Microstructure Dependence of the Mechanical and Thermal Behavior

Feb 18, 2010 - This article is part of the C: Barbara J. Garrison Festschrift special issue. ... simulations with the Brenner reactive empirical bond-...
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J. Phys. Chem. C 2010, 114, 5709–5714

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Microstructure Dependence of the Mechanical and Thermal Behavior of Pyrolytic Carbonaceous Char† Maxim A. Makeev and Deepak Srivastava* UniVersity Affiliated Research Center (UARC) at UC Santa Cruz, NASA Ames Research Center, MS 229-1, Moffet Field, California 94035-1000 ReceiVed: October 3, 2009; ReVised Manuscript ReceiVed: December 15, 2009

The mechanical and thermal properties of carbonaceous char, obtained through ultrahigh-temperature pyrolysis using molecular dynamics simulations with the Brenner reactive empirical bond-order potential, are reported in this work. The main focus is on the effects of microstructure and microtopology on the mechanical and thermal response properties of pyrolytic char. The Young’s modulus of char is found to follow an ∼(nj - nT)γ dependence on the average coordination number, nj, for nj values below ≈2.97, with nT being a threshold coordination number for transition between soft and solid phases of the material. This behavior is characteristic of a random network model of glassy systems, in which case γ ≈ 1.5. As the average coordination number nj increases from ≈2.97 to ≈3.10, deviations from the simple power-law behavior are observed. We show that these deviations are due to a microtopological transition associated with the formation of buckled graphenesheet-like microstructures in the char. The thermal conductivity of pyrolytic char is also investigated in a wide range of temperatures and shown to follow a behavior described by the modified Einstein heat transfer theory, corrected for the presence of buckled graphene-sheet-like features in the pyrolyzed char. I. Introduction High-temperature pyrolysis-induced transformations of materials are of great interest from both fundamental science and advanced technological application perspectives. In particular, the high-temperature pyrolysis of polymeric (or polymer-based composite) materials has been extensively investigated recently to understand the kinetics of char formation and microstructure of the resultant char.1-3 The chemical processes, involved in the high-temperature decomposition of polyethylene, have been studied in refs 1 and 2. In ref 3, the authors have investigated the microstructure of the industrial char specimens using reverse Monte Carlo modeling. Most noteworthy is their finding that pyrolytic char contains buckled graphene-sheet-like microstructures in the pyrolyzed material. Moreover, the ultrahightemperature pyrolysis of carbon nanotube (CNT) reinforced polymer-matrix composites has recently been performed with an emphasis on the microstructure of the resultant char. The studied nanocomposite materials were polymethyl methacrylate/ carbon nanotube4 and polypropylene/carbon nanotube5,6 composites. An increase in integral char yield, as compared with char residue from pristine polymer systems, was reported for both of these composite materials. The presence of structural units resembling buckled graphene-like sheets, such as the ones observed in ref 3, may cause modifications in the mechanical and thermal response properties of the corresponding char specimens. In the past, such property modifications have been established mainly for nanocomposite materials, including mechanical7 and thermal8 properties. It is thus of significant importance to investigate the structure-property relationships for disordered systems with complex microstructures, such as carbonaceous char containing embedded graphene-sheet-like units. In the present work, we investigate the mechanical and thermal properties of char, obtained by ultrahigh-temperature †

Part of the “Barbara J. Garrison Festschrift”.

Figure 1. The simulation setup used to model the pyrolysis of model PE at high temperatures. The model system is shown in initial stages of the charring process (for details, see ref 9).

pyrolysis of model polyethylene (PE), with an emphasis on the corresponding structure-property relationships. The remainder of the paper is organized as follows. In section II, we describe the simulation procedure. Section III is devoted to discussion of the microstructure of pyrolytic char. The mechanical and thermal response properties of the char are discussed in sections IV and V, respectively. Finally, in section VI, we summarize our principal findings and report the main conclusions of this work. II. Simulation Procedure To obtain molten-phase char specimens, a molecular dynamics simulation approach was devised in ref 9. The approach models the kinetics of temperature-mediated hydrogen removal from the system and high-temperature relaxation of the carbonaceous char. A brief summary of the method is as follows. Polyethylene chains (a total of 40 chains with 40 monomer units each) were randomly distributed over a simulation box with linear sizes ≈22 Å along the x- and y-directions (see Figure 1). Along the z-axis, the system size was fixed at ≈87 Å. The initial simulation setup corresponds to a density of amorphous PE of

10.1021/jp909507m  2010 American Chemical Society Published on Web 02/18/2010

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Figure 2. RDFs are shown for char specimens with nj being (a) 2.93 and (b) 3.10.

≈0.89 g/cm3 at T ) 800 K. Periodic boundary conditions were applied in all three dimensions of the simulation cell. Classical molecular dynamics simulations were performed using the Brenner reactive bond-order (REBO) potential for hydrogen and carbon atoms.10 In the past, this potential has been proven to be effective in capturing the essential physics of amorphous carbon formation.9 Moreover, it has been shown that this potential provides an adequate description for amorphous phases of carbon at elevated temperatures.11 A MD time step of 0.125 fs was used throughout the simulations. The process of pyrolysis was simulated at T ) 5000 K as well as at higher and lower temperatures to study the kinetics of hydrogen removal. In the process of dynamic evolution of the PE sample, the hydrogen atoms, broken off from the polymer backbone, were removed from the simulation cell. A hydrogen atom was considered free or broken off from the polymer backbone, if it moved away from the nearest carbon atoms to distances exceeding the C-H interaction range of the Brenner potential of ≈1.8 Å.10 After the hydrogen removal was completed at a given high temperature, the final carbon structure collapsed into a dense carbonaceous char. This final all-carbon configuration was subsequently relaxed and cooled to lower temperatures (T ) 5 K) for MD times varying from 100 to 250 ps. Char specimens differing in microstructure were generated due to the different annealing and cooling rates employed (thus being largely independent of the kinetics of PE decomposition). These char configurations were then investigated for mechanical and thermal response properties. We note that the process of the pyrolysis of PE specimens over long times could involve the transition from the 2-fold coordinated (polymer chain) structures to the 3-fold (graphitic) or 4-fold (diamond-like) coordinated materials. The simulation temperatures and relaxation times used in this work are suitable mostly for the 2-fold to 3-fold transition in the coordination numbers. Therefore, the range of average coordination numbers, obtained for our MD-generated char specimens, is limited to mainly within ≈2.95 to ≈3.10, which is suited for studying microstructural (microtopological) transition with graphene-sheet-like microstructural element development in carbonaceous char. III. Microstructure and Microtopology of PE Char Shown in Figure 2 are the radial distribution functions (RDFs) of the char specimens with the lowest (nj ≈ 2.93) and the highest (nj ≈ 3.10) average coordination numbers. For the purpose of microstructural analysis, a pair of atoms was considered bonded

Makeev and Srivastava

Figure 3. Ring-size distributions are shown for the five samples differing in microstructure. The values of average coordination numbers, nj, are shown in the figure legend.

if the interatomic distance between atoms of the pair is less than or equal to the corresponding cutoff distance of the Brenner interatomic potential (≈2.0 Å). As can be observed in the figure, the calculated RDFs of both char specimens have their first peak at ≈1.47 Å. A narrowing of the first peak occurs as the coordination number increases from ≈2.93 to ≈3.10. Note that the increase in average coordination number is correlated with a shift of the average C-C bond length toward the characteristic graphite value of ≈1.42 Å. The position of the second peak shifts by ≈0.04 Å toward smaller values in the sample with higher coordination. To understand the details of microstructural evolution, we analyzed the ring statistics as a function of the average coordination number. The ring-size distributions for char specimens with different values of the average coordination number (as indicated in the figure legend) are shown in Figure 3. As can be observed in the figure, all of the specimens demonstrate a minimum at the 4-membered ring and two maxima at 3- and 6-membered rings. The number of 6-membered rings grows as the average coordination number increases from 2.93 to 3.01 and 3.10. This can be explained by the formation of extended 6-member-ring structures, reminiscent of graphene sheets, where a smaller number of rings is observed on average but more 6-member rings are energetically favored. Note that, according to the theory of network glasses, the corrections to the elastic constraints come from rings with sizes ranging between three and five,12 whereas the graphene or buckled graphene layers would mainly consist of the 6-membered rings. Therefore, our choice of range of considered ring sizes is well justified. In the case when nj ≈ 3.10, the microtopology of the sample includes nanoribbon-like graphitic features. Thin slices of the two atomically resolved configurations with the lowest and largest nj are shown in Figure 4, to demonstrate the graphene-sheet-like microfeature formation and highlight the differences between possible char microstructures. IV. Mechanical Properties of Pyrolytic Char The mechanical response behavior of char was studied for char specimens with different atomic structural configurations varying in the average coordination numbers, ring-size statistics, and density of graphene-sheet-like structural units. The atomistic simulations of the mechanical response properties were performed as follows. The strain-loading was performed by rescaling z-coordinates of all C atoms in the simulation cell by a factor of 0.01, with a relaxation procedure employed between loading steps to mimic an adiabatic loading. To this end, the

Mechanical and Thermal Behavior of Carbonaceous Char

Figure 4. Two atomically resolved configurations of the simulated systems (XY and ZX cross-sectional views), illustrating the differences in the microtopology, shown for char samples with average coordination numbers, nj, of (a) 2.93 and (b) 3.10. Thin slices of ≈5 Å are shown in each case.

Figure 5. Elastic energy versus applied uniaxial strain, εzz, is shown for char samples with average coordination numbers, nj, of (a) 2.93 and (b) 3.10 (solid circles). The dashed lines in the two figures represent ∼εzz2 fits to the simulation data.

system was first relaxed at elevated temperature for no less than 5000 MD time steps and, subsequently, cooled down to T ≈ 5 K to obtain equilibrated strained-char atomic configurations. The total energy of the strained system, Ee, was computed after relaxation and cooling down procedures were completed. The j s ) Ee - E0, is the elastic energy of the char, excess energy, E where E0 is the original (unloaded) sample’s total energy at low temperature. The stepwise loading and relaxation were continued until the applied strain magnitude increased up to ≈10%. In Figure 5, the computed strain energies, Es, as a function of the applied strain are shown for char specimens with nj ≈ 2.93 (a) and ≈3.10 (b). For all char specimens studied, the strain energy was found to be a quadratic function of the applied strain. This behavior is characteristic for linear elastic response, with elastic energy varying as ∼εzz2. Consequently, the Young’s modulus, Y, can readily be extracted from the simulation data using the j s/εzz2), where Y is the Young’s modulus, formula Y ) (1/Ω)(∂2E Es is the elastic energy per atom, and Ω is the average atomic volume. In the following, we use the random network theory of glasses to understand and interpret the MD simulation results. The basic framework of the theory is the following. The theory is based on a uniform disordered system represented as a random network of atoms connected by bonds.13 Depending upon the local and global configurations of the random network, the state of the matter can be soft-polymer (floppy) or amorphous solid (rigid).14,15 The transition between the two states of matter occurs

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Figure 6. Shown in the figure is the Y2/3 dependence on nj. The dashed line represents the best fit to the experimental data of ref 20. “MTT” shows the point where a microtopological transition occurs from nearly random network behavior to that with elastic softening.

as the average coordination number increases above its threshold value, nT, with corresponding elastic responses being that of a soft matter below the threshold and that of a solid, when nT is above the threshold value. Within the framework of the random network theory, the elastic response of a perfect random network is defined by the number of constraints in the system. It has been shown that the threshold value for the floppy-to-rigid transition in an idealized perfect random network is nT ≈ 2.4.12,15,16 This result is based on the consideration of percolation of floppy (soft) and rigid (solid) regions in the disordered sample, and the threshold value of nT ≈ 2.4 holds only for perfect random networks. If rings and/or extended microtopological features, such as buckled graphene-sheet-like inclusions, are present in the system, corrections to the threshold value are introduced. Note that the random network theory has been used to describe the elasticity of ta-C 4-fold coordinated amorphous carbon structures,17 and found to provide an adequate description of the elastic behavior. We employed the random network theory to analyze the Young’s modulus dependence on the average coordination number by comparing our simulation results with the random network model predictions. Note that the presence of ring structures and/or extended microtopological features in the char introduces corrections to the network glass theory predictions and may lead to significant deviations in elastic properties.15,16 According to ref 18, the Young’s modulus dependence on the network microstructure is given by

Y ) YD

(

nj - nT nD - nT

)

3/2

(1)

where YD is the Young’s modulus of a perfect random network with connectivity nD and nT is the threshold value for the floppyto-rigid transition. In the case of amorphous carbon networks, nD corresponds to a diamond-like network (nD ) 4) and YD is the Young’s modulus of the tetrahedral (diamond-like) a-C. For a perfect random network with fixed values of nD, YD, and nT, the behavior of the Young’s modulus of the type similar to that of a perfect random network is expected to hold.18,19 In Figure 6, Y2/3 behavior is shown as a function of nj (solid circles). As can be observed in the figure, the simulated behavior differs

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significantly from the dependence expected for a perfect random a-C network. For comparison with the random network model predictions, in Figure 6, a linear fit to the experimental data of ref 20 (dashed line) is shown along with our MD simulated data. Note that in the fitting procedure we used nT ≈ 2.6, as suggested in ref 20. As can be observed in the figure, the expected Y ∼ (nj - nT)3/2 dependence holds for values of nj less than or equal to ≈2.97. When nj grows further, however, a sublinear behavior in the Y2/3 dependence on nj is observed. Furthermore, by treating nT as a fitting parameter in eq 1, we found that the MD simulated data points for nj less than or equal to ≈2.97 fall on the nearly overlapping curves with nT all being close to ≈2.6, whereas the data points corresponding to an average coordination number exceeding ≈2.97 fall on significantly different curves. Consequently, the simulated behavior differs from the random network model in two different aspects. First, the threshold value of nT ≈ 2.6 is obtained in contrast to the expected theoretical value of nT ≈ 2.4, reported for perfect random networks.12,15,16 Second, a saturation behavior for the Young’s modulus of the simulated char specimens with nj exceeding ≈2.97 is observed. We suggest that the latter is a signature of a microtopological transition, taking place at nj slightly above ≈2.97, but less than ≈3.01. The observed differences can be explained as follows. All char configurations, used in our simulation, contain a non-zero density of rings of sizes varying from 3 to 6 (see Figure 3). To account for the presence of rings, corrections to the nT threshold value must be introduced. As was shown in ref 12, if rings are present in the network, the threshold value shifts as nT ) 2.4 + δ, where δ is a linear combination of 3-, 4-, and 5-member ring densities, with the corresponding weighting factors given in ref 12. As follows from our analysis of the observed ring densities in Figure 3, the maximal correction factor δ gives nT ≈ 2.67 for the MD simulated values, in agreement with the observed experimental value of nT ≈ 2.6 reported in ref 20, and not nT ≈ 2.4 as suggested theoretically for the random network model. When nj is larger than ≈2.97, however, the observed deviations in behavior are all too large to be accounted for by ring-related corrections, even if nT ≈ 2.6 is used as the threshold value. Therefore, we propose that these deviations are caused by the presence of localized buckled graphene-sheetlike inclusions in the char, as illustrated in Figure 4. Indeed, the char specimen with the average coordination number ≈3.10 clearly has buckled graphene-sheet-like features, with many more 6-member rings in the material, as compared to the sample with an average coordination number ≈2.93. Consequently, the mechanical response behavior of such network configurations is affected by sliding of the graphene-sheet-like inclusions during the strain loading. This causes the softening of the Young’s modulus described above. Interestingly enough, a similar behavior has also been observed in a recent experimental study of the tetrahedral amorphous carbon, obtained via filtered cathodic vacuum arc deposition.21 V. Thermal Properties of Pyrolytic Char The thermal conductivity of char was investigated using the same set of char specimens as the ones described above, employing the direct method, as described in ref 22. The same simulation methodology has been previously used by us in a wide variety of thermal transport simulations of nanostructured materials, and further details of the method can be found in refs 23-25. A brief description of the method is as follows. For a given set of constant temperatures, a temperature gradient was created by adding (removing) heat to (from) thin boundary

Makeev and Srivastava

Figure 7. Thermal conductivity, λ, as a function of temperature shown for the char sample with nj ≈ 3.10 (solid circles). The solid line in the plot is the theoretical behavior, as predicted by eq 2.

layers of thickness δz ) 1.5 Å of the simulation cell, and a heat current flowing along the z-direction was computed using the relation Jz ) δε/(2LxLy∆t), where Lx and Ly are the linear dimensions of the simulation system along the x- and y-axis, and ∆t is the MD time step. The amount of heat added (removed) to (from) the system was fixed at δε ) 1.84 × 10-4 eV. Corresponding temperature gradients, ∂T/∂z, were computed for steady-state heat-flow conditions obtained at different temperatures. The thermal conductivities were computed using the Fourier law of heat transport: Jz ) -λ(∂T/∂z). The temperature dependence of the simulated thermal conductivity, λ(T), for the char specimen with nj ≈ 3.10 is shown in Figure 7. As expected, the thermal conductivity gradually increases over the whole studied temperature range. As the temperature grows above ≈300 K, however, a saturation takes place. This behavior is consistent with the theory of heat transport via incoherent atomic vibrations.26 The mean free path of incoherent atomic vibrations generally decreases with increasing temperature until it becomes comparable to the average interatomic distance. The saturation value of the thermal conductivity, λs, can be estimated as λs ) 1/3CVVsa, where CV is the heat capacity per unit volume, Vs is the velocity of sound, and a is the average interatomic distance. Furthermore, a comparison with the predictions of the theory can be made by numerical analysis of the expression for thermal conductivity behavior. For such analysis, we use a theoretical model developed in ref 26 to describe the thermal conductivity of disordered solids26 and amorphous carbon thin films.27 The model provides the following expression for the minimum value of thermal conductivity of a disordered solid:

λm )

π 6

1/3

()

3

kBn2/3

( )

2 x

Θ /T ∑ Vi ΘTi ∫0 (exx-e 1)2 dx i)1

2

i

(2)

where index i ) (1,..., 3) marks two transverse and one longitudinal sound modes, traveling with speeds Vi, Θi ) Vi(p/ kB)(6π2n)1/3 are the cutoff frequencies of the three sound modes (expressed in kelvins), and n is the number density of carbon atoms (expressed in 1/m3). As can be observed in Figure 7, eq 2 provides an adequate qualitative and quantitative description of the thermal conductivity behavior in the regime of intermediate and high temperatures. At low temperatures, deviations from the model behavior are observed. Those are due to the quantum

Mechanical and Thermal Behavior of Carbonaceous Char

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Figure 8. Thermal conductivity of char shown as a function of nj, at T ) 300 K (solid circles). The solid line shows numerical solution to eq 2.

corrections to the thermal transport,28 which are not included in our classical simulations. To understand the behavior of the thermal conductivity with the microstructure of char, we established a connection between the mechanical properties of char and the thermal response behavior. This is achieved by considering the dependence of the sound modes in eq 2 on the microstructure and microtopology. As we show above, the Young’s modulus of the char depends significantly on both the microstructure and microtopology, which in turn, leads to microstructure dependence of the two sonic wave speeds, Vt ) [Y/2(1 + V)]1/2 and Vl ) Vt[2(1 - V)(1 - 2V)]1/2, where Y is the Young’s modulus and V is the Poisson ratio. We solve eq 2 numerically, assuming that F ) -3.29 + 1.65nj, as according to ref 29. In ref 29, this relation was obtained by quenching liquid carbon structures, equilibrated at different densities, using a tight-binding molecular dynamics method, with allowance for volume relaxation after quenching. For the Young’s modulus dependence on nj, we use eq 1 with nT ) 2.6, the Poisson ratio is taken to be 0.3, as according to ref 27, and YD corresponds to mostly 4-foldcoordinated ta-C amorphous carbon structure.20 The analytical behavior of λ is shown in Figure 8 as a solid line, along with the MD simulation data (solid circles). Similarly to the mechanical properties, a good agreement between theory and simulations is observed for nj below ≈2.97. Also, the deviations from the predictions of eq 2 are observed, when nj increases above ≈3.0. Those are correlated well with the deviations in the elastic response behavior shown in Figure 6. Interestingly enough, a similar behavior follows from recent experiments on amorphous carbon a-C,27,30 where a deviation from the expected theoretical behavior is observed at nj ≈ 3.0. Thus, in ref 27, λ starts out at ≈0.251 W/mK at nj ≈ 2.20, increases to about 1.25 W/mK at nj ≈ 2.8, and drops off to 0.8 at nj ≈ 3.0 before starting to increase again. Similar treads are observed in ref 30. As follows from the discussion above on the elastic properties of char, this effect is due to the elastic softening caused by the presence of the graphene-sheet-like microstructures in the char specimens. Because of the small system sizes in our simulations, the density of these graphene-sheet-like microstructures is low, as compared to the real systems. The observed deviation of thermal conductivity from the expected theoretical behavior (≈6%) is comparable with estimated change, as predicted by eq 2.

In summary, we have used atomistic MD simulations to study microstructure, mechanical, and thermal response behavior of carbonaceous char prepared by high-temperature pyrolysis of model amorphous PE. The microstructure of pyrolytic char specimens is characterized in terms of radial distribution functions and ring statistics. It is shown that the density of rings as a function of the number of carbon atoms in a ring reaches a minimum at the 4-membered ring and maxima at the 3- and 6-membered ring for all of the char samples simulated and studied in this work. The peak height in the ring-size distributions at the 6-membered ring increases significantly as the average coordination number grows from ≈2.93 to ≈3.10, thus indicating the microtopological transition, with the formation of buckled graphene or graphitic microstructures in char. The elasto-mechanical response is investigated through tensile Young’s modulus using atomistic MD simulations and analyzed as a function of the average coordination number. It is shown that the Young’s modulus with an average coordination number less than or equal to ≈2.97 (non-graphitic type) follows a ∼(nj - nT)3/2 dependence on nj, which is similar to the behavior expected within the framework of the random network model. The deviations from the random network model occur for average coordination numbers larger than ≈3.01. These are explained in terms of microtopological transition with the formation of buckled graphene sheet or graphitic microstructure formation in the material. The average threshold percolation value in the random network model regime (average coordination number less than or equal to ≈2.97) is found to be ≈2.6, which agrees well with the experimental value reported on amorphous carbon. The thermal response properties of the same carbonaceous char samples are studied using the direct MD simulations of thermal conductivity. The computed thermal conductivity, as a function of temperature and average coordination number, is interpreted in terms of the incoherent vibrations model. We show that the essential features of its behavior are captured well within the framework of this model, provided that the char microstructure and microtopology are carefully taken into consideration. Acknowledgment. M.A.M. and D.S. gratefully acknowledge support from NASA through the Hypersonics project of the Fundamental Aeronautics program (NNX07AC41A). References and Notes (1) Nyden, M. R.; Forney, G. P.; Brown, J. E. Macromolecules 1992, 25, 1658. (2) Lomakin, S. M.; Brown, J. E.; Breese, R. S.; Nyden, M. R. Polym. Degrad. Stab. 1993, 41, 229. (3) Petersen, T.; Yarovsky, I.; Snook, I.; McCulloch, D. G.; Opletal, G. Carbon 2004, 42, 2457. (4) Kashiwagi, T.; Du, F. M.; Douglas, J. M.; Winey, K. I.; Harris, R. H.; Shields, J. R. Nat. Mater. 2005, 4, 928. (5) Kashiwagi, T.; Grulke, E.; Hilding, J.; Groth, K.; Harris, R.; Butler, K.; Shields, J.; Kharchenko, S.; Douglas, J. Polymer 2004, 45, 4227. (6) Kashiwagi, T.; Grulke, E.; Hilding, J.; Harris, R.; Awad, W.; Douglas, J. Macromol. Rapid Commun. 2002, 23, 761. (7) Buryachenko, V. A.; Roy, A.; Lafdi, K.; Anderson, K. L.; Chellapilla, S. Compos. Sci. Technol. 2005, 65, 2435. (8) Cahill, D.; Ford, W. K.; Goodson, K. E.; Mahan, G. D.; Majumdar, A.; Maris, H. J.; Merlin, R.; Phillpot, S. R. J. Appl. Phys. 2003, 93, 793. (9) Lawson, J. W.; Srivastava, D. Phys. ReV. B 2008, 77, 144209. (10) Brenner, D. W. Phys. ReV. B 1990, 42, 9458. Brenner, D. W.; Shenderova, O. A.; Harrison, J. A.; Stuart, S. J.; Ni, B.; Sinnott, S. B. J. Phys.: Condens. Matter 2002, 14, 783. (11) Marks, N. A.; Cooper, N. C.; McKenzie, D. R.; McCulloch, D. G.; Bath, P.; Russo, S. P. Phys. ReV. B 2002, 65, 075411. (12) Thorpe, M. F. J. Non-Cryst. Solids 1983, 57, 355. (13) Zachariansen, W. H. J. Am. Chem, Soc. 1932, 54, 3841.

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Makeev and Srivastava (24) Ponomareva, I.; Srivastava, D.; Manon, M. Nano Lett. 2007, 7, 1155. (25) Diao, J.; Srivastava, D.; Manon, M. J. Chem. Phys. 2008, 128, 164708. (26) Cahill, D. G.; Watson, S. K.; Pohl, R. O. Phys. ReV. B 1992, 46, 6131. (27) Bullen, A. J.; O’Hara, K. E.; Cahill, D. G.; Monteiro, O.; von Keudell, A. J. Appl. Phys. 2000, 88, 6317. (28) Lee, Y. H.; Biswas, R.; Soukoulis, C. M.; Wang, C. Z.; Chan, C. T.; Ho, K. M. Phys. ReV. B 1991, 43, 6573. (29) Mathioudakis, C.; Kopidakis, G.; Kelires, P. C.; Wang, C. Z.; Mo, K. M. Phys. ReV. B 2004, 70, 125202. (30) Shamsa, M.; Liu, W. L.; Balandin, A. A.; Casiraghi, C.; Milne, W. I.; Ferrari, A. C. Appl. Phys. Lett. 2006, 89, 161921.

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