Uncovering the intrinsic permittivity of the ...

JOURNAL OF APPLIED PHYSICS 109, 074107 (2011)

Uncovering the intrinsic permittivity of the carbonaceous phase in carbon black filled polymers from broadband dielectric relaxation M. Essone Mezeme,1 S. El Bouazzaoui,2 M. E. Achour,2 C. Brosseau,1,a) 1

Universite´ Europeénne de Bretagne, Lab-STICC, Universite´ de Brest, CS 93837, 6 avenue Le Gorgeu, 29238 Brest Cedex 3, France 2 Laboratoire des Syste`mes des Te´lećommunications et d’Ingeńierie de la Dećision, De´partement de Physique, Faculte´ des Sciences, Universite´ Ibn Tofail, B. P. 133, 14 000 Keńitra, Morocco

(Received 6 December 2010; accepted 22 January 2011; published online 5 April 2011) An outstanding experimental issue in the physics of composites concerns the reliable extraction of the intrinsic dielectric characteristics from effective permittivity measurements of heterostructures. Though recent analytical and numerical models have made progress in tackling this question, their applicability is typically limited by the lack of information about the structural organization of the filler phase. As a follow-up of our earlier work [S. El Bouazzaoui et al. J. Appl. Phys. 106, 104 (2009), we report in this paper a systematic study of the intrinsic permittivity e2 of the carbonaceous phase in carbon black (CB) loaded polymers. A variety of authors has suggested very early that e2 can be modeled with a simple free-electron (Drude) metal model with static disorder. Despite the interest in the physics of carbonaceous materials, there have been few experimental tests of this assumption, in part, due to the experimental challenge of measuring e2 . Here, this interpretation is questioned by an analysis of the frequency-dependent complex effective permittivity of these lossy conductor-insulator composites using the Hashin-Shtrikman bounds of the effective medium approximation. For the materials investigated over the range of frequencies explored (10–104 kHz) 0 0 00 00 it is found that e2 can be written as e2 ¼ e2 ie2 with e2 e2 . We critically evaluate the possibility that the estimates of e2 are related to Drude model. We found that the intrinsic permittivity of the carbonaceous phase dispersed in the composite materials investigated is consistent with the dielectric response described by the Drude metal model in a percolative morphology. The sensitivity of this method is fundamentally related to the complexity of the morphological changes which occur during mechanical mixing, i.e., interphase formation, CB particles aggregation. Such knowledge can be used to determine the role of the conducting states at C 2011 American Institute the interface between insulating polymer chains and carbonaceous phase. V of Physics. [doi:10.1063/1.3556431] I. INTRODUCTION

The study of carbon black (CB) loaded polymers is a rich subject with a venerable tradition in materials science and condensed matter physics with a rejuvenated activity in the context of percolative conductor to insulator transitions, localization, as well as mesoscopic phenomena. From a technological viewpoint, these materials are ubiquitous in the automotive and aerospace industries, the electrical engineering for the dissipation of electrostatic charges and electric shielding. The incorporation of CB into a rubber is of significant commercial importance since CB not only enhances the mechanical and electrical properties of the final products but also decreases the cost of the end product. Advances in both the synthesis and physical characterization of the dielectric relaxation in these materials have brought to the forefront of this research from both experimental and theoretical approaches.1–8 Correspondingly, understanding of the universal features of the dielectric relaxation behavior from the analysis of the experimentally determined complex effective

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]. Also at De´partement de Physique, Universite´ de Bretagne Occidentale.

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permittivity is one of the key challenges in developing a microscopic theory to understand the wave and charge transport in these materials.9–20 Despite the vast amount of literature available on carbonaceous materials7,8,20 many questions and ambiguities on the understanding of the transport mechanisms in these compounds remain. Historically, our understanding of the wave and charge transport in finely divided media has proceeded like essentially all studies of heterostructures–from simple systems to complex. The development of physics has given rise to a number of established phenomenological, i.e., mixing laws (the first step in this chain of inquiry), and statistical models, i.e., effective medium theories (EMT), percolation theory. Several theoretical papers and books addressed this issue.9,10,17,19,21 Rarely do mixing laws offer the opportunity for rigorous and systematic improvements in predictive performance, due to their specificity to a certain class of materials, or a lack of theoretical sophistication in their underlying principles. EMT or coarse graining models have been addressed in great detail in the literature. EMTs are conceptually rooted within self-consistent methods. Generally speaking, these approaches make use of the idea, that a medium which contains inclusion of another phase can be approximated as a homogeneous material. The underlying assumption, i.e., effective properties

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depend only on the materials constants of the pure phases and their volume fractions, is of course an approximation, and the quality of the theoretical predictions can hardly be controlled.9,10,21 Fundamental premises for these methods include (a) that the typical inhomogeneity length scale in the heterostructure is much smaller than the wavelength of the electromagnetic wave probing the system, (b) that only dipolar interactions contribute to the polarization, and (c) that a continuum representation of the system is effective. Depending on the character of the dipoles, interactions, and dimensionality of the system, a number of universal behaviors emerge, e.g., criticality, bounding methods.15,21 Various derivations of bounds have been considered.22 Remarkably, even the simplest models have defied analytical description and can be studied only numerically.9,10 However, the considerable achievements made in computing capabilities cannot be fully exploited to provide commensurate advances in our understanding of the dielectric relaxation properties of disordered materials until rigorous techniques for constructing an interface model between the phases in the mixture are established. It should be noted that Monte Carlo or molecular dynamics simulations11–13 gives different interface structures depending on initial conditions, and is often difficult to guarantee that the constructed interface is equilibrated. One of the crucial tasks in the physics of disordered systems is the identification of the model and disorder types in real materials, necessary to elucidate the microscopic mechanisms underlying macroscopic behaviors. On the experimental side, owing to the difficulty in the characterization of the carbonaceous phase, very few direct measurements of its permittivity e2 have been reported. As this experimental characterization phase is problematic, theoretical calculations have an important role to play in determining possible transport mechanisms. The ability of C atoms to form strong chemical bonds with a variety of coordination numbers leads to a remarkably wide range of physical properties of the condensed phases of C. Unlike diamond, the graphite phase is semimetallic, where the low density of states at the Fermi level, and its high anisotropy induces significant differences from conventional metals.23 Past work in the literature generally builds on the premise that e2 in CB loaded polymers can be described by the free-electron model of simple metals. In a previous study of the microwave dielectric response of such materials over a wide range of CB concentrations,3 a further simplification was made for modeling the results by taking only the imaginary component, i.e. e2 ir2 =2pFe0 . There is some arbitrariness in the manner how r2 is chosen. This originates in part from the fact that the structural knowledge of the CB powder is incomplete. While several in-depth studies have been performed on some aspects of the electrical properties of carbon materials,8,24–26 their systematic investigation is still lacking. Thus far, measurements have only been able to characterize r2 versus compaction pressure,25–35 and thus new measurements are needed. Many interesting effects have been observed when such particles are closely aggregated and start to interact with each other, see, e.g., Ref. 27. Theoretically, the crucial assumption yielding such simplification is that the displacement current may be neglected in the microwave range.36 Notice that r2 lies within the rather

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wide range of values used by other researchers.7,8,14 Other authors stress that the frequency dependence of the complex permittivity for percolating conductor-insulator composites can be matched to a heuristic EMT using a genetic algorithm fitting process to obtain values for the percolation threshold, percolation exponents, and the filler particle conductivity.14 The use of genetic algorithm techniques enables the uniqueness of the four fit parameters to be considered. It was also hypothesized that nonuniversal exponents could be offset by ambiguity in the filler conductivity. These advances motivate us to ask the following question: To what extent is valid this assumption? Verifying this assumption is important for the general study of EMT since many approaches to quantifying the effective permittivity of heterostructures do not include the emergence of the so-called dynamic heterogeneities, i.e., heterogeneous CB particle aggregates (or agglomerates) that rearrange in a correlated manner during mechanical mixing of the polymer and CB particles above the glass transition temperature. Recent work has searched for connections between dynamics and structure,20,37 suggesting multicontact short-range chain adsorption to the surface of the filler as a structural cause for dynamic heterogeneity, i.e., spatially varying characteristic time scales for relaxation. The fruits of this continued search, if attained, will be directly applicable to the understanding of the relative importance of inter- and intra-cluster polarization and conduction.38–40 The present work represents a continuation of an effort that has begun with a comprehensive study of the frequency dependence of the critical exponents characterizing the behavior of the effective complex permittivity close to the filler percolation threshold.2 The CB volume fraction and frequency dependencies of the effective complex permittivity of several CB filled polymer samples was measured over a broad range of frequencies.2 The CB particles differ by their average particle size, surface area, and commercial origin. In this paper, we take a different point of view and reexamine the data of Ref. 2 by using bounds of the effective medium approximation. Specifically, this paper addresses the question: given that one knows how such composite material responds to oscillatory inputs at discrete frequencies what bounds can one place on the complex intrinsic permittivity of the carbonaceous phase? This task is not trivial because the nature of this carbonaceous phase remains a contentious open question. Otherwise stated, the question raised by the present approach is whether the carbonaceous phase can be described by an idealized Drude metal for the frequency and CB volume fraction considered. Here our goal is to explore and constrain this and alternative models. Using the bounds method, it is found that the intrinsic permittivity of the carbonaceous phase dispersed in the composite materials investigated is consistent with the dielectric response described by the Drude metal model in a percolative morphology. However, interactions between local clusters are complicated and this model is likely to be at best a crude approximation for e2 . The sensitivity of this method is fundamentally related to the complexity of the morphological changes which occur during mechanical mixing, i.e., tendency of the CB particles to form conducting states (aggregates) at high volume fractions.


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We have divided this paper in to five sections. In Sec. II, we focus on several procedural points that underlie the percolation equations and the bounding methods. In Sec. III, we describe experimental details of the samples’ preparation and introduce the experimental methods. Based on the methods described in Sec. II, we then enter the discussion of intrinsic CB complex permittivity in Sec. IV. Finally, in Sec. V we offer a brief summary and conclusion. II. PRELIMINARIES

Before getting into the actual material, let us start with some remarks about notations and give a brief synopsis of two standard ways for exploring the dielectric properties of mixtures. None of these approaches are rigorous, but they are eminently reasonable and are likely to correctly predict the effective parameters of a heterogeneous material. Readers acquainted with the subject may proceed directly to the next section. More comprehensive expositions appear elsewhere as indicated in the citations. The general situation we consider in this work is depicted in Fig. 1(a). We consider a two-phase medium consisting of particles (phase 2), of volume fraction u2, embedded in a host matrix (phase 1) with volume fraction 1u2. The phases are assumed to be homogeneous and isotropic. Two fundamental parameters of interest here are the intrinsic (relative) complex permittivity ei

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and the electrical conductivity ri inherent in the phase itself. Let us reiterate the key general ideas, which of course are well known, in these two methods: (i) no interaction exists between the filler particles and the host matrix. (ii) The random distribution of particles within the polymer matrix is assumed to be uniform, i.e., the material is assumed to be statistically homogeneous. (iii) As mentioned above, the basic premise of the EMTs is the quasistatic assumption, i.e., the characteristic length scale f of the typical inhomogeneities in the medium which is small compared with the wavelength k of the electromagnetic wave pffiffi sets an upper limit on angular frequency x ¼ 2pF c=f e, where c and e denote the electromagnetic wave velocity in vacuum and the effective permittivity, respectively. A. Phenomenological static percolation equations

The theory of percolation (critical) phenomena is relevant to a very large number of physics problems. These include, on the condensed-matter side, the insulator-conductor transition.15,21,41 Going back to the critical behavior of percolative heterostructures can give rise to a bewildering array of physical effects,15,17,18,21 especially when they are combined with finite-size effects.11,41 The experimental activity15,41 has been steady. To describe how the effective conductivity versus inclusion volume fraction behaves close to the isotropic percolation threshold u2c of the conducting constituent the following formulas can be employed,41–43 r ¼ r1 ð/2c /2 Þs

for /2 < /2c ;

r ¼ r1 ðr2 =r1 Þt=ðsþtÞ ; r ¼ r2 ð/2 /2c Þt

FIG. 1. Schematic drawings of the composite systems considered. (a) Random two-phase material shown as white (polymer matrix) and black (filler particles) regions: no interaction exists between the filler particles and the host matrix. (b) Random three-phase composite system in which the interphase (dotted region) volume is dependent upon the characteristics of the filler component. Because of the physical adsorption of the polymer chains to the filler surface, the filler particles are “glued” in the matrix. Conductivity is determined by resistance of direct contact or by the properties of the interparticle layer where the tunnel effect allows electrons to pass through the thin layers. This contact resistance causes also a region of smearing to appear. The region of smearing represents the area of critical transition from the insulator sate to the conductor state.

for /2 > /2c ;

(1) (2) (3)

where s and t are critical exponents having the so-called universal values, i.e. s % 0.7 and t % 2 in three-dimensional (3D), r1 is the electrical conductivity of the insulator matrix and r2 is the electrical conductivity of the conductive fillers. Since the percolation model is independent of the nature of the filler particles and the nature of the insulating host matrix, it is possible to apply for most polymers filled with powdery materials. At this stage it is worthwhile stressing the fact that while such an approach can give valuable insights into the CB volume fraction dependence of the effective conductivity, the intrinsic value of the filling phase is generally unknown. Scher and Zallen44 predicted that the percolation threshold is close to 17 vol.% for conducting spherical particles distributed randomly within a 3D dielectric matrix.44,45 However, it is well established that the CB particles form aggregates with more or less irregular and tenuous shapes [Fig. 1(b). For “low structure” CB the aggregates are small and regular, while for “high structure” CB the aggregates typically contain a few hundreds of particles. Because of their often somewhat elongated shapes the probability of aggregate-aggregate contact is larger than it would have been if the aggregate had been a compact sphere. Hence, the conduction threshold is usually lower than the 17 vol.% of filler particles, as is the case of the composites investigated here.


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In addition, the insulator-conductor transition width is often large due to variations in the particle aggregate size, geometry, and orientation.46 B. Bounds of the effective medium approximation

For the purposes of the present work, we next examine working equations for evaluating the effective permittivity of real composite systems in a simple and transparent way. The quest for such equations has occupied theoreticians for many years. Bounds have been developed in order the casual researcher might interpret permittivity (or other physical quantities) results of inhomogeneous media without elaborate curve fitting, i.e., it was first shown by Wiener47 that the effective permittivity is bounded between the harmonic and arithmetic means. This prediction has been and still is influential in the area of random heterostructures. There have been significant advances not only in deriving optimal bounds, but also in describing the materials that attain these bounds. The boundary methods approach combines the universality character with the precision of a mathematical model.22 Finite element simulations have been used11 to clearly show that the fourth-order bounds provide an excellent estimate of the effective permittivity for a wide range of surface fractions, in accordance with the fact that the bounds become progressively narrower as more microstructural information is incorporated. To explore the applicability of this approach to the numerical data concerning CB filled polymers, we first summarize a priori estimates of upper, ðiÞ ðiÞ i.e., eU , and lower, i.e., eL , bounds for e which have been used for comparison with the experimental results. As noted in the Introduction, for such descriptions, a number of authors have developed various approaches delivering different levels of sophistication and predictive power. These estimates narrow the composition range of possible effective permittivity. Detailed treatments can be found in Refs. 11, 15 and 22. The loosest and simplest one-point bounds were first derived by Wiener43 ð1Þ

eL ¼ e1 e2 =½e2 ð1 /2 Þ þ e1 /2 ; ð1Þ

(4)

eU ¼ e1 ð1 /2 Þ þ e2 /2 : Equation (4) is used when only u2 is known but no geometrical information about the microstructure is available. These two bounds, which represent the harmonic and arithmetic means of the permittivities, correspond to plane capacitors that are connected in series or parallel in a circuit. Note that ð1Þ ð1Þ eL (resp. eU ) is the minimum (resp. maximum) for both e2 > e1 and e2 < e1 cases. Using variational principles, Hashin-Shtrikman (HS) derived the best possible two-point bounds48 on the effective permittivity for the set of all twophase isotropic three-dimensional composites with given phase volume fractions, in which e2 > e1 ð2Þ eL ð2Þ eU

¼ e1 þ /2 =

1 1 /2 þ ; 3e1 e2 e1 1 /2 : ¼ e2 þ ð1 /2 Þ= þ e1 e2 3e2

(5)

ð2Þ

ð2Þ

It turns out that eL and eU correspond exactly to Maxwell Garnett mixing law and inverse Maxwell Garnett obtained by interchanging e2 $ e1 and /2 $ 1 /2 , respectively.15,22 We note in passing that the use of bounds is not restricted to real-valued permittivities but can also be used for complex-valued quantities.15,22 In this case, these bounds are represented by limited domains in the complex plane inside which the allowed values of the permittivity exist. “Improved bounds” depending nontrivially upon correlation functions, i.e., containing information beyond that embodied in the volume fraction of the different constituents, have been proposed by several investigators, e.g., three-point and four-point bounds on properties of two-phase isotropic heterogeneous media, the general rule being that the bounds become progressively narrower as more microstructural information is included.15,22,49 But they will not be discussed in the current work. Our aim at this stage is not a one-to-one evaluation of experimental spectra using these bounds but rather to gain insights in the electric nature of the carbonaceous phase in CB filled polymers. III. MATERIALS AND MEASUREMENTS

For the current experiments, we have selected commercial (spherical) CB particles (Raven 2000, Raven 5000, and Raven 7000 obtained from the Columbian Chemicals Company, and Monarch 1100 obtained from Cabot Company) and diglycidyl ether of bisphenol-F (DGEBF) epoxy resin commercially available as Araldite XPY 306 (epoxide equivalent weight¼172) from Ciba Geigy Ltd., respectively.50 CB particles were mechanically stirred with DGEBF and an amine curing agent (4,9-dioxadodecan-1,12-diamine, equivalent weight¼81, supplied by BASF, and used without further purification). Resin was cured at room temperature for 24 h. The ratio by weight of the mixture of the DGEBF and CB particles to the amine curing agent was adjusted to achieve stoichiometry. The basic features of the CB particles are summarized in Table I. The neat DGEBF has a density of 1.19 gcm3 and a higher glass transition temperature than room temperature, i.e., Tg¼83 C. Four series of samples (DGEBF=Raven 2000, DGEBF=Raven 5000, DGEBF= Raven 7000, and DGEBF=Monarch 1100) were fabricated by mechanical mixing, as described elsewhere.1–3 The CB volume fraction within the composite is denoted as u2. Frequency-dependent measurements of the effective (relative) complex permittivity, e ¼ e0 ie00 , of the composite samples were carried out with an impedance analyzer (HP 4194 A) in the frequency range from 100 Hz to 15 MHz. The experimental setup and procedures for measuring e(F) are similar to those of our previous studies.1,2 The real, e0 , and imaginary, e00 , parts of e were deduced from the measurement of the capacitance and resistance of the sample, respectively. Note that the contribution of the Ohmic conductivity, i.e., rð/2 Þ=2pe0 F was removed to the raw e00 data. Concomitantly, all dc conductivity measurements were performed on the composite samples through four-point electrical measurements.1 The samples are cylinders with typical thickness 3 mm and diameter 12 mm. Prior to measurement, the circular surfaces of the samples were polished and covered with a


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TABLE I. The specifications of the CB materials examined in the current study from manufacturer product literature (Ref. 47) and extracted parameter data of the carbonaceous phase obtained from the comparison of the experimental and theoretical values using Eqs. (1)–(3). The neat DGEBF has a dc conductivity 1014 X1 m1. The curves in Fig. 2 show that the effective conductivity data of the CB filled samples measured at room temperature can be fitted using Eqs. (1)–(3).

CB type Raven 2000 Raven 5000 Raven 7000 Monarch 1100

Average particle sizea (nm)

Density (gcm3)

Surface areaa NSA (m2g1)

9.6 8 9 14

1.817 1.917 1.891 1.891

190 583

dc powder conductivitya (X1cm1) 6.56 3.50 17

/2c (vol.%) 3.6 6 0.2 10.0 6 0.5 2.75 6 0.15 8.0 6 0.4

r2 (X1m1)

s

t

0.017 6 0.003

0.7

2.1

5.12 6 0.73

0.7

2.0

a

Reference 47.

thin layer of silver paste to serve as electrodes. Conduction was Ohmic. To ensure that the results reported herein were due to the bulk composite materials, and not from the silver paint electrodes, control experiments with a contactless configuration were employed to eliminate the possibility that Schottky barriers at the contacts influence the data. IV. RESULTS AND DISCUSSION A. Dc conductivity and percolation

As a first step we tried to evaluate the static intrinsic conductivity of the carbonaceous phase. The results plotted in Fig. 2 obviously suggest the need to carry out a detailed study of the dc conductivity profiles using Eqs. (1)–(3). The results of this fit procedure are shown in Fig. 2. From the analysis of rð/2 Þ graphs, the four independent fit parameters /2c , s, t, and r2 were obtained for the composites containing Raven 2000 and 7000 and are listed in Table I. We attempted to fit the experimental conductivities for the samples with Raven 5000 and Monarch 1100 and were unable to obtain a good fit since the amount of data for compositions close to /2c is not significant. Observe the large region of smearing representing the area of critical transition from the insulator state to the conductor state in the case of Monarch 1100. This smearing may originate from the contact resistance of the inter-particle layer. This layer can act to separate the CB

FIG. 2. Experimental (symbols) dc conductivity data as a function of CB volume fraction (normalized to /2c ) and corresponding best fits (lines) of the data to the percolation equations, i.e., Eqs. (1)–(3), for CB filled DGEBF and different types of CB: (solid star) Raven 2000, (circle) Raven 5000, (square) Raven 7000, and (open star) Monarch 1100. Room temperature.

particles, potentially inhibiting conductance if the CB particles are too thick. The values of r2 extracted from fitting r versus /2 which are in the range 1021 X1 m1 are well below the standard values for a normal metal.51 It is instructive to observe that these extracted values are also much smaller than the corresponding manufacturer (powder) conductivity data 102 X1 m1 (see Table I). These findings corroborate the conclusions reached by Sańchez-Gonza´lez et al.27 that the electrical conductivity and its variation under compression are directly related to the density of the carbon particles. Our calculations also reveal additional features. The value of r1 is consistent with our earlier result 1014 X1 m1.1,2 There is a degree of scatter in the low /2 data. That the effective conductivity for the samples containing Raven 2000 and Raven 7000 and /2 0 since the complex permittivity is analytic in the upper half plane as required by causality36), and ð2Þ

e2U ¼

e1 ½2e1 ð1 /2 Þ eð2 þ /2 Þ : eð1 /2 Þ e1 ð1 þ 2/2 Þ

(6)

In Fig. 3, the cross-hatched region (inset) indicates the allowed values of the values of e02 and e002 for a sample (u2 ¼ 5.5 vol.%) of the series of Raven 7000 filled DGEBF and a specific value of the frequency (15 MHz). Details about the construction of these diagrams are found in Refs. 15, 21 and 22. This observation is consistent with the recent contributions of Youngs14 and Stoyanov and co-workers.49 The evolution of the allowed (lens-shaped) regions with the HS bounds was studied for different frequencies and u2. The synopsis of our results is shown in the form of the plots of the allowed values (represented by vertical bars) and average values (represented by symbols) of e02 and e002 versus frequency displayed in Figs. 4–7 for a range of CB volume fractions and the different series of samples. By examining these graphs, we identify a common rule shared by the spectra for all CB concentrations. We find that, when the CB volume fraction is well below the percolation threshold (Figs. 4(a)–4(b), Figs. 5(a)–5(b), Figs. 6(a)–6(b), and Figs. 7(a)–7(b), the “average value” of e02 is close to zero with very large fluctuations of both signs, and e002 shows a tend-

FIG. 4. (Color online) The allowed values (indicated by vertical bars) of e02 as a function of frequency for Raven 7000 filled DGEBF sample and u2 ¼ 2 vol.% (< /2c ). Each symbol refers to the average value of e02 and e002 . (b) Same as in (a) for e002 for u2 ¼ 2 vol.%. (c) Same as in (a) for e02 for u2 ¼ 4.5 vol.% (> /2c ). (d) Same as in (a) for e002 u2 ¼ 4.5 vol.%. (e) Same as in (a) for e02 for u2 ¼ 5.5 vol.% (>/2c ). (f) (d) Same as in (a) for e002 u2 ¼ 5.5 vol.%.

FIG. 3. (Color online) Calculated values of the region of the complex plane inside in which the allowed values (cross-hatched region in the inset) of the intrinsic complex (relative) permittivity of the carbonaceous phase exist from the HS bounds. Raven 7000 filled DGEBF sample with u2 ¼ 5.5 vol.% (> /2c ). F ¼ 15 MHz. Room temperature.

ency to decrease as a function of frequency. But, above the percolation threshold, Figs. 4(c)–4(f) show that the allowed values of e02 become mainly negative. The fact that e02 < 0 rules out the possibility that the carbonaceous phase is described by the classical oscillator (Lorentz) model. Over the range of frequencies explored (10104 kHz) it is found that the intrinsic complex permittivity of the carbonaceous


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FIG. 5. Same as in Fig. 4 for the Raven 2000 filled DGEBF sample with u2 ¼ 3 vol.%.

phase can be written as e2 ¼ e02 ie002 with e002 >> e02 . These estimates should be taken with some caution, as we consider an “average value” of the allowed values of e02 and e002 which are characterized by large dispersions. We should make a note about the fact that our analysis of the dielectric spectra is very general since it is based on universal bounds and does not rely on a specific EMT such as in Ref. 14. Particularly noteworthy is that this EMT has not been rigorously derived from first principles, does not contain any structural disorder information, and does not

FIG. 6. Same as in Fig. 4 for the Raven 5000 filled DGEBF sample with u2¼5.5 vol.%.

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FIG. 7. Same as in Fig. 4 for the Monarch 1100 filled DGEBF sample with u2 ¼ 3 vol.%.

account for interactions between the individual components of the composite. Recently, we found that this EMT is consistent with Monte Carlo simulations only for u2 < f2c, and its failure for u2 > f2c can be attributed to a poor representation of the various degrees of particle aggregation present in the equilibrium distributions where increased aggregation results in an enhanced multipolar interaction.11 Having justified our methodology, we now turn to possible physical explanations for our observations. It is of interest to determine whether our experimental data can be captured by the Drude model. According to the idealized Drude metal model, the frequency-dependent permittivity is e2 ðxÞ ¼ e021 x2p =xðx ixd Þ, where e021 denotes the limiting high-frequency permittivity of phase 2, xp is the bulk plasma frequency, and the parameter xd ¼ s1 ¼ vF =R1 bulk defines the scattering frequency that is used to account for dissipation of the electron motion; Rbulk represents the mean free path of the conduction electrons and vF denotes the velocity of the electrons at the Fermi energy. Although this conventional form is often assumed to be applicable to small particles we add, parenthetically, that xp s may be reduced from the bulk value by surface scattering.17 At a more fundamental level, we notice that a modification of the imaginary part of e2 ðxÞ is sometimes assumed in the literature in order to account for the enhanced rate of electron scattering due to particle size-dependent effects,52 i.e., when the particle size R is smaller than the mean free path in the bulk metal, conduction electrons are additionally scattered by the surface and xd may be modified to become size-dependent according to the phenomenological form xd ð RÞ ¼ xd ðbulkÞ þ CvF =R, where C is a constant that includes details of the scattering processes and its value can be determined from ab initio calculations or from independent experimental measurements. However, such electron confinement in conducting nanoparticle will not be taken into consideration here.


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pffiffiffiffi derived above with the bounds. Since xp / N , where N denotes the density of free electrons, this would mean that the density of free carriers in the carbonaceous phase is low. The scattering frequency xd / l1 , where l is the mobility of the carriers, is a long-wavelength property of the carbonaceous phase, and constraints the motion of electrons over long distance. The fact that the xd value is 104 times larger than that of an ideal metal is consistent with the fact a strong loss contribution due to collisions of electrons.5 (iii) The limited frequency range means that conclusions can only be drawn with care. C. DISCUSSION

FIG. 8. (Color online) (a) A fit of the frequency dependence of the average values (symbols) of the estimates of e002 obtained from HS bounds with the Drude metal model, for the Raven 7000 filled DGEBF sample with u2 ¼ 4.5 vol.% (>/2c ). (b) Same as in (a) for u2 ¼ 5.5 vol.% (>/2c ).

To put the results in perspective, we have also compared the bounds and the theoretical curves e02 ðFÞ and e002 ðFÞ obtained from the Drude metal model in Fig. 8 in a percolative morphology [corresponding to Figs. 4(c)–4(f)]. We applied the nonlinear Levenberg-Marquardt modeling routine to fit e02 ðFÞ and e002 ðFÞ.53 Three observations are immediately made: (i) First, we find that the Drude model is quite successful in reproducing the e02 ðFÞ and e002 ðFÞ spectra over the frequency range considered for /2 ¼ 5:5 vol.% > /2c [see Fig. 8(b)]. In that case the transport is dominated by paths of least resistance. This behavior is also seen in the change of Fig. 4(a), albeit in a less satisfactory manner, so this model is likely to be consistent with the estimates of e02 and e002 within the accuracy that can normally be achieved in such an analysis. (ii) The second observation is that differences do appear, however, with the case of an ideal metal. Table II presents the differences between the different CB filled polymer samples for xp and xd . Metals usually have xp 1016 rads1 and xd 1013 s1 (Ref. 51). This is to be contrasted with the xp about 103 times smaller in magnitude TABLE II. The values of the plasma frequency xp and scattering frequency xd estimated from the Drude model in the case of a percolative morphology of the carbonaceous phase (Raven 7000 filled DGEBF samples with 4.5 and 5.5 vol. %). For comparison, we have also indicated typical values of xp and xd for an ideal metal (gold) and SiC. Material

xp (rads1)

xd (s1)

Raven 7000 [4.5 vol.% DGEBF] Raven 7000 [5.5 vol.% DGEBF] Au [51] SiC [51]

2.1 1012 1.7 1013 1.4 1016 2.7 1014

2.1 1017 1.5 1018 3.8 1013 9 1011

It is worthwhile discussing several physical mechanisms which can be suggested to explain the deviation from the Drude model for nonpercolative morphologies in which the carbonaceous phase is dispersed in small disconnected regions. Here the motion along these clusters is rapid locally, but motion over long distances is slower. Another consideration is to consider that the filler phase is not just a homogeneous phase. To circumvent this problem an interphase model was introduced by Todd and Shi (TS).37 Lewis54 has also reported evidence for the role of interphase regions on the dielectric properties of bulk composite materials with small (nanoscale) filler particles. Most of the current applications of polymers filled with a variety of particles, e.g., carbon black and fibers, ferrite, necessitate that the filler particles should be homogeneously dispersed within the polymer matrix. However, depletion mixing prevents to realize a homogeneous particle distribution. The rigid surface of the particles does not allow for penetration of the polymer chains and therefore create an excluded volume with shell thickness of the same order of the radius of the polymer coil. If two or more particles aggregate, their excluded volume zones overlap, causing a decrease in total excluded volume and a corresponding increase in free volume accessible for the polymer chains. If the gain in entropy due to the increase in free volume is larger than the entropy loss due to aggregation of the filler particles, the entropic balance favors depletion mixing. The point of view may be taken that the filler phase consists of two distinct types of constituents: the carbonaceous phase with particle fillers which have rigid, noninterpenetrable surface, and a shell around each aggregate containing a gradient of properties describing the interphase region, i.e., interfacial bonding region [exhibited in Fig. 1(b)]. The estimated thickness of this preferential multicontact chain adsorption layer is in the range of 1–10 nm.20,37,54 If the aggregates can be modeled as spheres, which is a good approximation for many observations, then the spherical inclusion surrounded by a thin layer containing a spherically symmetric gradient of isotropic dielectric properties can be handled exactly. The TS model assumes that a composite system comprising two primary components (matrix and filler) and an interphase region may be treated as a unique three-phase composite system in which the interphase volume is inextricably dependent upon the characteristics of the filler component. This simplification is used because the available data are not sufficient to develop a fully anisotropic model. Key model parameters include the


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surface area of the filler, A, and the “thickness” of the interphase region, e. Within this model, the effective permittivity of the composite system can be phenomenologically described by e ¼ ð/2 eb2 þ ð1 /2 UÞeb1 þ Uebi Þ1=b , where U ¼ ð1 ð6/2 =pÞ3 ÞAeq is the interphase volume fraction, ei is the intrinsic permittivity of the interphase, and q is the density of the filler.37 Many experiments have found that ei is generally within 610% of the matrix phase.16 On the experimental side, an accurate determination of b has remained challenging. Technically, the values of the adjustable parameters ei and b can be chosen to fit the simulated values of the effective permittivity of the composite system to the experimental data. However, generally speaking, when such complex interface is involved such empirical rules are prone to giving wrong results. This indicates the perils of trying to get the experimental numbers through a model which is too simple in terms of details despite introduction of phenomenological parameters. For completeness, we mention that the Wiener and HS bounds for three-component materials can be found in the literature.55 For example, the upper HS is attained by a mixture of all different sized spheres of e1 and e2 each coated with e3 in the appropriate volume fraction; the lower bound being attained by the same mixture with e1 and e2 reversed. The study of the electromagnetic properties of an ensemble of CB nanoparticles embedded in a polymer matrix requires the determination of not only the intrinsic electromagnetic properties of the individual particles but also the type and strength of interparticle interactions, e.g., longrange dipole-dipole interactions, clustering, and matrix-particle interactions, e.g., multicontact chain adsorption to the surface of the filler. The fact that spatial inhomogeneities, i.e., clusters of CB particles, give rise to polarization phenomena and therefore to a frequency dependence of the effective permittivity has been known for many years.20 However, understanding how electrons are distributed in these CB aggregates remains one outstanding unresolved problem in mesoscopics of disordered materials, making it difficult to know which theoretical framework is most appropriate for understanding these compounds, e.g., localization by disorder is a common cause of insulating behavior in condensed matter systems. An epoxy polymer chain is made up of alternating single and double carbon bonds, which enable the formation of a spatially delocalized electron system. This electronic distribution is strongly affected by selective chain adsorption on the CB aggregates providing charge trapping sites.8,20 The effect has previously been studied experimentally in the context of another series of CB filled polymers.5 The conductivity of these systems relies on two distinct carrier types within the CB aggregates and agglomerates. Such a separation of carriers has been recently predicted by Brosseau et al.5 on the basis of electron spin resonance experiments. Their results suggest that the majority of the carriers are confined to the aggregate and agglomerate surfaces. Localization by disorder is a common cause of insulating behavior in these systems.5 Some open questions remain. For example, it would be desirable to know how aggregation effects of the CB particles create microstructures with multiscale and hierarchical

J. Appl. Phys. 109, 074107 (2011)

structures formed when the CB particles are immersed in the complex fluid environment formed by polymer chains.20 The heart of the problem is the strength of the van der Waals attraction among the particles. Once they come into contact, they form irregular clusters whose morphology is determined by the kinetics of aggregation. Another relevant question is how aggregation and agglomeration of CB particles affect the conductivity and the polarization mechanisms.56 The experimental characterization of CB particles cluster polarization and conduction is a nontrivial task.39 There is ample evidence that the electrical conductivity of a compacted powder of CB particles depends on intrinsic factors such as the type of CB particles, the number of effective contacts between the particles, their packing, the surface chemistry, and on extrinsic factors as the applied pressure.25,27–36,57 In particular, it has been shown that larger particle size and higher aggregate structure increase the electrical conductivity since the probability of contact between the CB particles is larger in both cases.25 It follows from the fact that under compression, the number of electrical contacts increases. However, the experimental determination of the 3D detailed structure of this carbonaceous phase can be complicated because it requires a difficult tomography analysis, which is often restricted in spatial resolution.15,56 Most data available is either very limited or performed in extremely diluted systems where the average dipolar interaction is very weak. In addition, most conventional analysis do not take spatial confinement factors into consideration. In this regard, important information of the electrical connection among the CB aggregates can be obtained by using local probes, e.g., electric force microscopy (EFM), capacitance probe microscopy (CPM), atomic force microscopy (AFM).58–61 Although there is abundance of publications on effective (macroscopic) electric conductivity,1–8,26 there is a lack of data related to the local measurement of the electrical resistance through thin (a few tens of nm) slices of the composite material. It has become clear in recent years that the two-parameter percolation equations, Eqs. (1)–(3), do not capture fully the story of charge transport, and attention has focused on the correlation length n, that measures the spatial extent of the largest clusters, which diverges on both sides of the percolation transition. AFM (Resiscope) results published on a series of acetylene CB filled elastomers show the importance of the knowledge of the cluster number, size and distribution, n, and local resistance distribution.61 V. SUMMARY AND CONCLUSIONS

To conclude, above we have attempted to understand how the intrinsic conductivity and permittivity of the carbonaceous phase in CB filled polymers can be determined from an experimental point of view. We have performed a systematic study of the transport properties in CB filled polymers using percolation theory and the HS bounds of the effective medium approximation. Several aspects are particularly remarkable. The standard percolation model can serve as a good approximation for the analysis of the dc conductivity of CB-polymer composites. Over the range of frequencies explored (10–104 kHz) it is found that the intrinsic complex


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permittivity of the carbonaceous phase can be written as e2 ¼ e02 ie2 with e002 >> e02 . The facts and arguments above suggest that the intrinsic permittivity of the carbonaceous phase is associated with a description of the polarization properties for the carbonaceous phase by a classic freeelectronlike metal (Drude form) in the case of a percolative morphology. This model provides a decent, though not perfect, description of the intrinsic permittivity of this phase. However, contrary to the popular view of this material as a “conducting phase”, the Drude model is not effective for nonpercolative morphologies in which the carbonaceous phase aggregate in small disconnected regions. To address this inconsistency we suggest that it is likely that embedding matrix effects are of crucial importance for evaluating the frequency dependence of the intrinsic complex permittivity for an arbitrary CB filled polymer sample. The preceding discussion showed us that the results presented in this work may constitute an important first step in developing a more complete modeling of the role of the interphase, i.e., the conducting states at the interface between insulating polymer chains, and can serve as benchmarks for further theoretical investigations. Some final comments: Besides the theoretical interest inherent in the CB filled polymers characteristics, it is worth pointing out that the above approach is especially meaningful for percolative composites and can be applied to other types of heterostructures, e.g., carbon nanotubes- and graphene-filled polymers. Within this context, it is worth pointing out that bounds on the effective tensor for anisotropic two-phase composite materials were recently derived by Engstro¨m.62 The rising interest of the scientific community in graphene is motivated by its special and unique physical properties which make it one of the most promising materials for plastic nanoelectronics.63–65 Areas requiring greater theoretical elaboration include the connection between spatial heterogeneity and mean-field (effective) approach, more rigorous examination of the aggregation of CB particles central to the interfacial states, and consideration of the physical basis by which charge and wave transport proceeds. Interestingly, use of polymeric materials is increasingly demanding nanofabrication applications require that an understanding of their dielectric behavior be developed at nanometer length scales. Future work will attempt to improve our understanding of these transport mechanisms by simulating particle aggregation and polymer chain adsorption,11,40 and calculating how microstructural changes much smaller than the electromagnetic wavelength can alter the effective permittivity and conductivity. Once these questions are adequately solved, it is believed that the foundations will be laid for rigorously and systematically understand the effective properties of these complex materials, especially in terms of frequency domain response. Given the experimental challenges involved in characterizing aggregate boundary structures, theoretical modeling will play a particularly important role. We hope that efforts to understand the phenomenon we report here will be useful in testing morphological models of the carbonaceous phase embedded in these heterostructures.

J. Appl. Phys. 109, 074107 (2011)

ACKNOWLEDGMENTS

This work was supported by Lab-STICC which is Unite´ Mixte de Recherche CNRS 3192. 1

M. E. Achour, C. Brosseau, and F. Carmona, J. Appl. Phys. 103, 094103 (2008); C. Brosseau and M. E. Achour, ibid. 105, 124102 (2009). 2 S. El Bouazzaoui, A. Droussi, M. E. Achour, and C. Brosseau, J. Appl. Phys. 106, 104107 (2009). 3 E. Tuncer, J. Bellatar, M. E. Achour, and C. Brosseau, Advances in Composite Materials, edited by I. Lorkovic (In-Tech, Vienna, Austria=Rijeka, Croatia, 2011). 4 C. Brosseau, J. Appl. Phys. 91, 3197 (2002). 5 C. Brosseau, P. Molinie´, F. Boulic, and F. Carmona, J. Appl. Phys. 89, 8297 (2001); C. Brosseau, F. Boulic, C. Bourbigot, P. Queffelec, Y. Le Mest, J. Loaec, and A. Beroual, ibid. 81, 882 (1997);F. Boulic, C. Brosseau, Y. Le Mest, J. Loaec, and F. Carmona, J. Phys. D 31, 1904 (1998). 6 A. Mdarhri, C. Brosseau, and F. Carmona, J. Appl. Phys. 101, 084111 (2007); C. Brosseau, A. Mdarhri, and A. Vidal, ibid. 104 074105 (2008); C. Brosseau, W. Ndong, and A. Mdarhri, ibid. 104, 074907 (2008); A. Mdarhri, P. Elies, and C. Brosseau, ibid. 104, 123518 (2008). 7 R. G. Geyer, J. Baker Jarvis, M. D. Janezic, and R. Kaiser, Natl. Inst. Standards Technical Note 1529 (2003). 8 J. B. Donnet and A. Voet, Carbon Black Physics, Chemistry and Elastomer Reinforcement (Marcel Dekker, New York, 1976); J. B. Donnet, Compos. Sci. Technol. 63, 1085 (2003); E. K. Sichel, J. I. Gittelman, and P. Sheng, Phys. Rev. B 18, 5712 (1978); Carbon Black Polymer Composites, edited by E. K. Sichel (Marcel Dekker, New York, 1982); R. H. Norman, Conductive Rubber and Plastics (Elsevier, New York, 1970). 9 C. Brosseau, J. Phys. D 39, 1277 (2006). 10 C. Brosseau and A. Beroual, Prog. Mater. Sci. 48, 373 (2003). 11 V. Myroshnychenko and C. Brosseau, Phys. Rev. E 71, 016701 (2005); J. Appl. Phys. 97, 044101 (2005); J. Phys. D 41, 095401 (2008); J. Appl. Phys. 103, 084112 (2008); IEEE Trans. Dielectr. EI 16, 1209 (2009). 12 E. Tuncer, B. Nettelblad, and S. M. Gubanski, J. Appl. Phys. 92, 4612 (2002); E. Tuncer, S. M. Gubanski, and B. Nettelblad, ibid. 89, 8092 (2001). 13 O. Pekonen, K. Ka¨rkkaïnen, A. Sihvola, and K. Nikoskinen, J. Electromagn. Waves Appl. 13, 67 (1999); K. Ka¨rkkaïnen, A. H. Sihvola, and K. I. Nikoskinen, IEEE Trans. Geosci. Remote Sens. 38, 1303 (2000). 14 I. J. Youngs, J. Phys. D 35, 3127 (2002). 15 S. Torquato, Random Heterogeneous Materials (Springer, New York, 2001). 16 A. K. Jonscher, Dielectric Relaxation in Solids (Chelsea Dielectric Press, London, 1987); see also Universal Relaxation Law (Chelsea Dielectric Press, London, 1996) and more recently J. Phys. D 32, R57 (1999). See also L. A. Dissado and R. M. Hill, Nature 279, 685 (1979); Proc. R. Soc. London A 390, 131 (1983); K. Weron, J. Phys.: Condens. Matter 3, 9151 (1993); 4, 10503 (1992); 4, 10507 (1992). 17 D. J. Bergman and D. Stroud, in Solid State Physics, Advances in Research and Applications, edited by H. Ehrenreich and D. Turnbull (Academic, New York, 1992), Vol. 46; see also D. Stroud and D. Bergman, Phys. Rev. B 25, 2061 (1982). 18 R. Landauer, in Electrical Transport and Optical Properties of Inhomogeneous Media, edited by J. C. Garland and D. B. Tanner, AIP Conf. Proc. N 40, (AIP, New York, 1978), p. 2; See also R. Landauer, J. Appl. Phys. 23, 779 (1952). 19 J. P. Calame, J. Appl. Phys. 94, 5945 (2003); J. P. Calame, A. Birman, Y. Carmel, D. Gershon, B. Levush, A. A. Sorokin, V. E. Semenov, D. Dadon, L. P. Martin, and M. Rosen, ibid. 80, 3992 (1996); J. P. Calame, D. K. Abe, B. Levush, and D. Lobas, J. Am. Ceram. Soc. 88, 2133 (2005). 20 C. Brosseau, Prospects in Filled Polymers Engineering: Mesostructure, Elasticity Network, and Macroscopic Properties, edited by C. Brosseau (Research Signpost, Trivandrum, 2008), pp. 177–227. 21 M. Sahimi, Applications of Percolation Theory (Taylor and Francis, Bristol, PA, 1994). 22 G. W. Milton, The Theory of Composites (Cambridge University Press, Cambridge, 2002); G. W. Milton, J. Appl. Phys. 52, 5294 (1981). 23 N. B. Brandt, S. M. Chudinov, and Y. G. Ponomarev, Modern Problems in Condensed Matter Sciences (North Holland, Amsterdam, 1988), Vol. 20.1; the construction of accurate and computationally efficient potentials capable of describing the wide range of interactions in C is still a challenge. 24 D. D. L. Chung, J. Mater. Sci. 39, 2645 (2004).


074107-11 25


A. Celzard, J. F. Mareche´, F. Payot, and G. Furdin, Carbon 40, 2801 (2002). 26 F. Carmona, Ann. Chim. (Paris) 13, 3995 (1988); Physica A 157, 461 (1989). 27 J. Sańchez-Gonza´lez, A. Macıás-Garcıá, M. F. Alexandre-Franco, and V. Go´mez-Serrano, Carbon 43, 741 (2005). 28 T. Noda, H. Kato, T. Takasu, A. Okura, and M. Inagaki, Bull. Chem. Soc. Jpn. 39, 829 (1966). 29 K. Kendall, Powder Technol. 62, 147 (1990). 30 S. Mrozowski, Studies of carbon powders under compression, in Proceedings 3rd Biennial Conf. Carbon (New York, Buffalo, 1957), p.495. 31 A. Voet, Rubber Age 95, 746 (1964). 32 K. J. Euler, H. Kirchoff, and A. Metzedorf, J. Mater. Chem. 4, 611 (1979). 33 F. I. Zorin, Inorg. Mater. 22, 47 (1986). 34 N. Deprez and D. S. McLachlan, J. Phys. D 21, 101 (1988). 35 A. Gervois, M. Ammi, T. Travers, D. Bideau, J. C. Messager, and J. P. Troadec, Physica A 157, 565 (1989); J. P. Troadec and D. Bideau, One Electr. 71, 30 (1991); J. P. Troadec, A. Gervois, D. Bideau, and L. Oger, J. Phys. C 20, 993 (1987). 36 S. Ramo, J. Whinnery, and T. van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1984). See also L. D. Landau, E. M. Lifchitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon Press, New York, 1984), J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975); G. W. Milton, D. J. Eyre, and J. V. Mantese, Phys. Rev. Lett. 79, 3062 (1997). 37 M. G. Todd and F. G. Shi, IEEE Trans. Compon. Pack Technol. 26, 667 (2003); J. Appl. Phys. 94, 4551 (2003); IEEE Trans. Dielectr. Electr. Insul. 12, 601 (2005). 38 Y. Song, T. W. Noh, S. I. Lee, and J. R. Gaines, Phys. Rev. B 33, 904 (1986). 39 M. F. Hundley and A. Zettl, Phys. Rev. B 38, 10290 (1988). 40 T. E. Doyle, A. T. Tew, R. Jain, and D. A. Robinson, J. Appl. Phys. 106, 114104 (2009). 41 D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor and Francis, London, 1994); the percolation phase transition on both lattices and continuum is a subject of intense study, as it provides a model for the onset of large-scale connectivity in random media, such as resistor networks, porous rocks, and even social networks. Although the set of Eqs. (1)–(3) is representative of a continuous transition, it was recently shown via numerical simulation that percolation transitions can be discontinuous, i.e.; D. Achlioptas, R. M. D’souza, and J. Spencer, Science 323, 1453 (2009). 42 A. L. Efros and B. I. Shklovskii, Phys. Status Solidi B 76, 475 (1976). 43 J. P. Clerc, G. Giraud, J. M. Laugier, and J. M. Luck, Adv. Phys. 39, 1 (1990). 44 H. Scher and R. Zallen, J. Chem. Phys. 53, 3759 (1970). 45 M. B. Heaney, Phys. Rev. B 52, 012477 (1995).

J. Appl. Phys. 109, 074107 (2011) 46

D. H. McQueen, K.- M. Ja¨ger, and M. Pelı˜skova´, J. Phys. D 37, 2160 (2004). 47 O. Wiener, Abh. Math.-Physichen Klasse Ko¨nigl. Sa¨csh. Gesel. Wissen. 32, 509 (1912). 48 Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962). 49 M. J. Beran, Statistical Continuum Theories (Wiley, New York, 1968); Nuovo Cimento 38, 771 (1965). 50 See http://www.cabot-corp.com/ and http://www.columbianchemicals.com/. 51 E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, 1998); L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (Institute of Physics, Bristol, UK, 2000); D. W. Lynch and W. R. Huttner, Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic, New York, 1985); P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972); see also the SOPRA database, http:// www.sopra-sa.com. 52 I. V. Antonets, L. N. Kotov, S. V. Nekipelov, and Ye. A. Golubev, SolidState Electron. 74, 24 (2004); S. L. Westcott, J. B. Jackson, C. Radloff, and N. J. Halas, Phys. Rev. B 66, 155431 (2002). 53 W. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, 2007). 54 T. J. Lewis, IEEE Trans. Dielectr. Electr. Insul. 11, 739 (2004); T. J. Lewis, J. Phys. D 38, 202 (2005). 55 K. Golden, J. Mech. Phys. Solids 34, 333 (1986). 56 R. Pelster and U. Simon, Colloid Polym. Sci. 277, 2 (1999); R. P. Kusy and D. T. Turner, Nature (London); Phys. Sci. 229, 58 (1971); W. J. Kim, M. Taya, K. Yamada, and N. Kamiya, J. Appl. Phys. 83, 2593 (1998)]; A. ¨ berall, and K. Chouffani, J. Stoyanov, B. F. Howell, E. C. Fischer, H. U ¨ berall, ibid. ibid. 86, 3110 (1999); A. J. Stoyanov, E. C. Fisher and H. U 89, 4486 (2001); A. V. Osipov, K. N. Rozanov, N. A. Simonov; S. N. Starostenko, J. Phys: Condens. Matter 14, 9507 (2002). 57 D. Pantea, H. Darmstadt, S Kaliaguine, and C. Roy, Appl. Surf. Sci. 217, 181 (2003). 58 D. Jeulin and L. Savary, J. Phys. I France 7, 1123 (1997). 59 F. Carmona, and J. Ravier, Carbon 40, 151 (2002); J. Ravier, F. Houze, F. Carmona, O. Schneegans, and H. Saadaoui, ibid. 39, 314; (2001); F. Houze, R. Meyer, O. Schneegans, and L. Boyer L, Appl. Phys. Lett. 69, 1975 (1996). 60 R. Viswanathan and M. B. Heaney, Phys. Rev. Lett. 75, 4433 (1995). 61 I. Balberg and J. Blanc, Phys. Rev. B 31, 8295 (1985). 62 C. Engstro¨m, J. Phys. D 38, 3695 (2005). 63 C. Brosseau (unpublished). 64 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 65 T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg, Science 313, 951 (2006).
