Modeling of Thermal Conductance at Transverse CNT−CNT Interfaces

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Modeling of Thermal Conductance at Transverse CNT-CNT Interfaces Vikas Varshney,*,†,‡ Soumya S. Patnaik,§ Ajit K. Roy,*,† and Barry L. Farmer† Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio, UniVersal Technology Corporation, Dayton, Ohio 45432, and Propulsion Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio ReceiVed: May 6, 2010; ReVised Manuscript ReceiVed: August 17, 2010

This article explores the transverse thermal conductance between two parallelsbonded as well as nonbondedscarbon nanotubes, embedded in an epoxy matrix using nonequilibrium molecular dynamics simulations. Here, we study the effect of different organic linkerssconnecting the two nanotubesson the thermal interface conductance and compare these results with those of nonbonded nanotubes. Our results suggest that incorporation of linker molecules significantly modifies overall interface conductance between nanotubes. Specifically, we find that the conductance increases with the number of linking functionality but shows an opposite trend with respect to the linker’s length, that is, the longer the linker is, the lower the conductance. We attribute this behavior to weakening of van der Waals interactions between carbon nanotubes in the case of longer linkers as well as possible scattering of thermal vibrations that occur along the linker molecules. Introduction 1-3

Since their discoveries, carbon nanotubes (CNTs) and related carbon architectures have generated a great deal of interest from research as well as the industrial community due to their outstanding physical properties and their utilization in many different macro- to nano-scale applications.2 Specifically, thermal conduction in these architectures is one of such emerging fields that has gained much attention over past decade.3 This is especially true in electronic and aerospace industries, where heat dissipation is a major issue that concerns the durability and working efficiency of the device components.4 In the literature, CNTs and graphene have been reported to possess very high thermal conductivitysas high as ∼6600 W/mKsby several experiments5 and computer simulations.6 However, because of their intrinsic geometrical anisotropy, the thermal conduction is highly directional in nature, primarily attributed to highly stiff C-C sp2 aromatic bonds and their ordered arrangement along axial and in-plane directions in CNTs and graphene, respectively. In corresponding orthogonal directions, thermal conduction is reduced by two or more orders of magnitude.6,7 This reduction becomes evident in thermal conductivity measurements of CNT-based composite materials as a lack of thermal percolation.8 Even at high loading of CNTs as fillerss well above geometric percolationsonly a minor enhancement in the overall thermal conductivity is observed9,10 for the composite system, which is well below what is predicted by the rule of mixture calculations.11,12 In the literature, this observation has led to two possible reasonings. First, the intrinsic conductivity of the fillers (CNT) decreases significantly in composites due to scattering of heat-carrying phonons by interactions with the surrounding matrix. Such a decrease has also been shown through simulations recently.13 Second, the * To whom correspondence should be addressed. E-mail: vikas.varshney@ afmcx.net (V.V.) or [email protected] (A.K.R.). † Materials and Manufacturing Directorate, Air Force Research Laboratory. ‡ Universal Technology Corporation. § Propulsion Directorate, Air Force Research Laboratory.

more important reasoning is attributed to interface thermal resistance, also known as Kapitza resistance,14 at the transverse CNT-CNT or CNT-matrix interface. This resistance causes significantly hindered heat flow across the interface, resulting in a large temperature drop or discontinuity. The inverse of this resistance, known as Kapitza conductance, is defined as

Λ)

J ∆T

(1)

where Λ, J, and ∆T correspond to Kapitza conductance, heat flux across the interface, and the associated temperature drop, respectively. In the case of a nanotube/polymer interface, a large temperature drop across the interface suggests poor interfacial conductance due to the weak heat exchange through van der Waals forces. In recent years, new hierarchical morphologies have emerged, motivated by the development of composite/hybrid systems with superior physical properties of interest by the establishment of new synthesis schemes. Some of these morphologies include the growth of CNTs on carbon fibers,15 carbon foams,16 etc. In these composite system morphologies, the grown nanotube arrays often intermingled with each other, giving rise to a significant number of CNT-CNT transverse interfaces. Such interfaces significantly affect the thermal properties of these systems. Other systems of interests where CNT-CNT transverse interfaces play a crucial role in thermal transport could include buckypaper,17 Velcro-like intermingled CNT array thermal interfaces,18 and thermal insulators based on random networks of CNTs.19 Molecular dynamics simulations provide an alternate route to experiments for studying such interfaces. Specifically, nonequilibrium molecular dynamics simulationssbased on Fourier law approach20shave recently been used to predict interface thermal conductance at CNT-matrix,21-23 CNT-argon,24 and CNT-CNT25,26 transverse interfaces. Specifically, Zhong et al.25 have modeled the thermal resistance for (10, 10) transverse

10.1021/jp104139x  2010 American Chemical Society Published on Web 09/16/2010

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CNT-CNT interfaces for a variety of overlap lengths and nanotube spacings for isolated nanotubes. Here, they simplified the multidimensional cylindrical contact area problem by treating this area between nanotubes as equivalent to the circumferential contact, resulting in the same effective area of contact for different values of overlap lengths. The CNT-CNT transverse interface resistance was revisited recently by Xu et al.,26 where they studied the transverse interface as a function of several overlap lengths between CNTs and with wrapping of polymer around nanotubes. They suggested that thermal resistance at CNT junctions can be significantly improved by modifying the molecular structure at the interfacesisolated vs wrapped by polymer chainssdue to matching of phonon spectra and phonon mode coupling. An alternative approach to modify the CNT-CNT interface is to link them through covalently bonded functionalization. From a molecular perspective, such a methodology should provide an opportunity to predict and tailor the interface thermal resistance by comparing different functionalized linkers and their degree of functionalization. This effective resistance could then potentially be employed by mesoscale Monte Carlo simulations27 or continuum modeling28 to predict effective thermal properties of macroscopic systems where transverse CNT-CNT interfaces play a significant role. We should point out that although the chemical functionalization of CNTs reduces their intrinsic thermal conductivity, this functionalization has successfully been shown to have a positive effect on reducing the interface thermal resistance, a severe bottleneck for thermal transport in nanocomposites, and hence assisting toward obtaining a higher overall thermal conductivity of such composite materials. With this motivation, this study investigates the effect of covalent functionalization between two nanotubes on the interface thermal resistance in a surrounding matrix environment using nonequilibrium molecular dynamics simulations. We have studied several linkers with different degrees of functionalization to appreciate the overall behavior of the interface resistance. The flow of the remaining article is as follows. In the next section, we present our system design along with important simulation details and analysis methodology. Then, we analyze our simulation results for various studied linkers and discuss them in terms of different linker parameters. We also compare our results with the published literature in appropriate segments. We finish our article by summarizing important findings of the work. Modeling Protocol Initial Setup and Simulation Details. Our base-representative matrix consisted of the epoxy resin (diglycidyl ether of bisphenol F; EPON-862) and cross-linker EPI-Cure-W (diethylenetoluenediamine; DETDA) and was generated using the Amorphous Cell module29 of Material Studio. The matrix consisted of randomly dispersed 512 EPON-862 molecules and 256 DETDA molecules (∼32000 total atoms). The choice of the matrix components was partly based on their experimental importance30 and partly on our previous studies.31 The system was equilibrated using a series of NVT and NPT (independent barostat in all directions) simulations prior to building the composite CNT-matrix system. Here, we would like to mention that our equilibrated matrix system was cuboidal (not cubic) with the Z-direction close to the length of the nanotube to be inserted (as discussed later). All simulations were performed with Consistent Valence Force Field (CVFF)32 using the LAMMPS molecular dynamics package as provided by Sandia National Laboratories.33 The CVFF force field was chosen as it

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Figure 1. Top: Schematic of different studied systems with CH2 linker(s). Bottom: Schematic of nanotubes linked through other studied linkers (for linkers except CH2, only the case for a single linkage is shown). Color scheme: carbon, cyan; hydrogen, orange; and oxygen, red.

successfully predicted thermal as well as other thermodynamic properties of interest for mentioned epoxy network systems,31 which was treated as the effective surrounding medium. A Nose-Hoover thermostat and barostat were employed to control the system temperature and pressure for all simulations, respectively. Long-range electrostatic interactions were simulated using the PPPM (particle-particle-particle-mesh) technique34 as employed in LAMMPS. Building Composite System. To build different CNT-matrix composite systems, first, a series of several two-nanotube-linked systems were created and minimized with respect to the total energy within Material Studio. For the current study, we used six different linker molecules between (10, 0) nanotubes. The length of the nanotubes was set to be ∼43 Å, corresponding to 10 unit cells of a (10, 0) nanotube. Overall, we investigated 25 such systems (four systems for each linker molecule along with a nonbonded pair of nanotubes). These different linkages are schematically shown in Figure 1. To incorporate these linked CNT structures within the matrix, first, a cylindrical hole of diameter ∼30 Å was created along the Z-direction (a vacuum) by incorporating a short-range cylindrical repulsive potential at the center of the matrix.35 After that, the linked nanotube structure was inserted in the center of the matrix, making sure that the inserted structure did not overlap with the matrix. Such insertions were done for each of the 25 studied systems, separately. Then, for each system, a constant pressure simulation of 50 ps (along X- and Y-directions, keeping the Z-direction fixed) was performed to bring the matrix back close to the linked structure. During this stage, the inserted CNT structure was kept fixed. Finally, a constant pressure simulation (independent barostat in all three directions) for all atoms was performed to further equilibrate the system for 100 ps followed by NVE simulation for 100 ps. It should also be noted that all studied systems were periodic in nature in X-, Y-, and Z-directions. One of the such equilibrated systemsa pair of nonbonded nanotubessis schematically shown in Figure 2 as embedded in the matrix. Thermal Simulation Module. After prior equilibration, thermal transport simulations were performed using constant

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Figure 2. Schematic of two nanotubes (nonbonded case) immersed in the epoxy matrix at the van der Waals separation. Color scheme for nanotubes: red, nanotube in which the energy is put in (hot); and blue, nanotube from which the energy is taken out (cold).

heat methodology in NVE ensemble. For each studied case, a constant energy was added and removed simultaneously at the source (red CNT) and the sink (blue CNT), respectively, at a rate of ∼4.98 ×10-8 J/s. Because the input energy from hot CNT would diffuse into both the matrix and the CNT (blue), two types of simulations were performed for each of the 25 studied cases; one with no constraints on the matrix (matrixmoving) and the other with the fixed constraints on the matrix (matrix-frozen). For latter case, all intra-matrix energetic interactions were excluded to increase computational efficiency. However, CNT-matrix interactions were taken into account, creating an effective rigid medium surrounding the CNTs. After the steady state was achieved, the temperature of each nanotube was calculated as Ni

∑

1 Ti ) mV2 3NikB k)1 k k

(2)

where Ni is the number of atoms in each nanotube i, kB is the Boltzmann constant, and mk and Vk represent the atomic mass and velocity of atom k, respectively. Each simulation was run over 1 ns for their equilibration and subsequent data collection. The calculated temperature was time-averaged over the last half a nanosecond in block averages for better statistical accuracy. For further verification of our results, we also performed simulations using constant temperature methodology. In these studies, the temperatures of hot and cold nanotubes were kept at 320 and 280 K, respectively. To keep both nanotubes at specified temperatures, energy was continuously added and removed from hot and cold CNTs using a velocity rescaling procedure every 100 fs. The energy required to maintain the temperature of thermostatted CNTs was tracked over the course of the simulations and was later employed for heat flux calculations. For ∆T, a constant value of 40 K (temperature difference between hot and cold nanotubes) was used. Finally, eq 1 was used to calculate interface thermal conductance using both methodologies. These results are also compared later for certain cases.

Figure 3. Plot of temperature evolution (as a running average) over time in hot and cold nanotubes for the nonbonded case. The inset shows the top-view schematic of the studied system. Temperature color scheme: red, hot nanotube; blue, cold nanotube; green, first epoxy shell in the vicinity of hot nanotube; and brown, first epoxy shell in the vicinity of cold nanotube (as shown in the inset and in Figure 2). The location of epoxy shells was diametrically opposite to van der Waals contact.

Results and Discussion Let us start with the results for our most simple casesnonbonded pair of nanotubes. For this case, Figure 3 shows the temperature evolution of the hot (source) and cold (sink) nanotubes as the function of simulation time. The steady-state equilibrium temperature values were calculated to be ∼380 and ∼211 K for the aforementioned amount of input energy (∼4.98 × 10-8 J/s). The figure also shows the temperature profiles of the surrounding epoxy matrix in the vicinity of the nanotubes (closest shell). These profiles were evaluated by averaging the temperature within a specific region (1/4 of CNT perimeter region with the width of 3 Å) as shown in the inset of Figure 3. The difference in the two equilibrated temperature values for epoxy matrixsin the vicinity of the nanotubessis attributed to the energy flow that occurs between the nanotube and the

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Figure 4. Plot of steady-state temperature values of hot (red) and cold (blue) nanotubes as a function of the number of CH2 linkers. The inset shows the actual temperature difference in steady-state conditions. The plot shows two different studied scenarios, representing the heat transfer from hot nanotube to cold nanotube when the surrounding epoxy matrix was (a) subjected to normal simulation conditions (solid lines); and (b) kept frozen (dashed lines). Here, a part of the plotted data (solid lines) was reproduced from Figure 10 that appeared in our recent publication.36

matrix (from hot nanotube to matrix and from matrix to cold nanotube) through van der Waals interactions. Due to the involvement of the epoxy matrix with energy transfer, it becomes very difficult to estimate the correct value of heat flux flowing from the hot nanotube to the cold one. To circumvent this problem, we also analyzed the results from matrix-frozen simulations. The resulting equilibrated (steady-state) temperature for such nonbonded (zero linkages) nanotube simulation is shown in Figure 4, along with other results. The difference in steady-state CNT temperatures for the matrix-frozen simulation (dotted lines) is significantly larger than matrix-moving simulations (solid lines). If we assume that the thermal interface resistance between CNTs is independent (or weakly dependent) of temperature in studied temperature range, this increase in ∆T is attributed to a higher amount of heat flow within nanotubes as all of the input energy in a hot nanotube is being carried to the cold nanotube. Figure 4 also shows the steady-state nanotube temperatures for a different number of CH2 linkages between them for both matrix-moving and matrix-frozen cases. The figure shows a nonlinear decrease in ∆T with respect to the number of linkages between nanotubes. This is more clearly shown in the inset of Figure 4. Although such short intertube linkages are extremely difficult to synthesize currently, it provides an idea of how a such functionality might affect the overall thermal conductance between nanotubes. The figure also shows that around 2-5 CH2 linkages, the difference in ∆T values as calculated from both types of simulations becomes negligible, suggesting a dominant exchange of input energy between the nanotubes and minimal exchange with the surrounding epoxy matrix. We believe that the results should provide an estimation of the approximate functionalization density of nanotubes such that dominant transport occurs through the connected nanotubes within the composite matrix. After steady-state was achieved, we also investigated the temperature distribution along the nanotubes. Figure 5 shows the temperature distribution for one of such cases for a single CH2-linked pair of nanotubes. The figure shows that the temperature profile is uniform in both tubes. However, a temperature difference of ∼5% does exist when comparing ∆T

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Figure 5. Temperature distribution along the two nanotubes in steady state for the case of single-linked nanotubes through CH2 linkage. Color scheme: red, hot nanotube; and blue, cold nanotube (as shown in Figure 2).

at the linking site with respect to the temperature at the boundaries of the system. We attribute such a deviation to the constrained nature of linked sp3 carbon atoms. Once the temperature difference, ∆T, is known from MD simulations, it is possible to calculate the Kapitza interface conductance for different studied systems using eq 1. However, as discussed previously, matrix-moving simulations do not provide us with correct values of energy exchange between nanotubes due to the heat loss into the epoxy matrix. Hence, to calculate the thermal interface conductance, we employed ∆T values from matrix-frozen simulations, where the input value of heat flux was known a priori. Here, we want to point out that we have calculated the Kapitza conductance in units of W/m-K (per unit length of nanotube), instead of W/m2-K or W/K as frequently reported in the literature due to following reasoning: (a) for the former case (in units of W/m2-K), one needs to calculate the correct cross-sectional area through which energy is being transported, which is ambiguous in nature for two cylindrical nanotubes in transverse contact with each other: (b) for the latter case, the conductance becomes an extensive property and its value will be a function of the length of the simulated nanotubes. However, in the context of comparing our simulated values with those of the reported literature for CNT-CNT van der Waals interface resistance, we calculated these resistances (in units of W/m2-K) for the nonbonded case, assuming a rectangular cross-section. The length of this rectangular cross-section was assumed to be the CNT length (the box dimension along Z-direction), while two different values were used for the width over which CNTs interact, corresponding to (a) van der Waals contact of 3.4 Å and (b) the diameter of the nanotube (7.8 Å). Table 1 shows several reported interface thermal resistance values for CNTs at different interfaces along with their methodology, where the CNT-CNT resistance values are emphasized in particular. As seen from the table, the resistance values range over 2 orders of magnitude, presumably due to differences in models, simulation methodology and force field, chirality of the nanotubes, and, more importantly, consideration of the nanotube area of contact, due to its cylindrical geometry. The table also shows our calculated range of 0.7-1.7 × 10-8 m2-K/W for cross-sectional areas discussed above, which is in good agreement with the published literature. Specifically, for CNT-CNT simulated contact resistances, our values are slightly lower than those predicted by Zhong et al. and Maruyama et

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TABLE 1: Comparison of Interface Thermal Resistance literature studies Gao et al. (CNT-oil/polymer) Foygel et al. (CNT-CNT) Bryning et al. (CNT-epoxy) Huxtable et al. (CNT-water suspensions) Cola et al. (CNT-CNT) Huxtable et al. (CNT-octane) Shenogin et al. (CNT-octane) Clancy et al. (CNT-polymer) Carlborg et al. (CNT-argon) Murayama et al. (CNT-CNT) Zhong et al. (CNT-CNT) Xu et al. (CNT-CNT) Current study (CNT-CNT)

thermal interface resistance (m2-K/W) 1.58 × 10-8 (theoretical)37 107-108 K/W (theoretical)38 0.24-2.6 × 10-8 (experimental)39 8.3 × 10-8 (experimental)11 0.2-7 × 10-8 (experimental)18 4.0 × 10-8 (simulation)11 3.3 × 10-8 (simulation)21 0.2-9.6 × 10-8 (simulation)23 40.0, 62.5 × 10-8 (simulation)24 6.5 × 10-8 (simulation)25 3.0-12.0 × 10-8 (simulation)25 0.01-0.5 × 10-8 (simulation)26 0.7-1.7 × 10-8 (simulation)

al.25 but higher than Xu et al.26 predictions. The predicted range is also in very good agreement with the theoretical fit of Cola et al.18 and Foygel et al.38 In terms of interface conductance per unit length of nanotube (as shown in Figure 6), our calculated value for the CNT-CNT interface, 0.46 W/m-K, is also in good agreement with the experimental value of 0.14-0.2 W/m-K for CNT-insulated substrate as predicted by Pop et al.40 Figure 6 shows the thermal interface conductance as calculated from eq 1 for several studies cases, including a nonbonded pair of nanotubes. For all plotted cases, the increase in conductance with an increase in the number of linking molecules is evident from the figure. Toward a higher linking density, we observe a deviation from linear behavior in the interface conductance for all studied cases. We attribute this deviation to possible interlink nonbonded interactions at higher linking functionality. Under these circumstances, the linker molecules are within the van der Waals interaction distance with respect to each other and could interfere with each other’s individual transmission. Figure 6 also suggests an inverse dependence of thermal conductance on the length of the linker molecules. In other words, for a given linking density, the conductance decreases with an increasing linker length (please see the figure caption for various cases). With the increase in linker length, the separation between the nanotubes increases, which leads to weaker van der Waals interaction between nanotube and thus

Figure 7. Comparison of constant heat (solid lines) and constant temperature (dotted lines) simulation methods for the calculation of thermal interface conductance. Color scheme: unbonded, black; CH2, red; benzene, green; and C5H10, brown. The plotted values are based on matrix-frozen simulations.

resulting in lower heat transfer through these nonbonded interactions. As a consequence, the dominant heat transfer channel becomes the linker molecule. In such cases, the heatcarrying vibrationssfrom hot to cold nanotubesare subjected to scattering along the linker molecule due to the soft and flexible (angle, torsional) nature of its bonds. The longer the molecule is, the stronger the scattering of heat-carrying vibrations are, resulting in low overall conductance. For long enough linker moleculesssuch as C5H10 and EPONsthe CNT-CNT nonbonded interaction essentially becomes negligible, and the heat transfer only occurs through the linker molecule. We observe that in such cases, a single linkage is not enough to compensate for the effect of weaker/negligible van der Waals interactions as their conductance values are lower than those of nonbonded cases, limiting their potential as thermal conductance enhancers. In the literature, two separate approachessconstant heat and constant temperature simulationssare employed to investigate thermal transport using nonequilibrium molecular dynamics simulations. Our previous discussion of Kapitza conductance was based on constant heat simulations in which nanotubes were subjected to constant heat flow. To compare and validate our results further, we also performed constant temperature simulations for the same systems of interest. For these simulations, the temperatures of the hot and cold nanotubes were kept at 320 and 280 K, respectively. In this context, Figure 7 shows the comparison of Kapitza conductance (normalized with respect to nanotube length) for CH2, benzene, and C3H6 linker molecules, as calculated from both simulation approaches. It is clear from the figure that both approaches indeed predict similar conductance values, along with correct trends for Kapitza conductance with respect to the degree of functionalization and the length of the linker molecules. Summary and Conclusions

Figure 6. Thermal interface conductance as a function of number of linkers for various studied linkers (constant heat simulations). Color scheme: unbonded, black; CH2, red; C3H6, blue; benzene, green; C5H10, brown; and EPON, violet. The plotted values are based on matrixfrozen simulations.

In this study, we have modeled the transverse interface thermal conductance between two parallelsnonbonded as well as bondedsCNTs in an effective epoxy matrix medium. The observed enhancement in conductance (with respect to nonbonded pair) is attributed to the additional bonded pathwaysin addition to van der Waals interactionssthat the linker molecule provides for carrying thermal vibrations between the nanotubes.

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