Computation of refractive index and optical ... - OSA Publishing

Vol. 25, No. 14 | 10 Jul 2017 | OPTICS EXPRESS 16409

Computation of refractive index and optical retardation in stretched polymer films JUNG-GU LIM,1,2,3 KEUMCHEOL KWAK,1,3 AND JANG-KUN SONG1,* 1 School of Electronics & Electrical Engineering, Sungkyunkwan University, Jangan-Gu, Suwon, Gyeonggi-do 16419, South Korea 2 Advanced Materials R&D Center, Dongwoo Fine Chem. Co., Ltd., Pheongteak, Gyeonggi-do, 17956, South Korea 3 These authors contributed equally to this work * [email protected]

Abstract: Functional polymer films are key components in the display industry, and the theoretical prediction of the optical properties of stretched polymer films is important. In this study, we try to establish the theoretical calculation process without an empirical database to predict the refractive index, including wavelength dispersions and optical retardation of stretched polymer films using several commercial simulation tools. The polarizability tensor and molecular volume for periodic units of polymers are accurately simulated, resulting in the accurate prediction of the mean refractive index and its dispersion for raw polymer materials. The birefringence of stretched films is also calculated to predict reasonably accurate optical properties of stretched films. The simulation method is an effective way that requires a relatively short time and low cost to develop new types of polymer films. © 2017 Optical Society of America OCIS codes: (160.5470) Polymers; (260.1440) Birefringence.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

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#297970 Journal © 2017

https://doi.org/10.1364/OE.25.016409 Received 12 Jun 2017; revised 28 Jun 2017; accepted 28 Jun 2017; published 30 Jun 2017


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1. Introduction Functional polymer films have been widely used in various applications, and particularly, polymer films are essential components in display devices. Polarizers and optical compensation films made of polymers are typical examples indispensably used in liquid crystal displays (LCDs) and organic light emitting diode (OLED) displays [1–3]. Polymer films have several exclusive advantages such as low weight, flexibility, low cost, and a simple manufacturing process. Most importantly, the optical tensor properties of polymer films can be precisely tuned to comply to the specific requirements for display applications, by optimizing the polymer materials, thickness of films, and the casting and elongation processes [4–6]. To be concrete, the selection of the film types such as positive and negative uniaxial films and biaxial films [7], the combination method of those films for a specific LCD mode [8], and the phase matching of wavelength dispersions with the employed LCD mode [9] are all important factors to obtain high performance such as wide viewing angle, high contrast ratio, and low color shift [10, 11]. Usually, developing a new polymer film is a lengthy and extremely expensive process, because from the material selection to the fabrication, the development and optimization processes are performed via numerous trial and error methods. To overcome these issues, several methods to predict and calculate the optical properties of polymeric films have been proposed. The molar refraction of polymers was estimated using the group contributions model by Krevelen [12]. Askadskii proposed semi-empirical equations for predicting some physical properties of polymers and copolymers [13]. Yang and Jenekhe proposed a new Lorentz and Lorenz molar refraction group contributions model for functional groups in polymers [14]. The effect of external stress on polymers and the resulting birefringence were also predicted by assigning an empirical stress-optical coefficient [15]. Although these methods provide rapid and inexpensive approaches to computationally estimate the material properties of polymer films, they rely on an empirical database of materials and are limited to systems composed of known materials, the material data of which is required to be collected beforehand. To overcome the aforementioned limitations, the theoretical quantitative structure property relationship (QSPR) approach was developed [16, 17]. In addition, the theoretical approaches to predict the refractive index of polymers have been further advanced or diversified using density functional theory [18]. However, these quantum mechanical calculations were designed to predict the optical properties of virgin polymer materials before applying external tension. In reality, the optical properties of polymeric films in the optical compensation process are mostly governed by the elongation process, namely, the stretching of the film to result in anisotropy in the optical refractive index [19]. Thus, the calculation and prediction of the optical properties of stretched polymer films using purely theoretical calculations are in high demand; however, the relevant study is at an early stage. In this study, we establish a series of theoretical calculation processes to predict the refractive index, optical retardation, and wavelength dispersion in both raw polymer films and stretched polymer films. The detailed calculation in each step is performed using commercial simulation tools; hence, one can readily follow the calculation steps. The order parameter of


monomers in the polymer films, molecular volume, and density variation are accurately calculated, giving the refractive index in the final stretched films. We compare the theoretical results with the experimental data; the refractive index and its wavelength dispersion for raw polymer materials before the stretching process obtained from the experiment are in excellent agreement with the theoretical values with R2 values of 0.9885, and the predicted birefringence of the stretched film is also in acceptable agreement with the experimental results albeit with some error. Although the theoretical calculation method may need further improvement, the method will allow one to predict and tailor the optical properties of the stretched polymer films in a relatively short time and at a low cost. 2. Experimental This study was designed to establish the theoretical simulation process for the refractive index and birefringence of the polymer films including wavelength dispersion and elongation effect, and so, we selected well known polymer materials for experiments: cyclic olefin polymer (COP), polyethylene terephthalate (PET), and polyvinyl alcohol (PVA). These polymers have been widely used especially in the optical compensation films (COP, PET) and polarizers (PVA) in LCDs and OLEDs. The COP, PET, and PVA materials and films were purchased from Zeon Company (Japan), Toyobo Company (Japan), and Kuraray Company (Japan), respectively. Each polymer film was treated further by a well-defined stretching process to achieve optical anisotropy. To determine the mean refractive index of the polymer film, a thin, spincoated polymer film was prepared. The three types of polymers were dissolved in respective solvents (DI water for PVA, and cyclohexanone for PET and COP), and the solution was coated on a silicon wafer. Then, the solvent was fully evaporated above the boiling point of each solvent. A reflectometer (ST4000-DLX, K-Mac, Korea) was used to determine the refractive index as a function of wavelength by measuring the reflectance of the films coated on the silicon wafer. An automatic birefringence analyzer (KOBRA-WPR, Oji Scientific Instrument, Japan) was used for measuring the birefringence and retardation properties of the polymers at varying wavelengths using 80-μm PET films and 40-μm COP before and after stretching. 3. Simulations According to the Vuks isotropic local field model [20], the principal refractive indices (ni) of the anisotropic media can be expressed as

ni 2 − 1 4π Nα i = , n2 + 2 3

(1)

where n, N, and αi are the mean refractive index ( = [n12 + n22 + n32]1/2), the molecular packing density (the number of molecules per unit volume), and the mean principal polarizability component in tensor form of a periodic unit (i.e., a monomer), respectively. The molecular packing density can be rewritten as ni2 (λ ) − 1 4π ρ N A 4π = α i (λ ) = α i (λ ), 2 3 MW 3Vmol n (λ ) + 2

(2)

where λ, ρ, NA, MW, and Vmol are the wavelength, density, Avogadro’s number, the molecular weight, and the molecular volume, respectively. Thus, by calculating the polarizability tensor of a monomer of polymer films and its molecular volume, one can predict the average refractive index of raw polymer films based on Eq. (2). On the other hand, the arrangement of monomers is significantly distorted in the stretched polymer films. Here, we assume that the molecular conformal structure of a monomer is almost unchanged, but the alignment of monomers is changed by the stretching. When the monomers in the polymer films align with


uniaxial ordering under the stretching stress along one direction, the principal polarizability tensor components of bulk polymer can be expressed using the order parameter as 2 



1

α X = α  + S α xx − (α yy + α zz )  , and 3  2  1 

1



αY = α Z = α  − S α xx − (α yy + α zz )  , 3  2 

(3) (4)

where the subscript, X, Y, and Z denote the axes of films, and x, y, and z denote the molecular axes of monomers, and S ( = ) is the uniaxial order parameter of molecular long axis, where θ is the angle between the elongation direction and the direction of a monomer unit. In this study, we consider only the uniaxial elongation of polymers, and hence, the biaxial order parameters are not considered. The refractive indices and retardation of stretched polymer films can be calculated by finding the polarizability tensor and the molecular volume in the modified Lorentz–Lorenz equation in Eq. (2). The simulation was composed of three parts, as described in Fig. 1. In the first quantum calculation part, which was performed using commercial software, Gaussian09, (setting Function: B3LYP, BasisSet: 6-31G), the molecular conformal structure and electron density distribution are calculated for a periodic unit of the polymer (Calculation 1). The polarizability tensor (αii) of a monomer was then calculated using the electron distribution function (Calculation 2). This step can be done by minimizing the ground state chemical energy (electrostatic potential energy). In the second part of the QSPR calculation, the arrangement of periodic units in the polymer are calculated using the MS-Synthia module on the Material Studio platform (Accelrys Company, USA). This quantitative structure-property relationship calculation module uses topological information, specifically, connectivity indices derived from graph theory, based on individual atoms and bonds, and can provide the molecular packing information of the polymer before the elongation process. In this step, the Young’s modulus and the glass transition temperature as a function of molecular weight of the polymer are also obtained (Calculation 3). To reduce the time in the next step of the simulation, we chose the minimum polymer molecular weight (MW) that did not deteriorate the accuracy of the simulation results. The minimum value of MW was chosen such that the calculated Young’s modulus and the glass transition temperature reached more than 95% of the saturation value at the high molecular weight (Calculation 4). Then, the molecular volume, Vmol, at an initial stage without elongation is estimated based on the molecular packing information obtained in the QSPRs simulation (Calculation 5), and the wavelength dispersion of refractive index of raw polymer materials is calculated based on the calculated polarizability tensor and Vmol (Calculation 6). In the last part of the simulation, molecular dynamic simulation is performed using the MS-Forcite module in the Material Studio® platform, which performs energy minimization and geometry optimization to find the geometrical structure of elongated polymers. For the elongated polymer, the Poisson’s ratio is calculated to deduce the change in film thickness, and the order parameter of unit monomers is calculated from the monomer distributions under the elongation stress (Calculation 7). The Poisson’s ratio influences the value of Vmol, the film thickness, and the density after elongation (Calculation 8). Then, the refractive indices along three principal axes at varying wavelengths are determined using the polarizability of monomer, reduced Vmol, and the order parameter.


Fig. 1. Simulation flow for optical properties of stretched polymer films.

4. Results and discussion

The electronic structure of the monomers was determined by minimizing the electrostatic potential energy of the monomers. In Fig. 2, the electronic structure optimization process of COP is shown by monitoring the electrostatic potential energy with an increasing iteration of calculations; in each iteration, the conformal structure is slightly modified to find the optimal structure of the monomer. The images on the right in Fig. 2 show the optimized conformal structures of PET and PVA, following the same procedure. Using the optimized electronic structure of the monomers, the polarizability of a monomer was calculated, and the results are shown in Table 1, where the x-axis is along the polymer main chain, and y- and z-axes are perpendicular to the x-axis. represents the mean polarizability of the two perpendicular components to the x-axis. When the monomers uniaxially align along the x-axis, the polarizability anisotropy can be defined as (αxx − ). The anisotropy of the polarizability tensor is directly related to the conformal structure of the molecules. While the polarizability tensors of the COP and PVA are rather isotropic, that of PET is highly anisotropic, as shown in Table 1.

Fig. 2. Electronic structure optimization for COP by monitoring electrostatic potential energy (left), and optimized structure of PET and PVA (middle and right).


Table 1. Molecular polarizability tensor calculated for COP, PET, and PVA. , : mean polarizability of y- and z-axes, and of all three primary axes, respectively. Materials COP

PET

PVA

λ (nm) 447 546 630 670 447 546 630 670 447 546 630 670

αxx 21.12 20.86 20.73 20.69 29.43 28.06 27.45 27.25 4.41 4.36 4.33 4.32

αyy 22.07 21.79 21.65 21.60 19.21 18.75 18.54 18.47 4.51 4.45 4.42 4.41

αzz 19.34 19.11 19.00 18.96 10.18 10.06 10.01 9.99 4.11 4.06 4.03 4.02

20.71 20.45 20.32 20.28 14.69 14.41 14.27 14.23 4.31 4.25 4.23 4.22

20.85 20.59 20.46 20.42 19.61 18.96 18.67 18.57 4.34 4.29 4.26 4.25

To choose an appropriate value of MW for the calculation economy’s sake, the dynamics of Young’s modulus (Fig. 3(a)) and the glass transition temperature (Tg) (Fig. 3(b)) were calculated as a function of MW. The value of MW for the rest of the simulations was determined at a value such that the Young’s modulus and Tg become 95% of the saturation values. The chosen molecular weights for COP, PVA, and PET were 5300 g/mol, 44.0534 g/mol, and 5000 g/mol, respectively. From the statistical calculation using MS_Synthia module, the molecular packing state and the density at the given molecular weights were calculated. The corresponding molecular volumes were calculated as 282.3 Å3, 58.7 Å3, and 249.0 Å3 for COP, PVA, and PET, respectively. Based on the calculated molecular packing density and the polarizability tensor, the mean refractive index was calculated for the three polymers. These were 1.5314, 1.5600, and 1.5325 for COP, PET, and PVA, respectively. Using the same procedures, the mean refractive indices for various polymers were calculated in order to verify the accuracy of the calculated values; for comparison, the experimental values of refractive indices in Table 1 in [21] were used. As shown in Fig. 4(a), the calculated mean refractive indices are in excellent agreement with the experimental data, and the corresponding R2 value was about 0.9885. This implies that the quantum calculation for the electronic structure in monomers and the statistical calculation for the molecular packing density by simply connecting the monomers provide excellent predictions for the refractive indices for polymers. Using the wavelength dispersion properties of polarizability shown in Table 1, the dispersion of the refractive indices for the chosen polymers was calculated, as shown in Fig. 4(b). The experimental data were obtained using ST4000. The simulated data accord well with the experimental data, confirming the accuracy of the simulation used to obtain the polarizability and the molecular packing density.

Fig. 3. (a) Young’s modulus and (b) glass transition temperature for COP as a function of MW.


Fig. 4. (a) Simulated refractive indices versus experimental data for various polymers. (b) Simulated and measured dispersion of refractive indices for COP, PET, and PVA films.

When a block of polymer is elongated, both the shape of the block and the arrangement of the individual monomer within the block are altered. These modifications were simulated for the reduced polymer having a reduced molecular weight. Figure 5(a) shows the morphological change of a COP block under the stretching stress, and the inset number in each image represents the stretching ratio, which is defined by the ratio of the length of the polymer block after and before the elongation, as illustrated in Fig. 5(b). The change of the molecular volume of the repetition unit because of the stretching can be described by the Poisson’s ratio (RP), which can be expressed as (−Δy/Δx) or (−Δz/Δx) for small deformation. However, for large deformation, the deformation relationship among three primary axes can be written as Δy Δz  Δx  = = 1 +  y z  x 

− RP

−1

(5)

Fig. 5. (a) Molecular packing state with increasing stretching ratio, (b) definitions for stretching ratio and Poisson’s ratio, (c)–(d) order parameters and molecular volumes as a function of stretching ratio for COP, PVA, and PET.

The order parameter of monomers was also calculated from the elongated polymer block, and the calculated order parameters of monomers are plotted as a function of the stretching ratio for COP, PVA, and PET in Fig. 5(c). The order parameter increases as the stretching ratio increases. That is, as the polymer block is stretched, the monomers rotate and rearrange to align along the stretching direction. In PVA, the order parameter is relatively low and saturates at large stretching ratios. The Poisson’s ratio was calculated to be 0.382, 0.417, and 0.424 for COP, PVA, and PET, respectively. From the values, the molecular volume for


stretched polymer can be obtained, as shown in Fig. 5(d). Although the variation of the molecular volume by the stretching is not significant for PVA, it is relatively large for COP and PET. Based on the relationship between Poisson’s ratio and the deformation of width and height of films under the stretching stress in Eq. (5), the relative film thickness and the density of polymer films were calculated as shown in Figs. 6(a) and 6(b). Both the film thickness and density gradually decrease as the stretching ratio increases. The density variation directly reflects the expansion of molecular volume in each polymer. Then, using the calculated order parameter, polarizability of monomer, and modified molecular volume, the retardations of the 10-μm-thick COP, PET, and PVA films were calculated for varying stretching ratios and varying wavelengths, as shown in Fig. 6(c). While the estimated optical retardation values are rather low for COP and PVA, those for the PET are relatively high. Because of the low anisotropy of the polarizability tensor for COP and PVA as shown in Table 1, the elongation of polymer does not generate large birefringence for COP and PVA. The retardation values for COP keeps increasing as the stretching ratio increases, but those for PET and PVA saturates or decreases at high values of the stretching ratio. The retardation of a film is defined as [Δn × (film thickness)]. For the PVA film, the increase in order parameter becomes gentle at high stretching ratios (Fig. 5(c)), while the decrease in the film thickness is still significant (Fig. 6(a)). Hence, the decrease in film thickness dominantly contributes to the retardation values, resulting in the decreasing retardation at the high stretching ratios.

Fig. 6. (a)-(b) Simulation for the relative film thickness and the density of polymer films as a function of stretching ratio. (c)Simulated retardation of COP, PET, and PVA films as functions of wavelength and stretching ratio (legend: stretching ratio).


Fig. 7. (a) Illustration of experimental set-up for polymer stretching, and PET and COP films before and after stretching. (b–c) Experimental and simulated birefringence for a 1.5- timesstretched PET (80 μm) and COP (40 μm) films at various wavelengths.

To confirm the validity of the simulation method for birefringence estimation, the birefringence of the stretched PET and COP films was measured and compared with the simulation result, as shown in Fig. 7. The experiment was performed using the in-house experimental set-up for stretching polymer films, as shown in Fig. 7(b). PET (80-μm thickness) and COP (40-μm thickness) films were stretched by 1.5 times. The simulation results roughly accord well with the experimental results as shown in Figs. 7(b) and 7(c), but some discrepancy between the experimental and simulated values is observed. Particularly, the error in the COP film is about 8.2%, which is much larger than the errors in raw polymers before stretching. Apart from the intrinsic inaccuracy of simulation tools, three are several assumptions that can cause an error in the simulation process. First, we assumed that the structure of periodic unit is not distorted under stretching stress, but in fact, it may be subjected to conformal change to some degree, and the ratio of conformal change may vary depending on polymers. Second, we ignored the surface effect and the vertical stress imposed during film fabrication process, which may produce initial anisotropy between the vertical axis (z) and the in-plane axes (x and y) even before stretching. The initial anisotropy may be much significant in thinner films. Third, for the elongation simulation, we used a reduced molecular weight to save simulation time, which can cause a small error. To improve the accuracy of the simulation method, those factors may be additionally considered. 5. Conclusion

We developed a computer simulation procedure for predicting the wavelength-dependent refractive indices and phase retardation of stretched polymer films based on purely theoretical calculations. The simulation procedure is composed of three steps of the quantum calculation for the electron distribution function, which are the QSPR calculation for the molecular volume and mean refractive index, the molecular dynamic calculation for the order parameter of periodic units in polymer after elongation, and the resulting birefringence calculation. The simulation results accord excellently with the experimental results in raw polymer film, and exhibit some error less than 10% in stretched films. The error in the stretched films may arise from several assumptions introduced to simplify the simulation procedure. Thus, further improvement remains as a future work. The method can be applied to develop new types of retardation films. For instance, a quarter wave plate (QWP) with a negative dispersion is highly in demand in OLED displays, because an anti-reflection film using a negative dispersion QWP can compensate perfectly the reflected external light in the entire range of visible wavelengths, ensuring an excellent contrast ratio in OLEDs. However, double layered compensation films are usually used to fully compensate in the entire range of wavelengths [22]. To develop a perfect QWP in a


single layered film [23], a new type of material design is necessary. Our simulation method will be especially important for finding an optimal molecular design for such a purpose. Funding

National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT, and future planning (NRF-2016R1A6A3A11934167).