( µ ) on µ (for any linear solvation theory), so that evaluation of the line shape function (Eq. (2.53)) is seen to produce a Gaussian band shape:

(

I (ω ) = 2π 1 2 ∆

(

δ = 2α s

)

−1

exp  − (ω − Ω − δ ) 

)(

2

D3 µ g ⋅  µe − µ g 

56

)

4∆ 2  

(2.59)

(2.66a)

Chapter 2

{(

∆ = α s µe − µ g

2

)(

2

D3 kT

)}

12

(2.66b)

Here it is seen that the Gaussian inhomogeneous width is quadratic in ∆µ , the difference between excited and ground state dipole moments, and inversely dependent on a cubic function D3 of the solute mean spherical size. Loring pointed out that for this band shape expression to hold, the solvent must be weakly coupled to its solvent environment. For strong coupling to the environment, the central limit theorem will no longer be valid and the band shape may be non-Gaussian. Loring gave the following expression for the solvent-dependent factor α s in

α ( µ ) , within the MSA model69:

αs =

8 ( ε − 1) 3

 2uR (1 − u )  3  1− u    + 2ε (1 + R (1 − 2u ) ) +  1 + R  1 − 2u    1 − 2u  3

3

(2.60)

d D

(2.67a)

3ξ 1 + 4ξ

(2.67b)

 1 9 1/ 3 −1/ 3  2 4+ f + f 

(2.67c)

R=

u=

ξ= 

f = 1 + 54ε

1/ 2

1/ 2   1   1 − 1 +    27ε 1/ 2    

(2.67d)

where ε is the matrix dielectric constant. Note that the term R is the only solute (dye) dependent parameter in the expression for α s .

2.8.6 Sevian and Skinner microscopic theory of inhomogeneous broadening

57

Chapter 2 Sevian and Skinner131 developed a microscopic theory of transition energy distributions for pairs of chromophores in liquid or glassy solid solvents by considering the mean square fluctuation F of the transition energy E ( R1 , …, R N ) for positions R i of the i -th solvent molecule:

 F=   

∑ i

 v ( Ri )   

2



∑v(R )

2

(2.61)

i

i

where v ( R i ) is the perturbation of the transition by a single solvent molecule a distance

R i from the solute molecule.

This analysis led to a Gaussian expression for the

transition energy probability distribution P ( E ) , where it was assumed that the transition dipole moment is independent of the solvent configuration:  ( E − A )2  1  P(E) = exp   2F  2π F  

(2.62)

The average absorption transition energy A in this expression is given by

A = E ( R1 ,..., RN ) = E 0 +

∑v(R ) i

(2.63)

i

where E 0 is the unperturbed (gas phase) transition energy. Note the similarity between this expression and Kador’s expression for the inhomogeneous band shape in Eq. (2.41), showing that the inhomogeneous width σ s is related to the mean square fluctuation F of the transition energy E . Theoretical predictions and Monte Carlo simulations by Sevian and Skinner131 based on this microscopic model showed that a large inhomogeneous width accompanied by a small red shift is brought about by large density fluctuations, as a culmination of blue shifting brought about by high density configurations and red shifting brought about by low density configurations. Calculations of correlation between transition energy fluctuations between solute pairs showed that the solute-solvent interaction is short-range 58

Chapter 2 (and described by a Lennard-Jones potential), while dipolar interactions arising from chemical or physical defects (as in the case of disordered crystals) can be much longer range. This interpretation is consistent with the Kador lattice irregularity theory66, in that the precise transition energy of each solute molecule depends on interactions with its local environment, so that lattice irregularities such as point defects lead to an inhomogeneously broadened absorption band.

2.8.7 Gonzalez – Liebsch treatment of inhomogeneous broadening in cermet topology Gonzalez and Liebsch74 theoretically calculated optical absorption band shapes and band shifts for small Drude-like metal particles in a dielectric matrix representing a cermet topology.

The influence of particle interactions on absorption spectra was

evaluated by considering the metal as randomly oriented, polarizable particles and applying a coherent potential approximation in conjunction with a lattice-gas model as a generalized extension of Maxwell-Garnett theory.

In this approximation, the

polarizability of metal particles of spherical shape and of size smaller than the optical wavelength is given by

ε (ω ) α (ω ) χ (ω ) − 1 = 4π n 4 0 ε1 (ω ) 1 − 3 π nα 0 (ω ) χ (ω )

(2.64)

where ε (ω ) is composite dielectric function, ε1 (ω ) is the matrix dielectric function,

α 0 (ω ) is the polarizability of a single metal sphere in the matrix, n is the volume density of particles, and χ (ω ) is a factor associated with the positional disorder of the particles. Defining a complex effective polarizability α eff (ω )

α eff (ω ) = α eff (ω ) cR3 = α1 (ω ) + iα 2 (ω )

(2.65)

where α eff (ω ) is an effective polarizability

α eff (ω ) = cα 0 (ω ) χ (ω )

59

(2.66)

Chapter 2 and c is the fraction of sites of a hypothetical crystal lattice occupied by particles, the optical absorption constant k (ω ) ( = Im ε (ω ) ) is expressed k (ω ) =

3ε1 f mα 2 (ω ) 2

1 − f mα1 (ω )  +  f mα 2 (ω ) 

2

(2.67)

where f m is the filling fraction of metal particles. For particles of non-uniform sizes, the single-particle polarizabilities are expressed

α i (ω ) = γ iα 0 (ω )

(2.68)

γ i = ( Ri R )

(2.69)

where 3

Ri is the radius of the i -th type of particle and R is the average radius for a particle of effective polarizability α eff (ω ) . The self-consistent effective polarizability of the nonuniform composite dielectric medium is then given by

α eff (ω ) =

∑ c α (ω ) i i

χ (ω )

(2.70)

i

Optical spectra were calculated from these expressions by Gonzalez and Liebsch74, and demonstrated that as the particle size becomes more non-uniform, the extent of red shifting is decreased, while the inhomogeneous bandwidth increases. Predicted absorption spectra vs. experimental spectra for silver particles in different dielectric media were shown to be in good agreement. This result is significant in the context of the present study, in that it shows that inhomogeneous broadening and solvatochromism follow the same trends for polarizable particles in a dielectric matrix, for widely different material systems following vastly different physical models, with absorption behavior strongly influenced by discontinuous phase polarizability in one extreme (Drude particles) vs. solute dipole moments in another extreme (merocyanine dyes). 60

Chapter 2

2.9 Solvent polarity-induced band shift relationships

Kawski and Bilot132 used a rigorous second-order quantum mechanical perturbation treatment, incorporating the Onsager dielectric continuum model, and accounting for contributions of the linear and quadratic Stark effects to describe solutesolvent dipolar interactions, to develop a general expression for the absorption and fluorescence band shifts ∆ν A and ∆ν F , respectively, of a solute molecule in solution, relative to the band energy of the free solute molecule in the vapor phase, ν A0 , F , when the relaxation time for solvent molecules is very small relative to the solute excited state lifetime: −1 −1 −1 ∆ν A, F = − mA, F (1 − α 0 f ′ )  f (1 − α 0 f ) − f ′ (1 − α 0 f ′ )   



µe2 − µ g2 2hc

( 2 − α 0 f ′) f ′ (1 − α 0 f ′)−2 (2.71)

where subscripts A and F refer to either absorption of fluorescence, µ e and µ g are solute permanent ground and excited state dipole moments, respectively, α 0 is the linear polarizability of the solute, h is Planck’s content, and c the speed of light in a vacuum. The prefactors mi are given by

mA =

mF =

µ g ( µe − µ g ) hc

µe ( µe − µ g ) hc

(2.72)

(2.73)

f and f ′ are referred as reaction field factors related to the solvent dielectric properties ( n and ε , and the solute shape and cavity radius: f =

2 ε −1 F ( ε , As ) abc 2ε + 1

61

(2.74)

Chapter 2

f′=

2 n2 − 1 F n 2 , As abc 2n 2 + 1

(

)

(2.75)

where 2a, 2b and 2c

are the major and two minor axes of an ellipsoidal solute cavity; F ( ε , As ) =

(

)

2

F n , As =

3 As (1 − As ) ( 2ε + 1)

(2.76)

2 ε − ( ε − 1) As 

(

)

3 As (1 − As ) 2n 2 + 1

(

)

(2.77)

2  n 2 − n 2 − 1 As   

where As is a solute shape factor given by

As =

abc 2



∫ 0

ds

(s + a ) (s + b ) (s + c ) 2

3/ 2

2

1/ 2

2

1/ 2

(2.78)

As the solute molecule becomes more spherical in shape, the value AS increases. As an approximation, the field factors can be expressed

(

)

(2.79)

(

)

(2.80)

f = 2 a3 Φ f ′ = 2 a 3 Φ′ where Φ=

Φ′ =

ε −1 F ( ε , As ) 2ε + 1

n2 − 1 F n 2 , As 2n 2 + 1

(

(2.81)

)

(2.82)

This general expression derived by Kawski and Bilot132 (Eq. (2.71)) takes into account the ellipsoidal geometry of the solute cavity, the solute polarizability and dipole 62

Chapter 2 moment, and the dielectric properties of the solvent, i.e. refractive index n and dielectric constant ε . A variety of dielectric models related to solvent polarity and solute dipole properties have been reported in the literature133-138 to account for the solvent shift. Most of these can be expressed in simple form as

(

)

(2.83)

(

)

(2.84)

)

( )

(2.85)

∆ν A − ∆ν F = m1 f ε , n 2 + C1 ∆ν A + ∆ν F = − m2ϕ ε , n 2 + C2 where C1 and C2 are constants, and

(

)

(

ϕ ε , n2 = f ε , n2 + 2 g n2

m1 =

m2 =

(

2 µe − µ g

)

2

(2.86)

hca03

(

2 µe2 − µ g2

)

(2.87)

hca03

where a0 is a dimension representative of the solute cavity size. Note that from Eqs. (2.83) and (2.84) above, the absorption band shift can be expressed ∆ν A =

m1 m f ( ε , n 2 ) − 2 ϕ ( ε , n 2 ) + C3 2 2

(2.88)

where C3 is a constant. The solvent shift relationships derived from different theories in the literature

(

correspond to different forms of the solvent polarity parameters f ε , n 2

)

and g ( n 2 ) ,

and whether or not simplifying assumptions about the solute cavity geometry and polarizability are invoked.

In the original theory of Kawski and Bilot132, these

parameters are expressed as 63

Chapter 2

(

)

f ε , n2 =

Φ − Φ′  2α 0   2α 0 ′  1 − 3 Φ  1 − 3 Φ  a a   

( )

g n2

(2.89)

2

 2α 0  1 − 3 Φ  a  = Φ′  2  2α 0 ′  1 − 3 Φ  a  

(2.90)

A simplified form of the Bilot-Kawski expression follows when the solute cavity is assumed to be spherical, of radius a0 , leading to a = b = c , AS = 1 3 , and F ( ε , AS ) = F ( n 2 , AS ) = 1 . These assumptions are frequently invoked by other solvent

band shift theories. When directly applied to the Kawski and Bilot expression, they lead

(

)

( )

to reduced solvent polarity parameters f ε , n 2 and g n 2 expressed as

(

)

f ε , n2 =

ε − 1 n2 − 1 − 2ε + 1 2n 2 + 1  2α 0 ε − 1   2α 0 n 2 − 1  1 − 3  1 − 3  a0 2ε + 1   a0 2n 2 + 1  

( )

g n2

 α 0 n2 − 1  1 − 3 2  n 2 − 1  a0 2n + 1  = 2 2n + 1  2α n 2 − 1 2 0 1 − 3  a0 2n 2 + 1  

2

(2.91)

(2.92)

Kawski and Bilot132, 139 originally demonstrated and Chamma and Viallet134 later independently confirmed that applying the above simplifications with the following approximation for isotropic polarizability arrives at highly simplified forms of the solvent polarity parameters: a0 = ( 2α 0 )

1/ 3

64

(2.93)

Chapter 2 Kawski133 showed that this assumption does not introduce significant error in the experimental determination of the excited state dipole moment µ e . With these combined simplifications, the solvent polarity parameters in Eq. (2.88) above, attributed to Chamma, Viallet, and Kawski, are fCVK ( ε , n ) =

2n 2 + 1  ε − 1 n 2 − 1  −   n2 + 2  ε + 2 n2 + 2 

gCVK ( n ) =

(2.94)

3 n4 − 1 2 n2 + 2 2

(

(2.95)

)

McRae135 developed a solvent band shift theory using second-order perturbation theory, treating the dipoles as point dipoles and invoking isotropic polarizability, leading to the expression

(

)

α 0g − α 0e m1 2 3µ g2 − 5µ e2 + 2 µ g µ e  f M2 ε , n 2 fM ε , n + ∆ν A − ∆ν F =  2 2hca06 

(

)

(

)

(2.96)

where the McRae solvent polarity parameter f M is

(

fM ε , n

2

)

 ε − 1 n2 − 1  = 2 − 2  ε +2 n +2

(2.97)

and α 0g and α 0e are ground and excited state polarizabilities, respectively. Applying the same simplifications as for the Chamma-Viallet-Kawski expression (above), the second McRae solvent polarity parameter in Eq. (2.88) above becomes

ϕM ( n) =

n2 − 1 2n 2 + 1

(2.98)

Bakhshiev137 developed a generalized expression for the solvent band shift by considering differences in energies of stabilization (between ground and excited states) determined by: 1) a static field, associated with both the orientation of the solvent 65

Chapter 2 molecules surrounding the solute and the inductive polarization of the dielectric; and 2) a dynamic field resulting from the polarization of the solvent arising from the induced solute dipole moment. Expressions for the solvent reaction fields from Onsager-Bottcher theory

Rg =

Re =

( µ g + α 0 Rg ) 2ε − 2 2ε + 1

a03

( µe + α 0 Re ) 2ε − 2 2ε + 1

a03

(2.99)

(2.100)

were used to derive the orientation, induction, and dynamic fields, leading to the expanded Bakhshiev expression ∆ν A = −

2 µ g ∆µ g

(

µe2

− µ g2

)

f Be ( ε , n ) −

m2 ′ ( n ) − C4 f Be ′′ ( n ) f Be ( n ) − C′3 f Be 2

(2.101)

where C4 is a measure of the dispersion shift (experimentally determined for each solute), C3′ = −e 2 f e 8π 2cmν A0 a03

(2.102)

C3′ = −e 2 f e 8π 2cmν A0 a03

(2.103)

( f e = oscillator strength, c = speed of light, h = Planck’s constant, m = electron mass, e = electron charge, ν A0 = transition frequency in vapor phase) 2

 2n 2 + 2   ε − 1 n 2 − 1  f Be ( ε , n ) =  2 − 2     n +2  ε +2 n +2

( 2n + 1) ( n − 1) (n) = ( n + 2) 2

2

f Be

2

66

(2.104)

2

3

(2.105)

Chapter 2 ′ (n) = f Be

n2 − 1 2n 2 + 1

(2.106)

′′ ( n ) = f Be

n2 − 1 n2 + 2

(2.107)

Applying the same simplifying assumptions for solute geometry and polarization as was used for the Chamma-Viallet-Kawski expression, the Bakhshiev solvent polarity parameters in Eq. (2.88) reduce to:

(

)

2 2n 2 + 1  ε − 1 n 2 − 1  f B (ε , n ) = −   n2 + 2  ε + 2 n2 + 2 

ϕB ( n) =

(2.108)

2n 2 + 1  n 2 − 1    n2 + 2  n2 + 2 

(2.109)

Liptay140, 141 rigorously applied classical Onsager formalism to account for both the polarization and dispersion interactions between a merocyanine solute and solvent, considering the solute polarizability and ground and excited state dipole moments, leading to the following expression for the Liptay solvent polarity parameters in Eq. (2.88): f L (ε , n ) = f K (ε , n ) =

Φ − Φ′

(

)

(

)

1 − 2α 0 a 3 Φ  1 − 2α 0 a 3 Φ′   

ϕ L (ε ) =

(

Φ

)

1 − 2α 0 a 3 Φ   

2

(2.110)

(2.111)

where Φ and Φ′ were defined in Eqs. (2.81) and (2.82) above. Applying the same simplifying assumptions for polarization and solute geometry as was used for the Chamma-Viallet-Kawski expression, the Liptay solvent polarity parameters in Eq. (2.88) reduce to:

67

Chapter 2  ε − 1 n2 − 1  f L (ε , n ) = f M (ε , n ) =  − 2  ε 2 + n +2 

ϕ L (ε ) =

ε −1 ε +2

(2.112)

(2.113)

Ooshika138 used perturbation theory in conjunction with a bond-length alternation treatment, and considered only the van der Waals dispersion interactions of merocyanine dyes in solution. Upon application of the simplifying assumptions used in the ChammaViallet-Kawski expression, the Ooshika solvent polarity parameters in Eq. (2.88) reduce to: fO ( ε ) =

2 ( ε − 1)

ϕO ( n ) =

2.10

2ε + 1 n2 − 1 n2 + 2

(2.114)

(2.115)

Theoretical and experimental treatments of correlated and anti-correlated

broadening distributions

To account for the experimentally observed pressure dependence of zero phonon holes burned in the inhomogeneous absorption bands of a series of polymethine and polycyclic arene chromophores in ethanol glass at 6 K, Renge142 developed a model treating the solvent shift as a superposition of repulsive, dispersive, electrostatic, and other possible interactions. The observed pressure shifts were properly accounted for by assigning each interaction a particular separation distance dependence, a Gaussian frequency distribution function, and correlations between different solvent shift and broadening mechanisms. Inhomogeneous broadening was modeled in terms of various microscopic shift mechanisms, such as reaction fields, cavity fields, hydrogen bonding, and chromophore conformational flexibility. In a second-order perturbation treatment of solvent shifts, the overall band shift can be taken as a superposition of individual shifts caused by dipole- dipole, induced

68

Chapter 2 dipole-dipole and dispersive interactions. The inhomogeneously broadened band can then be determined as a superposition of Gaussian frequency distribution functions associated with different solvent shift mechanisms displaying specific intermolecular distance dependences. The inhomogeneous bands are viewed as Gaussian statistical random distributions, and potential correlations between them taken into account, including perfect correlation, perfect anticorrelation, and statistically independent (uncorrelated).

To illustrate, consider two shift mechanisms I and II exhibiting

Gaussian probability distributions yI and yII given by142 yI = exp  −2 ln 2 (ν − ∆ν max I ) Γ 2I 

(2.116)

yII = exp  −2 ln 2 (ν − ∆ν max II ) Γ 2II 

(2.117)

of widths Γ I and Γ II , and shifts ∆ν max I and ∆ν max II relative to the vacuum transition frequency ν 0 . Representations of these two distributions (as given by Renge) for the cases in which they are ideally correlated, ideally uncorrelated, and uncorrelated are shown in Figure 2.7, for each of the cases of both peaks being red shifted (upper series) and one peak being blue shifted and one peak being red shifted (lower series). For ideal correlation, the probability distribution on the same side of the peak for mechanisms I and II are equal ( yI = yII ), and the net spectrum is the sum of the two bandwidths: Γ = Γ I + Γ II

(2.118)

For the ideal anti-correlation case, for which the probability distribution on the opposite sides of the peak for mechanisms I and II are equal ( yI = yII ), the net spectrum is the difference between the two band widths, leading to band narrowing142: Γ = Γ II − Γ I (if Γ II > Γ I )

(2.119)

Note that the correlated portion of the net spectrum is on the same side of the broader of the two anti-correlated distributions. Finally, for the case of no correlation, the net spectrum is a convolution of the two distributions: 69

Chapter 2

(

Γ = Γ 2I + Γ 2II

)

12

(2.120)

From Figure 2.7, it is seen that superposition of two red shifted spectra is even further red shifted, while that of one red shifted and one blue shifted spectrum is intermediate to the two spectra. While the narrowing expected for anticorrelation seems counter-intuitive, a plausible explanation is in the case of a closely packed environment is a large red shift caused by dispersion interactions ( ∆ν disp large blue shift ( ∆ν rep

0 ) in conjunction with a

0 ) due to repulsive interactions.

Renge found that the measured absorption bandwidths of polycyclic arene chromophores in EtOH glass at 6 K are linearly related to the solvent shifted band positions as142 Γ = (108 ± 8 ) − ( 0.087 ± 0.007 ) ∆ν max

(2.121)

and found that the broadening in EtOH glass is greater than that measured in a completely nonpolar glass, 3-methylpentane (3-MePe), for a number of dyes, which he ascribed to “third” interactions, defined as142 Γ III

(

=  Γ 2 − Γ disp + Γ rep 

)

2 1 2



(2.122)

This broadening component accounts both for additional electrostatic contributions and the distortion effects of flexible long-chain chromophores that can assume a wide variety of conformations exhibiting different vacuum transition frequencies. For example, a dicarbocyanine chromophore HIDCI (Figure 2.8) exhibited additional broadening Γ III of pentamethine chain.

333 cm-l due to the conformational flexibility of its

Note that both the conformational flexibility contribution to

inhomogeneous broadening and the linear Stark effect for the broadening of a dipolar transition in a disordered matrix can occur with little or no average shift in transition energy.

70

Chapter 2

Figure 2.7 Renge model of inhomogeneous broadening of an optical transition in a disordered matrix. Two solvent shift mechanisms (I and II) are shown, each exhibiting a solvent shift ∆ν max and inhomogeneous bandwidth Γ , for various correlations between distributions, for cases of two red shifts (upper series) and opposing shifts (lower series). “O” marks a tracer chromophore on the right side of each spectrum I and projected on spectrum II: on the same side in case of ideal correlation and on the opposite side in the case of ideal anti-correlation. Lines represent predicted pressure broadening slopes (from reference 142).

N +

N I

-

Figure 2.8 Structure of polymethine chromophore HIDCI studied by Renge142. Nakamura et al.143 used a combination of absorption and fluorescence spectroscopies to measure both homogeneous and inhomogeneous broadening of β carotene solutions in alcohol solvents at 175 K. Obtained values of inhomogeneous width decrease from 430 to 350 cm-1 with increasing alcohol chain length, while the homogeneous width is independently 290 cm-1. The homogeneous width is sensitive to the existence of -OH comparing the result of 3-methyl-l-butanol and 3-methylpentane solutions.

71

Chapter 2 The optical band shapes of both the absorption and fluorescence emission spectra are Gaussian, and implicitly assuming independent, uncorrelated homogeneous and inhomogeneous distributions, the total absorption bandwidth Wa (half-width at halfmaximum, HWHM) is Wa = Wh2 + Wg2 , where Wh and Wg are the homogeneous and inhomogeneous absorption band widths (HWHM). Nakamura et al. gave the expression for the peak fluorescence frequency Fp in terms of the fluorescence excitation frequency

ω1 , peak absorption frequency ω a , and absorption bandwidths as143 Wg2

Wh2 ω1 + 2 ω a − 2ω LR Fp = 2 Wh + Wg2 Wh + Wg2

(2.123)

Thus, absorption band broadening components Wh and Wg of β -carotene in each solvent were determined from the experimental slope of fluorescence peak position Fp vs. excitation energy ω1 and measured values of Wa .

The resulting values of

inhomogeneous width were found to decrease linearly (from 430 to 350 cm-1) with increasing alcohol chain length (from n = 1 to 4), while the homogeneous width was nearly constant with an average value of 290 cm-1 (Figure 2.9).

However, the

homogeneous width was found to be appreciably narrowed (by 50 cm-1) in an alkane solvent that substitutes -CH3 for –OH, with very little effect on the inhomogeneous width. From these results, the observed inhomogeneous broadening was assigned to slow solvent dynamics involving motions which disrupt and reorganize the local solute environment, while homogeneous broadening was associated with the fast solvent dynamics related to the high frequency motions of the alcohol -OH group.

72

Chapter 2

Figure 2.9 Homogeneous and inhomogeneous widths versus normal alcohol solvent chain length n of β -carotene solutions measured by Nakamura et al. (from reference 143).

73

Chapter 2 2.11 Quantum mechanical treatments of order and disorder effects on inhomogeneous broadening

Ovchinnikov and Wight144 performed model calculations of inhomogeneous broadening for randomly arranged chromophores in a regular lattice, based on a generalized tight-binding Hamiltonian expression for the inhomogeneous band: Hˆ =

1

∑c E c + 2 ∑cV c i

i i

i

(2.124)

i ij j

ij

where ci is the excited state coefficient for chromophore i , Ei is the chromophore 0-0 transition energy in a vacuum, and Vij is the excited state interaction energy of chromophore i with nearest neighbor solvent molecule j .

Within the laboratory

reference frame, the orientation of chromophore transition dipoles are distributed relative to the electromagnetic field, such that the diagonal disorder given by the first term is represented by a distribution of chromophore transition energies Ei , caused by local interactions of each chromophore with the surrounding matrix molecules. The second, nondiagonal disorder term represents pairwise interactions between chromophores, which for strongly dipole allowed transitions, as in the case of high- β guest-host polymers, can be related to a disturbance in the transition dipoles of surrounding molecules by an oscillating field set up by a disturbance in the chromophore transition dipole144. Diagonal inhomogeneous broadening of impurity spectra in a crystalline matrix has been shown by Stoneham145 to be induced by defects in the solid crystal, where at low defect densities dislocations were shown to cause a Lorentzian band shape and point defects were shown to yield a Gaussian band shape.

Skinner and co-workers146

predicted a Gaussian band shape for all concentrations of point defects. In the wings of the spectrum, both groups predicted asymptotic decay deviating from Gaussian, and described by a power law I (ω ) ∝ ∆ω − m

74

(2.125)

Chapter 2 where m = 5 2 for dislocations and m = 2 for point defects, assuming a random spatial distribution of point defects.

Kador, Horne, and Moerner75 demonstrated slowly

decaying tails on inhomogeneous bands even for single chromophore molecules in an organic crystal that extend out much farther than predicted by the Gaussian model, and speculated that this could be due to solute molecules present near dislocations or point defects that are highly strained and shifted a large amount from their equilibrium position. The second, nondiagonal term in the inhomogeneous band Hamiltonian Hˆ above is expressed as144:

Hˆ d =

∑ ij

(

)(

 µˆ µˆ µˆ r µˆ j rij  i j − 3 i ij 5 r 3 r ij ij 

)   

(2.126)

where µˆ i is the dipole moment operator of chromophore i , and rij is the distance vector between chromophores i and j . Ovchinnikov and Wight modeled the case in which the chromophore electronic transition dipole moment is sufficiently strong that nondiagonal, pairwise dipole broadening dominates.

By assigning both orientation and position

randomization of the chromophore to the Hamiltonian matrix, their simulations established that in the far spectral wings of the absorption, far removed from the diagonally broadened (Gaussian) wavelengths, the broadening is due nearly entirely to dipolar chromophore pairs separated by a distance much shorter than the average (based on the volume fraction of chromophores), and exhibits the same power law decay dependence as predicted by others145, 146 for point defects. This non-Gaussian broadening was assigned to vibrational coupling of the chromophore pair due to the strong dipole interaction144. 2.12 Optical band shapes in different spectral ranges in amorphous inorganics

The absorption spectrum of a broad range of amorphous compound semiconductors and glasses have a shape as shown in Figure 2.10, exhibiting a high absorption region “A” ( α > 104 cm -1 ), and exponential region “B” spanning four orders

75

Chapter 2 of magnitude of α , and a weak absorption tail region “C” ( α < 1 cm -1 ). Region A or an additional region intermediate to regions A and B is frequently referred to as a “near gap” power law region, in which the absorption coefficient follows a

(

ω)

m

dependence,

where m is a positive value.

Figure 2.10 Typical shape of absorption edge for amorphous materials, with characteristic regions A, B, and C (from reference 149).

Chew et al.147 used a deconvolution procedure to fit to a density of states (DOS) model developed for the conduction band, valence band, and gap states to calculate theoretical absorption spectra and describe the density and energy distribution of gap states in PE-CVD grown Si-rich amorphous hydrogenated silicon carbide films. They assigned region A to transitions involving extended states, region B to transitions related to tail states, and region C to transitions involving deep defect states. Valence band tail states are attributed to weak Si-Si bonds brought about by strain within the disordered network, magnified by C alloying, which introduces more structural disorder. The width of the valence band tail is ascribed to dangling bonds created disorder-induced bond breaking, and the defects deep within the energy gap are assigned to isolated σ defects, characterized as 3-fold coordinated ( sp 2 ) sites due to dangling bonds. Dasgupta et al.148 characterized these spectral regions for amorphous carbon and hydrogenated carbon semiconductors ( a − C , a − C : H ) in the same manner, ascribing

76

Chapter 2 the extended states in region A to long range correlation and symmetry, the tail states in region B to structural disorder, and the deep defect states in region C to structural defects. They developed a DOS model to account for optical absorption spectra and electron spin resonance measurements of sputter deposited films, For materials with a mixed σ and π bonding system, a sharp change from region A to B is typically not observed, and broad Gaussian π and π * peaks, symmetrically distributed about the Fermi level (defined as EF ≡ 0 , with Eπ = − Eπ * ) suitably account for absorption band intensity and shape. The absorption coefficient can be related to the refractive index n and imaginary part of the dielectric constant ε 2 ( E ) as a function of photon energy E as148

α (E) =

2π Eε 2 ( E ) hcn

(2.127)

where ε 2 ( E ) for indirect optical transitions at 0 K within the one-electron approximation is given in terms of the density of occupied states N o and unoccupied states Nu by

ε 2 ( E ) =  2π e

2

2

/ m  ( Q E ) ( 4 ν ρ A ) 2

E f +E



N 0 ( Z − E ) Nu ( Z )dZ

(2.128)

Ef

where, e is the electron charge, m the electron mass, Q the transition matrix element,

ρ A the atomic density, ν the valence number, and with the assumption of Gaussian band shapes  ( E ′ − E )2  π ′  N o ( E ) = Nπ ,max exp  − 2 2σ  

(

 E ′′ − E π* Nu ( E ′′ ) = Nπ ,max exp  −  2σ 2 

)

2

   

(2.129)

(2.130)

Making substitutions, the absorption coefficient α which can be written as148 77

Chapter 2 K α (E) = E



EF + E

EF

  2 E * − E  2  erf [ E 2σ ] N o ( Z − E )Nu ( Z ) dZ = K π σ Nπ ,max exp  −  π   E   2σ    2

(2.131) where

( 2π ) K=

3

e2

m 2 cn

3

Q2

4

(2.132)

νρ A

Thus, the DOS model predicts the absorption band shape to be a Gaussian centered at E = 2 Eπ * , the peak energy, with a width 2 σ , and with an error function term that introduces deviations from the Gaussian band shape at low energies148. 2.12.1 High absorption region A Tauc149 observed that in both the vibrational and electronic spectra of crystals, by symmetry arguments the sharp features are exclusively due to the presence of long-range order; for systems possessing comparable short-range order, however, the broader spectral features are less distinguishable. This led to the proposition that disorder has minimal effects on optical transitions corresponding to wave functions localized to about a lattice constant.

Moreover, covalently bonded systems with valence electron

wavefunctions distributed over long distances (delocalized) are also subject to smaller effects of disorder on optical spectra. Hence the electronic properties of semiconductors are much more sensitive to disorder than are their optical properties. In region A of the optical spectrum, the absorption coefficient of amorphous materials is frequently observed to follow149

ωα (ω ) ∼

(

ω − Egopt

)

r

(2.133)

where r is a positive constant and Egopt is called the “optical gap.” Tauc developed a model for optical transitions in amorphous semiconductors, using the Mott-CFO density of states construct (Figure 2.11), which relaxed the requirement for conservation of

78

Chapter 2 momentum that applies to crystals within the periodic boundary condition, and is restricted to transitions between localized and extended states or between weakly localized states. For the latter case, the transition states are closely approximated by Bloch functions for indirect transitions confined to a volume V ( E ) . The absorption coefficient for these two cases is expressed

α=

2π 2 e 2 ⋅ p 2 ⋅ V ( E )gi ( E ) g f 2 m cω n



( ω + E ) dE

(2.134)

where n is the refractive index, gi and g f are the densities of the initial and final states, respectively, and m and e are as defined earlier. In the case of indirect transitions between weakly localized states, V ( E ) is taken to be the volume of the unit cell Vcell . Region A of the spectrum is presumed to stem from transitions from localized valence band states below Ev to delocalized conduction band states above Ecm , and vice-versa. Taking the densities of states positioned sufficiently far from Ev or Ec as r

r

gv ∼ ( Ev − E ) 1 , gc ∼ ( E − Ec ) 2

(2.135)

and substituting into Eq. (2.134), it is seen that

ωα ∼

( ω − Egopt )

r1 +r2 +1

(2.136)

where Egopt = Ec − Ev . Tauc surmised that for localized or weakly localized transitions in amorphous semiconductors, r1 = r2 = ½ as in crystals, so that r = 2 . Since then, it has been observed that a wide range of amorphous semiconductors and glasses obey the empirical relation

(

α ω = B ω − Egopt

)

2

(2.137)

in the highly absorbing region A of the spectrum. Experimental values for the constant B has been reported between 105 and 106 cm-1eV-1. 79

Chapter 2 An interpretation of this empirical expression and the associated parameter Egopt , based on calculations showing a linear relationship between photon energy and the density of delocalized states within the valence band near its maximum150, 151, is that Egopt is the lesser of the energy difference between the localized states in the valence band and delocalized states in the conduction band or vice versa (the smaller of Ecm − Ev or Ec − Evm in Figure 2.11). Moreover, the expression is consistent with the parabolic

behavior of the electronic valence and conduction DOS near the Fermi level EF . However, Egopt should be regarded as an estimation given the uncertainty in DOS behavior near EF .

Figure 2.11 Mott-CFO density of states (DOS) model for amorphous semiconductors. g ( E ) is the electronic density of states at photon energy E . Evm and Ecm are the edges of the mobility gap (from reference 149).

Various forms of Tauc relationship that have be used to describe Region A in a variety of materials are152 1/ 2 (α ω ) = B1 ( ω − Egopt )

80

(2.138)

Chapter 2

(α )

1/ 2

= B2

(

ω − Egopt

)

1/ 2 (α ω ) = B3 ( ω − Egopt )

(2.139)

(2.140)

2.12.2 Exponential absorption region B Urbach first reported, in a meticulous study of the sensitivity spectra of various types of photographic emulsions153, a log-linear relationship between sensitivity and optical frequency for the simplest unsensitized emulsions of his study. The log-linear slope was observed to be approximately 1 kT in pure silver bromide emulsions near room temperature. A colleague of Urbach’s at Eastman Kodak, F. Moser, showed that, over the range 200 K to 620 K, optical absorption spectra of silver bromide crystals also follow a log-linear relationship at the absorption edge, with slopes closely approaching 1 kT .

Urbach surveyed the available literature of the time and found log-linear

absorption edge behavior for several other materials, including AgCl, Ge, TiO2, and CdS, exhibiting slopes within a factor of 2 of 1 kT . It was conjectured at that time that the thermal energy was contributing directly to the transitions responsible for the absorption edge, but no theoretical basis for this empirical observation was provided. Since that time, the exponential absorption tail (generally referred to as the Urbach tail or Urbach edge) has been reported for numerous materials, including crystalline and amorphous semiconductors, insulating glasses, and organic compounds154, corresponding the region B in Figure 2.10. Crystalline materials commonly display Urbach tails with thermal widths, and in amorphous glasses, the glass transition temperature, which in turn determines the Urbach tail thermal width, characterizes a “frozen-in” thermal disorder. Additional structural disorder contributes to the tail width that is temperature-independent and additive with the thermal width. Both absorption and fluorescence emission edges of organic molecules in solution exhibit Urbach tails with thermal widths. Given these empirical observations, the absorption coefficient in region B can be expressed as

81

Chapter 2

α (ω ) ∼ exp ( ω Ee )

(2.141)

where Ee is referred to as the Urbach energy or Urbach tail width. The Urbach tail in many materials occurs over the absorption coefficient range from less than 1 cm-1 to approximately 104 cm-1. At temperatures well below room temperature, the Urbach energy Ee shows little or no temperature dependence, and for a large range of amorphous semiconductors, falls within a narrow range of 50 – 80 meV. At higher temperatures, Ee increases with temperature, eventually reaching Ee ≈ kT . Tauc149 has suggested that this exponential dependence of the absorption edge is evidence for localized disorder-induced tail states extending into the optical gap from the tails of the band states (i.e. from the tail just above Ev to above Ecm , for ω < Egopt in Figure 2.11), and developed an expression for the absorption coefficient for transitions between tail states and extended band states, under the assumption that V ( E ) = Vcell

α=



π 2 e2 fVcell mcn

∫ ωg

tail

( Ei ) gc (

)

ω − Egopt + Ei dEi

(2.142)

E − opt g

where the energy Ei is measured as shown in Figure 2.11, g tail ( Ei ) is the density of states in the valence band tail, and the oscillator strength f = 2 p 2 m ω . An analogous expression can be written for transitions from the valence band extended states to the conduction band tail. For a total concentration N of electronic states lying between Ei = 0 and ∞ , and assuming g tail ( Ei ) = Et−1 N exp [ − Ei Et ]

(2.143)

where Et is constant, the absorption coefficient can be expressed

α=

π 5 2e2 h f 2mc

n

gc ( Et ) Vcell N exp [ ω Et ]

82

(2.144)

Chapter 2 which has the form of the empirical Urbach expression. Tauc149 ascribed this model of absorption to both regions B and C of the absorption edge, taking g c ( E ) as the free electron density in the case of amorphous semiconductors. Wood and Tauc155 applied this analysis to regions B and C of measured absorption spectra for a − As 2S3 , Ge28Sb12Se60 , and Ge35 As12Se55 , resulting in calculated total concentrations of states of ~1020 cm-3 and ~1016 – 1017 cm-3, respectively. These values can be rationalized against independent measurements for region C, but for region B, it was considered unlikely that the density of tail states of a wide range of amorphous semiconductors would produce such a narrow range of experimentally observed values of Et (~50 to 80 meV). An alternative form of the empirical relation describing the Urbach edge is

α (ω ) ∼ exp σ 

(

)

ω − Egopt (T ) kT *  , ω < Egopt (T ) 

(2.145)

where σ is a constant of the order of unity, referred to later as the “steepness coefficient,” Egopt (T ) the temperature dependent Tauc optical gap and T * an effective temperature that is nearly constant at low temperatures and proportional to T at high temperatures. Tauc attached the temperature dependence of the slope d log α d ω at higher temperatures to the evolution of internal electric fields brought about by thermal vibrations, and identified two components of internal potential fluctuations occurring at long wavelengths, an elastic deformation and component an electrostatic component. The deformation component creates fluctuations of the optical gap Egopt about a mean Egopt0 , which can be represented in terms of density fluctuations δρ ρ :

δ Eg = Eg − Eg 0 = ( Elc − Elv ) δρ ρ

(2.146)

where Elc and Elv are the conduction and valence band deformation potentials, respectively. Under these fluctuations, Tauc showed that at optical frequencies sufficiently below Eg 0 , the absorption coefficient would be expressed

83

ω

Chapter 2

α (ω ) ∼ exp  − ( ω − Eg 0 )

2



(

2 δ Eg

)

2

(2.147)



Since this Gaussian shape is not observed at the absorption edge of glassy semiconductors, density fluctuations are shown not to responsible for region B of the spectrum for these materials, and the electrostatic contribution are regarded as a more likely source of potential fluctuations. By analogy with longitudinal acoustic phonons, hypothetical frozen-in longitudinal optical phonons are modeled as characteristic of disorder in amorphous solids, which conceivably exhibit a Gaussian distribution as do real phonons156, with the ability to induce sufficiently large internal fields to account for the exponential absorption in Part B of the spectrum. This would account for exponential broadening of the optical edge of compound semiconductors possessing non-zero permanent electric dipole moments, as is observed in the sharp absorption edges in a-Ge and Si. In addition to structural disorder, fluctuations of chemical composition may contribute both to the deformation and electrostatic parts of the internal potential fluctuations. The wide variety of potential fluctuations sources in amorphous solids that can lead to Urbach broadening makes it remarkable that the experimentally observed Urbach

energies

are

so

narrowly

distributed

among

numerous

amorphous

semiconductors, which Tauc suggests may be due to some unknown universal mechanism that acts to bring the internal potential to within a restricted range. 2.12.2.1 Urbach tail broadening examples Numerous examples of Urbach tail broadening are reported in the literature for amorphous semiconductors, glasses, and C60 and C70 fullerene compounds spanning all

σ bonding and mixed σ and π bonding geometries. The examples given here are provided as illustrative experimental treatments and interpretations. Teo et al.157 measured an exceptionally high optical gap of 3.5 eV and a large Urbach energy of 600 meV for as-deposited tetrahedral amorphous carbon, which indicated the presence of a large concentration of tail states, measured as 7.5 × 1020 cm-3 by electron spin resonance (ESR). Annealing slightly increased the optical gap while decreasing the defect density by a factor of three and significantly reducing the stress with no change in sp3 fraction, and changed the shape of the low-energy region of the 84

Chapter 2 spectrum from exponential to parabolic. This change in the low-energy band shape and reduced stress upon annealing was attributed to a reduction in σ tail states responsible for the exponential band edge, and an increase in π − π * transitions between Gaussian shaped distributions of localized π and π * states, owing to more well aligned sp 2 pairs or chains in the film. Chew et al.147 found for PE-CVD grown hydrogenated amorphous silicon carbide ( a − Si1− x C x : H ), “amorphous silicon-like lattices” evolve which exhibit Si-like optical band edges. Small additions of carbon were shown to produce films with higher degrees of disorder, leading to Urbach tail broadening and a higher concentration of defect states. The valence band tail width Eυu was found to linearly correlate with the Urbach energy Ee : Ee = A0 Eυu + B0

(2.148)

and the corresponding valence band tail states were attributed to weak Si-Si bonds arising from strain within the disordered structure.

C alloying increased this effect by

introducing more structural disorder in terms of C and Si bond distances and strengths. The valence tail width, and hence the Urbach energy, correlated with defect density, such that a wider band tail leads to a higher overall defect density, A . Experimentally, A = e Eυu

46 meV

= e(

C0 Ee + D ) 46 meV

(2.149)

The defects and tail widths were attributed to dangling bonds brought about by breaking of bonds (to minimize energy locally), caused by the disorder-induced elastic strain field. Visible photoluminescence in porous silicon (PS) has been attributed by Koch et al.158, 159 to a highly localized surface defect in the form of a dangling bond. The defects had been previously shown to follow behavior similar to the dangling bond defects in hydrogenated amorphous silicon ( a − Si : H ), in which the paramagnetic spin-state defect density N s was found be related to the Urbach energy as

85

Chapter 2 N s = b0 e Ee

10 meV

(2.150)

The Urbach edges of anodically etched p + and n + PS were measured by So et al.160 by photothermal deflection spectroscopy (PDS), and by analogy with the behavior of a − Si : H , the defect densities N s were estimated using the above empirical expression, showing defects levels of 1017 − 1018 cm-3 for p + PS and at least 1019 cm-3 for n + PS. These experimentally observed differences in dangling bond defect densities were used to explain the presence of free carrier absorption in p + PS and absence in n + PS, in that the deep levels associated with dangling bonds could trap most of the free carriers in n + PS. The Urbach energy measured for the near-gap region of C70 thin films was interpreted by Zhou et al.152 as broadening of the intrinsic absorption edge due to disorder and related to transitions from extended band states to localized states in the opposing band tail. Increases in the PDS-measured Urbach energy Ee were attributed to greater compositional, topological or structural disorder, which could be brought about by contamination of the film surface by impurities or intercalation by O2 , structural defects such as stacking faults or irregularly oriented micro-crystalline boundaries, or domains introduced by C70 rotation that could destroy long-range order. Cui et al.161 found that the PDS-measured Urbach energies Ee for electron cyclotron resonance CVD grown films of hydrogenated amorphous silicon carbide ( a − Si1− x C x : H ) increased monotonically with increasing RF bias voltage applied to the substrate during growth. This was interpreted as greater incorporation of carbon atoms into the a − Si : H structure leading to a large internal potential fluctuation and greater disorder. Cui et al. pointed out that it is generally accepted that the Urbach energy Ee is related to the local structural disorder brought on by microstructural disorder due to alloying in these materials. Previous work162 has shown that Ee is strongly correlated with carbon fraction x for 0 < x < 0.6 , and is more weakly correlated for x > 0.6 .

86

Chapter 2 Zhou et al.163 used PDS to measure the Urbach edge and sub-gap defect absorption of undoped C60 thin films vs. single crystals. An appreciably higher Urbach energy was measured for an f.c.c. C60 thin film (62 meV) vs. an f.c.c. C60 single crystal (48 meV), indicating greater structural disorder in the thin film, and this was given as the reason for a slightly smaller optical gap in the single crystal. The lower disorder in single crystal C60 was attributed to a lower phonon state density. Compangnini et al.164 used PDS to study the absorption edge of ion-irradiated hydrogenated amorphous carbon films ( a − C : H ) for the effects of bound hydrogen on disorder-induced tail states, and compared this with the behavior of a − Si : H . Fits of the absorption spectra for films with decreasing hydrogen content (adjusted by irradiation), which all exhibited the exponential tail, showed a progressive shifting of the band edge to lower energy and a sharpening of the Urbach tail, expressed as lower values of Ee .

This steady sharpening of the Urbach edge with lowered H − content was

interpreted as an increase in the fraction of trigonal ( sp 2 -coordinated) carbon which forms in six-fold clusters, at the expense of tetrahedrally ( sp3 ) coordinated carbon, leading to a more ordered structure with lower strain.

Further, the introduction of

trigonal carbon into a hydrogenated amorphous carbon structure, exhibiting weak π − π * transitions, increases the density of weakly bound π -valence states into the gap very close to the Fermi level, thus lowering the energy gap, as was observed. Cody et al.165 suggested that for a − Si , Ee is a measure of thermal and structural contributions the disorder in the atomic structure of the material. If U 2

x

and U 2

T

represent structural- and thermally-induced of atomic positions from a perfectly ordered position in a lattice, the Urbach energy Ee can be expressed E0 (T , x ) = K  U 2  where K is a constant.

T

+ U2



x

(2.151)

Compangnini et al. [DO-29] found that only structural

fluctuations contribute appreciably to the Urbach energy of a − C : H .

87

Chapter 2 Kandil et al.166 fit the exponential part of the PDS spectra of amorphous GeSe2 chalcogenide semiconductor films to the Urbach rule, and found an optimum value of the film evaporation rate (130 Å/s) for which the Urbach energy Ee is minimized, interpreted as a more relaxed amorphous network, with fewer defects, under these growth conditions. 2.12.3 Weak absorption tail region C At energies below the Urbach absorption edge (region B) in amorphous inorganics a weaker absorption tail is typically observed (region C in Figure 2.10). For example, As2 S3 chalcogenide glasses display weak absorption tails at α < 1 cm -1 , even in the case of high purity glasses, with an empirical absorption coefficient with exponential energy dependence, analogous to the Urbach tail, expressed as

α = α 0 ehν

Ew

(2.152)

where Ew is on the order of ~ 200 meV167. The width and intensity of these tails have been seen to depend on purity, deposition conditions, and the thermal history of the film [DO-34]. Until the advent of sensitive photothermal spectroscopy techniques, this region was extremely difficult to characterize quantitatively.

Tauc et al.155,

168, 169

have

suggested that this behavior is characteristic of transitions between localized initial states and extended final states, and vice versa. Tauc performed an analysis analogous to the one discussed above for Urbach tails, applying the equation for absorption coefficient given above for transitions between tail states and extended band states, under the assumption that V ( E ) = Vcell

α=

π 5 2e2 h f 2mc

n

gc ( Et ) Vcell N exp [ ω Et ]

(2.153)

to experimental tail spectra collected on chalcogenide glasses, which showed that the oscillator strength f in this region was quite small (10-2 to 10-1). This was given as confirmation of decreased optical cross-sections in amorphous glasses relative to crystals

88

Chapter 2 for localized state - delocalized state transitions, due to correlation effects, which render glasses more transparent in this spectral region. Tauc pointed out that the density of localized states could be increased by defects and impurities, thus increasing the tail width and absorption in region C.

Moreover, strong photoluminescence behavior

measured for amorphous solids have been ascribed to the effects of localized states associated with this region on radiative recombination170, 171. Tanaka172 has suggested that the weak absorption tail width is controlled by the density of As-As bond defects, consistent with Raman scattering measurements in which the weak absorption tails correlated with the density of bonding defects in as-evaporated films (~10 atom %) vs. bulk (~1 atom %)173. Measurement of weak absorption tails in As x S1− x glasses by PDS as a function of composition167 showed a maximum in tail width and absorption at a stoichiometric composition of x = 0.40 , regarded as consistent with Tanaka’s model of the tail being dominated by As-As bond defects.

89

Chapter 2

2.13 Studies of disorder and defect states in band tail regions of amorphous inorganics In this section we provide a survey of studies of disorder and defect states by spectroscopic techniques on amorphous glasses, semiconductors, and C60 and C70 compounds, illustrating in some detail the relationships between structural disorder and optical band properties.

This includes systems with predominantly single-valence,

tetrahedrally coordinated σ bonding as well as those with mixed-valence σ , π bonding, which are judged as appropriate for encompassing the bonding states found in condensed phases of aromatic high polymers and organic molecules.

The optical band-edge

analyses reported in the literature for amorphous inorganic materials is much more extensive and rigorous than has been reported for organic materials, and therefore physical analogies are drawn from the results and interpretations of this larger body of work on inorganics for guidance and interpretations of band edges in the present studies of solid phase organics. In mixed σ , π bonding systems such as amorphous carbon, there are finite probabilities of both σ and π defect states, however, the π defect density is expected to control the overall distribution as a result of their lower energy174. Hyperfine ESR spectra are most commonly used to assess a defect state bonding configuration in amorphous solids with mixed σ , π bonding systems, via determination of the interaction of hydrogen nuclei with unpaired electrons in hydrocarbon radicals. The localization of

π electron spin density ρ , related to the localization of the defect wave function, can be related to the ESR line width a by174 a = Qρ

(2.154)

with Q = -23 G. Robertson174 has treated the dangling bond σ defect in amorphous carbon as a three-fold coordinated site, which is considered π -like in hydrogenated amorphous carbon ( a − C : H ), as opposed to sp3 -like as in a-Si:H, since the analogous molecular CH 3 radical, in a trigonal planar configuration, has the unpaired electron in a

π orbital. PDS has been extensively used in absorption measurements of amorphous semiconductors to derive the density of gap states (DOGS) distribution, due to its strong 90

Chapter 2 sensitivity to sub-gap absorption by all defect types, without being restricted by the position of the Fermi level or probe-energy range, as is the case for techniques such as field effect, capacitance, or photoconductivity147. In a detailed study of structural disorder and defects in a − GeSe2 films by Kandil et al.166, medium-range order extending beyond nearest neighbor bond lengths is exhibited, as evidenced by x-ray and neutron diffraction first sharp diffraction peak (FSDP) at Q ~ 1 Å-1 175, 176. The absorption spectrum follows power law dependence in the high absorbing region, due to transitions between valence and conduction band extended states. In the exponential region, optical transitions occur between extended states in one band and localized states in the exponential tail of the other band, assigned to structural disorder.

In the plateau at lowest energies, the transition is between

extended states and deep localized states, assigned due to defects. The high absorption region is described by the empirical Tauc formula: 1/ 2 (α ω ) = B ( ω − Egopt )

(2.155)

while the exponential (Urbach) region is described by α = α 0 ehν

Ee

.

The Urbach Ee

parameter is associated with the width of the more extended band tail part of the edge, and is considered a measure of the total disorder for a − GeSe2 materials. Additional absorption at the lower energies (down to 0.6 eV) is resolved by the high sensitivity of PDS, which shows a small but non-trivial deep defect density near the Fermi level. The authors arbitrarily characterized the density of these deep defects by the

α value at 0.7 eV, which showed at minimum at the same process condition as for the Urbach “disorder parameter” Ee , corresponding to a more relaxed amorphous network with fewer defects. A decrease in Ee accompanied by an increase in

Egopt is interpreted as a

reduction in disorder due to reorganization of elementary GeSe4 / 2 tetrahedra in order to relieve local strains. An increase in optical gap Egopt is related to a global decrease in bond distortion, which is compatible with the observed increase of the average bond

91

Chapter 2 strength. This process is accompanied by breaking of a few bonds, as evidenced by a small increase in defect absorption at low energy. In a study of hydrogenated and unhydrogenated polysilicon thin-film transistors by Khan and Pandya177, both deep level states and band tail states at the grain boundaries are shown to be a consequence of disorder, and the band tail density of states is modeled as − E−E g ( E ) dE = Ne ( v )

kT0

dE

(2.156)

where kT0 is a measure of the extent of these tails from the band edge toward the middle of the gap. Consistent with measured PDS spectra, their modeling shows that deep defect states lay 0.36 eV from the valence band edge. It was found through a combination of IV measurements and numerical analyses that the band tail states can be localized states at the grain boundary and potentially spread out as delocalized states throughout the bulk of the material as disorder, while deep defects are localized at the grain boundary only. Upon plasma hydrogenation, the concentration of band tail states was decreased to a level such that the deep level states dominate the TFT I-V characteristics, consistent with the results of transient photoconductivity measurements vs. gate voltage, and ESR measurements on hydrogenated and unhydrogenated films. The energy gap of undoped C60 has been modeled by Zhou et al.163 as that of a narrow-band semiconductor with an optically forbidden gap of 1.5-2.6 eV between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). The C60 exhibits optical absorption spectra very similar to those of amorphous semiconductor films163, where Region C (see Figure 2.10) is designated as the “sub-gap” region, and is associated with transitions from fully extended valence band states to defect states, or from defect states to fully extended conducting band states. While the optical gaps E gopt determined from Tauc plots for single crystal f.c.c. and orthorhombic forms of C60 are identical, the sub-gap region in orthorhombic C60 single crystals show higher absorption, possibly due to intercalation by solvent or impurity. A shoulder found at 1.45 eV in the PDS spectrum between regions B and C for orthorhombic C60 single

92

Chapter 2 crystals and f.c.c. C60 thin films was suggested to be a transition between sub-gap states, indicative of the distribution of these defect states. A proposed DOGS deconvolution model was developed by Chew et al.147 to describe the defect-induced optical absorption in PE-CVD grown Si-rich a − Si1− x C x : H films. The model takes into account optical transitions due to the presence of defect states near the Fermi level, which are increased by carbon alloying. Three categories of optical transitions are distinguished in the model: 1) transitions to the conduction band; 2) transitions to the conduction band tail; and 3) transitions to empty deep defects state; from each of: 1) the valence band; 2) the valence band tail; and 3) filled defect states; yielding a total of nine possible transitions (T1 − T9 ). A representation of the density of states (DOS) model developed for these materials by Chew et al. is illustrated in Figure 2.12.

Extended States

Density of States, N(ε)

Tail States

Gap state transitions Near-gap transitions Tail state (Urbach) transitions Deep defect transitions

EV

ED Ef

EC

Energy, ε Figure 2.12 Representation of DOS distribution model developed by Chew et al.147 for a − Si1− x C x : H . Ev is the valence band edge, ED the filled defect band, E f the Fermi level, and Ec the conduction band edge. The shaded region denotes filled states, and the unshaded region denotes unfilled states.

93

Chapter 2 The physical basis of the model is experimental observation of increased disorder and Urbach tail broadening upon slight C alloying of Si : H films, and Si-like band edges in Si-rich a − Si1− x C x : H films, suggesting that the deep defect states are due primarily to Si dangling bonds. The C alloying effect was thus taken into account be adjusting the defect state density, distribution, and energy levels. A parabolic distribution of the DOS for the conduction band N c ( E ) and valence band N v ( E ) is assigned to the extended states, as147 N c ( E ) = N c 0 ( E + E0 )

1/ 2

N v ( E ) = N v 0 ( E − Ev − E0 )

1/ 2

(2.157) (2.158)

where E = energy level with respect to Ec (conduction band), defined as E = 0 Ev (valence band edge) ≡ Eg (mobility gap) N c 0 , Nυ 0 = free electron values in the extended states E0 = constant = abruptness of N c ( E ) and N v ( E ) at the mobility edges The band tail states are given as exponentially decreasing expressions NCBt ( E ) = NCBt 0 e E

Ecu

− E−E NVBt ( E ) = NVBt 0 e ( υ )

(2.159) Eυ u

(2.160)

where NCBt 0 , NVBt 0 are conduction and valence band tail state densities at the mobility gap edges, and Ecu , Evu are conduction and valence bandwidths. The distribution of defect states deep in the gap is expressed by a distribution of neutral singly occupied dangling bonds, D 0 , of density N D1 ( E ) centered at E = ED1 , is expressed as147

N D1 ( E ) =

A e w 2π 94



( E − E D1 ) 2 2 w2

(2.161)

Chapter 2 and a second distribution N D 2 ( E ) of negatively charged dangling bonds, D − , centered at E = ED1 + U + ED 2 , where U is the positive correlation energy required to ionize a singly occupied defect with a second electron, as − A N D1 ( E ) = e w 2π

( E − ED1 −U )2 2 w2

(2.162)

A w 2π in these distributions is the amplitude and w is the r.m.s. standard deviation

from

( = σ ),

mean

so

that

the

distribution

full-width

at

half-

maximum FWHM = 2 w 2 ln 2 , and the integrated area of the distribution is A , the overall defect density.

In the one-electron approximation, invoking the Velicky

summation rule, the continuous convolution of densities of initial (filled) states and final (filled) states is used to express the absorption coefficient α ( ω ) at photon energy ω for transitions between energy levels E and E + ω , in terms of the initial and final state Fermi-Dirac distribution functions f ( E ) and f ( E + ω ) :

α ( ω) =

M

ω

∫ Ni ( E ) f ( E ) ⋅ N f ( E +

ω ) 1 − f ( E + ω ) dE

(2.163)

where M scales with the transition dipole matrix elements, taken to be a constant at all energies. Using the distributions N c ( E ) and N v ( E ) for N f and Ni in Eq. (2.163) for region A, NCBt ( E ) and NVBt ( E ) for region B, and N D1 ( E ) and N D 2 ( E ) for region C, Chew et al.147 calculated PDS spectra for a − Si1− x C x : H compositions with x varying from 0 to 0.36 that showed exceptional agreement with experimental spectra.

The

valence band tail widths Evu derived from the model fits were seen to correlate linearly with experimentally derived Urbach tail widths Ee and logarithmically with deduced values of defect density A , each of which increased with carbon alloying, leading to the correlation α ∝ exp ( Evu 46 meV ) .

95

Chapter 2 As an extension of a well-accepted model of weak bond-to-dangling bond conversion178, Chew et al.147 explained these correlations as a local energy minimization by breaking of weak bonds lying at a particular level above the valence band edge, to form dangling bonds, such that a wider valence band tail brings about a larger density of dangling bond defects; these defects can therefore be associated with disorder-induced strains which in turn dominate valence band (and Urbach) widths. The authors also demonstrated abrupt changes in the midgap defect distribution ED1 (by 0.3 eV toward the valence band), width W , and correlation energy U at x = 0.36, indicating a shift in the nature and distribution of deep defect states as a result of C alloying. A combination or ESR and light-induced ESR measurements showed that both neutral trigonally coordinated C and Si dangling bonds and positively charged trigonal Si and negatively charged trigonal C dangling bonds contribute to deep defects. The negatively charged dangling C bonds were reasoned to be responsible for the observed ED1 shift, and resultant deep defect redistribution, as a result of increased C alloying. Dasgupta et al.148 pointed out that in the terahedrally coordinated ( sp3 ) hydrogenated amorphous silicon ( a − Si : H ), different shapes of the density of states (DOS) exist above and below the band edges, as evidenced by a sharp change in slope of

α vs. ω on a log-linear scale near the optical gap E gopt . However, for the mixed σ , π bonding system a − C : H , a much less distinct change occurs near E gopt , the widths of the associated π and π * DOS peaks are closely related to the distribution of graphiticlike island sizes, and there is no difference in physical origin of DOS lying above and below the band edges. A DOS model of the absorption spectrum based on Gaussian distributions of the π and π * bands (given earlier) predicts that the tails of these bands cannot reach the Fermi level, and as such, defect densities measured by ESR must be due to contributions by other states. These were suggested to be due to the presence unpaired electrons of five- and seven-fold aromatic rings lying close to the Fermi level, which optical spectra will be insensitive to due to the broad nature of the Gaussian π and π * bands. Two orders of magnitude decrease in ESR-measured spin defect density upon

96

Chapter 2 hydrogenation was interpreted as relief of internal stress linked to formation of unpaired spins, or partial saturation of C dangling bonds. Fathallah et al.179 used PDS to study the formation of photoexcited defects in a − Si : H upon sub-gap illumination, showing an increase in sub-gap absorption

associated with Si dangling bonds, that is dependent on photon energy and doping, consistent with a metastable defect mechanism linked with recombination of trapped holes with excited state electrons. The change in defects resulting from illumination at various illumination photon energies was derived from PDS spectra, and this defect density parameter was plotted against illumination photon energy for undoped and doped samples. The onset energy of metastable defect generation for undoped samples (< 0.95 eV) is consistent with the excitation energy for a hole from the valence band edge to a doubly occupied dangling bond defect state.

The vibrational energy released upon

recombination of the trapped hole with an electron is sufficient to create the metastable defect. This mechanism was further confirmed in doped samples in which the Fermi level was shifted by 0.3 eV, in which the illumination energy onset for metastable defects increased to 1.2 eV. A proposed mechanism for the metastable defect generation was lattice rearrangement upon release of vibrational energy during recombination. Jiao et al.180 modeled the absorption spectrum of a − Si : H as a parabolic distribution of extended states and exponential distribution of localized tail states, leading to an extrapolation procedure for spectral measurements in the strong absorption region by transmission and reflection spectroscopy or spectroscopic ellipsometry measurements to non-overlapping sub-gap regions measured by constant photocurrent measurements (CPM) or dual beam photoconductivity (DBP). Calculated spectra using these functional forms for extended and localized states reproduced experimental spectra quite well, providing support for these mathematical models for the band regions. Silicon matrix disorder was studied by Schubert et al.181 for hydrogenated amorphous Si, SiGe, SiC, and SiN, using a combination of PDS to measure Urbach energies Ee and Raman spectroscopy to measure half-widths of Si-Si transverse optic (TO)-like vibrational modes. Bond angle distortions in the Si lattice in plasma-deposited a − Si : H as little as 8.5 ° were derived from Raman and PDS data showing mutually

decreasing Urbach energies and TO full width at half maximum (FWHM). Major lattice

97

Chapter 2 distortion assigned to three-fold N incorporation into the a − Si : H matrix was deduced from increased TO-FWHM and Ee at low N alloying. For a − SiC : H up to 35% C, TO-FWHM was unchanged indicating no lattice distortion while Ee increased in a manner characteristic of CH3 incorporation into the amorphous structure.

The TO-

FWHM did increase, however, upon sputter deposition and for diborane doping, revealing SiC matrix distortion and a change in C bonding coordination. Moderate changes in TO-FWHM were seen for Ge alloying up to 25% Ge in a − SiGe : H before leveling off, while Ee continued to broaden, indicating compositionally controlled internal potential fluctuations and a change in relaxation mechanism at higher Ge levels. The Urbach slope and band tail absorption, expected to relate strongly to defects, in freestanding porous silicon was characterized by PDS by So et al.160. A highly localized surface state of a dangling band defect in a − Si : H is analogous to porous silicon (PS), such that an increase in sub-gap absorption is seen after UV light irradiation. Zhou et al.152 categorized the absorption spectrum in C70 thin films into near-gap and sub-gap absorption regions as: A:

High absorption, Tauc region (near-gap), following the empirical behavior

1/ 2 (α ω ) = B ( ω − Egopt ) , as described earlier, where the parameter

B was regarded by

Zhou et al. as a constant related to the structural properties of the material, held to typically follow B ∝ B:

Exponential absorption region (near-gap), following classical Urbach tail

behavior: α = α 0 ehν C:

1 . valence band tail width

Ee

Sub-gap absorption region, dominated by transitions from extended valence band

states to defect states, or from defect states to extended conduction band states. These characteristics, as well as the overall structure of the PDS-measured absorption spectrum, are remarkably similar to those obtained for amorphous semiconductors, and the proximity of the value of Egopt derived from Tauc plots of C70 thin films, 1.66 eV, to a reported calculated HOMO-LUMO gap value of 1.65 eV, was taken to be evidence of the semiconductor nature of C70 . A shoulder in the sub-gap region at about 1.5 eV is

98

Chapter 2 quite similar to that seen in a multilayer amorphous semiconductor, a − Si : H / SiN , assigned to silicon dangling bonds in the case of a − Si : H / SiN . This was taken to be possible evidence for a carbon dangling bond in C70 , due in part to pregraphitic fragments known to possess radicals, possibly consisting of hydrocarbons, as well as possible incorporation of a few percent of C60 . In studies by Cui et al.161 of the effect of bias voltage on the optical and structural behavior of a − Si1− x C x : H , an increase in disorder due to an increase in hydrogen content and sp 2 bonding was found. Cui et al. held that a high value of B implies a smaller degree of overall structural disorder, and that the Urbach energy Ee is related to the local microstructural disorder induced by C alloying. A linear correlation identified between the Urbach energy Ee and the Tauc slope B in this study

(

E0 = 559 − 0.48B cm -1/2eV -1/2

)

(2.164)

provided evidence that upon alloying with carbon, the overall defect structure is determined by local defects related to microstructural disorder. Note also that the relative quantum efficiency (RQE) of photoluminescence (PL) was found to increase in a complex fashion with the empirical “ E04 ” gap, defined as the energy associated with an absorption coefficient of 104 cm-1, and closely related to the Tauc energy gap E gopt . For samples containing significant C-C bonding, the PL RQE increases exponentially with E04 , and can be accounted for by emission from radiative recombination localized to sp 2 clusters, and is consistent with experimental observations by Rusli et al.182 on undoped a − C : H . For near stoichiometric a − Si1− x C x : H compositions ( x ≈ 0.5 ), the observation of room temperature visible PL indicates that Si-C bonds instead of C-C bonds dominate the optical band structure of these materials, and that the density of Si dangling bonds, associated with PL quenching, is lower than in Si-rich samples. Compagnini et al.164 considered the progression in the density of weakly bound π valence electrons, due to the presence of trigonal C atoms lying near the middle of the

99

Chapter 2 gap, to interpret the results of studies of bonded H content on E gopt and disorder on the optical edge of hydrogenated amorphous carbon films ( a − C : H ). In this study, a monotonic decrease of Urbach energy Ee and the Tauc optical gap E gopt was seen with decreasing H content, while the Ee was found to be linearly correlated to E gopt . The latter was attributed to a decrease in both disorder and strain in the material, as a result of greater formation of trigonal carbon atoms, leading to an increased tendency to cluster in ordered graphitic domains. Lower coordination of bonding is considered to enhance the ability of amorphous networks to develop lower densities of structural defects and voids, with smaller fluctuations from the ideal bond lengths and angles. A progressive decrease of E0 with hydrogen content may be due to disorder and strain reduction in the film. This is due to an increased tendency to cluster in ordered graphitic domains, the ability of atoms with lower ordination to give rise to amorphous networks with lower amounts of structural defects and voids, and to a smaller departure from optimum covalent lengths and angles. Further, it will discussed later that clusters of six-fold benzoic rings are more thermodynamically stable in a − C : H

174

, which leads to medium-range order in

amorphous carbon networks exhibiting a high degree of sp 2 bonding coordination. Nesladek et al.183 developed an analytical model describing optical absorption coefficients in region C for CVD diamond using a numerical deconvolution procedure, considering optical transitions due to the presence of π -bonded carbon. Experimental spectra provided strong evidence to suggest that the main defect structure is associated with π -bonded carbon. As previously discussed, models of optical absorption developed by Tauc184 address materials exhibiting only short-range order, as in amorphous silicon ( a − Si ) and amorphous hydrogenated silicon ( a − Si : H ), invoking conservation of energy but not momentum. As was shown earlier in the discussion of the DOGS deconvolution model developed by Chew, in the one-electron approximation, with the Velicky summation rule, the absorption coefficient α ( ω ) can be expressed



α ( E ) = CE −1 gi ( ε ) f ( ε ) g f ( ε + E ) 1 − f ( ε + E )  d ε 100

(2.165)

Chapter 2 where gi and g f are the initial and final density of states (DOS), respectively, f ( ε ) and f ( E + ε ) are the Fermi-Dirac occupation functions for the initial and final states and the

constant C is proportional to the transition dipole matrix elements. The Nesladek et al. model of sub-gap optical transitions is based on an accepted description of the electronic behavior amorphous carbon185 and on their own molecular dynamics simulations, illustrating that π , π * states exist in the calculated density of electronic states. Taking the localized π bands as a Gaussian distribution of the density of states, centered at Ei ( i = 1, 2 ) with a half-width 2 w , with π and π * bands as symmetric about the Fermi

level, The π (i = 1) and π * (i = 2) density of states can be expressed

(

gi (i =1,2) = A 2π w

2

)

−1/ 2

 ( ε − E )2  i  exp  − 2 2w  

(2.166)

Substituting this into the one-electron convolution integral (Eq. (2.165)), the optical absorption coefficient, αππ * is found to be

(

αππ * ( E ) = A 2 Eπ w 2

2

) ∫ −1

 ( ε − E )2   ( ε − E )2  1 2  exp  −  dε exp  − 2 2 2w 2w    

(2.167)

where A is a constant. Upon integration,

(

αππ * ( E ) = A 2 E π 2

)

−1

(

 E −E * exp  − ππ 2  4w 

)

2

(

  E −E  = B E −1 exp  − ππ * 1   4 w2  

)

2

  (2.168)  

where B1 is a constant. The Gaussian shape of the absorption coefficient derives directly from the Gaussian distribution of localized states, centered at Eπ and Eπ * taken as symmetrical about the mid-gap. The convolution integral is maximized when the optical energy corresponds to the energy separation between the two ( π , π * ) DOS distributions. The resultant combined DOS is also Gaussian, but with a half-width of 4 w 183. The

101

Chapter 2 calculated absorption spectra using the model (Eq. (2.168)) agree very well with experimental PDS spectra for a − C : H , and diamond films ranging from nanocrystalline to well-faceted diamond films. Note that this model is an alternative approach to the Tauc model, which approximates the transitions between two parabolic π bands. Zammit et al.186 examined structural relaxation process in ion-implanted amorphous Si ( a − Si ) using photothermal deflection spectroscopy (PDS). The structure of a − Si has generally been viewed a random continuous network with tetrahedral coordination of covalently bonded Si atoms, and further, a structural relaxation process occurring during annealing has been treated as a continuous change in the average structural bonding parameters, such as covalent bond angle disorder, across every atom of the network. PDS measurements are regarded as sensitive to the density of sub-gap point defects in solids, while for optical absorption measurements at the optical absorption edge, the Urbach tail and optical band gap have been shown to depend on the average strain in the network165. In thin films (500 nm) of Si ion-implanted a − Si subjected to isochronal heat treatments at temperatures up to 793 K, Zammit and coworkers found a progressive increase in the energy-position of the absorption edge as well as a sharpening of both the Urbach edge slopes (seen as decreasing values of Ee ) and Tauc slopes. This behavior was attributed the structural relaxation phenomenon, leading to a reduction in strain, as confirmed by both Raman spectroscopy and calorimetric measurements of the relaxation enthalpy. A linear correlation was found for these materials between the Tauc optical gap Egopt and the Urbach energy Ee , and both trend in the direction of increased strain relaxation (sharper band edge, larger optical gap) with increased annealing temperature, in agreement with previous results obtained for a − Si:H . The integrated excess sub-gap absorption due to Urbach broadening of the absorption edge, α ex , has been related to the absolute concentration of dangling bond defects in a − Si:H over three decades of magnitude, where α ex can be related to the Urbach broadening energy Ee as

α ex ( E ) = α ( E ) − α 0 exp ( E Ee )

102

(2.169)

Chapter 2 Zammit et al. measured a nearly factor of five reduction in the integral of this value from PDS spectra for a fully relaxed (annealed) ion-implanted a − Si film as compared to an as-implanted film, indicating a commensurate reduction in dangling bond density upon amorphous network strain relaxation on annealing. An alternative expression of the Urbach rule has been used to describe optical absorption due to excitons in semiconductors near the fundamental absorption edge E′ − E   F ( E ) = A exp  −σ e kT  

(2.170)

where Ee′ sets the focal point of log-linear plots of absorption coefficient vs. photon energy, and σ is referred to as the steepness coefficient, which is a constant with respect to energy and temperature187. This expression has also been shown to apply to amorphous materials and molecular crystals, as well as to indirect edges. The Urbach rule for direct absorption has been described in terms of the influence of a phonon field on exciton translational motion. Schreiber and Toyozowa187 calculated the line shape of F ( E ) using a standard Hamiltonian, employing the Monte-Carlo method in conjunction with an averageoscillator-strength-per-state (AOSPS) approach f (E) = F (E) ρ (E)

(2.171)

where f ( E ) is the oscillator strength at energy E , F ( E ) the absorption intensity, and

ρ ( E ) the density of states. Line shape functions were determined for in this manner for indirect edges for simple one-, two- and three-dimensional lattices, which displayed classical Urbach edge behavior for all three dimensionalities over an absorption coefficient range of 2.5 – 4 orders of magnitude. The steepness coefficient σ was calculated from slopes of the calculated AOSPS spectra (on semi log scales) as a function of temperature, indicating it is not temperature dependent for two- and three-dimensional crystals, but follows T 1 3 dependence for one-dimensional lattices. Schreiber et al. have

103

Chapter 2 shown that the Urbach tail of an indirect edge involving exciton states can be ascribed to momentarily localized states, as is the case for the direct edge. The authors have introduced a parameter they define as the steepness index s , which relates the Urbach tail steepness coefficient σ to the exciton-phonon coupling constant g as

σ =s g

(2.172)

Since it has been established that the steepness coefficient σ is temperatureindependent for two- and three-dimensions, the exponent of the Urbach rule is seen to be inversely correlated with the exciton-phonon coupling constant.

Measurements of

steepness coefficients of mixed and pure crystal silver halides by Kanzaki et al.188 were shown to be on the correct side of the critical value for self-trapping, for crystals displaying either free exciton or self-trapped exciton luminescence. 2.13.1 Robertson theoretical treatment of defects and disorder in amorphous carbon Robertson and O’Reilly174 have performed quantum mechanical calculations on a variety of model configurations of sp 2 and sp3 bonding sites in order to characterize the electronic and defect structure of amorphous and hydrogenated amorphous carbon ( a − C and a − C:H ).

The sp 2 configurations give rise to electronic band structures and

concomitant effects of disorder that can be distinguished from the behavior of a − Si . Only three of the four valence electrons enter into σ bonds at an sp 2 bond, with the fourth lying normal to the trigonal σ bonding plane as a π orbital. The π orbital states lie sufficiently close to the Fermi level EF at the mid gap, due to the weak nature of the

π bond, that they form valence π and conduction π * bands in both a − C and a − C:H . Robertson’s174 model used a tight-binding Hamiltonian with an sp3 orthogonal basis set that includes nearest-neighbor potentials and some next-nearest-neighbor potentials, and applied a common set of molecular orbital energies and C-C and C-H interaction potentials for sp 2 and sp3 sites, derived from fitting of existing band structures, and experimental photoemission and optical gap measurements. Of the seven

104

Chapter 2 possible C-C interactions potentials ( V ( ss ) , V ( sp ) , V ( pσ ) , V ( pπ ) , V ( pzπ ) , V2 ( pzπ ) , V ( s * π ) ), the nearest-neighbor interaction V ( pzπ ) between two π states

and next-neighbor π interaction V2 ( pzπ ) were included and chosen in a manner to accurately represent the overall graphite π band width and experimentally observed asymmetry of π and π * bands. The resulting calculated band structures predict that single layer graphite is a zero-gap material with π bands merging at EF . Additional calculations of local DOS on random networks of sp 2 and sp3 in a − C using a recursion method found networks containing 100%, 91%, and 51% of sp 2 sites exhibited no gap at the Fermi level while the network with 0% sp 2 sites (100% sp3 ) is shown to possess a gap. Robertson174 showed that in order for a gap to exist there must be spatial correlation between sp 2 and sp3 sites. A notable aspect of Robertson’s DOS calculations174 of the random networks is that the π states are half-filled, and as such, formation of a gap at the Fermi level leads to overall structural stabilization due to a reduction of the total π electron energy per atom, which becomes the thermodynamic driving force for π -gap formation in a − C . As a result, the gap size is mainly dependent on the π electron density and distribution. These states were modeled by Robertson by locating the distribution of sp 2 and sp3 sites within an open network of π states, leading to a series of molecule-like clusters, using a one-electron Hückel Hamiltonian that includes only π orbitals and their first-neighbor interaction potentials β = V ( pzπ ) . The corresponding eigenvalues for an isolated planar ring with common bond lengths and N vertices are

 2π n  En = 2 β cos   , n=1,2,..., N  N 

(2.173)

which were used to calculated the π spectra of ethylene, N -fold rings for N = 5-8, fused rings, a single graphite layer, and single layer comprised of two five-fold and two sevenfold fused rings. These calculations revealed that rings of N = 5, 7, and 8 produce halffilled levels with energies near or coincident with the Fermi level. For N = 6, however

105

Chapter 2 (benzene) all levels are positioned away from the Fermi level, which, consistent with the Hückel aromatic stability rule, leads to thermodynamically favored six-fold ring structure with maximized interactions between neighboring π states. The optical gap for this case is predicted to be Egopt = 2 β

(2.174)

The ease of fusing multiple rings and double bonds with rings was suggested by Robertson174 to lead to a wide distribution of larger clusters, of which the most likely configurations give rise to the highest Etot β , expected to adhere to the following general principles: a) six fold rings are strongly favored, both as isolated or fused systems; b) closed rings of π states are favored; and c) larger, compact clusters of fused six-fold rings are favored over rows or rings, in agreement with calculations of Etot β vs. the cluster size (number of rings) M , which indicate that Etot β for the largest clusters is universally higher than that for rows. This suggests that large carbon clusters will be inclined to grow in graphite-like planes, promoting medium-range order in a − C . Calculations of the optical gap Egopt show it oscillating with ring number for smaller clusters (due to symmetry variation), while at higher cluster sizes it follows the expression Egopt = 2 β M −0.5

(2.175)

indicating that smaller optical gaps should occurs for larger clusters. Fused ring a − C clusters of 15-20 Å in size were calculated by Robertson to span 34-60 rings, consistent with a measured optical gap of 0.4-0.7 eV based on the above expression. Considering the broad nature of the a − C absorption edge, region A of the spectrum is dominated by the ~3% of larger clusters rather than those of average size. Robertson proposed that fused ring clusters are tied together by sp3 sites as opposed to being enclosed by dangling bonds, based on ESR spin density measurements showing N s is two orders of magnitude lower than expected for dangling bonds.

106

Chapter 2 Robertson174 also proposed that strain is relieved dramatically at the boundaries of fused ring clusters in lieu of redistributing as bond-angle distortion throughout the entire network, as a means of maximizing the π bond energy. This is in line with the cos φ dependence of π bonding energy, so that much more overall bond energy is gained through the close alignment of π orbitals in fused ring clusters vs. that gained from a broad distribution of π dihedral angles. Given the two mechanisms of strain relief in a − C , i.e. breaking π bonds or breaking σ bonds, breaking π bonds at the cluster

boundary takes less energy, depending merely on the presence of sp 2 sites with

φ ∼ 90° or sp3 sites free of π states, thereby relieving strain at the cluster edge and isolating its π electron states by confining π bonding to within the islands. The optical gap Egopt of a − C:H is also dependent on medium- rather than shortranged order, and based on the above analyses and Eq. (2.175) above, can be calculated by insisting that sp 2 sites cluster as either four or more fused aromatic rings, as a polyene chain of six or more atoms, or as a quinoid configuration. The experimental optical absorption edge of a − C:H is quite broad, suggesting, as for a − C , a large distribution of cluster sizes, and its optical gap depends on the 1-5% of largest clusters. As discussed earlier, for the mixed valence system of σ and π for materials such as amorphous carbon, π defects are expected to be the most abundant and control band behavior, as a result of their lower energy of formation. Further, the σ dangling bond defect is considered as an isolated trigonal site with π nature in a − C:H , due to the presence of the unpaired electron in a π orbital, and assigned no sp3 character as in the case of a − Si:H . A continuous distribution of defect states across the optical gap of a − C is seen experimentally, providing a mechanism for conduction by variable range hopping. For these states, a spin defect density of ~1018 cm-3 has been measured by ESR. Hydrogenation of a − C to a − C:H reduces the spin density to as low as 1016 cm-3 in well prepared films. Defect formation energies and band gaps for a − C:H were computed by Robertson174 as a function of cluster size, which when combined predict that an inverse

107

Chapter 2 relationship should exist between defect density and optical gap for these materials. Experimental determinations of optical gap and gap state densities in a − C:H deposited over a range of temperatures with resultant variations in hydrogen content have borne out this prediction, showing smaller gaps with higher defect densities as H content decreases. As a result, Robertson suggested that, rather than saturating dangling bonds, the defect density of a − C:H is decreased upon hydrogenation by reducing cluster sizes, which reduces the thermodynamic stability of defects by increasing the defect formation energy.

2.14 Optical bandshapes in different spectral ranges in organic materials As stated earlier in Section 2.11, Kador et al.66 demonstrated there are slowly decaying tails of inhomogeneous bands extending out much farther than expected based on a Gaussian broadening model for single chromophores in an organic crystal lattice. Kador et al. conjectured that this is could be due to solute (chromophore) molecules located near highly strained major lattice imperfections that are significantly displaced from their equilibrium positions. Also discussed above, Ovchinnikov and Wight144 demonstrated that dipole-dipole interactions could dominate broadening in the far spectral wings, in a manner consistent with broadening by point defects in crystals. Myers61 pointed out that a the fluctuating local environment of a chromophore in solution change with the configuration of the neighboring solvent molecules, such that the state-to-state transition frequency of the chromophore can experience attenuation or enhancement from the solvent’s molecular vibrations, rotations, or translational motions. These solvent modes can couple to the chromophore electronic transitions in a manner analogous to the Franck-Condon chromophore localized state transitions. The modes are seen as undamped phonon side bands on the chromophore main transition peak in solid phase materials.

2.14.1 Ulstrup bandshape formalism Ulstrup et al.79 invoked dynamic solute molecular bandshape theory to theoretically evaluate the optical bandshape of organic chromophores in solution. The

108

Chapter 2 dynamics of solvent polarization modes are considered as the principle origin of broadening of the vibronic transitions for solutes in which large intramolecular charge redistribution accompanies the transition. Closely related to the absorption coefficient (the measured quantity in absorption spectra) is the molar extinction coefficient, which for systems in which the ground and excited states are closely coupled to the solvent medium, is expressed

ν eg (ν ) =

2 2π hν M eg K (ν ) 3c ln (10 ) kT

where c is the velocity of light in the solvent, h Planck's constant,

(2.176)

= h 2 , M eg is the

transition dipole moment, k Boltzmann's constant, T the temperature, ν the optical frequency, and K (ν ) the bandshape function, which accounts for the excited and ground state solute-solvent interaction behavior.

Ulstrup et al.79 developed a theoretical

expression for K (ν ) using specific molecular and solvent vibrational models, first coupling to the entire vibrational frequency spectrum of the solvent, then incorporating the solute nuclear motion.

The solvent modes are treated in terms of dynamic

characteristics manifested as the solvent electronic and dielectric properties. In this model, the solvent is regarded as a continuous dielectric medium in vibrational equilibrium, linearly coupled to both the ground and excited state orbitals of the solute. The solvent coordinates are then treated as a collective, continuous distribution of undamped, harmonic oscillators. In this sense, the solvent nuclear wave functions may be evaluated by quantum mechanical techniques used to treat electronic transitions between harmonic potential surfaces, with temperature-dependent solute-solvent coupling parameters represented as free enthalpies, that can be related to macroscopic, continuum, solvent polarization components.

The resultant bandshape is determined by the equilibrium solvent

repolarization resolved as a coordinate shift that can be evaluated as solvent polarization wave functions with an associated repolarization free enthalpy. Ulstrup defined a solute-solvent system harmonic Hamiltonian H sys , assuming no perturbation of the optical field by the solute-solvent system, with components describing 109

Chapter 2 the undistorted solvent H sol , the vacuum solute molecule, H mol , and the interaction between these, H s,m : H sys = H sol + H mol + H s,m

(2.177)

such that the system is suitably evaluated by a first-order quantum mechanical perturbation approach. This Hamiltonian can be further separated into solvent ( m ) and solute (ν ) components Ns H sys = H mNs + H vNs

(2.178)

where H mNs =

1 2

H vNs =

∑ κ 1 2

∑ l



(

ωκ  qκ − qκs 0 

)

2



 s Ωl  QCl − QCl 0 

(

)

∂2  + Fs 2 ∂qκ 

2



(2.179)

∂2  2  ∂QCl 

(2.180)

where H mNs and H vNs are the solvent and local (solute) molecular mode Hamiltonians in the ground or excited state s ( = g , e ), respectively; qκ and QCl the nuclear coordinates, q = µω

⋅ R ( q = qk , Qcl ; ω = ω k , Ωl ) where µ is the effective mass of the coordinate

and R is the real coordinate; ωκ and Ωl the solvent and local mode frequencies, respectively; and F s is the electronic free energy, incorporating the solvation free enthalpy, where s = g or e . Ulstrup then evaluated a trace integral for K (ν ) :



K (ν k ) = −i dθ e

− βθ ( ∆F0 − hν )k

( exp ( βθ H ) exp ( − βθ H )) Ng sys

C

where β = 1 kT .

110

Ne sys

(2.181)

Chapter 2 Integration of Eq. (2.181) with the Hamiltonians as defined in Eqs. (2.178) – (2.180) yields K (ν k ) =





−∞

dt exp  −i β t ( ∆F0 − hν k ) − Fm ( t ) − Fv ( t )  ; ∆F0 = F e − F g

(2.182)

The molecular local modes Fv ( t ) , directly related to the solute molecular structure, are expressed Fv ( t ) = Fv(1) ( t ) + iFv(2) ( t ) Fv(1) ( t ) =

∑ ( ∆Ω

) cth ( 12 β

Ωl ) sin 2 ( 12 β Ωl t )

2

Cl

(2.183) (2.184)

l

Fv(2) ( t ) =

∑ ( ∆Ω

1 2

Cl

)

2

sin ( 12 β Ωl t )

(2.185)

l

i ∆ΩCl = ΩClf 0 − ΩCl 0

(2.186)

The solvent modes are given in terms of the dielectric induction change, ∆D ( r ) = De ( r ) − Dg ( r ) brought about by intramolecular charge transfer, and dielectric

permittivity ε (ω ) , accounting for the continuum solvent vibrational frequency dispersion, as Fm( ) ( t ) = 1

2



ωc

βh 0

2 Fm( ) ( t ) =

f (ω ) =



ω

2

1 βh

β 2π 2

f (ω ) cth ( 12 β ω ) sin 2 ( 12 β ω t ) ωc



0

ω2



f (ω ) sin ( 12 β ω t ) 2

∫ ∆D ( r ) dr

Im ε (ω )

ε (ω )

2

where ωc is a cut-off frequency to prevent analytical divergences in f (ω ) .

111

(2.187)

(2.188)

(2.189)

Chapter 2 The above expressions indicate that the bandshape K (ν k ) is strongly influenced by a combination of structural changes and vibrational frequencies of the solute, as well as the solvent mediated dielectric dispersion and electric induction changes. This form applies both to the spectra of impurities in crystals and liquid solutes in solution. Near the absorption maximum in the limit of strong coupling of the solute electronic transition to solvent vibrational modes, the bandshape integral is solved by a saddle point method, recognizing that K (ν k ) is zero everywhere except at Re ( t ) = 0 , yielding79

K (ν k ) =

2π Fm′′ θ *

( )

12

( )

exp  − βθ * ( ∆F0 − hν k ) − Fm θ *   

(2.190)

The imaginary saddle point is described by79

t = −iθ ; ∆F0 − hν k = β *

*

−1



ωc

0

(

)

sh  12 β ω 2θ * − 1   f (ω )  1 ω sh ( 2 β ω )



(2.191)

An analytical form is given by Ulstrup in the high temperature limit, β ω m

1,

where Fm ( t ) reduces to

(

)

Fm ( t ) ≈ β Es t 2 − it , t

2 β ωm

(2.192)

The solvent reorganization energy, Es is given by ωc

Es =

1

β

∫ 0



ω

f (ω ) =

(

1 −1 −1 εo − ε s 8π

) ∫ ∆D ( r )

2

dr

(2.193)

where ε o is the optical dielectric constant, and ω m is the frequency range corresponding to meaningful values of f (ω ) .

For this limit, representative of solutions at room

112

Chapter 2 temperature and above, the bandshape is expressed, by substitution of Eq. (2.192) for Fm ( t ) into Eq. (2.182), as  ( hν − hν )2  2 π max  ; ∆ s = 2 Es kT exp  − K (ν ) = 2 β∆ s ∆s  

(2.194)

which is seen to be a Gaussian with a maximum at hν max = ∆F0 + Es and inhomogeneous width ∆ s that increases as

T , both of which are in accord with Marcus theory (Section

2.8.1). In this theory, both the absorption maximum hν m and the width ∆ s are critically dependent on the dielectric dispersion of the solvent vibrational frequencies and the solute-solvent coupling strength. The more general form of the Gaussian bandwidth, without invoking the high temperature limit, is

∆ s = ( 2 Es kT )

1/ 2

  



ωc

0

12

 ω  f (ω ) coth  12  dω   kT  

(2.195)

The bandshape function K (ν ) (Eq. (2.194)) is attenuated when there is additional coupling of the solvent to local (solute) modes of vibrational frequency Ωc and equilibrium nuclear displacement ∆ c , which Ulstrup has shown to be expressed 2π 1/ 2 kT K c (ν ) = ∆s

 ( hν − hν + n Ω )2  m c  Φ n exp  − 2   ∆s n =−∞   ∞



(2.196)

where   2   1 Ω 1 2 ∆c  1 Ωc    exp  Φn = I n  − ∆ c coth     1 Ωc    2 kT    2 kT 2 sh   2 kT     

113

(2.197)

Chapter 2 and I n is the modified order n Bessel function. This bandshape is a convolution of transitions between each of the local mode vibronic components in the ground and excited electronic states, each of which are broadened by the basis bandshape function given in Eq. (2.194). 2.14.1.1 Bandshape asymmetry in the spectral wings Ulstrup79 showed that the above bandshape functions represent a Gaussian limit corresponding to small relaxation times t in Eq. (2.182), which is associated with optical energies near the absorption maximum at hν ≈ ∆F0 + Es and the high-temperature limit, such that hν − hν max

Es β ω m .

This bandshape can be seen to depart from Gaussian upon expansion of the solvent modes Fm ( t ) to higher orders: Fm ( t ) ≈ i −

1 i 6



(β )

ωc

f (ω ) ⋅ t + 12 β



ω d ω f (ω ) ⋅ t −

1 24

ω

0

2





ωc

0

3

ωc

0

dω f (ω ) cth ( 12 β ω ) ⋅ t 2

(β )

3



ωc

0

ω dω f (ω ) cth ( β ω ) ⋅ t 2

1 2

(2.198) 4

Upon substitution of this higher order expression into the general bandshape integral (Eq. (2.182), a new integral emerges defined by a saddle point t * given by

(

hν k − hν km hν k − hν km * t = −2i + 3iξ3 β∆ 2 β∆ 3

)

2

− 4iξ 4

( hν

− hν km

k

β∆ 4

)

4

(2.199)

where ∆ = 2 Es kT and hν km = ∆F0 + Es in the high-temperature limit, and the constants

ξ3 and ξ 4 are derived from by the prefactors A1 , A2 , A3 , and A4 to t , t 2 , t 3 , and t 4 , respectively, in the integrals defining Fm ( t ) in Eq. (2.198):

(

) (

ξ3 = A3 A23 2 ; ξ 4 = A4 A22 + 94 A32 A23 The bandshape function is then given by 114

)

(2.200)

Chapter 2

(

 hν − hν m k k 2 π K (ν k ) = exp  − 2 β∆ ∆  

)

2

+ ξ3

( hν

k

− hν km

)

3

∆3

− ξ4

( hν

k

− hν km ∆4

)

4

  (2.201)   

Given that ξ3 is a positive, real value, it can be shown that this bandshape function incorporated the higher terms decays more gradually on the high-energy side vs. the low-energy side of the peak, as is typically observed, and this asymmetry becomes stronger as temperature is reduced and vibrational frequencies increase. In the spectral wings, Ulstrup79 has shown that the bandshape function evaluated by the saddle point method is given by K (ν ) =

β2

 hν − ∆F0  hν − ∆F0  2π exp  − − 1   ln ωc  ω c Fm ω c ( hν − ∆F0 )   

(2.202)

This bandshape is seen to exhibit an exponential as opposed to Gaussian shape, following Urbach tail behavior typically observed for nonradiative relaxation processes at optical frequencies far from the absorption maximum. 2.14.1.2 Weak electronic-vibrational coupling limit Ulstrup79 demonstrated that the weak solute-solvent coupling limit corresponds to the condition t

2 β ω m , giving the bandshape function K L (ν k ) =

2πβ −2 a

( ∆J fi − hν k )

2

2

−2 2

−1

f (ω )

+π β a

(2.203)

where a=

1

βh

f ′ ( 0 ) = lim ( β ω ) ω →0

(2.204)

This has the form of a Lorentzian peak shape with position hν max = ∆F0 and width ∆ L = π kTa ≈ kTEs

ωm

kT , so that broadening is expected to linearly increase

with temperature. The Lorentzian width π kTa in this case is seen to be much smaller 115

Chapter 2 than contributions from solvent vibrational frequencies.

From this and the above

discussion, the weak and strong solute-solvent coupling limits correspond to Lorentzian and Gaussian bandshapes, respectively, and bandshapes intermediate to Lorentzian and Gaussian can emerge for conditions of intermediate coupling. Further, these behaviors are consistent with a solvent medium that is disordered to such an extent that the density of states falls off slowly at low frequencies. 2.14.2 Experimental Bandshape Measurements for N-pyridinium Phenolates Kjaer and Ulstrup67 performed absorption spectroscopy on N-pyridinium phenolate chromophores (“betaines,” see Figure 2.13) with various substituents in solution to gain more detailed insights into molecular structural effects on bandshape. Note that betaine-1, containing no substituents on the phenolate, is an extreme case with no screening of the electron donating phenolate from the solvent; betaine-22, with CH3 substituents at 2- and 6- phenolate ring positions, can induce a twist of the Npyridinium ring with respect to the coplanar phenolate ring; betaine-29 has a long chain dodecamethylene group which can partially screen the phenolate ring from the solvent; betaine-26 has a large tert-butyl substituent on the phenolate ring that can significantly screen it from the solvent. The measured absorption bandshape features were examined in the context of solute-solvent interactions relative to both solvent and betaine substituent structure.

+ N

O



-

CH2

+ N

O CH2

12

12

Figure 2.13 Illustration of solvatochromic transition in N-pyridinium phenolates as given by Kjaer and Ulstrup67. Specifically shown is “betaine-29.” The analogue “betainel” contains no substituents on phenolate rings, whereas the analogue “betaine-22” contains a CH3 substituent at 2- and 6- phenolate ring positions (referencing the N-

116

Chapter 2 pyridinium ring atom). The analogue “betaine-26” has a tert-butyl substituent on the phenolate ring.

Each chromophore was studied in normal alcohol solvents, and consistent with Ulstrup’s bandshape model discussed above, the measured main absorption peaks for betaine-1, betaine-22, and betaine-29 are well described as a single solute vibrational mode and a Gaussian solvent band in accordance with the bandshape function for strong coupling (Eq. (2.196)), exhibiting a marked degree of asymmetry, falling off more sharply on the low-energy side, suggesting high frequency solute modes are strongly coupled to solvent polarization modes. In independent absorption measurements on betaine-26, having a tert-butyl substituent on the phenolate donor group, the strong coupling bandshape is not adhered to, and is best represented using two solute modes with a cubic correction. Linear regression fits of the experiment spectra to suitable bandshape functions provide the bandshape parameters ∆ s , Ωc , and ∆ c for each of the dye-solvent combinations. Kjaer and Ulstrup67 attributed the differences in bandshape properties between betaine-26 and the three remaining betaines (-1, -22, and –29) to extensive screening of the phenolate donor group by the bulky tert-butyl substituent in betaine-26, which is nonexistent in the unsubstituted betaine-1, and unmatched by that of the two methylene groups in betaine-22 or the long, flexible methylene chain in betaine-29. As a result, the bandshapes of the three chromophores with little or no screening by the phenolate substituents are expected to exhibit appreciable structural contributions from the unperturbed solvent, as observed in the measured bandshapes. On the other hand, the ground and excited state electronic wavefunctions of the largely screened phenolate donor group, in the case of betaine-26, experience the solvent only as a structureless, isotropic, dielectric continuum. 2.14.3 Urbach tail behavior in organic systems Mullins at al.154 investigated the optical absorption edges of a large assortment of crude oils and asphaltenes from a range of oil fields and thermal maturation treatments, representing a wide range of fractions, from heavy to light. 117

Chapter 2 The absorption spectra of crude oils are strongly influenced by the concentration, size, heteroatoms, chelation, and complex formation of aromatic groups. The aromatic content is known to vary appreciably among different crude oils, leading to vast differences in their spectra. The asphaltenes are in solid form and exhibit the greatest degree of aromatic content among crude oil fractions. The band position for crude oils typically varies with the average size of the aromatic compounds present. Urbach tail behavior was seen for all crude oils (Figure 2.14) and asphaltenes studied, with progressive variation in the band position with aromatic fraction, from the near-UV for gas condensates, to the visible for medium crude oils, and to the near-IR for heavy oils and asphaltenes. Using fluorescence spectroscopy measurements, the observation of Urbach tail behavior was shown by Mullins et al.154 to be associated with the lowest energy electronic transitions in crude oils, dominated by π − π * transitions of the aromatic species. Notably, the exponential widths of the Urbach tails Ee reported for these petroleum fractions were relatively constant (± 11%) between the two asphaltenes and 22 crude oils studied, despite the large variation in band position. Moreover, the Urbach widths for the crude oils (270 meV) and asphaltenes (350 meV) are much greater those of inorganic semiconductors and glasses (30 – 80 meV), and are approximately an order of magnitude greater than thermal tail widths ( kT ). The excess tail width is attributed to additional, temperature-independent structural disorder in the material over and above that due to thermal disorder. A monotonically increasing absorption with increasing photon energy, seen for all crude oils and asphaltene, is ascribed to a monotonically increasing distribution of smaller chromophores. Assuming equal oscillator strengths for all energies of transition, the Urbach tail can then be assigned to an exponential size distribution of chromophores, such that the number density of aromatic molecules decreases exponentially with increasing size. The similarity of the Urbach tail widths among all of the complex crude oil compositions suggests that the thermal maturation process yields nearly identical ratios of molecular sizes, all of the same distribution of proportional sizes. This behavior can be accounted for by a thermally activated, rate-limited process involving the formation of larger aromatic molecules from smaller molecules during the thermal

118

Chapter 2 maturation treatment, consistent with lower relative fractions of larger species for a given treatment. As a result, the optical absorption edge width is determined by the size distribution of species present in a given crude oil composition, which is overwhelmingly larger than the sample thermal edge width. The tail widths among the different crude oils are therefore attributed to the similarity of size advancement rate during maturation, although the extent of this advancement, and hence the band position, varies with the temperature and duration of thermal maturation. The peaks spanning ~ 4000 – 9500 cm-1 (near-IR region) in Figure 2.14 represent the anharmonic overtones of fundamental C–H stretch, bend, and combination vibrations. These overtones are determined primarily by the presence of saturated CH2 and CH3 moieties with very little contribution by the aromatic C–H groups. The positions of these are seen to be nearly identical for all crude oils, and the intensity of those not attenuated by the Urbach tails are very similar. The intensity of both the overtones and the regions between them is determined primarily by the position of the Urbach edge and relatively little by differences in composition for tails not extending into this region.

It can

therefore be seen that, for absorption edges approaching the near-IR, the near-IR absorption is largely controlled by the size distribution of aromatic chromophores rather than the specific details of the CH2 and CH3 groups present. Urbach tail behavior has been reported for conjugated, aromatic polymer structures, such as polymethylphenyl-silane189 and polyacetylene190.

119

Chapter 2

Figure 2.14 UV-Vis absorption spectra measured by Mullins et al. for 22 different crude oils (from reference 154).

2.15 Theory and measurements of disorder and π − localization in linear organic polymers As was provided in the discussion of amorphous inorganics, we provide here a survey of studies of disorder and defect states by spectroscopic techniques on organic polymers, showing relationships between structural disorder and optical band properties for materials more closely related to those actually examined as part of this study. While these materials do not possess the extensive network structure with localized order approaching three-dimensional crystals, as is the case for vitreous glasses discussed in the previous section, they do possess mixed σ , π bonding states, and frequently possess short- or long-range one- or two-dimensional order. As we will see, studies of disorder of these materials examine many of the key features of band structure, disorder, and defect states that were described for amorphous inorganics in the previous section. The solid state of linear polymers may be described by a structural defect concept based on the metastable state normally found in polymers191. Two principle types of defects in molecular crystals of high polymers are chain disorder and amorphous defects. Examples of chain disorder are folds and imperfect alignment, while amorphous defects are typified by extensive chain disorder and three-dimensional imperfections, and has been characterized as a structural component, such as side chain, that decreases the crystallinity of the lattice by an amount larger than its own size or extent191. Most of the 120

Chapter 2 remaining discussion addresses defects associated with π bonding states of polymers and chromophores containing aromatic groups or π -conjugated segments as a significant fraction of their structures.

These defects relate to π delocalization and energetic

disorder of the π -state distribution. The family of highly π -conjugated polymers exhibiting either conducting or semiconducting properties, including poly(acetylene) (PA), poly(para-phenylene) (PPP), poly(para-phenylene vinylene) (PPV), or poly(alkyl-thiophene), possess defect and band behavior that occur in the limits of highly ordered local domains or even extended crystalline domains. These classes of polymers display luminescent behavior and tend to exhibit sharp optical band edges (compared with amorphous polymers), both of which can be linked to their defect structure within ordered domains. As such, the optical behavior of polymers appears to become more sensitive to the disorder present in the system as the degree of ordering increases. Thus, while none of the aromatic polymers examined in this study possess the degree of short-range or long-range order present in the highly conjugated semiconducting and conducting polymers, their band structure appears to follow a general sharpening with chain stiffness and apparent localized ordering, and it is postulated that defect states and extents of disorder become larger contributions to the optical spectra of the subject polymers with increased ordering, even for systems that are macroscopically amorphous. This supposition is closely aligned to treatments and measurements of band structure and defect states in amorphous inorganic materials, discussed in the previous section. The following review discusses previous work on optical behavior of highly conjugated polymers, from which analogies may be drawn for ordering and band structure trends in rigid, amorphous, aromatic polymers doped with highly conjugated, rigid chromophores, which were the subject of study in a series of polymers discussed in Chapter 6. To evaluate absorption and emission behavior of poly(para-phenylene vinylene) (PPV) polymers and model oligomers, Meskers and co-workers70 investigated the effect of disorder on the extent of delocalization of the π → π * photoexcitation over an aggregate of chromophore molecules arranged on a two-dimensional lattice.

121

Chapter 2 An excited state interaction energy V12 of a solvent molecule (1) with nearest neighbor chromophore solute (2) was defined, and the disorder was treated by assigning a Gaussian distribution of transition energies Ei of individual chromophores, of inhomogeneous width σ = D (diagonal disorder). Transition energies of the aggregate were obtained by diagonalizing the Hamiltonian matrix (Eq. (2.124)), and relative diagonal disorders D V12 over the range 5 to 0.375 were modeled. These models showed that at the low energy edge of the absorption band, localized photoexcited states can exist, consistent with widely reported experimental studies indicating strong exciton coupling between aggregates of π -conjugated oligomeric molecules. Measurements of the absorption band shape of these highly rigid polymers demonstrated that defects and disorder readily perturb interchain delocalization. Fluorescence measurements on these polymers revealed appreciable disorder (high D V12 ) for cases of excited states that are not delocalized over a large number of chains, and the results were consistent with a hopping mechanism for localized states, with significant delocalization of states between parallel, oriented chains. These fluorescence measurements also demonstrated migration of photo-excited states between disorderinduced trap sites lying in the low energy tail of the density of excited states. Low observed disorder in model oligomers was associated with delocalization of the photoexcitation over several molecules. For one-dimensional chromophore aggregates, strong intermolecular interactions V12 were found to correspond with changes in bandshape. Localization of photoexcited states and attendant disorder-induced broadening can be induced by variations of these interactions. Delocalization and reduced disorder-induced broadening is predicted when chromophores within the aggregate are in parallel alignment and exhibit finite intermolecular interactions (V12 > 0 ). Moreover, varying degrees of partially localized states are predicted for intermediate extents of disorder in these aggregates. Scherf and List192 examined the relationships between alkyl substitution pattern, solid state morphology, and optical and electronic properties in the polyfluorene (PF) class of rigid-rod and semi-rigid π -conjugated aromatic polymers (Figure 2.15a), exhibiting both ordered ( β ) and amorphous ( α ) phase behavior, and compared these 122

Chapter 2 with the closely related ladder-type poly-paraphenylene polymers (LPPP) (Figure 2.15b), each based on the fluorene monomer. These polymers exhibit strong electroluminescence (EL) and photoluminescence (PL), making them appealing for display applications. In general for highly aromatic polymers or supramolecular structures, the extent of π conjugation (or intramolecular delocalization) sensitively affects the electronic and optical properties, and this can be disrupted by structural defects, substituents along the chain, or out-of-plane distortion of the aromatic groups. The polyfluorenes and PPPs are favorably structured, with 9,9-substitution of the fluorene moiety, for conjugated polymers with supramolecular packing and cooperative behavior, including liquid crystallinity.

R1

R1 R1

a)

n

b)

R1

R3 R2

R2 R3

R1

R1

n

Figure 2.15 Structures of rigid aromatic polymers studied by Scherf and List 192: a) 9,9 dialkyl polyfluorene; b) planarized poly(paraphenylene)-type ladder polymers (LPPP, R1: -alkyl; R2: -aryl; R3: -methyl, -phenyl, or -H)

While the length and structure of the alkyl side chains in 9,9-dialkyl-PFs have small effects on optical and electronic behavior in dilute solution, they greatly influence packing behavior and optical properties in the solid state, even leading to crystalline ( β phase) formation for n-alkyl side groups due to side-chain ordering. Replacing n-alkyl side groups with branched side groups disrupts this ordered packing arrangement. Narrowly distributed ethylhexyl di-substituted PFs prepared by fractionation, display wormlike chains with a low persistence length (approximately 7 nm), which is a manifestation of the kinked and distorted configuration of the fluorene repeat unit192. The performance of devices fabricated from organic polymers, such as LEDs or photovoltaics, critically depends on density, energetic position and dispersion, and trap state of defects within and between the π − conjugated chains. Defect states in PFs,

123

Chapter 2 occurring as preferred centers for oxidation or photooxidation acting as trap sites for optical transitions or excitons, can be generated by small amounts of non-alkylated or only monofunctionalized monomers, and nearly defect-free materials are associated with a PF structures entirely free of benzylic hydrogens192. A red-shift and well-resolved vibronic progression of convolved solvent-mode peaks are seen in both the absorption and emission peaks of the β -phase of PF, while for the amorphous α -phase, the vibronic progression is well resolved for the emission spectrum only. By comparison, LPPP, which has a fully planarized main chain, displays absorption and emission behavior akin to the β -phase PF, suggesting that during the PF β -phase transition, the distorted and kinked backbone becomes planarized. PF films containing both α and β phases display a boost in polaron density upon increase of the energetic disorder of the bulk material. In contrast, polaron states in PFs with branched alkyl chains are low, owing to the relatively low trap state density and energetic disorder in these materials192. Quantum-mechanical calculations by Scherf and List using the intermediate neglect of differential overlap (INDO) Hamiltonian combined with a configuration interaction (CI) approach on keto-defect-containing monoalkyl-PF have illustrated that prior to exciton recombination, the fluorenone excited states are localized. Conversely, for dialkyl-PF, which can be synthesized as keto-defect-free materials, delocalization of the excited state over many fluorene units is predicted, and has been shown experimentally to produce PL emission. In monoalkyl-PF, a lower energy PL emission band forms due to the presence of the keto-defect sites, suggested by Scherf and List to behave as traps for singlet excitons on the PF chain arising from a dipole-dipole induced excitation energy migration (EEM)-assisted energy transfer process. These keto-defects can be incorporated as guests into the PF backbone during synthesis of monoalkyl-PF, or can be induced by photooxidative processes in monoalkyl- or dialkyl-PFs192. The shape of both the ground and excited state energy bands have been shown to be parabolic and very similar in LPPP and β -phase PF polymers, as a result of their highly related backbone structures. The absorption spectra of dilute solution and α phase PF (with branched alkyl side groups), however, show an appreciable difference in the parabolic shape of their ground and excited state bands, reflecting a marked change in 124

Chapter 2 configuration upon electronic transition for PF segments apparently isolated by disruptive side groups. The solid α -phase PF absorption band experiences broadening brought on by conformational disorder induced local deviations of π -overlap and delocalization192. Silbey et al.193 performed theoretical predictions using a modified neglect of differential overlap (MNDO) technique, with a band structure determined by a nonconsistent pseudopotential calculation employing a valence effective Hamiltonian (VEH), showing that the polyarememethide class of rigid aromatic conjugated polymers have degenerate bonding patterns in the ground state that support the formation of a neutral soliton (radical) defect. The defect site occurs at a –CH bridge between two phenyl rings, and has a formation energy of 0.33 eV, corresponding to half the band gap (0.53 eV, vertical excitation about the Fermi level) minus a configurational relaxation about the soliton site, reducing the total defect energy by 0.2 eV. The relaxed configuration is a quinoidal structure forming along the rings near the defect site, yielding allyl-like distortions of the resonant geometry almost totally localized to four aromatic rings. Further, a polaron (radical cation) defect can be introduced into the chain by removing one electron from the π -system, causing local lattice distortion (over five rings) due to coupling between the charged (polaron) and uncharged (soliton) defect. The polaron binding energy was calculated as 0.02 eV, comparable to those determined for polyacetylene (0.05 eV) and polyparaphenylene (0.03 eV). The localized spin density for isolated neutral solitons was calculated as 0.45

within six rings, while for interacting

soliton-polaron defects, the spin density was calculated as 0.395

distributed over seven

rings. Nakayama et al.189 used photothermal deflection spectroscopy (PDS) to measure the absorption edge of poly(methylphenyl-silane) (PMPS) films, known to exhibit both photoluminescence,

UV-induced

photodegradation,

and

upon

C60

doping,

photoconductivity . The absorption spectrum of PMPS displays a sharp Urbach tail following the classical exponential expression

α = α 0 e E Ee

(2.205)

where Ee was measured as 30 meV, representing a narrow exponential distribution. This band tail is ascribed to the so-called “energetic disorder” of the silane backbone, 125

Chapter 2 associated with the distribution of π -conjugated segment lengths and conformations, and is expected to significantly influence the carrier mobility. Low values of absorption coefficient in near-IR region are assigned to a low mid-gap density of defect states. The PDS spectrum is expected to be predominately due to the band tail of the HOMO in this system, given a steeper expected band tail of the LUMO. Varying the film exposure time to 325 nm UV light from 0 to 5000 sec. resulted in a gradual increase in the Urbach tail width (from 30 to 210 meV), due to photoscission of σ bonds leading to an increased density of σ defects. Weinberger et al.190 performed PDS measurements of doping-induced subgap absorption in poly(acetylene) (PA). Note that the absorption measurements at α < 104 cm-1 (regions B and C, over photon energies from 0.7 – 2.0 eV) taken by conventional transmission measurements were considered inaccurate, even when taking the specular reflectance into account. The measured absorption spectra of these materials followed classical Urbach tail behavior, with measured exponential widths Ee of ~70 meV, similar to that of amorphous semiconductors. The measured doping-induced subgap absorption is ascribed to transitions from neutral solitons formed around bond-alternation kinks. This behavior was not observed in either as-grown or NH 3 -compensated PA films, as confirmed by ESR measurements. Weinberger et al. associate the Urbach edge of PA with either photoexcited soliton pairs (as proposed by Heeger and MacDiarmid) or transitions from disorder-induced band tails.

2.16 Organic polymer and chromophore property prediction methods In this section we provide descriptions of quantitative structure-property relation (QSPR) correlation methods for polymer properties, following the generalized approach by Bicerano, and semi-empirical quantum mechanical calculations for organic NLO chromophore molecular and electronic properties following an accepted approach by Zerner. 2.16.1 Bicerano hydrogen-suppressed graph model QSPR methods for polymer properties

126

Chapter 2 Bicerano194 developed a highly correlated quantitative structure-property relationship prediction method for a wide range of properties of polymers, stemming from a hydrogen-suppressed graph topology model that incorporates the connectivity of atoms and bonds of the repeat unit, with corrections for special groups or structures to improve overall accuracy. The technique involves summation of the contributions from individual atoms and bonds of a polymer repeat unit, rather than by functional groups, as is the practice in the more conventional group contribution methods, and can be used to predict the properties of polymers containing one or more of the elements C, H, N, O, F, Si, S, Cl, and Br. The general form of the property correlations is given by:

Property =

( ∑ a χ ) + ( structure parameters ) + ( atomic group correction terms ) (2.206)

where a's are empirical correlation constants and χ 's are the connectivity indices of the atoms and bonds.

Four sets of connectivity indices are unambiguous and readily

computed from the repeat unit structure, given as the zeroth-order (atomic) electronic and valence indices,

0

χ and 0 χ v , respectively, and the first-order (bond) electronic and

valence indices 1 χ and 1 χ v , respectively. The atomic connectivity indices are defined as:

 1   , the zeroth-order (atomic) connectivity index, electronic (2.207) vertices  δ 

0

χ=

0

χv =



 1   , the zeroth-order (atomic) connectivity index, valence (2.208) v  vertices  δ 



where δ , the simple connectivity index, is defined as the number of nonhydrogen atoms bonded to a nonhydrogen atom at a vertex, and δ v is the “valence connectivity index,” defined as

127

Chapter 2

δv =

Z v − NH Z − Z v −1

(2.209)

where Z is the atomic number of the vertex atom, Z v is the number of valence electrons of the vertex atom, and N H is the number of hydrogen atoms bonded to the vertex atom. The bond connectivity indices are given by 1

χ=

1

∑ 

 1  , the first-order (bond) connectivity index, electronic edges  β 

(2.210)

 1  ∑  v , the first-order (bond) connectivity index, valence edges   β 

(2.211)

χv =

where the bond parameters β are

β ij = δ iδ j

(2.212)

β ijv = δ ivδ vj

(2.213)

for i th and j th adjacent atoms. An example of assignment of these indices is shown for a polyamic acid-based polyimide in Figure 2.16. Bicerano’s correlation expressions194 for prediction of polymer refractive index and for molar volume illustrates the calculation method:

Refractive index: n = 1.885312 +

0.024558 (17 0 χ v − 20 0 χ − 12 1 χ v − 9 N Rot + N Ref ) (2.214) N

where N Rot = Number rotational degrees of freedom N = Number vertices (atoms)

and 128

Chapter 2

129

Chapter 2

N Ref = −11N F − 3N Cl bonded to aromatic ring + 18 N S + 9 N fused ring + 12 N hydrogen bonds + 32 NSi-Si (2.215) Thus, the prediction of polymer refractive index involves calculating zero-order indices 0 χ and 0 χ v and first-order index 1 χ v from the repeat unit structure, counting the numbers of rotational degrees of freedom, fluorine atoms, chlorine atoms bonded to aromatic rings, fused rings, hydrogen bonds, silicon-silicon bonds, and total number of atoms in the repeat unit, calculating N ref , and substituting all quantities into Eq. (2.214) above.

Molar volume: V = 3.642770 + 0 χ + 9.798697 0 χ v − 8.542819 1χ + 0.978655N MV

(2.216)

where N MV = 24 NSi − 18 N( -S-) − 5 Nsulfone − 7 N Cl − 16 N Br + 2 N backbone ester

+ 3N ether + 5 N carbonate + 5 N C=C − 11N cyclic − 7 ( N fused − 1)

(2.217)

(last term used only when N fused ≥ 2 ) The molar volume is therefore predicted by calculating the zero-order indices 0 χ and 0 χ v and first-order index 1 χ v from the repeat unit structure, counting the numbers silicon atoms, thiol sulfur atoms, sulfur atoms as sulfone, chlorine atoms, bromine atoms, ether groups, carbonate groups, carbon double bonds, cyclic atoms, fused rings, and backbone ester groups in the repeat unit, calculating N MV , and substituting all quantities into Eq. (2.216) above. The properties of copolymers predicted by this method use the weighted averages of all extensive properties and specific functional forms for intensive properties expressed in terms of the extensive properties.

130

Chapter 2

Figure 2.16 Schematic illustration of assignment of atom vertices and bond edges in Bicerano hydrogen-suppressed graph topology model194. Shown is repeat unit structure of a polyimide. This QSPR technique has predictive capability with the versatility and accuracy to predict the properties of a broad spectrum of polymers comprised of arbitrary or exotic backbone structures within the nine-element set listed above, which are not handled by conventional group contribution methods. These properties range from fundamental parameters used to assess processibility, such as viscosity and solubility, to performance characteristics such as optical, electrical, thermal and mechanical properties.

The

technique has been codified, incorporating all of the empirical correlation constants, in the commercial software package SYNTHIA in CERIUS2 from Accelrys, Inc. 2.16.2 Seitz method of predicting polymer elastic modulus

Semi-empirical QSPR relationships for the prediction of polymer elastic properties were developed by Seitz195 starting from thermodynamic first principles, requiring only the repeat unit cohesive energy U coh , molar volume Vm , and length lm , and the polymer glass transition temperature Tg as input parameters.

Using group

contribution methods, as has been the usual practice, tables of fragments must be developed and correlated for each property and class of polymers of interest, without regard for relationships molecular properties and macroscopic properties. Thermodynamic equations of state, involving pressure, temperature, volume, and internal energy, are used to relate the polymer molecular properties to the elastic properties. Of relevance to determination of the mechanical properties are 131

Chapter 2  ∂S   ∂U  P=  −   ∂V T  ∂V T

(2.218)

 ∂S   ∂P    =  = Tα T B  ∂V T  ∂T V

(2.219)

where U is the internal energy, S is the entropy, and P , V , and T are pressure, volume, and temperature, respectively, and the coefficient of thermal expansion αT and bulk modulus B are expressed

αT = 1 V ( ∂V ) ( ∂T )  P

(2.220)

B = −V ( ∂P ) ( ∂V ) 

(2.221)

T

For small elastic deformations below the glass transition temperature at constant temperature, the entropy is nearly constant and  ∂U  P = TαT B −    ∂V T

(2.222)

 ∂U  At P = 0, TαT B =    ∂V T

(2.223)

Using the experimentally derived relationships between the molar volume and the van der Waals volume of polymers at the glass transition temperature and at 0 K, where the van der Waals volume is the virtually impenetrable volume occupied by a molecule, an empirical volume-temperature relationship for amorphous polymers in the glassy state has been shown by Seitz195 to be

  T Vg =  0.15 + 1.42  Vw   Tg  

(2.224)

The coefficient of thermal expansion below the glass transition is then found by differentiating the above equation: 132

Chapter 2

αT , g =

1 dVg 1 = Vg dT T + 9.47Tg

(

)

(2.225)

The elastic properties of amorphous, isotropic glasses possessing randomly oriented bonds are determined by rotational and intermolecular potentials. The internal energy U can be derived from an expression for the Lennard-Jones intermolecular potential:

  V 2  V 4  U = U 0 2  0  −  0     V   V  

(2.226)

where V is the molar volume, V0 is the molar volume at the minimum in the potential well, and U 0 is the depth of the potential well. Substituting into the equation of state (Eq. (2.222)) for pressure below the glass transition temperature gives 2 4 4U 0  V0   V0    ∂S  P= −    − V  V   V    ∂V T

(2.227)

The entropy term at zero pressure and constant temperature is 4U 0  ∂S    = VT  ∂V T

 V  2  V  4   0  −  0    VT   VT  

(2.228)

where VT is the molar volume at a reference temperature T , yielding

  1  1 1  1  P = 4U 0 V02  3 − 3  − V04  5 − 5   VT   VT V     V

(2.229)

Differentiation of this expression with respect to V in Eq. (2.221) gives the expression for bulk modulus as

133

Chapter 2

B=

4U 0 VT

2   V 4  V0   0 5 3 −    V        V 

(2.230)

Experimental pressure-volume data196 can be used to determine U 0 , and V0 is taken to be the molar volume of the glass at 0 K (=1.42Vw ). The elastic constant of interest, for investigations of rigid aromatic polymers in this study, is the tensile modulus E , related to the bulk modulus B as E = 3B (1 − 2ν )

(2.231)

The Poisson ratio ν can be related to the elastic strain as



V

ε = dε =

1

∫ (1 − 2ν )V dV

(2.232)

VT

For a unidirectional stress, the tensile modulus is found by substituting this strain equation and Eq. (2.230) for the bulk modulus into Eq. (2.231):

 V4 V2 E = 24 (1 − 2ν ) U 0 5 05 − 3 03  Vε   Vε

(2.233)

where Vε , the thermally induced volume-dependent strain, estimated from the coefficient of thermal expansion using Eq. (2.225).

The estimation of Poisson ratio uses an

empirical model based on the assumption that a unidirectional tensile modulus would varies with cross-sectional area normal to the stress direction as

ν = 0.5 − kν A

(2.234)

where A is considered the molecular cross-sectional area of a cylinder represented by the van der Waals volume of the repeat unit of length lm in its all trans configuration: A=

Vw N A lm

134

(2.235)

Chapter 2

where N A is the Avagadro number. Experimentally, Poisson’s ratio has been shown by Seitz195 to follow:

ν = −2.37 ×106 A + 0.513

(2.236)

A more convenient quantity representing the intermolecular potential is the cohesive energy U coh , which can be accurately predicted using other quantitative structure-property relationship (QSPR) models.

Seitz found experimentally for a

collection of 18 polymers that the average of the ratio of U 0 U coh was 2.06. Bicerano194 gives the following highly correlated relation for the cohesive energy U coh (with correlation coefficient 0.95) spanning a diverse set of 124 different polymers:

( )

(

U coh = 9882.5 1 χ + 358.7 6 N atomic + N group

)

(2.237)

where N atomic ≡ 4 N( -S-) + 12 N sulfone − N F + 3N Cl + 5 N Br + 7 N cyanide

(2.238)

N group ≡ 12 N hydroxyl + 12 N amide + 2 N non-amide ( -NH- ) unit  − N( alkyl ether -O-) − N C=C 



+ 4 N non-amide -( C=O )- next to nitrogen  + 7 N -( C=O )- in carboxylic acid, ketone or alehyde (2.239) 







+ 2 N other -( C=O )- + 4 N( nitrogen atoms in six-membered aromatic rings ) 



With use of Eqs. (2.225), (2.233), (2.235), (2.237), and (2.236), and making the substitution U 0 = 2.06U coh , the semi-empirical relationship for room temperature tensile modulus (for amorphous polymers exhibiting Tg

25°C ), in terms of the polymer repeat

unit properties U coh , Vm and lm , and the polymer Tg , is found from

135

Chapter 2

 Vm E ( Pa ) = 24.2 × 10 U coh 1 − 47.4 × 106  N A lm  6

   T  × 5     9.47T + Tg

4

  T  − 3    9.47T + Tg

  

2

  

(2.240) where Vm is the molar volume of a repeat unit. Seitz195 calculated the values of tensile modulus for the series of 18 polymers, which are seen to correlate well with experimentally determined tensile moduli in Figure 2.17. The semi-empirical QSPR method of Seitz for calculating polymer elastic modulus is codified, with all empirical correlation constants and QSPR methods of calculating U coh , Vm , lm , and Tg , in the commercial software package SYNTHIA in CERIUS2 from Accelrys, Inc.

Experimental Tensile Modulus (GPa)

5 4 3 2 1 0 0

1

2

3

4

5

Calculated Tensile Modulus (GPa)

Figure 2.17 Experimental vs. calculated values of polymer tensile modulus using the semi-empirical tensile modulus relationship developed by Seitz (from reference 195).

136

Chapter 2 2.16.3 ZINDO semi-empirical quantum mechanical model for chromophore molecular properties

Zerner et al.197 developed a linear combination of atomic orbitals – molecular orbital – self-consistent field – configuration interaction (LCAO-MO-SCF-CI) model as an extension of a model developed by Delbene and Jaffe, originally to predict the electronic spectra of pyrrole and azines. The method is referred to variously as INDO/1 for intermediate neglect of differential overlap, Version 1, or as ZINDO, for ZernerINDO. The basic approach is to find solutions of the molecular electronic Hamiltonian Hˆ ψ = Eψ

(2.241)

where ψ is a wavefunction corresponding to all n valence electrons of the system, approximated as a single Slater determinant of the molecular orbitals {φi } (MO’s):

ψ = φ1 (1) φ2 ( 2 ) ⋅⋅⋅ φn ( b )

(2.242)

The restricted Hartree-Fock (RHF) procedure is applied to solve this for a closed shell system, while an unrestricted Hartee-Fock (UHF) is used for open shell systems. Following the Roothaan-Hall procedure, the MO φi is expanded as a linear combination of atomic orbitals

φi =

∑ C α χα i

(2.243)

where { χα } is the orthogonal basis set of atomic orbitals (AO’s) used within the INDO approximation. In Zerner’s INDO/1 treatment, one-center exchange integrals are used to separate different terms from within a given configuration, and the one-center core integrals U µµ are obtained from ionization potentials. Nuclear attraction integrals are evaluated as VAB = Z Aγ AB , where γ AB is the appropriate two-center Coulomb integral between atoms A and B . Core integrals U µµ are determined from

137

Chapter 2

(

Z

U µµ = µ − 12 ∇ 2 − RA + V µ A

)

= I µ − ( Z A − 1) F 0 ( ss ) + 61m G1 ( sp ) for s AOs

(2.244)

2 = I µ − ( Z A − 1) F 0 ( ss ) + 61m G1 ( sp ) + 25 ( m − 1) F 2 ( pp ) for p AOs

where V is a pseudo-potential compensating for neglected inner shells, µ is the orbital centered on atom A , G1 ( sp ) and F 2 ( pp ) are semi-empirical Slater-Condon factors, and I µ is the ionization potential corresponding to the process s l p m → s l −1 p m + ( s ) or s l p m → s l p m−1 + ( p ) I µ , G1 ( sp ) , and F 2 ( pp ) are found in atomic spectral tables for hydrogen, carbon, and nitrogen. For a closed shell system, the INDO approach is as follows:

FC a = ε a C a

(2.245)

where F is the two-center Fock matrix, C a is the coefficient of MO a , and ε a the MO eigenvalue. If P is the charge and bond order matrix in the orthogonal basis set, the Fock and first-order density matrix elements are expressed as Fµµ = U µµ +

A

Pσσ ( µµ σσ ) − ( µσ µσ )  + ∑ ( P ∑ σ 1 2

BB

− Z B )γ AB , µ ∈ A (2.246)

B≠ A

Fµν = 32 Pµν ( µν µν ) + 12 Pµν ( µµ νν ) , µ ,ν ∈ A

(2.247)

Fµν = S µν ( β A + β B ) 2 − Pµν γ AB 2, µ ∈ A, ν ∈ B

(2.248)

Pµν = 2

MO

∑ Cµ Cν a

a

(2.249)

a

PAA =

A

∑µ Pµµ

138

(2.250)

Chapter 2

( µν σλ ) = ∫ dτ1dτ 2 χ *µ (1) χν (1) r1 χσ* ( 2 ) χ λ ( 2 )

(2.251)

γ AB = ( µµ νν ) , µ ∈ A , ν ∈ B

(2.252)

12

where µ is taken as the µ th AO of hypothetical s symmetry, β i is a bonding parameter descriptive of atom i , and S is a weighted overlap integral defined in terms of the ordinary overlap integral S as S ss′ = S ss′ = ( χ s χ s′ )

(

S sp′ = S sp′ = χ s χ p′

)

(

)

(

S pp′ = fσ Gσ S pσ p′σ + fπ Gπ S pπ p′π = fσ Gσ χ p χσ χ p′ χσ + fπ Gπ χ p χπ χ p′ χπ

) (2.253)

Gσ and Gπ are geometric coefficients used in a coordinate transformation from the local diatomic reference frame to the molecular reference frame.

fσ and fπ are empirical

scaling factors used to fit the model to experimental spectral measurements. The first row atoms consist of five different types of one-center exchange integrals, which are calculated using the empirical Slater-Condon factors:

( ss ss ) = ( ss pp ) = F 0

(2.254)

( ss pp ) = 13 G1 ( sp )

(2.255)

( px px

4 px px ) = F 0 ( pp ) + 25 F 2 ( pp )

(p p

2 p y p y = F 0 ( pp ) − 25 F 2 ( pp )

x

x

)

(p p x

y

)

px p y =

139

3 25

F 2 ( pp )

(2.256) (2.257)

(2.258)

Chapter 2 Calculation of spectra proceeds with determination of the ground state, producing molecular orbital coefficients and eigenvalues, followed by a configuration interaction calculation. A closed-shell (singlet) is used for organic compounds, and the “pure” configurations are produced by any MO’s with unassigned electrons (virtual orbitals). Singlet-singlet transition energies between pure configurations can be expressed ∆Eia = ε a − ε i − J ia + 2 Kia

(2.259)

where ε i and ε a are energies of orbitals i and a respectively, and the molecular Coulomb integral J ia is expressed



J ia = φi* (1) φi (1) φa* ( 2 ) φa ( 2 )

1 dτ 1dτ 2 = ( ii aa ) r12

(2.260)

The molecular exchange integral Kia is given by



K ia = φi* (1) φa (1) φa* ( 2 ) φi ( 2 )

1 dτ 1dτ 2 = ( ia ai ) r12

(2.261)

Within the INDO approximation, the molecular coupling integrals are expressed

( ij kl ) = ∑ Ciα C jβ Ckγ Clδ (αβ γδ ) δ ABδ CDδ AC + ∑ Ciα C jα Ckγ Clγ δ AC ∆αγ α ≠β γ ≠δ

+

αγ

∑ C α C β C γ C δ (αβ γδ ) i

j

k

(2.262)

l

αβγδ

where i, j , k , and l represent MO’s, α , β , γ , and δ represent atomic orbitals (AO’s), and ∆αγ ≡ (αα γγ ) − F 0 (αβ )

(2.263)

where F 0 is a semi-empirical Slater-Condon factor: F 0 = ( ss ss ) = ( ss pp )

140

(2.264)

Chapter 2 The first summation in the above expression accounts for the case of AO’s

α , β , γ , and δ located on the same center. Similarly, the second summation covers the case of AO’s α and γ occurring on the same center. The third term sums to zero if α and γ are a π and σ orbital, respectively. The diagonal elements of the configuration interaction (CI) Hamiltonian are given by the transition energies determined by these expressions. The off-diagonal CI Hamiltonian elements are expressed as

ψ 0 H 1ψ i→a = 0

(2.265)

ψ i→a H 1ψ j →b = 2 ( ai jb ) − ( ab ij )

(2.266)

1

1

The Coulomb integrals γ AB are parameterized following a modification of the Mataga-Nishimoto procedure:

γ AB =



2 fγ (γ AA + γ BB ) + RAB

(2.267)

where RAB is the distance (in Bohr radii) between the two centers, fγ is an empirical parameter set to 1.2, and

γ AA = F 0 ( AA ) = I A − AA

(2.268)

In evaluation of the Fock matrix elements Fµν = S µν ( β A + β B ) 2 − Pµν γ AB 2 Zerner first fit the product fπ β C to the experimental benzene absorption spectrum, followed by determination of uncoupled empirical parameters fπ , fσ , β C , and β N by fitting to experimental pyridine spectra. The value of β H was finally obtained by fitting to a combination of experimental spectra for benzene, pyridine, and pyrrole. The ZINDO model now serves as a semi-empirical electronic structure program parameterized for the spectroscopic properties of molecules. This method can be applied to a wide range of molecules including organic, inorganic, polymers, organometallics, metal clusters, and biological compounds. The model includes techniques for optimizing

141

Chapter 2 the geometry of the molecule, providing information regarding the valence electron density distribution within the system. ZINDO includes two valence-electron-only semi-empirical

procedures:

one

for calculating low-lying π − π * and n − π *

spectroscopic bands of molecules containing hydrogen and first- and second-row elements, and one dedicated to computing molecular conformations and structures. Current versions provide a variety of model Hamiltonians and electronic structure methods that, depending upon the desired property, can be chosen by the user, including EHT, CNDO/1, CNDO/2, INDO/1, INDO/2, SCF, PPP, and Extended Hückel. One can explore either ground state or excited state properties including UV-visible spectra, stationary geometries of molecules, or transition states of chemical reactions. ZINDO computes semi-empirical quantum mechanical values for molecules to compute the frontier molecular orbitals and orbital energies and coefficients, total energies of molecules, dipole and quadrupole moments, electron density distributions, partial charges, bond orders, ionization potentials, electron affinities, transition states, frequency-dependent molecular polarizabilities and hyperpolarizabilities, and grid data for constructing charge density, spin density, and molecular orbital contours. ZINDO also incorporates the self-consistent reaction field model to represent complex effects of solvation, and provides for application of an electric field and point charges. Owing to the approximate nature of the starting Hamiltonians, ZINDO calculations are carried out in a small fraction of the time needed to perform ab initio computations. ZINDO in Accelrys, Inc. Cerius2 is the most up to date version supplied by the Quantum Theory Project (QTP) of the University of Florida, and is parameterized for the first 30 elements (except for the noble gases He, Ne, and Ar), first- and second-row transition elements, and Br and I, to a limit of 250 atoms and 1000 basis functions. The accuracy of UV/visible spectra calculated by ZINDO generally increases with the size of the system being modeled.

ZINDO uses molecular computational science and

technology that is well-validated by a large volume of scientific publications.

142

Chapter 2

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1993, 5, 2533. (147) Chew, K.; Rusli; Yoon, S. F.; Ahn, J.; Zhang, Q.; Ligatchev, V.; Teo, E. J.; Osipowicz, T.; Watt, F. Journal of Applied Physics 2002, 91, 4319. (148) Dasgupta, D.; Demichelis, F.; Pirri, C. F.; Tagliaferro, A. Physical Review B: Condensed Matter and Materials Physics 1991, 43, 2131. (149) Tauc, J. in Amorphous and Liquid Semiconductors; Tauc, J., Ed.; Plenum Press: London, 1974. (150) Mott, N. F. Philosophical Magazine 1970, 22, 1. (151) Mott, N. F. Philosophical Magazine 1969, 19, 835. (152) Zhou, W.-y.; Xie, S.-s.; Qian, S.-f.; Zhou, T.; Zhao, R.-a.; Wang, G.; Qian, L.-x.; Li, W.-z. Journal of Applied Physics 1996, 80, 459. (153) Urbach, F. Physical Review 1953, 92, 1324. (154) Mullins, O. C.; Mitra-Kirtley, S.; Zhu, Y. Applied Spectroscopy 1992, 46, 1405. (155) Wood, D. L.; Tauc, J. Physical Review 1972, 5, 3144. (156) Dow, J. D.; Redfield, D. Physical Review 1972, 5, 594. (157) Teo, K. B. K.; Ferrari, A. C.; Fanchini, G.; Rodil, S. E.; Yuan, J.; Tsai, J. T. H.; Laurenti, E.; Tagliaferro, A.; Robertson, J.; Milne, W. I. Diamond and Related Materials 2002, 11, 1086. (158) Koch, F.; Petrova-Koch, V.; Muschik, T. Journal of Luminescence 1993, 57, 271. (159) Koch, F. Materials Reseach Society Symposium Proceedings 1993, 298, 319. (160) Chan, M. H.; So, S. K.; Cheah, K. W. Journal of Applied Physics 1996, 79, 3273.

154

Chapter 2 (161) Cui, J.; Rusli; Yoon, S. F.; Teo, E. J.; Yu, M. B.; Chew, K.; Ahn, J.; Zhang, Q.; Osipowicz, T.; Watt, F. Journal of Applied Physics 2001, 89, 6153. (162) Boulitrop, F.; Bullot, J.; Gauther, M.; Schmit, M.; Catherine, Y. Solid State Communications 1985, 54, 107. (163) Zhou, W.-y.; Xie, S.-s.; Qian, S.-f.; Wang, G.; Qian, L.-x. Journal of Physics: Condensed Matter 1996, 8, 5793. (164) Compagnini, G.; Zammit, U.; Madhusoodanan, K. N.; Foti, G. Physical Review B: Condensed Matter 1995, 51, 11168. (165) Cody, G. D.; Tiedje, T.; Abeles, B.; Brooks, B.; Goldstein, Y. Physical Review Letters 1981, 47, 1480. (166) Kandil, K. M.; Kotkata, M. F.; Theye, M. L.; Gheorghiu, A.; Senemaud, C.; Dixmier, J. Physical Review B: Condensed Matter 1995, 51, 17565. (167) Tanaka, K.; Gotoh, T.; Yoshida, N.; Nonomura, S. Journal of Applied Physics

2002, 91, 125. (168) Tauc, J. Materials Research Bulletin 1970, 5, 721. (169) Tauc, J.; Menth, A. Journal of Non-Crystalline Solids 1972, 8/10, 569. (170) Kolomiets, B. T.; Mamontova, T. N.; Babaev, A. A. Journal of Non-Crystalline Solids 1970, 4, 289. (171) Kolomiets, B. T.; Raspapova, E. M. Fizika i Technika Poluprovodnikov 1970, 4, 157. (172) Tanaka, K. Journal of Optoelectronics and Advanced Materials 2001, 3, 189. (173) Frumar, M.; Polak, Z. C.; Frumarova, B.; Wagner, T. Chemical Papers 1997, 51, 310.

155

Chapter 2 (174) Robertson, J.; O'Reilly, E. P. Physical Review B: Condensed Matter 1987, 35, 2946. (175) Fuoss, P. H.; Eisenberger, P.; Warburton, W. K.; Bienstock, A. Physical Review Letters 1981, 46, 1537. (176) Uemara, O.; Sagara, Y.; Muno, D.; Satow, T. Journal of Non-Crystalline Solids

1978, 30, 155. (177) Khan, B. A.; Pandya, R. IEEE Transactions on Electron Devices 1990, 37, 1727. (178) Street, R. A.; Winer, K. Physical Review B: Condensed Matter 1989, 40, 6236. (179) Skumanich, A.; Fathallah, M.; Amer, N. M. Applied Physics Letters 1989, 54, 1887. (180) Jiao, L.; Chen, I.; Collins, R. W.; Wronski, C. R.; Hata, N. Applied Physics Letters 1998, 72, 1057. (181) Schubert, M. B.; Mohring, H. D.; Lotter, E.; Bauer, G. H. IEEE Transactions on Electron Devices 1989, 36, 2863. (182) Rusli; Robertson, J.; Amaratunga, G. A. J. Journal of Applied Physics 1996, 80, 2998. (183) Nesladek, M.; Meykens, K.; Stals, L. M.; Vanecek, M.; Rosa, J. Physical Review B: Condensed Matter 1996, 54, 5552. (184) Tauc, J. in Amorphous and Liquid Semiconductors; Tauc, J., Ed.; Plenum Press: London, 1974. (185) Robertson, J. Philosophical Magazine B 1992, 66, 199. (186) Zammit, U.; Madhusoodanan, K. N.; Scudieri, F.; Mercuri, F.; Wendler, E.; Wesch, W. Physical Review B: Condensed Matter 1994, 49, 2163.

156

Chapter 2 (187) Schreiber, M.; Toyozawa, Y. Journal of the Physical Society of Japan 1983, 52, 318. (188) Kanzaki, H.; Sakuragi, S.; Sakamoto, K. Solid State Communications 1971, 9, 999. (189) Fujii, T.; Pan, L.; Nakayama, Y. Japanese Journal of Applied Physics, Part 1: Regular Papers, Short Notes & Review Papers 2000, 39, 3627. (190) Weinberger, B. R.; Roxlo, C. B.; Etemad, S.; Baker, G. L.; Orenstein, J. Physical Review Letters 1984, 53, 86. (191) Wunderlich, B. Polymer 1964, 5, 125. (192) Scherf, U.; List, E. J. W. Advanced Materials (Weinheim, Germany) 2002, 14, 477. (193) Boudreaux, D. S.; Chance, R. R.; Elsenbaumer, R. L.; Frommer, J. E.; Bredas, J. L.; Silbey, R. Physical Review B: Condensed Matter and Materials Physics 1985, 31, 652. (194) Bicerano, J. Prediction of Polymer Properties; Marcel Dekker, Inc.: New York, 2002. (195) Seitz, J. T. Journal of Applied Polymer Science 1993, 49, 1331. (196) Kaeble, D. H. Rheology 1969, 5, 223. (197) Ridley, J.; Zerner, M. C. Theoretica Chimica Acta 1973, 32, 111.

157

Chapter 3

Experimental Procedures

3.1 Introduction The materials and spectroscopic techniques used in the studies of Chapters 4 – 7 will be described in this chapter. The first section describes, in general, the polymers and dyes used to form the dye – polymer guest – host mixtures or copolymers. The details of materials synthesis and processing of mixtures and copolymers will be provided in the proceeding chapters, as well as any characterization of the component dyes and polymers. The second section will discuss the motivation for using photothermal deflection spectroscopy (PDS), PDS measurement principles, the design and development of a research PDS test bed, instrument control and data acquisition, specimen preparation, the PDS experimental procedure, complementary UV-Vis spectroscopy procedure, and the data reduction procedure.

3.2 Materials The polymer materials studied are grouped into three categories: Bis-phenol A polycarbonates (Figure 3.1), rigid amorphous aromatic polymers (Figure 3.2), and aliphatic main-chain polymers (Figure 3.3). Two distinct classes of NLO dyes were studied:

(2-(3-cyano-4-{2-[5-(2-{4-[ethyl-(2-methoxyethyl)amino]phenyl}vinyl)-3,4-

dialkylthiophen-2-yl]vinyl}-5,5-dimethyl-5H-furan-2-ylidene)malononitrile),

closely

related to the well-characterized dye FTC-2 first synthesized by Dalton and co-workers1, where the dialkyl group at the 3,4-position of the thiophene moiety was varied from two to six carbons per alkyl (Figure 3.4); and the monazo dyes 4-[Ethyl(2hydroxyethyl)amino]-4'-nitroazobenzene (Disperse Red 1, CAS 2872-52-8, Figure 3.5a)

158

Chapter 3 and 4'-[(N,N-Dihydroxyethyl)amino]-4-nitroazobenzene (Disperse Red 19, CAS 273452-3, Figure 3.5b). Chapters 4 – 6 cover studies of the FTC-like dyes in aromatic polymers (polycarbonates and rigid amorphous) as binary guest – host mixtures. In all cases the dye – polymer mixtures were prepared in solution and the dye concentration was varied over larges concentration ranges from as low as 0.05 wt% up to a maximum of 25 wt%. Mixing solvents were chosen on the basis of solubility parameter matching between dye and polymer, with boiling points of > 100°C, to prevent rapid solvent evaporation and crystallization of the dye out of the polymer during film spinning. All solvents were either dried with molecular sieve or stored in an argon glove box. Host polymer stock solutions were prepared by weighing into solvent (from 4 – 16 wt% solids, adjusted for appropriate viscosity for spin casting) and mixing overnight on a rotator. Solid guest dye was weighed into aliquots of the host polymer stock solutions, and then mixed on a rotator overnight. All solutions were inspected for complete dissolution by first ensuring no solid residues formed on container side walls, then by trial film spins on 2000 Å aluminum-sputtered silicon wafers, to ensure no aggregates, crystallites, or clumps formed in the films. Solution identifications, polymer identities and pedigrees, dye identities and pedigrees, dye loading levels, solvents, polymer solids contents in solution, and mixing dates and approximate times were recorded and maintained in for all polymer and dye – polymer solutions in laboratory notebooks. Chapter 7 covers studies of Disperse Red dyes (functionalized and nonfunctionalized) both as guest – host mixtures in various aliphatic backbone polymers, and covalently bonded to the polymers, forming dye side-chain and main-chain copolymers. Structures of the functionalized dyes and dye-copolymers are shown in Figures 3.6 and 3.7. As for the FTC-like guest – host mixtures described above, dye-copolymer spin solutions and polymer host stock solutions were prepared in solvents with boiling points of > 100°C and with solubility parameter matched between dye and polymer (from 3 – 25 wt%, adjusted for appropriate viscosity for spin casting), and mixed overnight on a rotator. Guest dye was weighed into aliquots of polymer stock solutions, and then mixed overnight on a rotator. Complete dissolution of dye-polymer guest host solutions and

159

Chapter 3 dye-copolymer spin solutions was confirmed by the same procedure described above for FTC-like dye – polymer guest – host systems.

O

a)

O n

O

O

O

O

O

O

O y x

b) O

n

O

O

n

c)

O O

O

O

O

O 0.25

d)

0.75

n

Figure 3.1 Bisphenol A polycarbonate polymer hosts discussed in Chapter 4: a) Bisphenol A polycarbonate homopolymer; b) Amorphous polycarbonate (APC): poly[Bisphenol A carbonate]x-co-[4,4’-(3,3,5-trimethylcyclohexylidene)diphenol carbonate]y, with x ≈ y; c) General Electric Red Cross polycarbonate; d) General Electric Bisphenol A-co-fluorenone polycarbonate.

160

Chapter 3

O S

O

O

n

a)

O

0.75

b)

O N O

c) F F O

d)

F F

0.25

n

O O

O

N O

n

F F O

n

Figure 3.2 Rigid amorphous aromatic host polymers discussed in Chapter 5: a) polysulfone; b) Mississippi Polymer Technologies Parmax 1200 substituted poly(1,4phenylene); c) General Electric Ultem 1000 polyetherimide; d) perfluorocyclobutane (PFCB) biphenylvinylether

161

Chapter 3

a)

n O

O R

b)

OH

n

N HO

O

O

OH n

c)

Figure 3.3 Aliphatic main-chain polymers studied as hosts and dye-copolymers, CH2 CH2CH2 ; b) or discussed in Chapter 7: a) poly(R-methacrylate), R = poly(4-vinyl phenol); c) linear aliphatic epoxy Bisphenol A N-hydroxyethylether-N’ether-co-N,N-bis(2-hydroxypropylaniline)

x x N CN H3CO

S

CN O

CN

Figure 3.4 Homologous series of FTC-like nonlinear optical guest dye 4x6m: 2-(3cyano-4-{2-[5-(2-{4-[ethyl-(2-methoxyethyl)amino]phenyl}vinyl)-3,4-dialkyllthiophen2-yl]vinyl}-5,5-dimethyl-5H-furan-2-ylidene)malononitrile. Dyes are designated: 4E6m (x = 0, diethyl spacer); 4P6m (x = 1, dipropyl spacer); 4B6m (x = 2, dibutyl spacer); 46m (x = 4, dihexyl spacer).

162

Chapter 3

OH

OH

N

N

N

N

NO2

a)

OH

N

N

NO2

b)

Figure 3.5 Azo dyes discussed in Chapter 7: a) Disperse Red 1: 4-[Ethyl(2hydroxyethyl)amino]-4'-nitroazobenzene; (b) Disperse Red 19: 4'-[(N,NDihydroxyethyl)amino]-4-nitroazobenzene

O

O

N

O

O

O

N

a)

O

N

N

N

NO2

b)

N

NO2

Figure 3.6 Esterified monoazo dyes discussed in Chapter 7: a) acyl-Disperse Red 1 (OAc-DR1); b) Bis-acyl-Disperse Red 19 (Bis-OAc-DR19)

163

Chapter 3 OH

x

O

m O

O

O

n R

N

N

N

N

N

NO2

a)

y n

b)

O

N

NO2

O N O HO

N O

O

N

x

OH

OH y n

N

c) O2N

Figure 3.7 Monoazo dye copolymers discussed in Chapter 7: a) poly(DR1-co-acrylate): CH2 CH2CH2 ; b) poly(DR1-co-4-vinyl phenol); c) poly(DR19 or R = Bisphenol A N-hydroxyethylether-N’-ether-co-N,N-bis(2-hydroxypropylaniline).

164

Chapter 3

3.3 Spectroscopy 3.3.1 Motivation for spectral characterization by PDS In this section we provide the motivation for spectral characterization of optical absorption of these NLO polymers, and in particular, the rationale for selection of photothermal deflection spectroscopy as the preferred characterization technique. 3.3.1.1 Spectral absorption vs. single-wavelength optical loss measurements The preference of spectral absorption over single wavelength optical loss measurements is due to two key considerations: 1) The study objective is to understand effects of component materials structure, intermolecular interactions, host environment, and disorder and defect states on the nearIR absorption loss, which are readily elucidated by measurements of changes in spectral features (as shown by the numerous theoretical treatments and experimental studies described in Chapter 2). A single wavelength loss measurement can provide diagnostic relationships between optical loss and structure, while revealing comparatively little insight into physical mechanisms for loss behavior. 2) Single wavelength loss measurements most often involve the coupling of an optical mode into a film of the material or a delineated channel waveguide of the material, and measuring either a decay of scattered light along the optical path in the material, or loss of intensity from an outcoupling point. Some of these include CCD imaging of scattered intensity, waveguide cutback insertion loss, waveguide mode surface refractometry (Metricon), sliding prism, and oil immersion.

Many of these

measurements are highly sensitive to operator technique or assumptions about baseline levels of loss, leading to large variations between operators and between measurement techniques.

Further, all of these are subject to large contributions from extrinsic

scattering from film or waveguide defects induced by fabrication, or from intrinsic scattering in the material, neither of which can be decoupled from the intrinsic optical absorption in the single wavelength loss measurement.

165

Chapter 3 3.3.1.2 Sensitive spectral techniques vs. standard reflectance / transmittance techniques Conventional UV-Visible-Near-IR absorption spectroscopy uses detection of optical power transmitted through or reflected from the sample. For measurement of the weak absorption in the near-IR due to gap states and anharmonic vibrational overtones of NLO polymers with suitably small optical loss for practical devices, differences of < 10 ppm between incident and transmitted (or reflected) intensities must be resolved, which for a 2 µm thick film, translates to measurement of absorption coefficients α down to 0.2 cm-1 (< 1 dB/cm). This is well below the lower limits of accurate detection by the standard reflectance and transmittance techniques, which are limited by contributions due to scattering from the material, and reflections and refraction at film and substrate surfaces and interfaces. For example, as pointed out in Chapter 2, Weinberger et al.2 found that UV-Vis-NIR transmission measurements of absorption spectra of poly(acetylene) were inaccurate for α below 104 cm-1, in spite of attempts to analytically correct for specular reflectance. Thus, the required detection limits for the NLO polymer of this study are several orders of magnitude smaller than achievable by conventional UV-Vis-NIR measurements. These limitations of conventional absorption spectroscopy have led to elaborate extrapolation methods and models to estimate the weak near-IR absorption behavior. As discussed in Chapter 2, Jiao et al.3 modeled the absorption spectrum of a − Si : H as a parabolic distribution of extended states and exponential distribution of localized tail states, leading to an extrapolation procedure for spectral measurements in the strong absorption region by transmission and reflection spectroscopy or spectroscopic ellipsometry measurements to non-overlapping sub-gap regions measured by constant photocurrent measurements (CPM) or dual beam photoconductivity (DBP). As will be shown in Chapter 4, we have found that extrapolation of standard UV-Vis absorption spectra to near-IR wavelengths (following the Voight profile method of Stegeman4, Chapter 2) is not accurate, and does not correlate with material structure or properties, or with direct measurements of near-IR absorption for the NLO polymers of this study. In the 1970’s, rapid developments in both optical communications and highpower lasers prompted the need for the measurement of small optical absorption losses in transparent solids, with significantly greater sensitivity than standard spectroscopic

166

Chapter 3 techniques5. It has long been established that upon absorption of optical energy by a substance, some or all of the excited state electronic or vibrational will relax nonradiatively to produce thermal energy.

This process provided the basis for the

emergence of various sensitive photothermal measurement techniques, including thermal lensing (TL), photoacoustic spectroscopy (PAS), photothermal displacement, and photothermal deflection spectroscopy (PDS)6.

Each of these techniques probes a

thermally induced change of a physical property in response to optical absorption and subsequent thermal relaxation. TL measures a curvature of refractive index, PAS a pressure wave, photothermal displacement a thermal expansion, and PDS a change in refractive index. The deflection of a probe beam due to heating-induced refractive index variation had been realized for many years, but had not been exploited for spectroscopic measurements until Jackson, Amer, Fournier, and Boccara6, 7 first reported a theoretical treatment and experimental application of PDS for studies of condensed phase materials (a-Si:H, Cs3Cr2Cl9, Nd2(MoO4)3) in 1980. Here, absorption of a modulated optical pump beam by the sample causes a thermally-induced gradient in refractive index in the surrounding medium, and the deflection of a probe laser through this index gradient is measured by synchronous ac detection by a photodetector. A schematic representation of photothermal detection given by A. Mandelis8, in terms of various manifestations of photothermal diffusion waves, is shown in Figure 3.8. In PDS, a thermal wave is generated upon illumination of the surface by a modulated pump beam (green); the synchronous ac deflection of a probe laser beam (pink) as it traverses parallel to the surface and through a refractive index gradient is called the mirage effect. Thermoelastic deformation of the surface during oscillating laser heating and associated thermal expansion creates a surface bump, which also acts to deflect an incident probe beam (purple). In infrared photothermal radiometry (PTR), the thermal IR radiation (red) is detected by a microbolometer in front of the surface.

In

photopyroelectric spectroscopy (PPES), a thermal wave field of skin depth L (ω ) propagates within the film, and is detected on the sample surface by a pyroelectric sensor.

167

Chapter 3

Figure 3.8 Schematic illustration of various photothermal detection mechanisms (from reference 8). See above text for description.

Each of these techniques have orders of magnitude greater sensitivity and dynamic range over conventional spectroscopy, with significantly higher fidelity in determining optical attenuation by the material due purely to absorption, by making the sample itself the detector.

Absorption measurements on optically inhomogeneous,

opaque, and scattering samples of any type of solid (crystalline, powder, amorphous, gel) is made possible by these techniques, whereas in conventional spectroscopy, these sample conditions can pose extreme difficulties.

For example, the magnitude of

deflection φ of the probe beam in PDS, derived from the heat diffusion equation with one-dimensional radiation from an isotropically scattering medium, is

168

Chapter 3

φ ∝ (1 − R − T )

(3.1)

where R is the reflection coefficient and T the transmission coefficient for diffuse and specular beams, respectively9. Jackson and Amer9 showed that for a scattering product

λα l up to unity (where λ is the fraction of light scattered, α is the total extinction coefficient (scattering plus absorption), and l is the film thickness), the PDS deflection φ is does not vary with scattering. In PAS, conversion of thermal energy to an acoustic wave is due solely to the optical absorption, with comparatively very little contribution by scattering10. Another significant advantage of photothermal techniques over other techniques is the ease of separation of background signals due to substrate and surrounding through use of the synchronous ac phase information7. A phase retardation of 45° with respect to pump beam chopper occurs for the substrate signal, and a positive phase angle of 135° occurs for the signal from the surrounding fluid, providing a straightforward means of determining the absorption signal associated with the film of study.

3.3.1.3

PDS vs. photoconductivity, electrical, interferometric, and calorimetric

techniques Other techniques that have been used to circumvent the limitations imposed by conventional absorption spectroscopy are photoconductivity (constant photocurrent measurements (CPM) or dual beam photoconductivity (DBP)), interferometric, and calorimetric

techniques

for

measuring

optical

absorption,

and

capacitance,

photoconductivity, and field effect techniques for determining sub-gap density of defect states.

Interferometric techniques measure optical path length differences upon

absorption. Calorimetric techniques directly measure a heat flux from the material. Photoconductivity techniques measure an induced current in the material upon absorption, and require an assumption of nonvarying mobile carrier efficiency-mobilitylifetime product ηµτ with optical frequency, in order to extract α from the expression

(

σ ∝ ηµτ 1 − e−α l

169

)

(3.2)

Chapter 3 Jackson and Amer7 found that for undoped a-Si:H, photoconductivity measurements are inaccurate for determining absorption spectra at photon energies below 1.5 eV (wavelengths > 830 nm), and that localized-to-localized transitions corresponding to the band tail produce no photocurrent, leading to an optical frequency dispersion of

ηµτ .

Photoconductivity measurements are far less sensitive than calorimetric,

interferometric, and photothermal methods, and cannot handle opaque or diffuse samples11. Calorimetric techniques have the disadvantage of requiring the sample to be contained in a cell, which can contribute signal artifacts to due scattering of the pump beam from the container walls. Interferometric techniques are more complicated and less direct and efficient than PDS11. For measurements of sub-band density of defect states, PDS has significant advantages, compared to capacitance, field effect, and photoconductivity techniques, in that it possesses sub-band sensitivity to all classes of defects, without being constrained by Fermi level position or energy ranges12. 3.3.1.4 PDS vs. other sensitive photothermal detection techniques PDS has several notable advantages over other sensitive photothermal techniques. When using monochromatic broadband light sources, a sensitivity of < 1 x 10-8 W of absorbed power can be readily achieved, which for a typical absorbed optical power of 1 mW, corresponds to absorptance resolution of C–H stretching-plus-bending combination overtone at 1390 nm; and 4) a ν0,2 C–H stretching overtone at 1660 nm4. In all cases, the main peak displays a Gaussian peak shape when plotted against photon energy, and obeys Beer-Lambert Law peak intensity vs. concentration behavior. The ν 0,2 C–H overtone at 1660 nm is insensitive to dye concentration, while the strengths of the ν 0,3 C–H stretching overtone (1140 nm) and |2,1> C–H stretching + bending overtone (1390 nm) peaks vary with concentration. The 1660 nm C–H peak behavior indicates the overall concentration of C–H moieties is fairly constant and independent of dye doping levels, while the 1140 nm and 1390 nm peak variations suggest that a low energy tail feature on the main absorption peak grows with concentration. This behavior for the 1390 nm peak is in contrast to the relatively invariant 1390 nm peak intensity seen for the 4E6m/polycarbonate series in Chapter 4. The experimental spectra were reasonably repeatable for each 4E6m/ amorphous aromatic (non-carbonate) guest-host series at each dye concentration. As seen in Figure 5.1, the absorption loss values at the key telecommunication wavelengths of 1060 nm, 1300 nm, and 1550 nm increase with dye concentration. The analysis of structure effects on absorption loss for this series of polymer hosts examined three primary features of the spectra: 1) the concentration dependence of loss at the transmission wavelengths; 2) the broadening and energetic shifting behavior of the dye electronic absorption peak as a function of host polymer structure; and 3) the nature of the low energy tail of the main absorption peak as a function of host polymer structure.

260

205 µmol/g

205 µmol/g

100000

96 µmol/g

α (dB/cm)

α (dB/cm)

100000

1060 nm

1000

46 µmol/g

1300 nm 1550 nm

Undoped

10

96 µmol/g 0.9 µmol/g

1000

1300 nm 1550 nm

10 Undoped

0.1

0.1 0

500

1000

1500

a)

0

2000

Wavelength (nm)

1060 nm

500

b)

205 µmol/g 96 µmol/g 1060 nm

1000

1300 nm 1550 nm 0.9 µmol/g

10

0.1

2000

96 µmol/g 1060 nm

1000

1300 nm 1550 nm 0.9 µmol/g

10

0.1 0

c)

1500

205 µmol/g

100000

α (dB/cm)

α (dB/cm)

100000

1000

Wavelength (nm)

500

1000

1500

Wavelength (nm)

2000

0

d)

500

1000

1500

2000

Wavelength (nm)

Figure 5.1 Representative concentration series of PDS absorption loss spectra for 4E6m/amorphous aromatic (non-carbonate) polymer guest-host materials: a) 4E6m/polysulfone; b) 4E6m/Ultem; c) 4E6m/Parmax; d) 4E6m/PFCB-BPVE.

The concentration dependence of absorption loss minima is plotted in Figures 5.2a – 5.2c for 4E6m in all four rigid aromatic polymers at each wavelength. Strong linear behavior of the absorption minima with concentration is exhibited for 4E6m doping of each polymer host at 1060 nm ( R 2 range 0.93 – 0.99). Near linear to strongly linear dependence of loss on concentration is seen at 1300 nm ( R 2 range 0.85 – 0.99). Weakly linear to strongly linear loss – concentration dependence is observed at 1550 nm ( R 2 range 0.51 – 0.995).

261

10

a) 1060 nm α (dB/cm)

α (dB/cm)

b) 1300 nm

3

8 6 4

2

1

2 0

0 0

100

200

0

Dye Concentration (mmol/g polymer)

200

Dye Concentration (mmol/g polymer)

c) 1550 nm

3

α (dB/cm)

100

2

1

0 0

100

200

Dye Concentration (mmol/g polymer)

Figure 5.2 Measured PDS absorption loss vs. dye concentration for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials at: a) 1060 nm; b) 1300 nm; and c) 1550 nm. Polymer hosts denoted by: for Ultem; for for PFCB. for polysulfone; and Parmax;

As seen in Chapter 4 for the polycarbonate host series, the intercept values for loss vs. concentration at 1060 nm are in close agreement among all four polymers, while loss increases linearly with concentration, consistent with the Beer-Lambert Law. The loss – concentration dependence at 1060 nm is controlled by dye electronic absorption peak broadening at higher concentrations, due to closeness of this spectral minimum to the 4E6m main absorption peak wavelength. Also in accord with the trends observed in Chapter 4, the measured variation in loss values corresponding to zero- or near-zero dye concentrations at 1300 and 1550 nm are comparable to those measured at 1060 nm among the four polymers, but this variation is closer in magnitude to measured variation of loss with dye concentration, consistent with a much weaker relative contribution of the dye absorption peak at these wavelengths

262

and expected variation in near-IR loss among neat (undoped) polymers5. We adopt here the approach discussed in Chapter 4 of examining the variations in loss vs. concentration slope, rather than simply evaluating the loss magnitude for a given dye concentration. This property is held to be associated with a spectral response characteristic of the host structure-induced environment, and will be shown in Chapter 6 to directly correlate with dye – polymer excess mixing free energies within the framework of Marcus theory. With this approach, the loss – concentration slope is handled as a fundamental material property for each dye-polymer pair as an input in the analyses of the structure – property relationships. The observed variation in neat polymer near-IR loss for this host series, in conjunction with the observation of polymer structure-dependent variation of loss with concentration, reinforces the material selection guideline for loss minimization in NLO guest-hosts discussed in Chapter 4, i.e. choice of a low-loss host polymer is a necessary but insufficient condition for achieving low loss at the high dye doping levels required for practical device applications. The role of the polymer environment on loss evolution with concentration can be of equal or even greater consequence. Representative UV-Vis absorption spectra, showing the dye main 0-0 transition peak, are shown in Figure 5.3 for each dye-polymer combination at the highest common dye concentration studied, 205 µmol per gram of polymer. As seen in Figure 5.3, strong solvatochromic shifts of the dye absorption peak are exhibited for these four rigid amorphous aromatic polymers. The strongest red shift is seen for 4E6m-doped Parmax. The solvatochromism displayed by 4E6m in these rigid aromatic polymers is considerably larger than that shown for the Bisphenol A polycarbonate series reported in Chapter 4. To examine the effects of the broadening and shifting behavior of the dye electronic absorption peak on near-IR loss in the rigid aromatics, the UV-Vis main absorption peak for each dye-polymer combination at the highest common dye concentration studied, 205 µ mol per gram of polymer, was fitted to a Gaussian form on an energy scale. The parameters full-width at half-max (FWHM = Γ s ) and ν max of the dye main absorption peak for each dye-polymer pair were determined from these fits and plotted against loss – concentration slope dα / dC at each wavelength. No meaningful 263

correlations were found between Γ s and loss at these wavelengths (not shown). An exceptional correlation ( R 2 = 0.995) is seen for dα / dC at 1060 nm vs. ν max for the combined polymers of Chapter 4 and this chapter (polycarbonates and non-carbonate aromatics, Figure 5.4a), neglecting the 4E6m/PFCB datum as an outlier. The outlying

ν max – loss behavior for 4E6m/PFCB at 1060 nm is not inconsistent with the behavior found for the amorphous non-carbonate aromatic series at 1300 nm and 1550 nm. As seen in Figures 5.4b and 5.4c, the dominant mechanism for loss at 1300 and 1550 nm for the rigid amorphous aromatic series appears to be completely different from and opposite to that acting at 1060 nm for these polymers, except for 4E6m/PFCB, for which the red tail behavior is sufficiently strong to carry over to 1060 nm. The observed loss – ν max behavior at 1060 nm for all polymers studied in Chapter 4 and this chapter appears to be due to the nearness of the 1060 nm minima to the dye main absorption peak, such that any shifting of the energy position of the main peak to lower energy directly (linearly) translates to higher loss a 1060 nm.

a) Aromatic non-carbonates

BPA-fluorenone

Arbitrary Units

Arbitrary Units

Parmax Ultem Polysulfone PFCBBPVE

1.2

1.6

2.0

Bisphenol A polycarbonate APC Red Cross polycarbonate

1.2

2.4

Photon Energy (eV)

b) Polycarbonates

1.6

2.0

2.4

Photon Energy (eV)

Figure 5.3 UV-Vis absorption spectra for 4E6m guest-host polymer series at a dye concentration of 205 µmol per gram of polymer: a) rigid amorphous aromatic (noncarbonate) polymer; b) Bisphenol A polycarbonate series discussed in Chapter 4. Structures and full names of polymers in (b) are given in Chapter 4.

264

dα/dC (dB-g/cm-µmol)

0.05

a) 1060 nm

0.04 0.03 0.02 0.01

1.78

1.82

1.86

1.90

0.012

0.012

b) 1300 nm

dα/dC (dB/cm)/(µmol/g)

dα/dC (dB/cm)/(µmol/g)

νmax (eV)

0.008 0.004 0.000 1.78

1.82

1.86

1.90

νmax (eV)

c) 1550 nm

0.008

0.004

0.000 1.78

1.82

1.86

1.90

νmax (eV)

Figure 5.4 Measured PDS absorption loss-concentration slope dα dC vs. main absorption peak position ν max for 4E6m/polymer guest-host materials at a dye concentration of 205 µ mol per gram of polymer, at: a) 1060 nm; b) 1300 nm; c) 1550 nm. Polycarbonate hosts of Chapter 4 represented by in (a); rigid non-carbonate aromatic hosts of present study represented by in (a) – (c). Combined data for in (a). is polycarbonates and rigid non-carbonate aromatic hosts represented by an outlier corresponding to PFCB-BPVE in (a). Trend lines for 4E6m/polycarbonate in (b) and (c). series shown in Chapter 4 are represented by

The dα / dC versus ν max plots at 1300 nm and 1550 nm for the 4E6m-doped non-carbonate aromatic polymers (Figures 5.4b and 5.4c) show strongly linear ( R 2 = 0.97) and approximately linear ( R 2 = 0.83) correlations, respectively, of opposite slope to the trend at 1060 nm. As shown in Figures 5.4a and 5.4b, the trends for 4E6m/non-carbonate aromatic polymers at 1300 and 1550 nm are also of opposite slope to those exhibited at the same wavelengths by the 4E6m-doped polycarbonate series discussed in Chapter 4, with the trend lines crossing at intermediate values of ν max . 265

These opposing near-IR loss behaviors for the two classes of host polymers will be discussed in greater detail in the Discussion section. At the intermediate ν max values, polysulfone is seen to fall on both correlation trend lines near the crossing point at both wavelengths. This could be a result of polysulfone having structural attributes of both the Bisphenol A family and the amorphous non-carbonate aromatics.

5.4 Discussion As discussed in the Results section, the 4E6m-doped rigid amorphous (noncarbonate)

aromatic

polymers

(Figures

3.2a



3.2d)

demonstrated

stronger

solvatochromism than for the 4E6m-doped polycarbonates reported in Chapter 4, with near-IR loss linearly decreasing with red shifting of the dye transition peak. This loss -

ν max behavior for the non-carbonate aromatics is opposite to that of the polycarbonates, and as discussed above, the experimental 4E6m/rigid aromatic ν max values could not be correlated to continuum dielectric solvent polarity models. As described in Section 2.8.1, the Marcus model of inhomogeneous broadening predicts a linear relationship between Γ 2s (square-inhomogeneous width) and ν max (solvent shift), while a model based on

Marcus theory, due to Obato, Machida and Horie (OMH)6, discussed in Section 2.8.4, predicts a linear relationship between Γs and the geometric mean solubility parameter

δ dye δ polymer . While two of the four rigid aromatic polymers fall on the Γ 2s - ν max trend line, neither the width Γs nor the shift in ν max was correlated to the solubility parameters for 4E6m/non-carbonate aromatics, in contrast to the 4E6m/polycarbonates. Moreover, the near IR loss vs. dye concentration slope dα dC for the non-carbonate aromatic polymers showed no correlation with the inhomogeneous width Γs (not shown). All of these observations lead to the interpretation that the 4E6m transition peak in the non-carbonate aromatic polymers is shifted by a distinctly different physical mechanism than that of the polycarbonates. The observation of near-IR loss decreasing with red shifting, and no clear correlation with peak width, runs counter to the simple view of loss being controlled by either dye transition peak broadening or its proximity to the near-IR. Inspection of the backbone structures in Figures 3.1a- 3.1d and 3.2a – 3.2d

266

shows that for these non-carbonate aromatic host polymers, while the overall rigidity in terms of rotational freedom and steric hindrances appears comparable to that of the polycarbonate series, there is greater aromatic character along the backbones of the noncarbonate host polymers, due to lack of interruption by carbonate moieties. It was postulated that for these backbone structures, the observed variations in peak shifting and counter-intuitive near IR optical loss behavior are more dominated by the stiffness of these aromatic chains and influence on dye electronic defect states, rather than by the dielectric properties of the medium, as was the case for the polycarbonate hosts. Thus, the near-IR loss behavior was expected to correlate with the bulk mechanical properties of the polymer hosts. To test this assertion, the polymer host Young’s modulus was predicted using Accelrys Cerius2 SYNTHIA quantitative structure-property relation (QSPR) model based on the topology method of Bicerano7. Shown in Figure 5.5 is a plot of measured peak transition frequencies ν max (at the reference dye concentration of 205 µmol per gram of polymer) vs. predicted polymer Young’s moduli, showing a strong linear correlation for 4E6m-doped rigid aromatics, but not for the polycarbonates discussed in Chapter 4.

The polymer Young’s modulus is expected to be highly

correlated with the polymer backbone stiffness. The observed ν max trends with this property are consistent with red shifting increasing with polymer backbone stiffness.

Note that reported modulus values,

available for five of the eight polymer hosts studied in Chapters 4 and 5, show a linear correlation with predicted values ( R 2 = 0.90). The amorphous non-carbonate aromatic

ν max values were also found to correlate with SYNTHIA-predicted values of bulk modulus, shear modulus, Poisson ratio, coefficient of thermal expansion, and shear yield stress (not shown), each of which is expected to correlate with polymer backbone stiffness. The Young’s modulus will be referred to throughout the remainder of the text as a suitable indicator of polymer macroscopic stiffness.

267

νmax (eV)

1.88

1.84

1.80

1.76 2000

3000

4000

Young's Modulus (MPa)

Figure 5.5 Measured main absorption peak position ν max at 205 µ mol dye/g polymer vs. SYNTHIA-predicted polymer Young’s modulus for 4E6m/rigid amorphous (noncarbonate) aromatic guest-host materials.

The experimental loss vs. concentration slopes dα dC were plotted against predicted Young’s modulus at 1060 nm, 1300 nm, and 1550 nm. No correlation is seen at 1060 nm, while dα dC for the 4E6m-doped rigid aromatics increases linearly with increasing Young’s modulus (stiffness increasing) at both 1300 and 1550 nm (Figures 5.6a and 5.6b). This behavior suggests that a spectral feature at optical energies below 1060 nm (longer wavelength) relates to the polymer chain stiffness, in support of the proposal by Kador, Horne, and Moerner8 (briefly discussed in Section 2.14) that near IR loss could be strongly influenced by disorder induced tail states. 5.4.1 Band structure and weak absorption tails associated with disorder and defect states As discussed in Section 2.15, defects in a dye-doped polymer system include dye conformational states, configurations within the surrounding polymer network9, 10 and π orbital localization11. For an amorphous polymer with rigid backbone segments, local ordering may occur between parallel chain segments, inducing alignment of the rigid dye molecules with the rigid polymer segments within this local nematic field.

268

0.011

dα/dC (dB/cm)/(µmol/g)

a) 1300 nm 0.008

0.005

0.002 2200

2700

3200

3700

4200

Predicted Young's Modulus (MPa)

b) 1550 nm

dα/dC (dB/cm)/(µmol/g)

0.006

0.003

0.000 2200

2700

3200

3700

4200

Predicted Young's Modulus (MPa)

Figure 5.6 Measured PDS absorption loss-concentration slope dα dC vs. SYNTHIA-predicted polymer Young’s modulus for 4E6m/rigid non-carbonate aromatic guest-host materials at: a) 1300 nm; b) 1550 nm.

The quantum mechanical description of disorder effects on spectral behavior in terms of diagonal and off-diagonal matrix elements of the tight-binding Hamiltonian12 (Eq. (2.124)) discussed in Section 2.11 will be shown later in the Discussion section to be a useful construct for considering the range and nature of disorder in the guest – host system. As outlined in Section 2.15, Meskers et al.11 showed that disorder induces strongly localized excited states, which can exist at the low energy tail of the absorption peak11. Low disorder implies delocalization of excited states over several molecules, and in polymers, the intermolecular delocalization is readily perturbed by the presence of defects. The delocalization of excitations between neighboring molecules, which can be

269

brought about by parallel alignment, can lead to a red shift. Localized states can be induced by fluctuations in the intermolecular interaction energy, and intermediate diagonal and off-diagonal disorder can result in the formation of partially localized states. Thus, the density of excited states in this region can be broadened by the degree of local disorder, brought about by misalignment between neighboring dye and polymer molecules. Most of the studies of disorder and defect states in the band tail regions have been reported for fullerenes ( C60 and C70 )13,

14

, amorphous semiconductors15-24, such as

a − Si : H , a − Si1− x C x : H , a − C : H , a − Si1− x Gex , and diamond films25, each of which

were described in Section 2.13. Many similarities exist between the band structures of these inorganics and amorphous polymers. For example, as discussed in Section 2.15, sub-gap absorption in polyacetylene26 and polymethylphenyl silane27 has sharp band tails as seen in amorphous semiconductors and glasses28, with an exponential distribution27 given as

α ( E ) = α 0e E Ee

(2.141)

where α 0 is a constant, α ( E ) is the absorption coefficient at energy E , and Ee is referred to as the Urbach energy and indicates the width of the exponential distribution. Further,

for

amorphous

polymers29

as

well

as

fullerenes13,

14

,

amorphous

semiconductors19, and glasses28, the extended state energy levels (valence and conduction band in the amorphous semiconductors and glasses, π -state HOMO and π *-state LUMO in the organic and amorphous carbon materials) can be represented as a Gaussian distribution. The disorder in the polymer main chains, stemming from interruption of π conjugation, length distribution and conformation variations, gives rise to the observed band tail for the polymers. As mentioned in Section 2.13, if the band tail of the HOMO is less steep than that of the LUMO in polymers, the measured PDS spectrum largely represents the HOMO band tail. Low levels of absorption in the near IR region imply reduced density of states in the sub-gap region19. The weak absorption tails in Region C of the spectrum (Section 2.12) reported for chalcogenide ( As2 S3 ) glasses28, follow exponential behavior analogous to Urbach tails 270

α = α 0ehν

Ew

(2.152)

where the tail width can be assigned to deep defects in the amorphous glass structure. The relationship developed by Dasgupta et al.17 (Section 2.12) for the absorption band shape   E − E 2  erf [ E 2σ ] α ( E ) ∝ exp  −  ππ   E   2σ   *

(5.1)

where Eππ is the chromophore peak transition energy, E is the photon energy and σ is *

the Gaussian peak width, is in accordance with the suggestion by Kador, Horne, and Moerner8 (Section 2.14) and Kjaer and Ulstrup30 (Section 2.14.1) that slowly decaying wings of the inhomogeneous peaks, extending much farther into the near-IR than predicted by a Gaussian band shape, could be due to chromophore defect states8. A typical near-IR-to-visible absorption spectrum is shown on an energy scale in Figure 5.7 for 4E6m-doped PFCB at a dye concentration of 205 µmol per gram of polymer. The five spectral regions of interest (A – E) for 4E6m doped non-carbonate amorphous aromatic guest-host polymers can be defined in terms of the possible optical transitions based on a proposed density of states (DOS) model proposed by Rusli etal.16, as illustrated in Figure 5.8. The optical transitions are classified by considering three principal electronic ground states: 1) valence band ( π states); 2) valence band tail; and 3) filled defect states; and three principal excited states: 1) conduction band ( π *  states); 2) conduction band tail; and 3) empty defect states. The valence and conduction bands correspond to extended states and are each well described by a Gaussian shape; the band tails can be assigned to structural disorder, and the deep defect states can be assigned to structural and electronic defects. Analogous to the description given in Section 2.12, area “A” in Figure 5.7 is treated as the above-gap spectral region due to transitions between extended states; area “B” is a high absorption near-gap (“Tauc”) region between band tail and extended states; area “C” is a near-gap region between band tails; area “D” is the “sub-gap” region, dominated by transitions from extended valence band states to defect states, or from defect states to extended conduction band states; finally, area “E” is the

271

near-IR region dominated by anharmonic overtones of fundamental stretching and bending vibrations of C–H, N–H, or O–H bonds, with contributions by transitions from mid-gap defect states13, 31.

α (dB/cm)

B

C

100000

A

1000 E D 10

0.1 0.5

1.5

2.5

3.5

4.5

hν (eV) Figure 5.7 Measured PDS absorption spectrum for 4E6m/PFCB at dye concentration of 205 µmol/g polymer showing five spectral regions of interest (A-E) Extended States

Density of States, N(ε)

Tail States

Gap state transitions Near-gap transitions Tail state (Urbach) transitions Deep defect transitions

EV

ED Ef

EC

Energy, ε

Figure 5.8 Proposed density of states (DOS) model for 4E6m doped rigid amorphous aromatic (non-carbonate) guest-host polymers and possible optical transitions. HOMO (valence) band edge represented by EV, filled defect band by ED, Fermi level by Ef, and LUMO (conduction) band edge by EC.

272

The extensive use of PDS to measure optical spectra and to determine the sub-gap DOS distribution in amorphous materials13,

15, 16, 18, 19, 21, 24, 26, 27

owes to its extreme

sensitivity, allowing measurement of sub-gap forbidden transitions as well as absorption due to all of the defects, independent of the Fermi level position, defect type, or probebeam energy. The red tail structures of the PDS spectra obtained from each of 4E6mdoped rigid amorphous polymers was examined in the spectral energy range 1.0 – 1.2 eV, corresponding to Region “D” above. Red tail spectra for each host are shown for the reference dye concentration of 205 µmol per gram of polymer in Figure 5.9. The shoulder at ~ 1.08 eV in each of the tail spectra is due to a ν 0,3 C–H stretching overtone and exhibits very similar shape and energetic position for all for polymer hosts.

100

α (dB/cm)

Parmax Ultem

10

PFCB-BPVE

Polysulfone

1 0.9

1.1

1.3

hν (eV) Figure 5.9 Comparison PDS spectra of absorption red tail (region D) for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials.

Significant differences in the overall slopes of the tail spectra are seen among the four polymer hosts. Inspection of Figure 5.9 shows that the tail slope in this region has direct consequences for the strength of the absorption in the neighboring lower energy spectral area “E,” which is the region of interest for optical transmission in electro-optic devices. To demonstrate this quantitatively, the log-linear fit of the red tail spectra of each 4E6m-doped polymer host was determined for each replicate sample and run at the

273

reference dye concentration, from which the exponential width Ew was determined, according to Eq. (2.152). Reasonable log-linear dependence was exhibited for all red tail spectra examined ( R 2 = 0.88 – 0.94). The near-IR loss vs. concentration slope dα / dC was plotted against the resulting values of red tail width Ew at each of 1060, 1300, and 1550 nm. Poor correlation between dα / dC and Ew was observed at 1060 nm, as expected from the poor correlation of dα / dC vs. bulk mechanical properties at 1060 nm. As shown in Figure 5.10, strong linear dependence ( R 2 = 0.98) of dα / dC on Ew was seen at 1300 nm, due to the close proximity of the red tail to 1300 nm, while nearly linear dependence ( R 2 = 0.77) was found at 1550 nm. The red tail width Ew is inversely related to the slope in this region, so the observed increase of dα / dC with Ew is consistent with near-IR loss increasing with decreasing red tail slope.

dα/dC (dB/cm)/(µmol/g)

0.012

0.008

0.004

0.000 0.05

0.09

0.13

0.17

0.21

EW (eV)

Figure 5.10 PDS-measured loss-concentration slope at 1300 and 1550 nm vs. PDS spectral sub-band tail width Ew at a dye concentration of 205 µmol per gram of polymer for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials. 1300 nm data represented by

; 1550 nm data represented by

.

Within the construct of the proposed defect DOS model, local disorder leads to a greater distribution of π–state localization, which results in a wider distribution of subband defect states. A wider sub-band state distribution leads to a greater distribution of

274

valence band-to-sub-band transition energies, yielding a less steep low-energy red tail (larger Ew ).

Figure 5.11 Idealized representation of effects of polymer chain stiffness and local order on π -electron localization and density of states: a) rigid polymer chains with fewer degrees of rotational freedom of dye molecules having their dipole axes in alignment with a local polymer nematic field director; b) less rigid polymer chains with greater rotational degrees of freedom of dye molecules between polymer chains.

The relationship between the distribution of localized defect states and the polymer chain stiffness is illustrated conceptually in Figures 5.11a and 5.11b.

For a

guest-host system comprised of a polymer with rigid chain segments, a dye molecule interposed between two parallel polymer segments has fewer degrees of rotational freedom and its dipole axis is in alignment with a local polymer nematic field director (Figure 5.11a).

In this case there is significant overlap of π –orbitals between

neighboring dye and polymer molecules, imparting greater π -electron delocalization and a narrow distribution of localized states with a small tail width Ew . For a polymer with less rigid chains (Figure 5.11b), the dye has greater rotational degrees of freedom

275

between polymer chains, with less efficient π –orbital overlap. This situation leads to a broader distribution of localized π -states, of varying degrees of π -overlap and dyepolymer configurations, giving rise to more sub-gap defect states and a larger tail width Ew . We therefore assert that higher local order (and narrower distribution of defect DOS) ensues from greater polymer chain stiffness, and as such, lower values of Ew are ascribed to greater chain stiffness. This assertion is supported by the strong linear correlation of Ew with predicted polymer modulus (Figure 5.12).

0.20

EW (eV)

0.15

0.10

0.05

0.00 2000

2500

3000

3500

4000

4500

Young's Modulus (MPa)

Figure 5.12 Measured PDS weak absorption tail width Ew at dye concentration of 205 µ mol/g polymer vs. SYNTHIA-predicted polymer Young’s modulus for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials.

To examine the relationship between defect DOS and dye-polymer orientation, according to the above conceptual description, ab initio calculations of dye electronic transition energies were performed using a density functional theory model, Materials Studio DMOL3, from Accelrys, Inc. Model calculations were performed* on the dye molecule LMCO-46, analogous to 4E6m but with no pendant spacer groups, situated

*

Dr. Peter Bedworth of Lockheed Martin Advanced Technology Center performed the DMOL3 simulations.

276

between two parallel rigid polymer segments of polysulfone at dye orientations of 0°, 45°, and 90° to the polymer segments, as shown in Figure 5.13. The DMOL3-predicted transition intensities in the near-IR region associated with weak tail states (>1030 nm) were used to compute the density of states (DOS) at each energy interval with the expression32 g ( E ) dE = N ( E ) e − E

Ew

dE

(5.2)

where N ( E ) , the number of states at energy E, is taken as proportional to the transition dipole moment at E, given by the DMOL3-calculated intensity. The number of near-IR states was computed by numerical integration of the DMOL3-derived DOS over all energy intervals in the near-IR, for each dye-polymer orientation, with the results shown in Figure 5.14. The number of states is seen to be at a maximum for a ~57° orientation, with a small decrease at 90° and significantly fewer states at 0° (parallel alignment). According to the chain stiffness controlled defect state model proposed above, more rigid polymer backbones are expected to have localized orientations approaching the 0° dye orientation case (Figure 5.13a), for which the number of localized tail states is fewer than the case of less rigid polymer backbones with orientations further away from 0°, expected to have more weak tail states.

a)

b)

c)

Figure 5.13 Accelrys, Inc. Materials Studio DMOL3 simulation of LMCO 46 dye molecule positioned between two aligned rigid polymer molecules: a) dye approximately perpendicular to polymer molecules; b) dye at approximately 45° angle to polymer molecules; c) dye approximately parallel to polymer molecules. 277

Relative Number of States

12 9 6 3 0 0

45

90

Dye orientation (°)

Figure 5.14 Integrated density-of-states of LMCO 46 dye molecule positioned between two aligned rigid polymer molecules at three orientations of dye relative to polymer segments, using results of Accelrys, Inc. Materials Studio DMOL3 simulation for predicted intensities over energy intervals in the near infrared region corresponding to weak tail states (>1030 nm).

5.4.2 Weak absorption tail steepness and loss vs. polymer chain stiffness To more precisely examine the relationship between tail states and polymer structure, a polymer stiffness structure parameter was calculated, in the manner of Bicerano7 for correlating the chain stiffness contribution of the glass transition temperature of 320 polymers, using the following definitions for structural component contributions: X 1 = # of rigid or fused rings along shortest path on backbone in para position (contributes to stiffness) X 2 = # of rigid or fused rings along shortest path on backbone in meta position (x 1) or in ortho position (x 5) (compromises stiffness relative to backbone rings in para position) X 3 = # of rigid aromatic rings on backbone surrounded by rigid units on both sides (significantly contributes to stiffness), minus the number of rigid rings surrounded on both sides by any combination of ether (-O-), thioether (-S-), or methylene (-CH2-) groups

278

X 4 = # of groups extending out from a rigid ring in the backbone, for two or more groups; or # of groups extending out from rigid ring of fused ring (contributes to stiffness) X 5 = # of single bonds on backbone not part of a ring (compromises stiffness) X 6 = # of single bonds in side groups not in a ring (compromises stiffness) X 7 = # of symmetrically substituted groups for hydrogen on carbon or silicon backbone atoms, except when located in rings or when flanked on both sides by rigid backbone rings and/or backbone hydrogen bonded groups (compromises stiffness) X 8 = # of tertiary carbon or silicon groups in a side chain (contributes to stiffness) minus the number of methylene units in a side chain (compromises stiffness); methylene units in rings of side groups and terminal =CH groups are not counted X 9 = # of asymmetrically disubstituted backbone carbon atoms (contributes to stiffness), including a backbone carbon attached by two bonds to a ring in which all of the other atoms in it are in a pendant group X 10 = # of non-hydrogen atoms residing in a multiple-ring unit attached to a backbone carbon atom by two bonds (contributes to stiffness) X11 = # of backbone C=C bonds for which both carbons are in the backbone, but not part of a ring (compromises stiffness) X 12 = # of fluorine atoms attached to non-aromatic atoms in side groups (compromises stiffness) The overall chain stiffness structural parameter σ cs was then determined from

σ cs = 11.1 + 15 X 1 − 4 X 2 + 23 X 3 + 12 X 4 − 8 X 5 − 4 X 6 − 8 X 7 + 5 X 8 + 11X 9 + 8 X10 − 11X 11 − 4 X 12 N

(5.3)

279

where N is the number of atoms (excluding hydrogen) in the repeat unit, as prescribed by Bicerano7. Here, larger values of σ cs connote greater polymer chain stiffness. The chain stiffness component is one of several factors that determine the glass transition temperature; other contributions include interchain cohesive forces, numberaverage molecular weight, morphological effects (such as crystallinity), crosslinking, orientation, conformational factors (such as tacticity), additives, fillers, unreacted monomers, and thermal history. Bicerano’s overall correlation and corrections for the glass transition temperature takes several of these factors into account. The value 11.1 in Eq. (5.3) is a constant in Bicerano’s correlation between chain stiffness structure parameters and Tg , which we retain here. We have made use solely of the chain stiffness parameter from Bicerano’s Tg correlation, and as such, would not expect it alone to accurately account for variations in Tg as a function of the repeat unit structure. The chain stiffness parameter is, however, expected to be a representative measure of the polymer’s internal resistance to large-scale cooperative motion of chain segments under a mechanical stress field, as reflected by the Kuhn segment length33, 34, and manifested macroscopically as the tensile modulus. The benefits of this parameter for correlation purposes are in its complete generality to polymer repeat unit structure, for any structure comprised of the nine elements C, N, O, H, F, Si, S, Cl, and Br, obviating the need to develop group contribution correlations for new or exotic structures, and ease of computation. We have calculated stiffness parameters for a series of aliphatic and aromatic polymers by this method, and compare these against experimentally determined values of Young’s modulus reported in the literature in Figure 5.15. From this plot, it can be seen 280

that the relationship between tensile modulus and calculated chain stiffness parameter

σ cs is somewhat linear for the aliphatic series of polymers, while Young’s modulus shows an apparent exponential dependence on the stiffness parameter for the aromatic series. For the rigid aromatic polymer hosts of this study, the chain stiffness parameters are on the high end of the scale of the series shown, where the tensile modulus is shown to be most sensitive to this parameter. We compare in Figure 5.16 values of SYNTHIA-predicted Young’s modulus (following the method of Seitz35) vs. calculated chain stiffness parameter, both for the non-carbonate rigid aromatic hosts of this study* and for the poly(bisphenol A carbonates) discussed in Chapter 4. For both Parmax and poly(bisphenol A carbonate25co-fluorenone carbonate75) (BPA-FL), the average value of σ cs for the random copolymers was calculated using

σ cs ( copolymer ) =

1 ( xA σ cs ( A) ) + ( xB σ cs ( B ) )

(5.4)

where x A and xB are the mole fractions and σ cs ( A ) and σ cs ( B ) are the calculated stiffness parameters for comonomers A and B , respectively. A linear correlation ( R 2 0.915) is seen between predicted Young’s modulus and calculated chain stiffness parameter in Figure 5.15, showing correspondence of the molecular chain stiffness parameter to the macroscopic mechanical stiffness property Young’s modulus) calculated by the semi-empirical procedure of Seitz35 described in

*

For Parmax, the comonomer ratio of substituted to unsubstituted phenylene assumed in the calculation was 75:25, based on a private communication with Dr. Nick Malkovich of Mississippi Polymer Technologies, Inc. (the manufacturer of Parmax 1200™), with and the value of σ cs calculated for the unsubstituted comonomer was the average of values calculated for para and meta bonding configurations.

281

Section 2.16.2, for the class of aromatic polymers represented by the polycarbonates of

Reported Young's Modulus (GPa)

Chapter 4 and the amorphous non-carbonate aromatics of this chapter.

7 6

Aromatic Aliphatic

5 4 3 2 1 0

0

2

4

6

8

10

12

14

Calculated Stiffness Paramter, σ

cs

Figure 5.15 Reported values of tensile modulus vs. calculated values of the stiffness structure parameter σ cs for an aliphatic and aromatic series of polymers. Polymers in aliphatic series, represented by are: high-density polyethylene36, polyisoprene37, polychloroprene37, polyvinylalcohol38, polypropylene35, poly(4-methyl-1-pentene)38, poly- ε -caprolactam37, polyvinylidene fluoride36, poly(ortho-chlorostyrene)35, poly(paratert-butylstyrene)35, poly- α -methylstyrene35, polystyrene35, polymethylmethacrylate35, polyvinyl chloride39, polyvinylidene chloride40, and polyvinyltoluene35. Polymers in are: poly[(2,2-propane-bis{4-(2,6aromatic series, represented by 41 dimethylphenyl)carbonate}] , poly[1,1-dichloroethylene-bis{(4-phenyl)carbonate)}]41, poly(bisphenol A carbonate]35, polysulfone35, polyphenylene oxide39, polyether sulfone42, Ultem43, PFCB-BPVE44, poly(etheretherketone)36, and Parmax 120045. (References correspond to sources of reported tensile modulus data.) As an aside, this correlation suggests that for polymers falling within this class of linear aromatic polymers, the stiffness parameter is a suitable indicator of macroscopic stiffness, and is a more conveniently computed parameter than the modulus by the Seitz model. We also see that the range of calculated stiffness parameters (molecular and macroscopic) for the polycarbonate series of Chapter 4 and the amorphous aromatic non-

282

carbonate series of the present chapter are significantly overlapping, indicating that the molecular stiffness does not exclusively determine the nature of the dα dC vs. ν max relationship. We propose that carbonate moiety of the polycarbonate series acts to geometrically and electronically disrupt the degree of neighboring chain alignment, and hence electronic delocalization, that must occur for the disorder-driven process to dominate the weak tail structure and near-IR loss behavior, as seen in the non-carbonate

Predicted Young's Modulus (GPa)

aromatic guest-host systems of this study.

5 4 3 2 1 0 0

4

8

12

Calculated Stiffness Parameter, σcs

Figure 5.16 SYNTHIA-predicted values of tensile modulus vs. calculated values of the stiffness structure parameter σ cs series of polycarbonates studied in Chapter 4, represented by , and aromatic non-carbonates studied in present chapter, represented by . Line corresponds to best fit through all data.

Both the tail width Ew and the dye peak position ν max display strong linear dependence ( R 2 = 0.90 and 0.99, respectively) on the polymer chain stiffness parameter

σ cs (Figure 5.17), demonstrating the clear relationship between spectral behavior and polymer structural features. The tail width decreases with increasing values of σ cs , consistent with the above assertion that greater polymer chain rigidity results in more

283

local order and fewer deep defect states.

The linear decrease of peak absorption

frequency ν max with chain stiffness is in agreement with the Meskers et al.11 finding (Section 2.15) that parallel orientation between neighboring chromophore and solvent molecules brings about delocalized excitations, and that intermediate levels of disorder lead to formation of partially localized states. The most rigid of the host polymers in this series, Parmax 1200, falls within the substituted poly(p-phenylene) (SPPP) class of polymers, widely reported to exhibit electroluminescence (EL) and photoluminescence (PL)46-57. The sharpness of the electronic band edge, and associated number of defect states that can act as carrier trap sites, are closely related to internal luminescence quantum yields in EL and PL materials. It is proposed that the dependence of the band edge shape and local disorder on chain stiffness in the rigid, π -conjugated, semiconducting polymers displaying EL and PL is completely analogous to that seen in this study. As such, the chain stiffness parameter introduced here may have applicability to the investigation of internal quantum efficiencies of EL and PL polymers.

0.16

0.12

1.87

0.08

EW (eV)

νmax (eV)

1.92

1.82

0.04

0.00

1.77 9

10

11

12

13

σcs

Figure 5.17 Peak position ν max and sub-band tail width Ew at dye concentration of 205 µ mol/g polymer vs. calculated polymer stiffness structure parameter σ cs for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials. ν max represented by

; Ew represented by

.

284

The tail width is shown to linearly decrease with increasing solvatochromic shifting of dye main absorption peak ν max in Figure 5.18, suggesting an influence of the local order on the local dielectric properties of the polymer medium. It is proposed that the polymer stiffness-induced nematic alignment of dye molecules, parallel to surrounding stiff polymer segments, creates a dipole – induced dipole interaction that increases the local polarity of the polymer medium, which in turn acts back on the dye as a self-consistent reaction field (SCRF), inducing stronger solvatochromism.

0.20

EW (eV)

0.15

0.10

0.05

0.00 1.78

1.80

1.82

1.84

1.86

νmax (eV)

Figure 5.18 Measured PDS weak absorption tail width Ew at dye concentration of 205 µ mol/g polymer vs. main absorption peak position ν max for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials.

As expected on the basis of the tail width dependence on the polymer stiffness structure parameter σ cs at energies below 1030 nm, the loss vs. concentration slope dα / dC (at the reference dye concentration 205 µmol per gram of polymer) shows

strong linear dependence on σ cs at 1300 nm ( R 2 = 0.97), and reasonably linear dependence at 1550 nm ( R 2 = 0.90) (Figure 5.19), with no linear correlation seen at 1060 nm (not shown). Given these observed experimental correlations at 1300 nm and 1550 nm, the polymer stiffness structure parameter σ cs (Eq. (5.3)) may serve as a useful predictive tool for absorption loss behavior of NLO dyes in rigid amorphous aromatic

285

polymer hosts. Further investigation with various merocyanine dye structures in a wider range of rigid amorphous aromatic hosts is warranted to verify this observation.

dα/dC (dB/cm)/(µmol/g)

0.012

0.009

0.006

0.003

0.000 9

10

11

12

13

σcs

Figure 5.19 PDS-measured loss-concentration slope dα dC at 1300 and 1550 nm vs. calculated polymer stiffness structure parameter σ cs for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials. 1300 nm data represented by . nm data represented by

; 1550

5.4.3 Tauc analysis of “near-gap” band tail To further examine disorder effects on the absorption loss behavior in 4E6m doped rigid aromatic polymer hosts, the empirical disorder model proposed by Tauc for optical spectra in amorphous glasses23, 58 (Section 2.12) was explored:

α E = B ( E − Eg )

2

(2.137)

where Eg ≡ “Tauc optical energy gap” B ≡ “Tauc slope” The Tauc relation is empirically consistent with the mathematical forms of the near-Fermi level DOS in the valence and conduction bands23. The Tauc slope B has 286

been generally associated with the overall long-range disorder in an amorphous material22, 23, where high values of B imply lower disorder. Here it is proposed that B corresponds to the distribution of transition energies Ei arising from the diagonal broadening term of the Hamiltonian (Eq. (2.124)) of the dipole-allowed transition. The analysis by Robertson59 using a Hückel tight-binding molecular orbital calculation for amorphous carbon ( a − C and a − C : H ), discussed in Section 2.13.1, shows that the Tauc gap Eg can be related to the dimensions of graphitic clusters: M =4 β

2

Eg2

(2.175)

where M = number of graphitic rings in the largest clusters of the amorphous matrix

β = nearest neighbor interaction energy between π − orbitals Further, Robertson59 and Zammit et al.60 (Section 2.13) showed that an increase in Eg is indirectly related to a reduction in strain in hydrogenated amorphous carbon ( a − C : H ) and silicon ( a − Si : H ), respectively. Plots of (α E )

1/ 2

vs. E (Tauc plots) are shown for 4E6m in each rigid amorphous

aromatic polymer host (at the reference doping level of 205 µmol per gram of polymer) in Figures 5.20a – 5.20d. The Tauc parameters B and Eg for each polymer host were derived from the slope and intercept of these plots. Each of these parameters is plotted against the polymer stiffness structure parameter σ cs in Figure 5.21, and plotted against each other in Figure 5.22. A strong linear decrease of Eg with σ cs ( R 2 = 0.96) and nearly linear decrease of B with σ cs ( R 2 = 0.84) is seen in Figure 5.21, while a strong linear increase of Eg with B is seen in Figure 5.22 ( R 2 = 0.96).

The B – σ cs

dependence suggests that the overall, long-range order of the guest-host system decreases as the polymer chain stiffness increases. By analogy with Robertson graphitic cluster theory59 (Eq. (2.175)), the Eg – σ cs behavior is ascribed to increasing size of locally

287

ordered domains in a higher strain configuration as polymer chain stiffness increases. Recall that for these dye-polymer systems, a domain is comprised of a localized region of short- to medium-range order, consisting of dye in parallel alignment with neighboring polymer chain segments within the amorphous glass network, analogous to that of amorphous inorganic glasses and semiconductors. The positive correlation between Eg and B implies that rigid amorphous aromatic guest-host systems with smaller domain sizes possess lower overall disorder, in a lower strain configuration, due perhaps to a more uniform distribution of domain sizes.

c)

b)

d)

(αhν)1/2

a)

1.3 1.8 1.3 1.8 1.3 1.8 1.3 1.8

hν (eV) Figure 5.20 Tauc plots for near-band spectral region C for 4E6m/rigid amorphous (non-carbonate) aromatic guest-host polymers at 205 µmol dye/g polymer: a) 4E6m/Parmax; b) 4E6m/Ultem; c) 4E6m/polysulfone; d) 4E6m/perfluorinated cyclobutane.

288

1.39

1.35 1400 1.31 1200 1.27

1000

1.23 9

Figure 5.21

Eg (eV)

B (dB/eV-cm)

1/2

1600

10

11

12

σcs

13

Experimental Tauc disorder parameter B and optical gap Eg at dye

concentration of 205 µ mol/g polymer vs. calculated polymer stiffness structure parameter σ cs for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials. Eg represented by

; B represented by

.

1.40

Eg (eV)

1.36

1.32

1.28 1000

1200

1400 1/2

B (dB/eV cm)

Figure 5.22

Experimental Tauc optical gap Eg vs. disorder parameter B and at dye

concentration of 205 µ mol/g for 4E6m/rigid amorphous aromatic (non-carbonate) polymer guest-host materials.

Next we examine relationships between the Tauc parameters and the weak tail width Ew , shown in Figure 5.23. The disorder parameter B increases nearly linearly with Eg ( R 2 = 0.87), and the Tauc gap Eg increases reasonably linearly with Eg

289

( R 2 = 0.91). This B – Ew dependence suggests that rigid amorphous (non-carbonate) aromatic guest-host systems with less short-range order, of lower chain stiffness, exhibit greater long-range order, of smaller domain sizes. We assign the short-range disorder, described by the weak tail width Ew , to the off-diagonal, pairwise matrix elements of the Hamiltonian (Eq. (2.124)), and the long-range disorder, related to the Tauc parameter B , to the diagonal elements. The observed Eg – Ew behavior suggests that domains of smaller sizes, associated with less rigid polymer chains, possess more short-range (offdiagonal) disorder.

1.41

1.37 1150

Eg (eV)

B (dB/eV-cm)

1/2

1350

1.33

950 0.04

1.29 0.08

0.12

0.16

EW (eV)

Figure 5.23

Experimental Tauc disorder parameter B and optical gap Eg at dye

concentration of 205 µ mol/g polymer vs. PDS spectral sub-band tail width Ew for 4E6m/rigid amorphous aromatic polymer guest-host materials. B represented by ; Ew represented by

.

Finally, we show correlations for loss – concentration slope dα / dC vs. the Tauc disorder parameter B at 1300 and 1550 nm in Figure 5.24.

Reasonably linear

dependence of dα / dC on B is seen at both wavelengths ( R 2 = 0.88 and 0.93 at 1300 and 1550 nm, respectively); no correlation was found at 1060 nm, as expected due to its proximity to the main absorption peak, and dominant solvatochromic behavior (not shown). These correlations, combined with those of dα / dC with Ew in Figure 5.10,

290

indicate that loss actually increases as the overall long-range disorder of the guest-host system decreases in the rigid amorphous aromatic polymer, and is controlled by the shortrange order between dye and parallel chain segments, and associated π – orbital delocalization, as polymer chain stiffness increases. It is inferred from the Tauc analysis that increased polymer chain stiffness leads to larger, more ordered domains of less uniform size distribution than those of less rigid polymers.

dα/dC (dB/cm)/(µmol/g)

0.012

0.008

0.004

0.000 1050

1150

1250

1350

B (dB/eV-cm)1/2

Figure 5.24 PDS-measured loss-concentration slope dα dC at 1300 and 1550 nm vs. Tauc disorder parameter B at dye concentration of 205 µ mol/g polymer for 4E6m/rigid amorphous (non-carbonate) aromatic guest-host materials. 1300 nm data represented by . ; 1550 nm data represented by

5.5 Conclusions The concentration dependence of optical absorption spectral behavior was investigated by photothermal deflection spectroscopy for glassy, amorphous nonlinear optical dyepolymer guest-host materials based on the nonlinear optical dye 4E6m, a structural analog of the well-known nonlinear optical dye FTC, incorporated into a series of rigid amorphous aromatic polymers. Linear or approximately linear dependence of absorption loss vs. dye concentration at three spectral minima, at 1060 nm, 1300 nm, and 1550 nm, was observed for all guest-host materials studied, indicating a lack of specific or strongbinding interactions between the dye and polymer.

Determination of the loss-

concentration slope behavior at each spectral minimum, in conjunction with red-shifting 291

behavior of the main absorption peak, is shown to be a valuable approach for revealing structure – property relationships based on variations in polymer backbone rigidity. The loss – concentration slope in the near-IR decreased with increasing red shift, in opposition to the behavior seen for 4E6m doped in a series of Bisphenol A polycarbonates in a previous study, indicating loss for the 4E6m-doped rigid (noncarbonate) aromatic polymers follows a distinctly different physical mechanism. The solvatochromic behavior of these guest-host materials was well correlated with predicted macroscopic polymer stiffness properties, such as Young’s modulus. The near-IR loss was shown to be controlled by the red-tail width in the 1.0 – 1.2 eV range, which in turn is well correlated with the predicted modulus. The red-tail width was shown to be associated with disorder in localized domains of dye molecules in parallel alignment between neighboring polymer chain segments. A defect density-of-states model was proposed to account for this local disorder, based on increasing extent of π – delocalization as polymer chain stiffness increases, with qualitative confirmation by ab initio calculations on various dye-polymer orientations. An empirical polymer chain stiffness structure parameter based on Bicerano’s structure parameters for prediction of glass transition temperatures was introduced, which accounts for the rotational degrees of freedom and steric hindrances in the polymer repeat unit, and which correlated with red-tail width, peak red-shift, loss – concentration slope, and semi-empirical predictions of macroscopic Young’s modulus of the host polymer. This structure parameter has the potential to serve as a predictive property for near-IR absorption loss for NLO guest-host materials comprised of rigid non-carbonate amorphous polymers, and may also prove useful for investigations of internal quantum efficiencies of electro- and photoluminescent materials. Analysis of the strong near-gap absorption region using the empirical Tauc model revealed that a long-range order parameter and optical energy gap are also correlated with the polymer chain stiffness and near-IR loss, but in opposite direction to those of the red-tail width associated with shortrange order. These long-range disorder effects are qualitatively consistent with cluster theory for amorphous glasses. The combined short-range and long-range disorder analyses are interpreted such that increasing polymer chain stiffness leads to larger, more ordered local domains of poorer size uniformity and higher strain, with greater π –

292

delocalization and lower near-IR loss. The near-IR loss is thus shown to ultimately be controlled by the short-range, non-diagonal disorder, dominated by pairwise, dipoleinduced dipole dye-polymer interactions, in the rigid amorphous aromatic polymer-based guest-host materials.

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298

Chapter 6

Effects of High β Guest Chromophore Dialkyl Spacer Length With a Polycarbonate Host

6.1 Introduction We showed in Chapter 4 how solvatochromism in a series of polycarbonate hosts controls near-IR loss at transmission wavelengths, such that higher polarity hosts yield higher near-IR loss. In Chapter 5 we showed that localized matrix disorder controls nearIR loss in a series of rigid amorphous non-carbonate aromatic polymer hosts, with both disorder and near-IR loss increasing as chain stiffness decreases. In neither of these studies were dye structural differences investigated. In this study, a homologous series of spacer lengths attached to NLO chromophores, of the family to which 4E6m belongs, as guest-host mixture with amorphous polycarbonate (APC) is characterized by a combination of PDS and UV-Vis spectroscopy to examine the effects of the spacer length on near-IR loss. The spacer length is expected to control both the dye-polymer solubility and the steric hindrance to dye-dye dipole interactions, each of which can influence broadening of absorption peaks and hence, the value of absorption at near-IR transmission wavelengths (1060 nm, 1300 nm, and 1550 nm). Both the absorption loss at spectral minima between C–H overtones peaks (at 1060 nm, 1300 nm and 1550 nm) and the dye electronic absorption inhomogeneous peak width are evaluated as a function of molar concentration of dye in APC for each dye spacer length. The near-IR loss–concentration slope is evaluated with respect to the main dye absorption peak spectral features and thermodynamic mixing nonidealities based on Marcus theory. A conceptual of model of the spectral features is presented in terms of a

299

Chapter 6 balance between dye solubility and size, founded on Reichardt’s classification of geometric and interaction energy contributions to the Gibbs solvation free energy. A physical model of the these spectral features and low wavelength dα / dC (where α is loss and C is dye concentration) is developed based on dye-dye dipole interactions and dye-polymer van der Waals interactions, considering models of dipole interactions due to Loring1 and to Kador2, and a geometric mean cohesive energy density for van der Waals interactions. The convolution of these two terms is determined and compared with the experimental spectral band features and loss-concentration slope vs. spacer length behavior, considering both Gaussian and non-Gaussian components of the band tail.

6.2 Experimental Procedures A structural representation of the homologous NLO dye series LMCO-4x6m used in the guest-host materials of this study is shown in Figure 3.4. The diethyl spacer analog in this series, 4E6m, is the same dye used in the evaluation of polymer host structure variations in Chapters 4 and 5. The remaining three dyes, having dipropyl, dibutyl, and dihexyl spacer moieties, are designated 4P6m, 4B6m, and 4H6m, respectively, and are identical to 4E6m with the exception of the spacer lengths. Each dye has three main components, an aminobenzene electron donor, a π -conjugated substituted thiophene ring bridge component, and an electron-accepting tricyanovinyldihydrofuran ring containing a gem-dimethyl substituent.

They are synthesized* by the Wittig reaction of 4-(2-

methoxyethylethyamino)benzyltriphenyl phosphonium iodide with 3,4-dialkylthiophene2-carboxaldehyde to form a coupled product, which is then formylated to give the coupled aldehyde product.

Reaction of each coupled aldehyde analog with 2-

dicyanomethylene-3-cyano-4,5,5-trimethyl-2,5-dihydrofuran (FTC acceptor) yields the product analog 4x6m, where x corresponds the thiophene pendant dialkyl length (see Figure 6.1). The polymer host of this study, APC, is a random, amorphous polycarbonate copolymer, poly[Bisphenol A carbonate]x-co-[4,4’-(3,3,5-trimethylcyclohexylidene)diphenol carbonate]y, and its structure is represented in Figure 3.1b. APC is one of the four *

All dye synthesis for this study was performed by Joseph Epstein and Dr. Peter Bedworth of Lockheed Martin Advanced Technology Center.

300

Chapter 6 polycarbonates discussed in Chapter 4, and has been the subject of several studies of NLO guest-host materials with FTC-like chromophores3-12.

x x N CN H3CO

S

CN O

CN

Figure 6.1 Homologous series of nonlinear optical guest dye 4x6m: 2-(3-cyano-4{2-[5-(2-{4-[ethyl-(2-methoxyethyl)amino]phenyl}vinyl)-3,4-dialkyllthiophen-2yl]vinyl}-5,5-dimethyl-5H-furan-2-ylidene)malononitrile. Dyes are designated: 4E6m (x = 0, diethyl spacer); 4P6m (x = 1, dipropyl spacer); 4B6m (x = 2, dibutyl spacer); 46m (x = 4, dihexyl spacer).

The 4E6m/APC guest – host films samples discussed in this Chapter are the same as those discussed in Chapters 4. See Section 4.2 for a description of guest – host solution preparation and film preparation and thicknesses of these samples.

The

dipropyl-spacer dye 4P6m was doped into APC at concentrations ranging from 0 to 330 µ mol dye per gram of APC, using 2:1 by weight mixtures of cyclopentanone and Nmethyl-2-pyrrolidone (NMP)*, at a polymer solids content of 15% by weight. The dibutyl- and dihexyl-spacer FTC dye analogs 4B6m and 4H6m were doped into APC at concentrations ranging from 0 to 500 µ mol dye per gram of APC, using 3:1 by weight mixtures of cyclopentanone and diglyme, at a polymer solids content of 15.5% by weight. Film samples† for spectral characterization were prepared on 1” diameter IR transparent fused silica substrates, which were described in Section 3.3.3.1.

All

*

Both 4E6m and 4P6m have significantly lower solubility in APC than 4B6m and 4H6m, and required a more polar solvent mixture to effect dissolution. The poorer solubility of 4E6m and 4P6m in APC compared with that of 4B6m and 4H6m was the reason for the lower concentration ranges of 4E6m and 4P6m in APC. †

Angelina Moss, formerly of Lockheed Martin Advanced Technology Center, prepared the 4B6m/APC and 4H6m/APC films for this study.

301

Chapter 6 substrates were pre-cleaned* prior to film deposition following the procedure given in Section 3.3.4.

All samples were solution spin cast, baked, and inspected by the

procedures of Section 3.3.4. All films were baked using a lower bake temperature of 100°C. 4P6m/APC films were baked using an upper bake temperature of 215°C† to drive off NMP and cyclopentanone. 4B6m/APC and 4H6m/APC films were baked using an upper bake temperature of 180°C to drive off diglyme and cyclopentanone. Film thickness was measured following the procedure described in Section 3.3.4. Average thicknesses for films of 4P6m/APC ranged from 1.38 to 4.30 µm . Average thicknesses for films of 4B6m/APC ranged from 2.46 to 2.94 µm . Average thicknesses for films of 4H6m/APC ranged from 2.25 to 2.98 µm . Spectral characterization of all films samples by PDS and Cary UV-Vis was performed as described in the procedures of Sections 3.3.5 and 3.3.6, respectively. An initial low-energy PDS scan from 1830 nm to a minimum of 870 nm was immediately followed by a second, high-energy scan from 1830 to < 640 nm for all samples. For all reported spectra, one-to-one correspondence of PDS and Cary UV-Vis measurements on the same sample was maintained. Reduction of Cary UV-Vis and PDS data into final combined spectra was performed as described in Section 3.3.7.

6.3 Results Representative PDS-Cary spectral overlay of the concentration series is shown for each of the dialkyl spacer functionalized dyes doped in APC in Figures 6.2a – 6.2d. As seen in the data of Chapters 4 and 5, each of the spectra are comprised principally of four peaks: 1) a strong dye π − π * main electronic absorption peak in the visible region, λmax 660 – 672 nm, which varies with dye concentration; 2) a ν

0,3

C–H stretching overtone

centered at ~1140 nm; 3) a |2,1> C–H stretching-plus-bending combination overtone at 1390 nm; and 4) a ν

0,2

C–H stretching overtone at 1660 nm13. For each dye/APC

concentration series, the main peak shows a Gaussian shape (on a photon energy scale) *

Some of the substrate pre-cleaning was performed by Gil Mendenilla of Lockheed Martin Advanced Technology Center, and Angelina Moss, formerly of Lockheed Martin Advanced Technology Center.



The higher upper bake temperature for 4E6m/APC and 4P6m/APC films was required to drive off the higher boiling NMP solvent ( Tb 202°C for NMP, vs. 162°C for diglyme and 131°C for cyclopentanone).

302

Chapter 6 with a linear increase in peak intensity with concentration, consistent with the BeerLambert Law.

330 µmol/g

330 µmol/g

100000

205 µmol/g

α (dB/cm)

α (dB/cm)

100000

1060 nm

1000

1300 nm 1550 nm

96 µmol/g 46 µmol/g

10

0.1

205 µmol/g 1060 nm

1000 46 µmol/g

10

0.1 0

a)

500

1000

1500

2000

0

b)

Wavelength (nm)

500

α (dB/cm)

α (dB/cm)

1060 nm

1000

205 µmol/g

1300 nm 1550 nm

96 µmol/g

10

1500

2000

501 µmol/g

100000

330 µmol/g

1000

Wavelength (nm)

501 µmol/g

100000

330 µmol/g 1060 nm

1000

205 µmol/g

1300 nm 1550 nm

96 µmol/g

10

46 µmol/g

46 µmol/g

0.1

0.1 0

c)

1300 nm 1550 nm

96 µmol/g

500

1000

1500

Wavelength (nm)

2000

0

d)

500

1000

1500

2000

Wavelength (nm)

Figure 6.2 Representative concentration series of PDS absorption loss spectra for 4x6m/APC guest-host materials: a) 4E6m/APC; b) 4P6m/APC; c) 4B6m/APC; d) 4H6m/APC. An unusual bistable peak intensity behavior was noticed for the 1390 overtone peak in 4B6m/APC films, in which the measured peak intensity was either near 3.5 dB/cm or 14 dB/cm (see Figure 6.2c, peak lying between 1300 nm and 1550 nm). At concentrations of 96 and 205 µ mol/gram, only the higher intensity peak was observed in replicate measurements of each. No other peak intensity values either between or outside (within uncertainty limits) of these two values were measured. Note that this bistability was seen at the lowest concentration (46 µ mol/gram) measured for 4H6m/APC, as well, but only the higher intensity was measured for all other concentrations of 4H6m/APC at 1390 nm in two samples at 96 and 205 µ mol/gram, six replicate runs at 330 µ mol/gram (including one replicate sample), and three samples at 501 µ mol/gram (including one 303

Chapter 6 replicate run). The higher intensity peak for 4H6m/APC in all measurements (except one), and in several measurements for 4B6m/APC, is in contrast to the lower intensity peak only seen for all measurements on 4E6m/APC and 4P6m/APC (compare peak lying between 1300 nm and 1550 nm in Figures 6.2c and 6.2d with those of Figures 6.2a and 6.2b). This observation prompted a re-examination of the 1390 overtone peaks in all PDS spectra measured for 4E6m/polymer guest – host materials discussed in Chapters 4 and 5. With the exception of one sample measured for 96 µ mol/gram 4E6m/Red Cross (showing the high intensity overtone), all of the 1390 nm overtones for the 4E6m/polymer materials were of the lower intensity. We suggest that the bistability in 4B6m/APC overtones at 1390 nm is due to a transition in the stretching + bending anharmonic resonance in increasing from the dialkyl spacer length of three to six carbons per alkyl group, from a weak resonance at three carbons to a strong resonance at six carbons*. We have carefully examined the influence of the 1390 nm overtone intensity on the values of the absorption minima at 1300 nm and 1550 nm (or correlation to with main peak spectral features), and it was found to be negligible in all cases. We therefore included all 4B6m/APC and 4H6m/APC spectra in the remaining analyses, while noting differences in the 1390 nm peak intensities. As seen for the 4E6m/polycarbonate host series in Chapter 4, the strengths of the 1140 nm C–H overtone peak varies with concentration, while the 1660 nm C–H overtone peak is insensitive to dye concentration. Relatively constant overall concentration of C– H moieties with dye doping level is suggested by the 1660 nm peak intensity vs. concentration behavior, while changes in the 1140 nm overtone peak with dye concentration suggest that a low energy tail feature on the main absorption peak is increasing with concentration. As in the study of Chapter 4, the analysis of dye spacer length effects on the absorption loss at these wavelengths focused on three characteristics of the spectra: 1) the concentration dependence of near-IR absorption loss at the spectral minima at 1060 nm, *

Dr. Carleton Seager at Sandia National Laboratories has suggested that variation in intensity signal at 1390 nm may be due to a “lateral saturation” of the PDS signal due to spatial inhomogeneity of the dye in the film. We believe this may account for random variations in overtone intensity, but not for the observed bistability.

304

Chapter 6 1300 nm, and 1550 nm; 2) the broadening behavior of the dye electronic absorption peak; and 3) the energetic shifting behavior of the dye absorption maximum. The latter two of these spectral features will be addressed in the Discussion section. The concentration dependence of absorption loss minima at each of 1060 nm, 1300 nm, and 1550 nm is plotted in Figures 6.3a – 6.3c, respectively. Linear behavior of the absorption minima with concentration is exhibited most cases, with the best correlation at 1060 nm ( R 2 range 0.94 – 0.99), followed by 1300 nm ( R 2 range 0.63 – 0.99), and finally by 1550 nm ( R 2 range 0.37 – 0.96). The poorest correlations were observed for the dibutyl-spacer dye 4B6m, which is attributed to the weaker concentration dependence of loss at each wavelength for this system. Following the approach of Chapters 4 and 5, we again treat the slope of the PDS loss minimum versus concentration plot as a characteristic material property for each dye-APC guest-host combination, analogous to a molar absorptivity, at each wavelength minimum, and use this as an input to the analyses of the structure – property relationships. The intercept values for loss vs. concentration at each wavelength are in close agreement between dyes and are relatively low compared to doped APC loss values. The loss-concentration behavior at each wavelength is consistent with the BeerLambert Law, and appears to be dominated by dye electronic absorption peak broadening with increased concentration. Note that the dye peak position is less red-shifted as concentration increases for each dye, in agreement with a perturbative analysis of dyesolvent system by Sevian and Skinner14 and a coherent potential approximation treatment of cermet topology by Liebsch and Gonzalez15.

305

Chapter 6 3

8

b) 1300 nm α (dB/cm)

α (dB/cm)

a) 1060 nm 6

4

2

2

1

0

0 0

200

400

0

600

200

400

600

Dye Concentration (µmol/g polymer)

Dye Concentration (µmol/g polymer) 3

α (dB/cm)

c) 1550 nm 2

1

0 0

200

400

600

Dye Concentration (µmol/g polymer)

Figure 6.3 PDS-measured loss vs. dye concentration for 4x6m dye dialkyl spacer 4E6m (diethyl length series in APC: (a) 1060 nm; (b) 1300 nm; (c) 1550 nm 46m 4P6m (dipropyl spacer); 4B6m (dibutyl spacer); spacer); (dihexyl spacer).

6.4 Discussion In Chapter 4 we examined, for the diethyl-spacer FTC analog 4E6m in a series of polycarbonate hosts, inhomogeneous broadening and solvatochromism associated with host structure, and showed that solvatochromism dominates the measured loss behavior16. Here we evaluate effects of inhomogeneous broadening and solvatochromism for the case of the four dyes in the same polymer host. Gaussian fits of the UV-Vis-PDS spectra at the highest common equimolar dye concentration (330 µ mol per gram of polymer) is shown in Figure 6.4. Very little difference is seen in the wavelength position of the peak maximum among the four dyes, indicating independence of solvent shift with dye side chain length. Instead, significant variation in inhomogeneous broadening is observed among the four dyes.

306

Arbitrary Units

Chapter 6

Dipropyl spacer Dihexyl spacer Dibutyl spacer

Diethyl spacer

0.8

1.8

2.8

hν (eV)

Figure 6.4 Gaussian fits to UV-Vis-PDS spectra at dye concentration of 330 µ mol per gram of APC as overlays for diethyl, dipropyl, dibutyl, and dihexyl spacer groups on LMCO-4x6m.

The dye with dibutyl spacer (4B6m) was found to exhibit the smallest inhomogeneous width Γ s

(full-width at half-maximum) at this and all other

concentrations studied. In Figure 6.4, it is observed that greater broadening occurs on the high-energy side of the main absorption peak. However, detectable broadening is seen on the low energy side, as well. The overall inhomogeneous width Γ s can be associated with a distribution of microscopic dye states in the surrounding polymer, such as dye orientations relative to neighboring dye and polymer molecules, and corresponding dyedye and dye-polymer interactions. It is proposed that this overall broadening behavior accounts for the observed variation in near-IR loss and associated loss vs. concentration dependence. To examine these assertions, the PDS-measured near-IR loss α at transmission wavelengths (1060 nm, 1300 nm, and 1550 nm) are plotted against the inhomogeneous width Γ s determined from Gaussian peak fits for all dyes at all concentrations in Figures 6.5a – 6.5c.

307

Chapter 6

α (dB/cm)

8

4

a) 1060 nm 0 0.34

0.36

0.38

Γs(eV)

α (dB/cm)

3

2

1

b) 1300 nm 0 0.34

0.35

0.36

0.37

0.38

Γs(eV)

α (dB/cm)

3

2

1

c) 1550 nm 0 0.34

0.35

0.36

0.37

0.38

Γs(eV) Figure 6.5 PDS-measured absorption loss vs. inhomogeneous 0-0 absorption band width Γ s for 4x6m dye-doped APC over entire concentration range at wavelengths of: (a) 1060 nm; (b) 1300 nm; and (c) 1550 nm. Curve shown in each plot is best fit through combined data. 4E6m (diethyl spacer); 4P6m (dipropyl spacer); 4B6m 46m (dihexyl spacer). (dibutyl spacer);

308

Chapter 6 As shown in Figures 6.5a – 6.5c, at each wavelength, the combined data can be fit to a single trend line. At 1060 nm, absorption loss α shows a strong linear dependence on Γ s , while at 1300 nm and 1550 nm, loss shows quadratic dependence on Γ s . Correlation of α with Γ s at 1550 nm is poorer than that of the other two wavelengths. We suggest that this is due to a broadening mechanism at 1550 nm that has more important contributions from non-Gaussian, sub-band broadening, as will be discussed later. The linear α – Γ s behavior at 1060 nm is most likely due to the close proximity of the main absorption peak to 1060 nm, such that a change in inhomogeneous broadening directly translates to higher loss. The quadratic dependence of α on Γ s at 1300 nm and 1550 nm may be due to higher order terms in the band shape function in the spectral wings, as proposed by Kjaer and Ulstrup17, and Kador et al. 18. This “universal” dependence of near-IR loss α on inhomogeneous broadening Γ s for the homologous series of dyes provides a framework for a more detailed physical description of loss mechanisms in these merocyanine NLO guest-host materials. The structure – property relationship for inhomogeneous width vs. dialkyl spacer length is shown in Figure 6.6, for Γ s measured at the reference dye concentration of 330 µ mol per gram of polymer, showing a distinct minimum in Γ s for a dialkyl spacer length of four carbons per alkyl. The structure – property relationships for loss-concentration slope dα / dC vs. dialkyl spacer length are shown in Figure 6.7 for measurements at wavelengths of 1060 nm, 1300 nm, and 1550 nm. These, too, show a minimum in near-IR loss corresponding to the dibutyl spacer, with the profiles at 1060 nm and 1300 nm closely mimicking the Γ s - spacer length profile. The minima in broadening and near-IR loss seen for the dibutyl spacer is fortuitous in practical terms, as this dye has also been found to exhibit superior thermal stability of dye orientation in d.c. field-poled APC films relative to the baseline di-hexyl dye at high concentration (500 µ mol per gram of polymer); the two shorter diakyl length dyes (diethyl and dipropyl) have insufficient solubility in APC to make stable singlephase films at this concentration. These findings for near-IR loss have therefore provided

309

Chapter 6 a compelling rationale for continued technological development of the dibutyl dye in polymer films for practical waveguide devices.

0.38

Γs (eV)

0.37

0.36

0.35 0

2

4

6

8

Dialkyl Spacer Length (# carbons/alkyl)

Figure 6.6 UV-Vis-PDS-measured inhomogeneous width Γ s (at a dye concentration of 330 µ mol per gram of APC) vs. dye dialkyl spacer length for 4x6m – APC guest – host materials. In Chapter 4 a parametric analysis of the Kawski solvent polarity relation suggested that a more spherically symmetric dye molecule should exhibit less solvatochromism, and hence, lower near-IR loss in polycarbonate hosts. Of the four dyes in the homologous series reported here, the dihexyl spacer possesses a shape closest to spherical symmetry, with a dialkyl spacer length of 8.0 Å vs. an axial length of 22.2 Å (based on minimized structures using Accelrys, Inc. Cerius2). The structure – property relationship measured in the present study (Figure 6.7) show that the loss is not minimized for most spherically symmetric dye in the case of inhomogeneous broadening. Rather, loss appears to be controlled by a balance of physical mechanisms related to side chain structure in this case.

310

Chapter 6 0.025

dα/dC (dB/cm)/(µmol/g)

a) 1060 nm

0.015

0.005 0

2

4

6

8

Dialkyl Spacer Length (No. Carbons/alkyl) 0.0050

dα/dC (dB/cm)/(µmol/g)

b) 1300 nm

0.0025

0.0000 0

2

4

6

8

Dialkyl Spacer Length (No. Carbons/alkyl) 0.0050

dα/dC (dB/cm)/(µmol/g)

c) 1550 nm

0.0025

0.0000 0

2

4

6

8

Dialkyl Spacer Length (No. Carbons/alkyl)

Figure 6.7

PDS-measured loss – concentration slope dα / dC vs. dye dialkyl spacer

length for 4x6m – APC guest – host materials at wavelengths of: a) 1060 nm; b) 1300 nm; and c) 1550 nm.

311

Chapter 6 6.4.1 Thermodynamic treatment of inhomogeneous broadening We now consider the concentration dependence of the inhomogeneous broadening measured in these materials.

We described Marcus theory for

inhomogeneous broadening and solvatochromism in Part I. Marcus19 showed that the mean-square fluctuation term σ s2 in the Gaussian band shape expression I G = ( 2πσ s )

−1/ 2

 (ν −ν )2  s  exp  − 2 2σ s   

(6.1)

(where ν s , the “solvent shift,” is the displacement of ν max from the vacuum absorption 0 frequency) is related to the Gibbs free energy of solvation ∆Gsolv and polar contributions

to the Helmholtz free energies of the solute (dye) ground and excited states, F0 and F1 , respectively, as

(

0 σ s2 = 2kT 2∆Gsolv + F0 − F1

)

(6.2)

Since the experimental peak transition energies ν s were nearly invariant with spacer length, the Helmholtz energy difference F1 − F0 may be treated as approximately constant: F1 − F0 = h0 = constant

(6.2a)

Thus,

(

0 σ s2 = 2kT 2∆Gsolv − h0

)

(6.3)

The inhomogeneous peak width Γ s is related to the mean-square fluctuation term

σ s through σ s = Γ s 2 2 ln 2

312

(6.4)

Chapter 6 Substituting into (3) above: 0 Γ 2s = ( 32 ln 2 ) kT ∆Gsolv − (16 ln 2 ) kTho

(6.5)

Reichardt20 defines the free energy of solvation of a solute as 0 0 0 ∆Gsolv = ∆Gsoln − ∆Glattice

(6.6)

where 0 ∆Gsoln = Gibbs free energy of the solute in solution 0 ∆Glattice = Crystal lattice energy of unmixed solute, defined as the work required

to separate the solute from its equilibrium position in a crystal lattice to infinity at 0 K. This definition is equivalent to the mixing energy of component A, ∆GAmix , in a binary A-B mixture: ∆G0solv ≡ ∆G Amix = kT ln a A

(6.7)

where a A is the activity of component A (solute) in solution. We can then re-write Eq. (6.5) in a more compact form Γ 2s = a0 ∆G Amix + b0

(6.8)

where a0 = ( 32 ln 2 ) kT = constant b0 = − (16 ln 2 ) kTho = constant

From Eq. (6.8), the square-inhomogeneous broadening is predicted to be linearly related to the mixing free energy. The concentration dependence of Γ 2s can now be

313

Chapter 6 found from a treatment given by Lupis21 for binary mixtures, by taking the Taylor series expansion of the activity coefficient: ln γ A = ln γ A∞ + ε A x A

(6.9)

where

γ A = activity coefficient of component A (solute), given by ln γ A ≡ ∆G E / kT = ln a A − ln x A

(6.10)

∆G E = excess Gibbs free energy of mixing (deviation from ideal mixing)

x A = mole fraction of solute (dye)

γ A∞ = infinite dilution solute activity coefficient, also referred to as Henry’s Law constant

ε A = solute first-order interaction coefficient Defining the factor k F′ = exp ( −b0 / a0 kT )

(6.11)

and taking limits as x A → 0 for Henry’s Law behavior, it can be shown from Eqs. (6.8) – (6.10) that Γ 2s = Γ 02 + ε ′A x A

(6.12)

where

{

(

{

)

Γ 02 = a0 kT ln γ A∞ k F′ − ln ( 2 a0 kT ) − ln Γ s ( d Γ s dx A )  x A →0

ε ′A = a0 kT ε A

314

}}

(6.12a)

(6.12b)

Chapter 6 The interaction coefficient ε A was introduced by Wagner22, and is a measure of how the solute activity coefficient increases with increasing solute concentration. ε ′A is referred to herein as an apparent first order dye interaction coefficient. The ratio γ A∞ k F′ is an effective Henry’s Law constant, and is measure of the dissimilarity of the solute and solvent species; higher values of γ A∞ k F′ imply lower affinity. The quantity a0 kT is 1.36 , x 108 J2/mol2 at room temperature. The third term in Eq. (6.12a),  Γ s ( d Γ s dx A )  x A →0 is found from the slope and intercept of a plot of Γ s vs. dye mole fraction x A . Thus, from Eq. (6.12), the square-inhomogeneous width Γ 2s should be linear in mole fraction with slope ε ′A , and intercept Γ 02 , from which the apparent Henry’s Law coefficient

γ A∞ k F′ can be calculated. Γ 2s is plotted against dye mole fraction in APC for the four dyes in Figure 6.8. From this plot, we see that the square-inhomogeneous width is indeed linear in mole fraction, consistent with Marcus theory and Lupis’ first-order expansion of the activity coefficient, with variations in slope and intercept as dialkyl spacer length changes.

6 -2 2 Γs (10 cm )

9.4

8.4 Diethyl spacer Dipropyl spacer Dibutyl spacer Dihexyl spacer

7.4 0.00

0.05

Xdye

0.10

0.15

Figure 6.8 Square-inhomogeneous width Γ 2s of 4x6m dye 0-0 absorption peak vs. dye mole fraction.

315

Chapter 6 The Γ 2s - x A trend for the dibutyl spacer in Figure 6.8 shows the smallest broadening over all concentrations, with the lowest slope and intercept. The Γ 2s - x A trend for the dihexyl spacer is shifted by ~ 0.04 eV above the trend for the dibutyl spacer, with a slightly higher slope. The diethyl and dipropyl spacers have similar slopes and intercepts, each of which are higher than those of the dihexyl and dibutyl spacers. From the intercepts of the Γ 2s - x A plots in Figure 6.8, and from the slopes and intercepts of Γ s vs. x A plots (not shown), the effective Henry’s Law constant γ A∞ k F′ for each dialkyl spacer length was calculated following Eq. (6.12a). The slopes in Figure 6.8 are equal to the apparent dye interaction coefficients ε ′A . These experimentally derived apparent Henry’s Law constants and effective dye interaction coefficients are plotted against dialkyl spacer length in Figure 6.9. Each of these thermodynamic parameters shows a structure-property profile similar to those of the inhomogeneous width Γ s (Figure 6.6) and near-IR loss (Figure 6.7). The smaller dialkyl spacers (dipropyl and diethyl) are seen to exhibit the highest interaction coefficients ε ′A and Henry’s Law constants γ A∞ k F′ . The higher interaction coefficients imply greater deviations from ideal mixing (excess free energies), while the higher Henry’s Law constants are interpreted as lower intrinsic affinity between dye and polymer, for dyes with the shortest spacers. This is expected based on predictions of differences between dye and APC solubility parameters, as discussed below. Intuitively, as the dialkyl spacer becomes shorter, it presents more polar nature to its local surroundings. For the relatively nonpolar APC polymer host, the dye-polymer interactions are weaker, and dye molecules are more likely to interact with each other at shorter spacer lengths. The differences between the thermodynamic parameters ( ε ′A and

γ A∞ k F′ ) are seen to be negligible between the dipropyl and diethyl spacer, but are quite pronounced when compared with those of the dibutyl spacer. This sharp decrease in broadening between a dialkyl spacer length of three and four is not predicted by solubility parameter differences, but is postulated to be brought about by a sudden change in balances between dipole interactions (dye-dye) and dispersion interactions (dye-polymer) at this critical spacer length range. 316

Chapter 6 10

ε'A

6

-4

10 x γ∞/k' f

0.09

0.06

2

0.03

0

2

4

6

8

Dialkyl Spacer Length (# Carbons/alkyl)

Figure 6.9 Structure – property relationship for effective Henry’s Law constant ∞ γ / k ′f and apparent interaction coefficient ε ′A vs. 4x6m dye dialkyl spacer length, derived from inhomogeneous width – dye mole fraction correlations.

The thermodynamic parameters ε ′A and γ A∞ k F′ show a less dramatic upward inflection in going from a dialkyl spacer length of four to six. This change is also not expected from calculated solubility parameter differences. This is postulated to arise from a more gradual change in balances between dipole and dispersion interactions as the dye presents more nonpolar character to its surroundings in going from four to six carbon spacer lengths. The broadening-derived thermodynamic parameters ε ′A and γ A∞ k F′ are plotted against experimental near-IR loss – concentration slopes dα / dC at wavelengths of 1060 nm and 1300 nm in Figures 6.10a and 6.10b. Strong linear correlations are seen for both parameters vs. measured loss at 1060 nm, with somewhat linear correlations seen at 1300 nm. Poor correlation was seen for these data at 1550 nm (not shown). The correlations at 1060 nm and 1300 nm suggest that that the near-IR loss can be attributed in large part to inhomogeneous broadening brought about by varying degrees of mixing nonidealities. The lack of correlation at 1550 nm is believed to be due to a change in broadening behavior from fully Gaussian to not fully Gaussian at this lower energy range.

317

Chapter 6

a) 1060 nm

0.10

ε'A

10-4 x γ∞/kf'

5 4

0.08 3

Henry's Law Constant Interaction Coefficient

2 0.008

0.06 0.012

0.016

0.020

dα/dC (dB/cm)/(µmol/g) b) 1300 nm

0.10

ε'A

10-4 x γ∞/kf'

5 4

0.08 3

Henry's Law Constant Interaction Coefficient

2 0.001

0.003

0.06 0.005

dα/dC (dB/cm)/(µmol/g) Figure 6.10 Effective Henry’s Law constant γ ∞ / k F′ and apparent interaction coefficient ε ′A vs. PDS-measured loss-concentration slope dα / dC at wavelengths of: a) 1060 nm; b) 1300 nm.

6.4.2 Conceptual physical model of inhomogeneous broadening A physical model for the changes in inhomogeneous broadening and near-IR loss with dialkyl spacer length is now presented in terms dipole and dispersion interactions. An analysis by Sevian and Skinner suggests that broadening is due to density fluctuations, which arise from a less uniform distribution of dye molecules in a polymer host14. Two primary changes are expected to occur as the spacer increases from shorter to longer lengths: the solubility parameter mismatch between the relatively nonpolar polymer and more polar dye should decrease, as discussed above; and the cavity volume occupied by the dye should increase.

Both the molar volume Vdye and solubility

parameter mismatch ∆δ p are plotted vs. 4x6m dialkyl spacer length in Figure 6.11. 318

Chapter 6

16

3 1/2

∆δp ((J/cm ) )

570

3

Vm (cm /mol)

620

14 520

470

12 0

2

4

6

8

Dialkyl Spacer Length (# carbons/alkyl)

Figure 6.11 Predicted 4x6m dye molar volume and polar component solubility parameter difference (with respect to APC) vs. dialkyl spacer length. Molar volumes Vm are calculated using Accelrys, Inc. SYNTHIA; Hansen polar solubility parameters δ p are calculated using empirical Karim-Bonner relation. denotes solubility parameter difference.

denotes molar volume;

In this plot, the dye and polymer solubility parameters are the Hansen polar cohesion parameters δ p calculated using the empirical Karim-Bonner relation23:

δ p = 102.5µ / V 3 / 4

(6.13)

with δ p in (J/cm3 )1/ 2 , where µ is the ground state dipole moment and V is the molar volume. The dye ground state dipole moment µ g was calculated by the semi-empirical intermediate neglect of differential overlap Hamiltonian version 1 (INDO/1), using the commercial package Cerius2 ZINDO from Accelrys, Inc. (see Section 2.16.3), applying a self-consistent reaction field representative of APC dielectric properties. The effective dipole moment µ M of APC was calculated by the highly correlated non-hydrogen graph connectivity method of Bicerano24 (described in Section 2.16.1), using Accelrys Cerius2 SYNTHIA. Molar volumes of dye and APC, Vdye and V polymer , were also calculated following Bicerano’s connectivity method using SYNTHIA. The curves for Vdye and

319

Chapter 6 ∆δ p in Figure 6.11 have opposing slopes, and cross at a point close to a spacer length of 4 carbons per alkyl. 0 as the change in Gibbs free Reichardt20 defined the solvation energy ∆Gsolv

energy upon transferring a molecule from the gas phase into a solvent, which is the superposition of four principal energy components: 1) the cavitation energy required to form a hole in the solvent to be occupied by the dissolved solute molecule; 2) the orientation energy due to partial reorganization of dipolar solvent molecules upon incorporation of the dissolved solute; 3) the isotropic interaction energy associated with non-specific intermolecular interactions (electrostatic, polarization, dispersion) over long interaction distances; and 4) anisotropic interaction energy associated with specific interactions, e.g. hydrogen bonds or electron pair donor/acceptor bonds. We restrict our discussion to first three of these, due to absence of any known specific interactions in these materials. The first two of these energies (cavitation and orientation) relate to geometric effects, while the third term corresponds to the relative strengths of dye-dye vs. dye-polymer interactions, as reflected in the solubility parameter. We can view the 0 increase in ∆Gsolv to the right of the crossover point in Figure 6.11, ascribed to excess

free energy of mixing through the activity coefficient, as being dominated by both of the geometric energy contributions, due to increased dye cavity volume and polymer 0 to the left of reorganization. The mixing nonidealities responsible for increasing ∆Gsolv

the crossover point are interpreted as due to a combination of the isotropic interaction energy term and the orientation energy term, due to increased dye dipole-dipole interactions and greater cooperative rearrangement of the surrounding polymer imposed by the higher aspect ratio dye. A conceptual model for these energy contributions as they relate to dispersion and broadening is put forth in Figure 6.12. (We put forth this model is as a framework for discussion purposes only, not to be construed as a rigorous accounting of the measurements and ensuing structure property relations.)

320

Chapter 6

+

+

-

+

-

+

-

-

-

+

-

+

+ -

+

-

-

+

-

-

+ + +

Figure 6.12 Depiction of cavity, dye-dye and dye-polymer configurations proposed for: a) short (diethyl and dipropyl) spacers; b) intermediate (dibutyl) spacer; and c) long (dihexyl) spacer. Gray rods represent dye main chains, and alkyl spacers are represented by dark “fuzz” units extending outward from dye. In (a), cavitation is accompanied by polymer organization, and configuration states are dominated by dipole-dipole interactions; in (c) cavitation energy is larger due to larger size of dye molecules, and both dye-dye and dye-polymer (dipole and dispersion, respectively) interactions contribute to configuration states; in (b) dye molecules are well isolated, cavitation energy is minimized, and only dye-polymer (dispersion) interactions contribute to configuration states.

In Figure 6.12a for short dialkyl spacer lengths, we propose that cavitation is accompanied by polymer organization, and configuration states are dominated by dye intermolecular dipole-dipole interactions. The dye may take on a distribution of quasiaggregation states in this case, leading to larger density fluctuations and hence, more broadening.

Note that neither differential scanning calorimetry nor atomic force

microscopy measurements revealed evidence of distinct aggregates in these materials. In Figure 6.12c for long dialkyl spacer lengths, cavitation energy is higher due to the larger size of dye molecules, and dye-polymer interactions significantly contribute to the distribution of configuration states, increasing broadening. In Figure 6.12b for intermediate spacer length, dye molecules are well isolated, cavitation and orientation energies are minimized, and the combined dye-dye and dye-polymer interactions are minimized.

321

Chapter 6 6.4.3 Quantitative models of inhomogeneous broadening The relationships between inhomogeneous broadening and dialkyl spacer length are examined quantitatively by considering separately the dye-dye and dye-matrix attractive interactions. Central to this analysis is development of a model that accounts for the abrupt decrease in broadening as the dialkyl spacer length increases from three to four carbons per alkyl unit, and the more gradual increase in broadening as the spacer length increases from four to six carbons, seen for measured Γ s and for experimental dα / dC values at 1060 nm and 1300 nm. Moreover, we endeavor to account for the

change in characteristic broadening vs. spacer length behavior exhibited by measured dα / dC values at 1550 nm.

6.4.3.1 Treatment of broadening and loss-concentration behavior at 1060 and 1300 nm We consider two inhomogeneously broadened distributions of electronic frequencies, one due to dye intermolecular permanent dipole-dipole attractive interactions of width D , and the second due to dye-matrix van der Waals attractive dispersion interactions, of width W . A general expression for the convolution of two uncorrelated statistical distributions is Γ 2 = Γ 2I + Γ 2II

(6.14)

where Γ I and Γ II are the widths of distributions broadened by uncorrelated interactions I and II , respectively25. We treat the D and W distributions as uncorrelated, and express their convolution more generally as Γ 2s = Wv2 + Dv2

where Wv = vW Dv = vD v = volume fraction of dye in guest-host mixture

322

(6.15)

Chapter 6 The van der Waals attractive potential can be expressed by the cohesive energy density CED 23, defined in terms of the molar interaction energy ∆Ev (the free energy of vaporization) and the molar volume Vm as CED = ∆Ev Vm = ( ∆H v − RT ) Vm

(6.16)

where ∆H v is the enthalpy of vaporization. By definition23, the solubility parameter δ is

δ = CED

(6.17)

In the solution theories of Hildebrand and Scott26 and Scatchard27, the geometric mean of the dye and polymer CED is used to approximate the dye-matrix intermolecular potential 2 2 CEDdp ≅ δ dye δ polymer = δ dyeδ polymer

(6.18)

The W term in Eq. (6.14), defined as the total van der Waals interaction energy, can then be found from 2 2 W = CEDdpVdp = VdyeV polymer δ dye δ polymer

(6.19)

where Vdp = geometric mean of dye and polymer molar volumes: Vdp = VdyeV polymer The W

(6.20)

broadening term was computed for each dye-APC combination using

Vdye and V polymer (APC) as calculated by SYNTHIA, and δ dye and δ polymer taken as the Hansen polar cohesion parameters δ p as described above.

323

Chapter 6 To examine the dye electronic behavior associated with the dipole broadening term D , the ground and excited state dipole moments were calculated for the four dyes using ZINDO, as described above. The results are shown in Figure 6.13.

∆µ (D)

6.5

6.0

5.5

5.0 0

2

4

6

8

Dialkyl Spacer Length (# Carbons/alkyl)

Figure 6.13 Predicted differences between excited state and ground state dipole moments ( µ e − µ g ) vs. dialkyl spacer length, calculated using ZINDO for 4x6m dyes in an APC dielectric medium.

The ZINDO-predicted values of ∆µ show a small decline as spacer length increases from two to three carbons, followed by a sharp decline from three to four carbons, and another slow decline from four to six carbons. We suggest that this ∆µ – spacer length profile accounts in large part for the observed broadening – structure trends seen at 1060 nm and 1300 nm, which we ascribe to classical Gaussian broadening. The Loring developed model1 for a Gaussian inhomogeneous distribution for an electronic two-level solute-solvent system dominated by dipole interactions was described in detail in Section 2.8.5. Recall that this treatment shows that for any linearized theory of 0 solvation of dipolar hard spheres, the equilibrium solvation energy ∆Gsolv is quadratic in

solute dipole moment µ , leading to a Gaussian broadening distribution of width ∆ given by

324

Chapter 6

α s ( ∆µ ) kT 2

∆=

2

(2.66b)

Rc3

0 to µ 2 , k is the Boltzmann where α s is a proportionality constant relating ∆Gsolv

constant, T is the absolute temperature,

is Planck’s constant, and Rc is the solute

(dye) cavity radius, and the solvent-dependent factor α s is defined in Section 2.8.5. Values of the matrix molecular radius σ M used in the calculation of α s and Rc used in Eq. (2.66b) were derived as roots of the polymer (APC) and dye molar volumes already calculated using SYNTHIA. The dielectric constant ε of APC used in the calculation of α s was predicted by the Bicerano connectivity method, also in SYNTHIA. Using these values along with the ZINDO-predicted values of ∆µ shown in Figure 6.13, the dipole broadening width ∆ was calculated using Loring’s expressions (Eqs. (2.66b) and (2.67)). Shown in Figure 6.14 is a function of ∆ , ( ∆ − ∆ 0 )v , which will be useful in 2

later comparisons:

( ∆ − ∆ 0 )v = ∆ v − ∆ 0 v

(6.21)

where ∆ v = v∆ ∆ v 0 = v∆ 0 ∆ 0 = constant We see in Figure 6.14 (right axis) that this dipole broadening term has the exact form of the ground-to-excited state dipole moment difference ∆µ shown in Figure 6.13 for values of ∆ 0 near 200 cm-1. We attribute this quantity to a homogeneous dipole broadening term that is invariant with dye spacer length. This value is comparable to values of homogeneous broadening reported for other dye – solvent systems28. Further, the analysis by Renge25 described in Section 2.10 shows that when two transition energy

325

Chapter 6 distributions are anti-correlated, i.e. correlated on opposite sides of the distributions, the bandwidths are linearly subtractive resulting in band narrowing: Γ ac = Γ IIac − Γ Iac

(6.22)

where Γ ac is the resultant convolved anti-correlated bandwidth, Γ IIac is the larger of the two anti-correlated bandwidths, and Γ Iac is the smaller of the two anti-correlated bandwidths.

-2

2000

2

16,000

3

2

-2

ρRc D (cm )

Kador Dipole Term Loring Dipole Term

(∆−∆0)v (cm )

24,000

1000

8,000

0

0 0

2

4

6

8

Dialkyl Spacer Length (No. Carbons/Alkyl)

Figure 6.14

Comparison of Kador model dipole broadening term ( ρ Rc3 D 2 ) with

Loring model dipole broadening term (∆ − ∆ 0 )v2 vs. dialkyl spacer length applied to 4x6m – APC guest–host materials using predicted values of solubility parameters, molar volumes, and dipole moments, with Rc defined as the dialkyl spacer C–C length and ∆ 0 = 203 cm-1.

We suggest that the dipole component of the observed Gaussian broadening follows this distribution, such that the net dipole broadening term is ∆ = ∆ inh − ∆ hom

(6.23)

where ∆ inh is the inhomogeneous dipole broadening width, and ∆ hom is the homogeneous dipole broadening width, and assert that the two dipolar distributions are 326

Chapter 6 anti-correlated. These dipole-broadening distributions are interpreted as originating from correlated pairwise interactions between dye molecules, as described by the theoretical work of Sevian and Skinner14 (Section 2.8.6). The lattice irregularity theory developed by Kador2, discussed in Section 2.8.3, gives an exact form of the inhomogeneous distribution of dopants in a transparent matrix, following a stochastic analysis due to Stoneham based on Markoff statistics, and is generally applicable to both optical and nuclear magnetic resonance spectra. Applying this exact expression to dipole-dipole interactions, without invoking the Gaussian approximation, Kador gave the following expression for the strength of interactions between permanent dipoles: D = 2 µ M ∆µ 4πε 0 hcRc3

(6.24)

where ε 0 is the permittivity of free space, h is Planck’s constant, and c is the speed of light in a vacuum, and µ M is the effective matrix permanent dipole moment. Recall that the extension of Kador’s analysis by Obata, Machida, and Horie29 discussed in Section 2.8.4 showed that the convolution of the dipole-dipole and van der Waals broadening interactions can be expressed as ΓOMH = A0 ρ R03 D 2 + ρ R03W 2

(2.48)

where A0 is a constant, ρ is the solute (dye) number density, and R0 is the interaction cavity radius. The Kador dipole broadening term ρ Rc3 D 2 in Eq. (2.48), using µ M for APC predicted by SYNTHIA, is displayed in Figure 6.14 (left axis) for comparison with the Loring dipole broadening term ( ∆ − ∆ 0 )v . We see in Figure 6.14 that the Kador dipole 2

width is dominated by the 1 Rc3 dependence, without the characteristic features of the ∆µ – spacer length profile seen for the Loring dipole width. Substituting the Loring dipole broadening term ( ∆ − ∆ 0 )v for Dv2 into the general 2

broadening relation in Eq. (6.14), the net convolved inhomogeneous width is written 327

Chapter 6 Γ L = A0′ Wv2 + ( ∆ − ∆ 0 )v

2

(6.25)

The previously calculated van der Waals component Wv2 and Loring dipole component ( ∆ − ∆ 0 )v are shown separately as functions of the dye spacer length on the 2

same plot in Figure 6.15a. The dipole interaction term ( ∆ − ∆ 0 )v shows the slow decline 2

from a spacer length of two to three carbons followed by a fast decline from three to four carbons to the left of the crossing point at four carbons, while the van der Waals term Wv2 increases more slowly than the dipole term from three to four carbons. To the right of the crossing point, from four to six carbons, the van der Waals term increases more rapidly than the dipole term. The net result, given by Eq. (6.25) and shown in Figure 6.15b, accurately reproduces the experimentally observed Gaussian broadening profiles, as evidenced by the strong linear Γ L – Γ s and 1300 nm Γ L – dα / dC correlations, and reasonably linear 1060 nm Γ L – dα / dC correlation shown in Figures 6.16 and 6.17.

0

200 2

4

6

8

90

88 0

2

4

6

8

Dialkyl Spacer Length (No. carbons/alkyl)

Dialkyl Spacer Length (No. carbons/alkyl)

Figure 6.15

2

-1

210

92

2

1000

0

Sqrt[Wv +(D-D0)v ] (cm )

2

2

220

-2

-2

(∆−∆0)v (cm )

2000

b)

230

Loring Dipole Term van der Waals Term

Wv (cm )

a)

General Gaussian inhomogeneous broadening model with Loring

broadening function ( ∆ − ∆ 0 )v for the dipole term, applied to 4x6m – APC guest–host 2

materials using predicted values of solubility parameters, molar volumes, and dipole moments: a) Loring inhomogeneous dipole term ( ∆ − ∆ 0 )v with ∆ 0 = 203 cm-1 and van 2

der Waals broadening term Wv2 vs. dialkyl spacer length; b) total broadening function

(W

2 v

+ ( ∆ − ∆ 0 )v

2

) with ∆

0

= 203 cm-1 vs. dialkyl spacer length.

328

Chapter 6

Γs (eV)

0.38

0.37

0.36

0.35 88

90

92

2

2

-1

Sqrt[Wv +(D-D0)v ] (cm )

Figure 6.16 Inhomogeneous width Γ s measured for 4x6m – APC guest–host materials (at a dye concentration of 330 µ mol per gram of APC) vs. general broadening function Wv2 + ( ∆ − ∆ 0 )v using Loring dipole broadening function ( ∆ − ∆ 0 )v with ∆ 0 = 203 cm2

1

2

.

0.0050

dα/dC (dB/cm)/(µmol/g)

dα/dC (dB/cm)/(µmol/g)

b) 1300 nm

a) 1060 nm

0.020

0.016

0.012

0.0025

0.0000

0.008 88

90

88

92

Figure 6.17

90

92

2

Sqrt[Wv2+(∆−∆0)v2] (cm-1)

2

-1

Sqrt[Wv +(∆−∆0)v ] (cm )

Loss – concentration slope dα / dC vs. general broadening function

Wv2 + ( ∆ − ∆ 0 )v using Loring dipole broadening function ( ∆ − ∆ 0 )v with ∆ 0 = 203 cm2

1

2

for 4x6m – APC guest–host materials at wavelengths of: (a) 1060 nm; and (b) 1300 nm.

6.4.3.2 Treatment of broadening and loss-concentration behavior at 1550 nm Lastly, we turn our attention to the dα / dC vs. spacer length profile measured at 1550 nm. As seen in Figure 6.5c, the PDS-measured loss α at 1550 nm is less well correlated with experimental Γ s values than at 1060 nm and 1300 nm, and measured 329

Chapter 6 dα / dC

values at 1550 nm were not correlated with the broadening-derived

thermodynamic mixing parameters (in contrast with those at 1060 nm and 1300 nm). Moreover, the dα / dC – spacer length relationship at 1550 nm is exhibits a distinctly different profile from the Gaussian width-correlated profiles for 1060 nm and 1300 nm, due to a precipitous decrease in dα / dC measured for a spacer length of three carbons at 1550 nm. This change in loss – concentration slope for the dipropyl spacer at 1550 nm is shown to be statistically significant. This change in the dα / dC – spacer length profile at 1550 nm is taken to be due to a deviation in broadening behavior from purely Gaussian. It is well established that broadening mechanisms can have different physical origins over different regions of the spectrum2, 17, 18, 30-35. For example, in Chapter 5, we showed that for rigid polymer hosts, broadening in the strong absorption (Gaussian) region could be attributed to the overall, diagonal disorder of the guest-host system, while in the weakly absorbing tail region, broadening was assigned to localized, non-diagonal disorder. Kador and co-workers observed that the absorption band in the spectral wings decays more gradually than predicted by the Gaussian line shape18, and Ovchinnikov and Wight30 show this to be consistent with dipole broadening, with a pairwise distance dependence of 1/ rij3 . Ovchinnikov’s and Wight’s model calculations showed that the absorption in the spectral wings can occur almost exclusively due to pairs of molecules with intermolecular spacings much shorter than the average pair spacing. This has direct implications for the effect of the spacer length on broadening in the extreme wing of the near-IR. We propose that the dipole broadening of the main absorption band in this region, at 1550 nm, is governed by this dependence, as given by the dipole interaction potential term derived by Kador, given by Eq. (6.24), yielding a net inhomogeneous width expressed by the OMH model in Eq. (2.48). Using this form of the dipole broadening function, as we showed in Figure 6.14, the predicted dipole broadening profile depends predominantly on the spacer length through the 1/ R03 dependence, when R0 is defined as the spacer length. Further, this profile is absent of any features due to ZINDO-predicted dipole moment variation. This form of the dipole broadening term ρ R03 D3 is compared with the van der Waals

330

Chapter 6 interaction term ρ R03W 3 , using previously calculated values of W , in Figure 6.18a, and combined using the Obata, Machida, and Horie relation (Eq. (2.48)) in Figure 6.18b. Note that these W and D terms, as originally derived by Kador, do not impose the mean spherical approximation, and as such, the definition of the interaction cavity radius R0 as the dye spacer length in the OMH expression is considered to be relevant to the dyespacing dependent dipole interaction.

500

24,000

2

2

-1

0

3

3

a)

-2

8,000

2

8,000

3

16,000

ρRc W (cm )

16,000

2

-2

ρRc D (cm )

Kador Dipole Term van der Waals Term

Sqrt[ρRc (W +D )] (cm )

24,000

3

6

300

200

0 0

400

0

9

Dialkyl Spacer Length (No. carbons/alkyl)

2

4

6

8

Dialkyl Spacer Length (No. carbons/alkyl)

Figure 6.18 Obato, Machida and Horie (OMH) model of inhomogeneous broadening applied to 4x6m – APC guest–host materials using predicted values of solubility parameters, molar volumes, and dipole moments, and with Rc defined as the dialkyl spacer C–C length: a) dipole ( D ) and van der Waals ( W ) broadening components vs. dialkyl spacer length; b) total OMH broadening function

(

ρ Rc3 W 2 + D 2

)

vs. dialkyl

spacer length. Comparing Figure 6.18b to Figure 6.7, this broadening model is shown to accurately reflect the experimental dα / dC – spacer length profile, as evidenced by the strong linear correlation shown in Figure 6.19.

We therefore attribute this loss –

concentration slope behavior at 1550 nm to a non-Gaussian dye intermolecular dipoledipole interaction broadening dominated by the dye spacer length convolved with the Gaussian van der Waals dye-matrix dispersion interaction. This change in broadening mechanism at 1550 nm represents a shift in physical origin from dipole interactions

331

Chapter 6 dominated by the dye dipole moment ∆µ to those dominated by the closest pairwise approach between dye molecules.

dα/dC (dB-g/cm-µmol)

0.004

1550 nm

0.002

0.000 200

300 3 2 2 Sqrt[ρRc (W +D )]

Figure 6.19

400 -1

(cm )

Loss – concentration slope dα / dC vs. OMH

(

broadening function

)

ρ Rc3 W 2 + D 2 for 4x6m – APC guest–host materials at a wavelength of 1550 nm.

6.5 Conclusions The concentration dependence of near-infrared absorption behavior was investigated by photothermal deflection spectroscopy for glassy, amorphous dye-polymer guest-host materials based on a homologous series of nonlinear optical (NLO) dyes 4x6m of dialkyl spacer lengths ranging from two to six carbons per alkyl group. These structural analogs of the well known NLO dye FTC were incorporated an amorphous polycarbonate co-polymer APC.

Linear or approximately linear dependence of

absorption loss vs. dye concentration at near-IR transmission wavelengths of 1060 nm, 1300 nm, and 1550 nm was observed for all guest-host materials in the series reported here, indicating a lack of specific or strong-binding interactions between the dye and polymer. Determination of the loss – concentration slope behavior at each spectral minimum, in conjunction with inhomogeneous broadening behavior of the main absorption peak, is shown to be a valuable approach for revealing structure – property

332

Chapter 6 relationships based on systematic variations in dye structure.

Knowledge and

minimization of the polymer host near-IR optical loss is not suitable for the understanding and control of loss for dye-doped polymers. No solvatochromism was observed due to dye spacer length in these systems. However, measurable variations of inhomogeneous broadening of the dye main absorption peak were observed as a function of the spacer length, which are assigned to differences in distributions of microscopic dye states. These states are due to varying degrees of dye-dye and dye-polymer interactions imparted by changing spacer length. A universal dependence of the PDS-measured absorption loss on experimental inhomogeneous peak width is seen at each near-IR transmission wavelength, although the dependence at 1550 nm is weaker than at the lower wavelengths. Structure – property relationships found for inhomogeneous widths and for lossconcentration slopes dα / dC at 1060 nm and 1300 nm vs. dialkyl spacer length show very similar behavior, with an abrupt decrease as spacer length increases from three to four carbons, and a gradual increase as spacer length increases from four to six carbons. A distinct change in structure – property behavior is seen at 1550 nm, in that the broadening-induced loss dα / dC decreases monotonically from two to four carbon spacer lengths. The minimum in broadening seen at a spacer length of four carbons is of higher aspect ratio than that of a sphere, which Dalton and Robinson predict will minimize dipole interactions under a poling voltage. These complementary relationships have direct consequences for optimizing the performance of E–O waveguide devices in terms of activity, thermal stability, and loss. A rigorous thermodynamic treatment based on Marcus theory shows that the solute Henry’s Law constant and first-order interaction coefficient, derived from the concentration dependence of the inhomogeneous width and representative of mixing nonidealities, are linearly related to near-IR loss dα / dC at 1060 nm and 1300 nm, but not at 1550 nm. This shows that the loss at the lower two wavelengths is controlled by mixing free energies, while a change in broadening mechanism occurs at 1550 nm. The lower wavelength (1060 nm and 1300 nm) loss dα / dC can be modeled as a convolution of Gaussian dye-polymer van der Waals interactions, dependent on the dye mean spherical size, and Gaussian dye-dye dipole interactions, based on the difference

333

Chapter 6 between ground and excited state dipole moments, based on a Loring model of dipole broadening.

The dipole broadening component is apparently narrowed by anti-

correlation of dye-dye homogeneous and inhomogeneous broadening interactions. The broadening-induced loss dα / dC at 1550 nm is modeled by an extension of Kador’s treatment of dipole and van der Waals interactions by Obata, Machida, and Horie, as a convolution of Gaussian dye-polymer van der Waals interactions and non-Gaussian dyedye dipole interactions, each governed by the closest approach between dye molecules as given by the dye spacer length.

6.6 References (1)

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(2)

Kador, L. Journal of Chemical Physics 1991, 95, 5574.

(3)

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(5)

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(6)

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(7)

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(9)

Zhang, H.; Oh, M.-C.; Szep, A.; Steier, W. H.; Zhang, C.; Dalton, L. R.; Erlig, H.; Chang, Y.; Chang, D. H.; Fetterman, H. R. Applied Physics Letters 2001, 78, 3136.

(10)

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(12)

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(13)

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(14)

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(15)

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(16)

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(17)

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(19)

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(20)

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(21)

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(22)

Wagner, C. Thermodynamics of Alloys; Addison-Wesley: Reading, MA, 1962.

(23)

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(24)

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(25)

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(26)

Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes; Reinhold: New York, 1949.

(27)

Scatchard, G. Chemical Reviews 1931, 8, 321.

(28)

Nakamura, R.; Ishizumi, A.; Watanabe, A.; Nakahara, J. Journal of Luminescence

1998, 76 & 77, 571. (29)

Obata, M.; Machida, S.; Horie, K. Journal of Polymer Science, Part B: Polymer Physics 1999, 37, 2173.

(30)

Ovchinnikov, M. A.; Wight, C. A. Journal of Chemical Physics 1995, 103, 9563.

(31)

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(35)

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337

Chapter 7

Effects of Bonding and Molecular Environment in Monoazo Chromophore-Polymer Materials

7.1 Introduction In Chapters 4 – 6, we reported studies of near-IR loss behavior in dye-polymer guest-host materials based on FTC-like NLO dyes, showing effects of dye spacer length and polymer structure on solvatochromism and inhomogeneous broadening of the main dye electronic transition peak. These processes were shown to correlate with near-IR loss vs. dye concentration behavior. The number density of the highly polar chromophores incorporated into the polymer system is a key determinant of the E-O device sensitivity, which can be constrained by E-O attenuation due to dipole interactions on close approach1, as discussed in Section 2.2, or by dye solubility limits in the polymer, exacerbated by polarity mismatch between dye and polymer. To increase chromophore loading and alignment stability, researchers have investigated covalent attachment of the chromophore to the polymer as either a pendant group2-27 or as a co-monomer28-42, crosslinking the active moiety into the polymer system43-49, or constructing covalently self-assembled superlattice structures containing oriented chromophores50-55.

These

approaches have had mixed success in improving chromophore loading, and have frequently led to higher device insertion loss. The near-IR spectral structure can provide a rich description of the relative contributions to absorption loss. One such class of contributions arises from weak subelectronic bands due to donor-acceptor56-61, acid-base62,

63

, or charge transfer

complexes64-66 or defect states67-71. Complexes associated with these states can arise from specific interactions between neighboring molecules, and the strength and breadth 338

Chapter 7 of these bands gives an indication of the distribution and relative population of such states. The shape, position and magnitude of these states will have direct consequences on loss in the near-IR, as discussed in Sections 2.2.12 and 2.14.3. Another significant contribution to near-IR loss is Urbach tail broadening72, associated with sub-gap absorption in amorphous materials exhibiting an exponential band tail distribution given by

α ( E ) = α 0e E Ee

(2.141)

where α 0 is a constant, α ( E ) is the absorption coefficient at energy E , and Ee is referred to as the Urbach energy and indicates the width of the exponential distribution. The Urbach width has a thermal contribution of magnitude

kT , as well as contributions

from temperature-independent disorder and distribution of defects in the material73. In this study, we investigate three structural strategies for incorporation of an NLO dye system into polymers for spectral near-IR absorption loss behavior: 1) covalent attachment of the dye to the polymer as either a co-monomer or as a pendant group; 2) solution mixing of the unmodified dye as a guest in the polymer host; 3) solution mixing of the dye as a guest in the polymer host, with substitution of potentially charge-transfer forming –OH groups on the dye by ester groups. The well-characterized monazo dyes 4[Ethyl(2-hydroxyethyl)amino]-4'-nitroazobenzene (Disperse Red 1, CAS

2872-52-8,

Figure 3.5a) and 4'-[(N,N-Dihydroxyethyl)amino]-4-nitroazobenzene (Disperse Red 19, CAS 2734-52-3, Figure 3.5b) copolymerized as a pendant group and as a co-monomer, respectively. These dyes are close analogs of each other, differing only in the number of donor ethyl –OH groups, with Disperse Red 1 (DR1) possessing one –OH group and Disperse Red 19 (DR19) possessing two –OH groups. Their structures fall within the merocyanine class of NLO dyes, rendering them representative of the FTC and CLD dye families, albeit with significantly lower E-O activity.

This, combined with their

commercial availability, makes them convenient model chromophores for studies of optical loss behavior associated with various dye-polymer structural strategies. DR1 is incorporated into two different polymers as a pendant group, poly(Rmethacrylate) and poly(4-vinylphenol) (Figures 3.3a and 3.3b).

339

The copolymer

Chapter 7 poly(DR1-methylmethacrylate) at 10 mer% DR1 has been widely studied and shown to exhibit relatively low absorption loss (< 0.5 dB/cm74).

Preliminary UV-Vis

measurements indicated that poly(4-vinylphenol) (PVP) has high transparency in the visible to near-IR, and the acidic pendant –OH group lends itself to pendant dye attachment. DR19 is incorporated as a comonomer into a linear aliphatic Bisphenol A epoxy

derivative,

Bisphenol

A

N-hydroxyethylether-N’-ether-co-N,N-bis(2-

hydroxypropylaniline (“epoxy,” Figure 3.3c). High transparency in the visible to near-IR was found for the neat epoxy derivative in initial UV-Vis measurements. Each of these copolymers is investigated at three dye loading levels. For each of the dye – polymer combinations discussed above for studies of covalently attached dyes, analogous guest – host materials are studied at comparable molar concentrations of dye, both for unmodified dye and with the –OH groups protected by esterification.

For DR1/acrylate and DR19/epoxy systems, the –OH group is

substituted with acetate (OAc), while for the DR1/PVP system, the –OH group is substituted with benzoate (OC6H5). The choice of –OH protecting groups in each dye – polymer system is intended to more closely mimic the molecular surroundings of the dye in the covalently attached state. We show the benefits of covalent dye-polymer attachment in reducing near-IR loss in all of the dye-polymer systems, and effectiveness of dye donor group ester protection for reducing near-IR loss in DR1/acrylate and DR19/epoxy guest – host systems. Charge transfer is shown to be an important effect in DR1 – PVP and, to a lesser extent, on DR19 – epoxy systems. Urbach tail broadening is important for all systems studied.

7.2 Experimental 7.2.1 Synthesis of covalently attached dye-polymer materials 7.2.1.1 Covalently attached DR1-polyalkylmethacrylates Poly(Disperse Red 1 methacrylate-co-methylmethacrylate) at a Disperse Red 1 (DR1) doping level of 1010 µmol per gram of polymer (10 mer%) was received from IBM Almaden Research Labs* and used as-received (CAS No. 119989-05-8, Figure 3.7a, *

Dr. Dan Dawson of IBM Almaden Research Laboratories provided DR1-PMMA material for this study.

340

Chapter 7 with R = methyl, Mw 160,000, polydispersity 5, “DR1-acrylate”).

The structure of

unmodified poly(methylmethacrylate) (PMMA) is shown in Figure 3.3a, with R =

CH2

.

Poly(DR1 methacrylate-co-ethylmethacrylate) at DR1 doping levels of 560 and 1710 µmol per gram of polymer (Figure 3.7a, with R = ethyl, “DR1-acrylate”) was prepared* as follows. Synthesis of monomer 1 (Scheme 1). To a solution of DR1 (6.00 g, 18.1 mmol) in dry methylene chloride (20 mL) was added triethylamine (2.7 mL) and methacryloyl chloride (1.77 mL, 1.90 g, 18.1 mmol). The resulting mixture was stirred at room temperature for 4 h. The methylene chloride solution was extracted with water to remove water-soluble impurities. The crude product was purified by silica gel chromatography eluting with hexane/methylene chloride (1:1 to 1:3) to afford a red solid (4.85 g, 70%). 1H NMR (300 MHz, CDCl3, ppm) δ: 1.27 (t, J = 7.2 Hz, 3H), 1.94-1.95 (m, 3H), 3.56 (q, J = 7.5 Hz, 2H), 3.74 (t, J = 6.3 Hz, 2H), 4.36 (t, J = 6.3 Hz, 2H), 5.60 (m, 1H), 6.11 (s, 1H), 6.83 (d, J = 9.3 Hz, 2H), 7.90 (d, J = 6.6 Hz, 2H), 7.93 (d, J = 6.3 Hz,

2H),

8.33

(d,

J

=

9.3

Hz,

2H).

Synthesis

of

DR1-containing

poly(ethylmethacrylate)s (Scheme 1). The polymerization was carried out in dimethylformamide (DMF) solution under nitrogen atmosphere at 65oC in the presence of 1 wt% of 2,2’-azobisisobutyronitrile (AIBN) for 24 h. The resulting polymer solution was cooled and poured into methanol to precipitate the polymer. The precipitated polymer was filtered, redissolved and reprecipitated, filtered and finally dried at 60oC under reduced pressure overnight. Copolymer 1 (560 µmol DR1/g polymer): Mw = 88,700, polydispersity 2.46. 1H NMR (300 MHz, CDCl3, ppm) δ: 0.87 (br), 1.03 (br), 1.24 (br), 1.81 (br), 1.91 (br), 3.53 (br), 3.70 (br), 4.02 (br), 4.14 (br), 6.81 (br), 7.89 (br), 8.31 (br). Copolymer 2 (1710 µmol DR1/g polymer): Mw = 74,200, polydispersity 2.50. 1

H NMR (300 MHz, CDCl3, ppm) δ: 0.88 (br), 1.04 (br), 1.25 (br), 1.82 (br), 2.00 (br),

3.56 (br), 3.70 (br), 4.03 (br), 4.04 (br), 4.15 (br), 6.84 (br), 7.91 (br), 8.32 (br).

*

Dr. Hong-Zhi Tang, presently at the Department of Chemistry, North Carolina State University, synthesized and performed confirming 1H NMR of the DR1-poly(ethylmethacrylate) materials for this study.

341

Chapter 7

m O

O OH

O

N

O

N

O

O n

N

O N

DR1

+ N

NO2

Cl

O Et3N CH2Cl2 N N

1

O AIBN, DMF

N

N

NO2

NO2

Copolymer 1: 560 µmol DR1/g polymer Copolymer 2: 1710 µmol DR1/g polymer

Scheme 1

7.2.1.2 Covalently attached DR1-polyvinylphenol Poly(DR1 styrene-co-4-vinylphenol) (Figure 3.7b, “DR1-PVP”) was prepared* at DR1 doping levels of 560, 1180, and 2120 µmol per gram of polymer via a Mitsunobu condensation (Scheme 2) between the pendant hydroxyl group on DR1 chromophore and the phenol group on poly-4-vinylphenol (PVP, Figure 3.3b). The loading level of the DR1 chromophore was estimated from the integration of the 1H NMR spectra of the materials. All of the chemicals were purchased from Aldrich. For each composition (560, 1180, and 2120 µmol DR1 per gram of polymer), DR1 was recrystallized from acetone, and tetrahydrofuran (THF) was distilled under nitrogen from sodium benzophenone ketyl prior to use. All the other chemicals were used as received unless otherwise mentioned. To a THF solution of poly(4-vinylphenol) (1.0 g, 8.32 mmol), DR1 at appropriate molar quantities (0.636 mmol, 1.66 mmol, and 2.50 mmol, respectively for 560, 1180, and 2120 µmol/gram) and triphenylphosphine (1.7:1 molar

*

Dr. Jingdong Luo of the Department of Materials Science and Engineering, University of Washington, synthesized and performed confirming UV-Vis and 1H NMR characterization of the DR1-PVP materials of this study.

342

Chapter 7 ratio to DR1) was added dropwise to diethyl azodicarboxylate (DEAD, 1:1 molar ratio to triphenylphosphine). The reaction mixture was allowed to stir under nitrogen at room temperature for 36 h. Then the filtered solution was added dropwise into diethyl ether. The collected red precipitation was further purified by Soxhlet extraction with methylene dichloride for 72 h and dried at 50°C under vacuum overnight to afford the product as the red solid (Figure 3.7b). Copolymer 3 (560 µmol DR1/g polymer): 1H NMR (300 MHz, acetone-d6, ppm): δ 8.27-8.42 (br s), 7.82-8.18 (br m), 6.2-7.13 (br, m), 4.06-4.33 (br s), 3.78-3.98 (br s), 3.53-3.74 (br s), 0.59-2.40 (br m). Molecular weight: Mw = 27,950, polydispersity 2.05. UV-Vis λmax (in acetone): 480 nm. Copolymer 4 (1180 µmol DR1/g polymer): 1H NMR (300 MHz, acetone-d6, ppm): δ 8.23-8.46 (br s), 7.77-8.20 (br m), 6.16-7.17 (br, m), 4.01-4.31 (br s), 3.72-3.96 (br s), 3.43-3.71 (br s), 0.44-2.39 (br m). Molecular weight: Mw = 23,500, polydispersity 1.87. UV-Vis λmax (in acetone): 483 nm.

Copolymer 5 (2120 µmol DR1/g polymer): 1H NMR (300 MHz, acetone-d6, ppm): δ 8.15-8.40 (br s), 7.72-8.10 (br m), 6.17-7.09 (br m), 3.95-4.28 (br s), 3.36-3.93 (br m), 0.44-2.30 (br m). UV-Vis λmax (in acetone): 484 nm.

OH

N x

1-x

n +

N

N

DEAD/PPh3

OH

O

OH

N

DR1 PVP

DR1-PVP

NO2

N

Copolymer 3: 560 mmol DR1/g polymer Copolymer 4: 1180 mmol DR1/g polymer Copolymer 5: 2120 mmol DR1/g polymer

Scheme 2

343

N

NO2

Chapter 7 7.2.1.3 Copolymerized DR19-aliphatic epoxy Poly(Disperse Red 19 Bisphenol A N-hydroxyethylether-N’-ether-co-N,N-bis(2hydroxypropylaniline) (Figure 3.7c, “DR19-epoxy”) was prepared* at DR19 doping levels of 530, 1010, and 1630 µmol/g as follows. Epoxide functionalized Disperse Red 19 (DR19) chromophore was synthesized from the azo coupling reaction of N,N-Bis(2,3Epoxypropyl)aniline and p-nitroaniline. Carefully matching the stoichiometry of epoxy groups, from epoxide-DR19 and bisphenol A-diglycydyl ether, with the amine (from anilines), all monomers were dissolved in anhydrous dioxane in a drybox. The solution was heated to 90°C and stirred for 30 min., then heated to 110 C for 36 hours to complete polymerization in the drybox to form a linear epoxy polymer with loading of DR19 at 530, 1010, and 1630 µmol/g (Figure 3.7c). Next the solution was pumped down to remove the dioxane and the polymer was titrated with methanol 3X (until the methanol from the titration was clear), and the polymer product was thoroughly dried.

The

undoped linear Bisphenol A epoxy backbone polymer is depicted in Figure 3.3c. The weight-average molecular weight Mw, of the epoxy backbone polymer was determined to be 10,300 by GPC against a polystyrene standard. 7.2.2 Ester protection of dyes 7.2.2.1 Acylation of azo dyes DR1 and DR19 For evaluation of protected DR1/acrylate and protected DR19/epoxy guest – host materials, the ethanolamine moieties of DR1 and DR19 were esterified to form acetateprotected DR1 and DR19 analogs (OAcDR1 and bis-OAcDR19, respectively, Scheme 3). DR1 was used as received from Aldrich. DR19 was 1X recrystallized in acetone. To 1.5 mmol DR1 or DR19 (0.5 g) was added 75.7 mmol acetic anhydride (7.1 ml) and 28.6 ml anhydrous pyridine. The mixture was stirred overnight under an argon blanket, then rotovapped at 40°C for 1 hour. The red crystalline crude product was washed with ethanol then heated into solution at 90°C and filtered through a fritted filter funnel. The filtrate was dissolved in chloroform and 0.2 µm membrane filtered, followed by *

Dr. Michael Lee, presently of APIC Corporation, and Dr. Albert Ren, formerly at the Department of Chemistry, University of Southern California, synthesized the epoxy backbone polymer and DR19-epoxy copolymers of this study.

344

Chapter 7 recrystallization in ethanol to form the purified product (compounds 2 and 3).

Compound 2 (OAc-DR1): 1H NMR* (200 MHz, CDCl3) δ 8.33 (d, J = 9.0 Hz, 2H), 7.957.88 (m, 4H), 6.8 (d , J = 9.0 Hz, 2H), 4.30 (t, J = 6.2 Hz, 2H), 3.68 ( t, J = 6.0 Hz, 2H), 3.50 (q, J = 7.0 Hz, 2H), 2.67 (s, 3H), 1.26 (t, J = 7.0 Hz, 3H). Compound 3 (Bis-OAcDR19): 1H NMR (200 MHz, DMSO d6) δ 8.34 ( d, J = 8.8 Hz, 2H), 7.92 (d, J = 8.8 Hz, 2H), 7.83 (d, J = 8.8 Hz, 2H), 6.97 (d, J = 9 Hz, 2 H), 4.20 (br t, J = 5.8 Hz, 4 H), 3.74 ( br t, J = 5.2 Hz, 4 H), 2.47 ( br s, 6H).

OH

O

R

N

R'

N

O

N

DR1: R = H DR19: R = OH

N

Acetic anhydride, Pyridine Ar Blanket

N

N

NO2

NO2

2: R' = H 3: R' = OCOCH3

Scheme 3

7.2.2.2 Benzoylation of azo dye DR1 For evaluation of benzoyl-protected DR1/PVP guest – host materials, the ethanolamine of DR1 was esterified with benzoate† to form benzoyl-DR1 (Scheme 4). DR1 was received from Aldrich and recrystallized in acetone. Dichloromethane was distilled over phosphorus pentoxide and THF was distilled over sodium/benzophenone prior to use. To the mixture of 1.0 g of DR1 (3.18 mmol) and 0.466 g of benzoic acid (Aldrich, 3.816 mmol) in dichloromethane/THF (2:1, v/v, 50 mL) was added 0.224 g of DPTS (4-(dimethylamino)-pyridinium 4-toluenesulfonate, 0.763 mmol) and then 0.866 g *

Dr. Peter Bedworth of Lockheed Martin Advanced Technology Center performed confirming 1H NMR characterization of the acyl-DR1 and acyl-DR19 dyes of this study. †

Dr. Jingdong Luo of the Department of Materials Science and Engineering, University of Washington, synthesized and performed confirming 1H NMR characterization of the benzoyl-DR1 dyes of this study.

345

Chapter 7 of DCC (1,3-dicyclohexylcarbodiimide, Aldrich, 4.197 mmol). The reaction mixture was allowed to stir at room temperature overnight. After filtration to remove the white precipitation of urea, the crude product (from the filtrate) was purified by column chromatography using dichloromethane/hexane as eluent (30:1, v/v) to afford 1.2 g of product (compound 4). Compound 4 (benzoyl-DR1) 1H NMR (200 MHz, CDCl3, ppm): δ8.32 (d, J = 9.15 Hz, 2H), 7.95-8.07 (m), 7.93 (d, J = 8.78 Hz, 2H), 7.92 (d, J = 9.16 Hz, 2H), 7.38-7.64 (m), 6.87 (d, J = 9.16 Hz, 2H), 4.55 (t, J = 6.23 Hz, 2H), 3.83 (t, J = 6.22 Hz, 2H), 3.60 (q, J = 6.96 Hz, 2H), 1.28 (t, J = 6.96 Hz, 3H).

OH

O

N OH + N

DR1

O

N

O DCC/DPTS N

N

NO2

4

N

NO2

Scheme 4

7.2.3 Guest – host mixture spin solution preparation All solvents used in guest – host solutions were dried with molecular sieve. DR1* and DR19† from Aldrich were 1X recrystallized in acetone. With the exception of DR1/PVP and benzoyl-DR1/PVP guest – host solutions, all guest – host solutions were prepared as dilute solutions (low polymer solids contents in solvent), due to the generally poor solubility of the azo dyes. An exhaustive dye/polymer solubility screening study

*

Dr. Jingdong Luo of the Department of Materials Science and Engineering, University of Washington, purified DR1 dye for this study. †

Prof. Shahin Maaref of the Department of Chemistry, Norfolk State University, purified DR19 dye for this study.

346

Chapter 7 was performed in a wide range of solvents for each azo dye – polymer pair to determine solvent and concentrations at which stable solutions could be prepared. DR1/PMMA guest – host solutions were prepared by adding DR1 to a dilute stock solution of PMMA (Scientific Polymer Products, CAS 9011-14-7, Mw 75,000) of 6.7 wt% polymer solids in N-methylpyrrolidone, at DR1 doping levels of 560, 1060, and 1710 µmol per gram of polymer. DR1/PVP guest – host solutions were prepared by adding DR1 to a stock solution of PVP (Aldrich, CAS 24979-70-2, Mw 20,000) of 24.0 wt% polymer solids in a 3:1 by weight mixture of dimethylacetamide and cyclopentanone, at DR1 doping levels of 560, 1060, and 1710 µmol per gram of polymer. DR19/epoxy guest – host solutions were prepared by adding DR19 to a dilute stock solution of the epoxy backbone polymer (Section 7.2.1.3, Mw 10,300*) of 8.4 wt% polymer solids in N-methylpyrrolidone, at DR19 doping levels of 530, 1010, and 1630 µmol per gram of polymer. OAcDR1/PMMA guest – host solutions were prepared by adding OAcDR1 (compound 2) to a dilute stock solution of PMMA (described above) of 6.7 wt% polymer solids in a 1:1 by weight mixture of dimethylformamide and cyclopentanone, at OAcDR1 doping levels of 560, 1060, and 1710 µmol per gram of polymer. Benzoyl-DR1/PVP guest – host solutions were prepared by adding benzoyl-DR1 (compound 4) to a stock solution of PVP (described above) of 14.8 wt% polymer solids in a 11:7 by weight mixture of cyclopentanone and N-methylpyrrolidone, at benzoyl-DR1 doping levels of 420, 560, 1060, and 1720 µmol per gram of polymer. The 1720 µmol/gram solution was diluted to 7 wt% polymer solids with 11:7 cyclopentanone:Nmethylpyrrolidone and boiled for 5 seconds. Bis-OAcDR19/epoxy guest – host solutions were prepared by adding bisOAcDR19 (compound 3) to a dilute stock solution of the epoxy backbone polymer (described above) of 4.9 wt% polymer solids in N-methylpyrrolidone, at bis-OAcDR19 doping levels of 430, 540, 800, 1000, and 1300 µmol per gram of polymer. *

Dr. Anthony Cooper of Lockheed Martin Advanced Technology Center performed the molecular weight determination by gel permeation chromatography against a polystyrene M w calibration standard.

347

Chapter 7

7.2.4 Covalently attached dye-polymer spin solution preparation DR1-co-acrylate polymer spin solutions were prepared by dissolving the copolymers in a 3:1 by weight mixture of cyclopentanone and diglyme at 22 wt% polymer solids content.

DR1-co-PVP polymer spin solutions were prepared by

dissolving the copolymers in cyclopentanone at polymer solids contents of 22.1 wt%, 21.1 wt%, 20.1 wt%, and 18.1 wt% for attached DR1 loading levels of zero (undoped), 560, 1180, and 2120 µmol per gram of polymer, respectively. DR19-co-epoxy polymer spin solutions were prepared by dissolving the copolymers in cyclopentanone at polymer solids contents of 21.7 wt%, 34.1 wt%, 25.0 wt%, and 25.1 wt% for attached DR19 loading levels of zero (undoped), 530, 1010, and 1630 µmol per gram of polymer, respectively. 7.2.5 Film preparation Film samples for PDS spectral characterization were prepared on 1” diameter IR transparent substrates, which were described in Section 3.3.3.1. The substrates were precleaned* following the procedure described in Section 3.3.4. 7.2.5.1 Film preparation from dilute guest – host solutions All dilute guest – host solutions (DR1/PMMA, OAcDR1/PMMA, DR19/epoxy, and bis-OAcDR19/epoxy at all dye concentrations, and benzoyl-DR1/PVP at a dye concentration of 1720 µmol per gram of polymer) were filtered in place at 0.2 µm, and cast by manual spreading† 1-5 drops over the substrate surface. Each of these films was baked under flowing dry nitrogen in a dark box to remove the solvent immediately after spreading the solution, using a bake schedule‡ involving a lower hold temperature of 100 *

Some of the substrate pre-cleaning was performed by Gil Mendenilla of Lockheed Martin Advanced Technology Center, and Angelina Moss, formerly of Lockheed Martin Advanced Technology Center.



An effective means of manually spreading the dilute dye/polymer solutions involved using either the side of a disposable Wheaton polypropylene 10 mL pipet tip, or the concave surface of a slightly bent, solventcleaned 1/2” O.D. polyethylene tube, ~ 4” long, gently applying to the drop and smoothing over the substrate.



Established from iterative film spin and bake trials, using the film quality criteria described in Section 3.3.4.

348

Chapter 7 – 130°C for 1 – 2 minutes, ramping to an upper hold temperature in 2 – 4 minutes, dwelling at the upper temperature for 5 – 7 minutes*, then cooling to 50°C in