Chemical Physics Letters 508 (2011) 248–251

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Chemical Physics Letters

journal homepage: www.elsevier .com/locate /cplet t

Heat transfer in high volume fraction CNT nanocomposites: Effectsof inter-nanotube thermal resistance

Khoa Bui, Brian P. Grady, Dimitrios V. Papavassiliou ⇑School of Chemical, Biological and Materials Engineering, University of Oklahoma, Norman, OK 73019, USA

a r t i c l e i n f o

Article history:Received 9 February 2011In final form 1 April 2011Available online 14 April 2011

0009-2614/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.cplett.2011.04.005

⇑ Corresponding author. Fax: +1 405 325 5813.E-mail address: dvpapava@ou.edu (D.V. Papavassi

a b s t r a c t

Monte Carlo simulations were employed to investigate the effective thermal conductivity of carbon nano-tube–polystyrene (CNT–PS) nanocomposites at different wt.% of CNT loadings and temperatures. By val-idating with experimental data, values of phonon transmission probability as well as Kapitza resistance atthe CNT–PS interface and at the CNT–CNT contact were calculated. In addition, the observed poorenhancement of thermal conductivity at high CNT concentrations was found to be the result of theCNT–CNT thermal resistance and of the bundle configuration.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Due to the high specific surface area of carbon nanotubes (CNTs)(i.e., 1315 m2/g for single-walled CNTs [1]), the transfer of heat inCNT-based composites is controlled by the thermal boundary resis-tance (also known as Kapitza resistance) at the nanotube–polymerinterface [2–4]. As a result of this resistance, whilst the electricalconductivity of SWNT composites has been reported to increasedramatically at percolation thresholds as low as 1.5 wt.% [5], thereis no radical enhancement of the thermal conductivity even abovethe percolation threshold [6,7]. Can one bypass this poor perfor-mance by increasing the volume fraction of the nanotubes? Peterset al. [8] measured the effective thermal conductivity (Keff) of sin-gle-walled CNT–polystyrene (SWNT–PS) composites at varioustemperatures above and below the glass transition temperature,and various weight fractions of SWNTs using differential scanningcalorimetry. As the wt.% of CNTs increased from 10.0% to 30.0%, Keff

of the composite was expected to increase by a factor of 3, but themeasurements [8] showed an increase of �15%. The reason for thisbehaviour could be a poorer SWNT dispersion at high wt.% and/orthe presence of an additional thermal boundary resistance be-tween the CNTs, when they are closely packed.

Currently available data from molecular dynamics (MD) simula-tions indicate that the CNT–CNT thermal boundary resistance canbe higher than the thermal boundary resistance between a CNTand a polymer matrix [9,10]. If in fact this is the case, then an in-crease in the volume fraction of CNTs in a composite will not im-prove its thermal properties, unless there are chemical bondsconnecting adjacent CNTs [11–14]. Experimental evidence sup-porting such results is needed. In the present work, off-latticeMonte Carlo (MC) simulations were used to replicate the thermal

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liou).

behaviour of the SWNT–PS composites of Peters et al. [8] at differ-ent volume fractions and at various temperatures. In addition tovalidating the simulation, the thermal boundary resistance be-tween SWNT–PS and SWNT–SWNT was back-calculated by fittingthe simulation results to the experimental data.

2. Computational methods

Details of the MC algorithm have been published in previouswork [15–18]. Similar algorithms have been used for the simula-tion of heat or mass transfer in convective flows [19,20] and in por-ous media [21,22]. In brief, the transport of heat is considered to bethe result of the Brownian motion of discrete heat walkers. Thismovement in each space direction is obtained as a random jumpfrom a normal distribution with zero mean and a standard devia-tion that depends on the thermal diffusivity, Dm, and the timeincrement, Dt, and is given as

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DmDt

pð1Þ

When a heat walker crosses the CNT–matrix interface, it is al-lowed to enter the inclusion based on the phonon transmissionprobability fm–CN, or it is forced to remain in the matrix with aprobability (1 � fm–CN). Once inside a CNT, the walkers are distrib-uted uniformly due to the high thermal conductivity of the CNTscompared to that of the matrix. A walker is allowed to exit theCNT based on another probability fCN–m that is related to fm–CN sothat the thermal boundary resistance is the same for heat to enterthe CNTs and for heat to transfer from the CNTs to the matrix[15,16]. The simulation box was a cube with sides of 100 nm.The CNTs were modelled as cylinders with 0.4 nm radius (similarto the CNT radius used in experiments [8]) placed randomly insidethe domain and were not connected to each other (see Figure 1).The aspect ratio (L/D) of the CNTs in these simulations equals to

Figure 1. CNTs randomly placed inside the computational domain as single cylinders at low volume fraction (Figure 1a) or arranged to form randomly oriented bundles athigher volume fraction (Figure 1b).

Table 1Properties of PS and number of CNTs used in the simulations.

Temperature Sound velocity CPS qPS DPS

�C m/s[28]

J/kgK[8]

kg/m3

[29]� 103 cm2/s[30]

�25 2395* 1095 1110* 1.60*

25 2325 1275 1070 1.2580 2200 1510 1030 1.05

No. CNTs (wt.%, vol.%) 470 (5.0%, 4.0%) 830 (10.0%, 8.2%) 2250 (30.0%, 25.7%).* extrapolated value from references.

K. Bui et al. / Chemical Physics Letters 508 (2011) 248–251 249

80, which was chosen based on the theoretical work of Nan et al.[23] and the computational work of Duong et al. [24]. The ratioL/D will affect the Keff of the system through the geometry factoras defined by Nan et al. [23]. This geometry factor is almost con-stant and does not affect the thermal conductivity once L/D is over20. Furthermore, Duong et al. [24] studied the CNT length scale ef-fect on the effective thermal conductivity of SWCNT suspensionand concluded that Keff was almost constant as L/D is P80. In everytime step of the simulation, there were 90,000 heat walkers re-leased from one side of the domain and an equal number of cold(negative) heat walkers from the opposite side. The temperaturewas obtained by creating a 300 � 300 � 300 mesh in the cubic do-main and counting the number of heat walkers in each grid cellafter attaining steady-state. The ratio between the temperaturegradients with and without the inclusions (only PS polymer pres-ent) provided the ratio of the thermal conductivity of the PS (KPS)divided by Keff for the composite. For each simulated case, 6 runswith different initial geometry were carried out in order to quan-tify the uncertainty associated with the random placement of theCNTs (error bars in Figure 2). Table 1 is a presentation of the phys-ical properties of the CNTs and the PS used I the simulations.

Figure 2. Keff of CNT–PS composites. The experimental data (points designated asEx), are from Refs. [8,26]. At each temperature, a single value of Kapitza resistancewas used (Table 2) to simulate the thermal conductivity at various volume fractions(points designated as Si). The error bar was calculated by six different simulationsat each point.

3. Results and discussion

3.1. Kapitza resistance and thermal conductivity of nanocomposites atvarious CNT loadings and temperatures

The value of fm–CN is an input for the MC simulation. Whenexperimental results are available, fm–CNT can be varied until thecomputational thermal conductivity becomes equal to that of theexperiment, Keff_exp/Keff_MC = 1. During these simulations, the CNTsare randomly oriented and well-dispersed inside the domain toreplicate the experimental results. Using this inverse calculationmethod, we found the values of fm–CNT at different temperaturesat 5.0 and 10.0 wt.% of CNT. According to the acoustic mismatchmodel (AMM) [25], the relation between fm�CN and the Kapitzaresistance, Rbd, is

fm—CN ¼ 4=ðqmCmmmRbdÞ ð2Þ

where the subscript m refers to the matrix material in contact withthe CNT, q is the density, C is the specific heat and m is the velocityof sound. Our inverse approach leads to the calculation of Rbd, whichlies between the two extremes found by AMM and the diffuse mis-match model (see Table 2). Furthermore, using the value of Rbd cal-culated based on the Peters et al. [8] data, the Keff found with MCsimulations at 0.75 and 1.0 wt.% CNT loadings agree very well withexperimental data obtained by another laboratory using the longi-tudinal steady state heat flow method (Jakubinek et al. [26], see Fig-ure 2). Based on Table 2, we obtained the behaviour of the Kapitzaresistance with temperature – the Rbd increased whilst the phonontransmission probability decreased with temperature.

3.2. Inter-nanotube thermal resistance

However, the question remains: why is Keff of the 30 wt.% SWNTcomposite not increasing as much as anticipated? Our simulationsfor 610.0 wt.% CNTs were carried out assuming that only Kapitza

Table 2Phonon transmission probability, fm–CNT � 103, and Kapitza resistance, Rbd � 108(W/m2 K)�1, calculated from AMM and DMM [25] and from MC simulation. The value of fm–CNT

designated as accepted was used in the MC simulations from now on.

Temp. fm–CNT AMM Rbd AMM Rbd DMM fm–CNT

5.0 wt.%fm–CNT

10.0 wt.%fm–CNT accepted Rbd accepted

�25 �C 2.30 59.83 0.04 59.00 65.00 63.00 2.1825 �C 2.05 61.65 0.04 56.00 58.00 57.00 2.2180 �C 1.69 69.22 0.04 44.00 53.00 47.00 2.48

Table 3Thermal conductivity of CNT–PS composites at 300 K with 10.0 wt.% of CNTs(8.0 vol.%). The total number of CNTs is 1000. The CNTs that remained after placingthem into bundles were randomly dispersed inside the simulation box, as seen inFigure 1b.

No. of bundles 50 50 50 25 10 0No. of CNTs per bundle 15 10 4 15 15 0Keff, W/m K 0.33 0.38 0.45 0.42 0.54 0.57

250 K. Bui et al. / Chemical Physics Letters 508 (2011) 248–251

resistance at the SWNT–PS interface is important (Figure 2). Be-cause the simulation results agree with the experimental dataand because the experiments were carried out with care to dis-perse the CNTs, we can conclude that the Rbd at the SWNT–PSinterface was the dominant factor and the CNT–CNT resistancehad negligible effects up to 10.0 wt.%. Our simulation attempts toplace randomly oriented cylinders inside the simulation box at30.0 wt.% CNT (or 25.7 vol.%) indicated that this is impossible, asexpected from Onsager’s theory that suggests that random place-ment of cylinders can be accomplished for vol.% inversely propor-tional to the aspect ratio. However, arranging the cylinders inbundles and in contact with each other at high wt.% could be away to fit 30 wt.% of CNTs in the domain. Nan et al. [23] suggestedthat Keff, when the CNTs are randomly oriented, is the same as thatof parallel CNTs with an angle, h, relative to the direction of theheat flux such that cos2h = 1/3. Our results indicated that this angleis valid at low vol.%, but it changes slightly at higher volume frac-tion of CNTs. Therefore, we placed the CNTs parallel to each otherwith an angle to the direction of heat flux. The value of Keff in thiscase was expected to be the thermal conductivity of SWNT–PScomposites at 30.0 wt.% of CNT loading that included the sameRbd as in the case of 0.75–10.0 wt.%. As seen in Figure 2, the Keff ob-tained from our simulation was about 150% higher than the corre-sponding experimental data for all temperatures. That means thatthe CNT–PS resistance is not dominating but, instead, the effect ofCNT–CNT resistance must be considerable, to the point of sup-pressing the Keff of the 30 wt.% composites.

The CNT–CNT resistance can be calculated based on the resultsof Duong et al. [27], who varied the ratio of the Kapitza resistanceof the CNT–CNT interface (Rbd–C) relative to that of the CNT–poly-mer (Rbd–p) to obtain values of the ratio of the effective thermalconductivities in these two cases. Duong et al. generated data fordifferent vol.% of CNT (see Fig. 7 in Ref. [27]). Using the ratio of Keff

resulting from the present simulations and the value obtained byPeters et al. [8] that includes both CNT–PS and CNT–CNT resistanceat 25.7 vol.% CNT, we can find the ratio of Rbd–C/Rbd–p by extrapola-tion of the data in Duong et al. [27] to 25.7 vol.% of CNT (see Sup-plementary material for details). The value of Rbd–p is taken as2.21 � 10�8 (W/m2 K)�1 for CNT–PS, as seen in Table 2. The resultof this calculation for the CNT–CNT resistance is Rbd–

C = 12.15 � 10�8 (m2 K)/W. This is higher than 0.11 � 10�8 (m2 K)/W obtained from AMM [25], but it is comparable to MD results,Zhong et al. [10] reported 8 – 11 � 10�8 (m2 K/W) and Maruyamaet al. [9] found 24.8 (m2 K/W).

3.3. Effect of bundle configuration on thermal conductivity

In order to examine further the effect of CNT bundling on Keff ofhigh vol.% composites, additional MC simulations were carried outat 10.0 wt.% of CNTs in the PS composite (Table 3). Kapitza resis-tance at the CNT–CNT interface was neglected. In the simulations,each bundle of CNTs was generated by first creating a large cylin-der having the same length as one CNT and then the CNTs wereplaced randomly inside this cylinder (see Figure 1b). As can be seenin Table 3, both the number and the size of the bundles had a det-rimental effect on Keff. This behaviour is expected if we consider

each bundle that contains several CNTs, as a single cylinder witha smaller aspect ratio than a single CNT.

4. Summary

In conclusion, Monte Carlo simulations when validated withexperimental data can be used to calculate the Kapitza resistanceat the CNT–PS interface as well as at the CNT–CNT interface. TheKapitza resistance increased whilst the phonon transmission prob-ability decreased with temperature. Comparison to experimentsand inverse calculation of the CNT–CNT resistance provided the va-lue of 12.15 � 10�8 (W/m2 K)�1. Furthermore, the Keff of a nano-composite was found to decrease dramatically with the presenceof bundles due to not only the inter-nanotube resistance but alsothe bundle configuration.

Acknowledgement

This work was supported by the DoE-funded Center for Applica-tions of Single-Walled Carbon Nanotubes (ER64239 0 012 293). Wealso acknowledge support by DOD-EPSCOR: FA9550-10-1-0031.Computations were carried out at the University of OklahomaSupercomputing Center for Education & Research.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.cplett.2011.04.005.

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