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Thermal transport phenomena and limitationsin heterogeneous polymer composites containingcarbon nanotubes and inorganic nanoparticles

http://dx.doi.org/10.1016/j.carbon.2014.07.0070008-6223/� 2014 Elsevier Ltd. All rights reserved.

* Corresponding author.E-mail address: mpedhm@nus.edu.sg (H.M. Duong).

Feng Gong a, Khoa Bui b, Dimitrios V. Papavassiliou c, Hai M. Duong a,*

a Department of Mechanical Engineering, National University of Singapore, 117576 Singapore, Singaporeb Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843, United Statesc School of Chemical, Biological, and Materials Engineering, University of Oklahoma, Norman, OK 73019, United States

A R T I C L E I N F O

Article history:

Received 25 March 2014

Accepted 5 July 2014

Available online 10 July 2014

A B S T R A C T

An Off-Lattice Monte Carlo model was developed to investigate effective thermal conduc-

tivities (Keff) and thermal transport limitations of polymer composites containing carbon

nanotubes (CNTs) and inorganic nanoparticles. The simulation results agree with experi-

mental data for poly(ether ether ketone) (PEEK) with inclusions of CNTs and tungsten disul-

fide (WS2) nanoparticles. The developed model can predict the thermal conductivities of

multiphase composite systems more accurately than previous models by taking into

account interfacial thermal resistance (Rbd) between the nanofillers and the polymer

matrix, and the nanofiller orientation and morphology. The effects of (i) Rbd of CNT–PEEK

and WS2–PEEK (0.0232–115.8 · 10�8 m2K/W), (ii) CNT concentration (0.1–0.5 wt%), (iii) CNT

morphology (aspect ratio of 50–450, and diameter of 2–8 nm), and (iv) CNT orientation (par-

allel, random and perpendicular to the heat flux) on Keff of a multi-phase composite are

quantified. The simulation results show that Keff of multiphase composites increases when

the CNT concentration increases, and when the Rbd of CNT–PEEK and WS2–PEEK interfaces

decrease. The thermal conductivity of composites with CNTs parallel to the heat flux can

be enhanced �2.7 times relative to that of composites with randomly-dispersed CNTs.

CNTs with larger aspect ratio and smaller diameter can significantly improve the thermal

conductivity of a multiphase polymer composite.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Polymer composites with carbon-based inclusions have

attracted substantial interest in both academe and industry

because of the prominent electrical, thermal and mechanical

properties of these inclusions [1,2]. The nanoscale size of the

inclusions leads to an increase in interfacial area, so that

improved properties can be observed at low mass additions

(less than 5 wt%) of nanofillers (e.g., nanoparticles, nano-

tubes, graphene nanoribbons, nanoplates), comparable to

those achieved by much higher loadings of microfillers

(20 wt% � 40 wt%) [3–8]. Among different nanofillers, carbon

nanotubes, especially single-walled carbon nanotubes

(SWCNTs), are very effective additives [9–11]. Two-phase

carbon nanotube (CNT)–polymer composites have been

well-studied previously. It is accepted that a homogeneous

306 C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6

dispersion of CNTs, as well as strong interaction and adhe-

sion at the CNT–polymer interface, are prerequisites to fabri-

cate high quality composites [12]. To achieve these

conditions, complicated processes (e.g., surfactant function-

alization, long time sonication) are usually involved in the

experiments, which lead to costly fabrication of two-phase

CNT–polymer composites.

During recent years, the incorporation of solid lubricant

nanoparticles into CNT–polymer composites without surfac-

tant functionalization is found to improve the homogeneity

of CNTs in polymer matrix, achieving outstanding electrical,

thermal and mechanical properties of the hybrid composites

[7,13,14]. For instance, Naffakh et al. [13] incorporated envi-

ronment-friendly inorganic tungsten disulfide nanoparticles

(WS2) into SWCNT–poly(ether ether ketone) (PEEK) compos-

ites to fabricate a new type of composite. Because of lubrica-

tion provided by the WS2 nanoparticles and the synergistic

effects between WS2 and SWCNTs, a more homogeneous dis-

persion of SWCNTs with enhanced thermal, electrical and

mechanical properties was obtained, when compared to

two-phase SWCNT–PEEK composites. Due to a much lower

cost of WS2 nanoparticles compared to SWCNTs, the fabrica-

tion of this SWCNTs/WS2/PEEK composite provides a cost-

effective, environment-friendly and process-accessible way

to produce high quality composites. Improved thermal con-

ductivity, in particular, makes this new multiphase composite

a promising candidate as a thermal interface material. Other

novel multiphase composites, such as CNTs/fibers/polymer

[15] and CNTs/nanosheets/polymer [16,17] composites, have

also been fabricated and investigated for their capability to

combine the advantages of both inorganic and organic mate-

rials within a hybrid system. For example, since the thermal

conductivity of CNTs is very high, the incorporation of CNTs

can contribute to a much higher effective thermal conductiv-

ity of the new composites compared with neat polymers. An

increase in thermal conductivity can induce a reduction in

cycle times during melt-processing, such as injection mold-

ing, thereby resulting in a higher processing efficiency [13].

Although a few experimental studies on multiphase com-

posites have been reported, there is no comprehensive inves-

tigation of the mechanism and limitations of the effective

heat transfer properties. Effective medium theory for two-

phase composite has been used to predict the effective ther-

mal conductivity of multiphase composites [18]. However,

this practice is not appropriate for multiphase composites

considering the composition of multiphase composites and

the interactions between any two phases [14]. In two-phase

nanofiller/polymer composites, the interfacial thermal resis-

tance (often referred to as the Kapitza resistance) at the nano-

filler–polymer interface dominates the heat transfer and,

therefore, controls the effective thermal conductivity of the

composite. Both the effective medium theory developed by

Nan et al. [19,20], and the Monte Carlo approach developed

by Duong et al. [21,22] predict effective thermal conductivity,

showing a good agreement with experimental results when

the interfacial thermal resistance is taken into account. In

the case of multiphase composites [14], interfacial thermal

resistance still plays a significant role in the heat transfer

properties. One needs to consider the interfacial thermal

resistance between any two components in the case of

multiphase composites in order to predict the effective ther-

mal conductivity more accurately.

To achieve the above target, a 3-dimensional mesoscopic

model was developed in the current work, by means of Off-

Lattice Monte Carlo simulation. The SWCNT/WS2/PEEK com-

posite was chosen as a case study in this work [13]. The

effects of interfacial thermal resistance at the SWCNT–PEEK

and the WS2–PEEK interfaces on the effective thermal con-

ductivity of the composite were studied quantitatively. The

model was validated by comparing simulation results with

measured effective thermal conductivities of SWCNT/WS2/

PEEK composites. Effects of the morphology of the SWCNTs

(e.g., diameter between 2 and 8 nm and aspect ratio in the

range 50–450) and dispersion pattern (e.g., 0.1–0.5 wt% of

mass fraction and nanofiller orientation of parallel, random

and perpendicular to the heat flux) were quantified and inves-

tigated. The contributions of this paper include: (a) providing

an effective model to study the heat transfer mechanism in

multiphase composites and predicting the effective thermal

conductivity; (b) suggesting an efficient way to back-calculate

the interfacial thermal resistance based on the measured

thermal conductivity of multiphase composites; and (c) pro-

posing a method to fabricate multiphase composites with

high thermal conductivity.

2. Simulation methodology

A 3-dimensional model was built based on the SWCNT/WS2/

PEEK composite fabricated by Naffakh et al. [13]. A homoge-

nous dispersion of SWCNTs was achieved experimentally tak-

ing advantage of the lubricant role played by the spherical

WS2 nanoparticles. A 0.5/0.5/99.0 composition of SWCNT/

WS2/PEEK composite was chosen to simulate, in which the

mass fractions of SWCNT, WS2 and PEEK matrix were

0.5 wt%, 0.5 wt% and 99.0 wt%, respectively. Based on the

morphology and composition of this SWCNT/WS2/PEEK com-

posite, one spherical WS2 nanoparticle with diameter equal to

110 nm was placed in the center of a PEEK cube with a side

length of 925 nm, as shown in Fig. 1(a). SWCNTs with 2 nm

in diameter and 500 nm in length were randomly distributed

and oriented in the PEEK matrix without contact with the WS2

nanoparticle. The simulated cube is assumed to be a repre-

sentative volume element (RVE) of the composite, which can

be repeated to replicate the experimentally synthesized

composite.

In the current work, the heat transfer is considered as the

result of the behavior of large number of discrete thermal

walkers (both hot and cold walkers). All the thermal walkers

have the same absolute value of energy (a hot walker has

positive energy while a cold walker has negative energy)

and travel randomly within the composite based on Brownian

motion [23–26]. The Brownian motion can be described by

changes in the position of thermal walkers in each time step.

These random changes of position in each of the three space

directions take values from a normal distribution with a zero

mean and a standard deviation, r, expressed as [27]

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiDmDt

pð1Þ

where Dm is the thermal diffusivity of the PEEK matrix and Dt

is the time increment of the simulation. Inside a WS2

Fig. 1 – Schematic plot of the computational model: (a) one WS2 nanoparticle with diameter of 110 nm is located in the center

of a PEEK cube with a side length of 925 nm, while 317 SWCNTs (2 nm in diameter and 500 nm in length) are randomly

distributed in the PEEK matrix. The plane with uniform red dots is the hot surface and the opposite plane with green dots

represents the cooled surface. (b) Composite with SWCNTs orientated parallel to the direction of the heat flux; (c) composite

with SWCNTs perpendicular to the heat flux; (d) side view of the composite depicted in (b) along the heat flux direction. The

black dots are SWCNTs and the red sphere is the WS2 nanoparticle. The mass fraction of the WS2 nanoparticle and the

SWCNTs in all the above models is 0.5 wt% and 0.1 wt%, respectively. (A color version of this figure can be viewed online.)

C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6 307

nanoparticle, thermal walkers jump randomly similar to the

way they travel within the PEEK matrix, but with a different

thermal diffusivity. While inside the SWCNTs, due to the bal-

listic phonon transport and the ultrahigh thermal conductiv-

ity [28] of the SWCNTs, thermal walkers are assumed to travel

with an infinite speed. Thus, they can be anywhere in a

SWCNTs within a single time step.

In order to estimate the effective thermal conductivity of

the multiphase composite, constant heat flux was applied

through the composite by continuously releasing a large

quantity of hot walkers (40,000) in each time step from one

side of the computational domain, at x = 0 (see the plane with

red dots in Fig. 1(a)). Meanwhile, an equal number of cold

walkers with negative energy were released from the opposite

side, representing a cooled surface (plane with green dots in

Fig. 1(a)). A walker was bounced back to the computational

domain when it jumped outside of the computational domain

in the x direction. Periodic boundary conditions were applied

in the other two space directions. All the thermal walkers

traveled randomly, starting from their plane of initial release.

Once a thermal walker in the matrix crossed the interface

between the PEEK matrix and a SWCNT, it was allowed to

either jump into the SWCNT with a probability of fm-SWCNT,

related to the interfacial thermal resistance, or still remained

in the matrix with a probability of (1 � fm-SWCNT). This proba-

bility fm-SWCNT can be calculated based on the acoustic mis-

match theory, which interprets the interfacial thermal

resistance by an average phonon transmission probability at

an interface, as follows [29]:

fm-SWCNT ¼4

qmCPmvmRbdð2Þ

where qm, CPm, vm and Rbd are the density of the PEEK matrix,

the specific heat capacity of the PEEK matrix, the speed of

sound in PEEK, and the interfacial thermal resistance at the

PEEK–SWCNT interface, respectively. All the parameters used

in the simulation are listed in Table 1. Because of the infinite

speed of a thermal walker inside the SWCNTs, it would exit a

SWCNT based on another probability, designated as fSWCNT-m,

from a randomly chosen point on the SWCNT surface. In the

Table 1 – Material properties and simulation parameters.

PEEK WS2 SWCNT

Basic propertyDensity (kg/m3) 1320 7400 2100Specific heat capacity (J/(KgK)) 1136 330 841Thermal conductivity (W/(mK)) 0.23 1.675 �3500Speed of sound (m/s) 2300 4000

GeometryComputational domain size (nm3) 925 · 925 · 925WS2 sphere nanoparticle diameter (nm) 110SWCNT diameter (nm) 2, 4, 6, 8SWCNT length (nm) 100, 300, 500, 700, 900SWCNT aspect ratio 50, 150, 250, 350, 450WS2 mass fraction (%) 0.5SWCNT mass fraction (%) 0.1, 0.2, 0.3, 0.4, 0.5Number of WS2 nanoparticle 1Number of SWCNT 317, 634, 951, 1268, 1585

Interfacial thermal propertyRbd at the SWCNT–PEEK interface, RSWNT–PEEK (·10�8 m2K/W) 0.1158, 1.158, 11.58, 115.8Rbd at the WS2–PEEK interface, RWS2–PEEK (·10�8 m2K/W) 0.0232, 0.232, 2.32Phonon transmission probability from PEEK to SWCNT, fPEEK–SWNT 1.0, 0.1, 0.01, 0.001Phonon transmission probability from PEEK to WS2, fPEEK–WS2 0.5, 0.05, 0.005Thermal equilibrium factors C CWS2–PEEK ¼ 0:285, CSWNT–PEEK = 0.35

Simulation parametersNumber of computational cells 300 · 300 · 300Number of thermal walkers 40,000Time increment (ps) 3Number of time steps 1,000,000

308 C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6

thermal equilibrium state, the heat flux that exits a SWCNT

should be equal to that entering the SWCNT in each time step.

This is the way to maintain a constant temperature within

the SWCNT. While all the thermal walkers inside a SWCNT

may jump into the surrounding PEEK matrix (owing to their

infinite speed in SWCNTs), only thermal walkers close to

the SWCNT surface may travel from the PEEK matrix into

the SWCNT due to Brownian motion jumps. Therefore, in

order to maintain the heat flux that exits and enters a SWCNT

balanced, the two probabilities, fSWCNT-m and fm-SWCNT, are

related as [23]

VSWCNTfSWCNT-m ¼ Cf-SWCNTrASWCNTfm-SWCNT ð3Þ

where VSWCNT and ASWCNT are the volume and surface area

of a SWCNT, and Cf-SWCNT is a thermal equilibrium factor

at the PEEK–SWCNT interface that depends on the geometry

of the SWCNTs and the interfacial area. At the PEEK–WS2

nanoparticle interface, thermal walkers from either the PEEK

side or the WS2 side behave similar to the walkers crossing

from the PEEK phase at the PEEK–SWCNT interface. How-

ever, the relation between fm-WS2and fWS2-m is different than

in Eq. (3), due to the different movement of walkers inside

the WS2 nanoparticle and inside the SWCNTs. Here fm-WS2

and fWS2-m are the walker traveling probability from the PEEK

matrix to the WS2 nanoparticle and the reverse, respectively.

As thermal walkers travel in the WS2 nanoparticle with a

Brownian motion, fm-WS2and fWS2-m should satisfy the follow-

ing relation:

fWS2-m ¼ Cf-WS2

ðrþ rmÞ3 � r3

r3 � ðr� rWS2Þ3

fm-WS2ð4Þ

where rm and rWS2are the standard deviations of the dis-

placement distributions used to model the Brownian motion

in the PEEK matrix and in the WS2 nanoparticle, respectively.

The radius of the WS2 nanoparticle is r, and Cf-WS2is the ther-

mal equilibrium factor at the PEEK–WS2 nanoparticle inter-

face. The thermal equilibrium factors Cf-SWCNT and Cf-WS2

were numerically determined to be 0.35 and 0.285 based on

our previous work. They were calculated by assuming that

the temperature of the whole system, including PEEK, WS2

and SWCNTs at thermal steady state should be constant

[30–32].

The computational domain was divided by 300 mesh

points on each side leading to 300 · 300 · 300 computational

cells in total. The temperature distribution was obtained by

counting the number of hot walkers in each computational

cell and then subtracting the number of cold walkers. Since

constant heat flux was applied through the composite, the

temperature profile along the heat flux direction should be a

straight line with a slope inversely proportional to the effec-

tive thermal conductivity of the composite [33]. To estimate

the effective thermal conductivity of the composite, a model

of neat PEEK matrix without the SWCNTs and the WS2 nano-

particle was built. Under the same constant heat flux and

boundary conditions, the temperature distribution along the

heat flux (x-direction) in the composite and the neat PEEK

matrix are related as

q00 ¼ �KeffdTn

dx¼ �Km

dTm

dxð5Þ

wherein, q00, Tn and Tm are the applied constant heat flux, the

temperature in the composite and the temperature in neat

C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6 309

PEEK matrix, respectively. The thermal conductivity of neat

PEEK, Km, was taken as 0.23 W/mK [34]. As seen in Eq. (5),

the effective thermal conductivity of the composite, Keff, can

be obtained from the temperature profiles in the composite

and the neat PEEK matrix.

To summarize, the assumptions made in the model are

listed below:

(1) The transfer of thermal energy is passive.

(2) The interactions between thermal walkers are ignored.

(3) The thermal properties of all components (e.g., density,

thermal conductivity, specific heat capacity) do not

change with the temperature over the modeled temper-

ature range.

(4) The thermal conductivity of SWCNTs (�3500 W/mK) is

much greater than that of PEEK (0.23 W/mK) and WS2

(1.675 W/mK), and hence thermal walkers uniformly

distribute once inside the SWCNTs. Walkers in the

WS2 nanoparticle have a similar Brownian motion to

that in PEEK matrix, but have a different thermal

diffusivity.

(5) SWCNTs are assumed to be well dispersed and without

contact with the WS2 nanoparticle, so direct interac-

tions between SWCNTs and WS2 are not taken into

consideration. The likelihood of a WS2–SWCNT contact

is much lower than the likelihood of attaining only

WS2–PEEK interfaces, due to the comparatively small

surface area of WS2.

(6) The interfacial thermal resistances are identical at the

same interface for thermal walkers entering or exiting

a component. The SWCNT–SWCNT contact interfacial

thermal resistance is ignored in the current model for

simplification and this is justified because of the low

SWCNT concentration.

(7) Walkers bounce back when jumping outside the

computational box in the x-direction, while periodic

boundary conditions are applied in y and z direction.

The computational domain has the same initial

temperature.

3. Results and discussion

3.1. Model validation with experimentally measuredthermal conductivities of SWCNT/WS2/PEEK composites

The developed model was validated with the experimentally

measured thermal conductivity of SWCNT/WS2/PEEK com-

posites with different composition. As reported by Naffakh

et al. [13], the measured effective thermal conductivities of

three types of SWCNT/WS2/PEEK composites (0.1/0.9/99.0,

0.5/0.5/99.0 and 0.9/0.1/99.0) are 0.35, 0.57 and 0.52 W/mK,

respectively. Here the different compositions are represented

by the mass fraction of SWCNT, WS2 nanoparticle and PEEK

matrix in the composite. For instance, composite 0.1/0.9/

99.0 represents a composite with 0.1 wt% of SWCNTs,

0.9 wt% of WS2 nanoparticles and 99.0 wt% of PEEK matrix.

The models for different compositions were built by main-

taining the same dimensions of SWCNTs (2 nm diameter

and 500 nm length) and WS2 nanoparticle (110 nm diameter),

but varying the side length of the PEEK cube used in the com-

putations based on the WS2 mass fraction. A lower mass frac-

tion of WS2 nanoparticle corresponds to a larger PEEK cube. In

the current work, values of the Kapitza resistance at the

PEEK–SWCNT and PEEK–WS2 interfaces were required as

inputs of the simulation. At the PEEK–WS2 interface in room

temperature (300 K), the Kapitza resistance was estimated to

be around 2.32 · 10�9 and 0.773 · 10�10 m2K/W, based on the

acoustic mismatch theory and on the diffusion mismatch

theory, respectively [25,29]. The much lower Kapitza resis-

tance at the PEEK–WS2 interface compared to that at the

PEEK–SWCNT interface (i.e., �10�8 m2W/K [35]) can be

ascribed to the similar thermal properties of the WS2 nano-

particle and the PEEK matrix (the thermal conductivity of

WS2, PEEK and SWCNT is 1.6, 0.23 and �3500 W/mK, respec-

tively). Our simulation results show further that the Kapitza

resistance at the PEEK–WS2 interface does not enhance or

impede the effective thermal conductivity of the composites

(detailed discussion can be found in Section 3.2) – the role

of the WS2 is to achieve a homogeneous dispersion of

SWCNTs. Based on the above finding, in our model validation,

the Kapitza resistance at the PEEK–WS2 interface was kept

constant (0.116 · 10�8 m2K/W) and the PEEK–SWCNT Kapitza

resistance was varied to match the calculated Keff with the

experimentally measured Keff. The PEEK–SWCNT Kapitza

resistance was back-calculated to be 1.425 · 10�8 m2K/W by

varying the phonon transmission probability at the PEEK–

SWCNT interface until the calculated Keff matched the mea-

sured value in one of the measured cases, the case of the

0.5/0.5/99.0 composite. The interfacial thermal resistance at

the PEEK–SWCNT interface obtained in this manner was then

utilized as input in the simulations for the other two compos-

ites (0.1/0.9/99.0 and 0.9/0.1/99.0) to obtain Keff and to com-

pare it with the published experimental results.

As presented in Fig. 2(a), there is agreement between the

simulated and the measured Keff, validating the developed

model. As noted in Fig. 2(a), a comparatively bigger difference

between the calculated and measured Keff occurred for the

0.9/0.1/99.0 composite. As reported by Naffakh et al. [13],

higher concentration of SWCNTs in the 0.9/0.1/99.0 composite

induced a lower Keff, compared with that of the 0.5/0.5/99.0

composite. This may be ascribed to the SWCNT contacts

and the formation of SWCNT bundles [13]. Higher concentra-

tion of SWCNTs is more possible to produce SWCNT contacts

and bundles that exhibit additional thermal resistance and

reduced thermal conductivity than SWCNTs. Such effects

can decrease the effective thermal conductivity of compos-

ites. The effects of SWCNT contacts and bundles were not

taken into account in the current model, which can explain

why Keff obtained by the simulation has a higher value than

the measured value.

In double-phase polymer composite, effective medium

theories (EMTs) can predict the effective thermal conductiv-

ity, showing good agreement with experimental data

[36–39]. For comparison, two well-known EMTs, the Max-

well–Garnett, MG, EMT [38,40] and the model developed by

Nan et al. [20], were utilized to calculate Keff of SWCNT/

WS2/PEEK composite. In MG-EMT, Keff is given as

(a)

(b)

0.1/0.9/99.0 0.5/0.5/99.0 0.9/0.1/99.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5 Experimental data, Naffakh et al. [13] Our model MG-EMT [39] Nan's model [20], Rbd=1.0×10-8m2K/W

Eff

ectiv

eTh

erm

al C

ondu

ctiv

ity, K

eff (

W/m

K)

Composite (SWCNTwt%/WS2wt%/PEEKwt%)

0.1/0.9/99.0 0.5/0.5/99.0 0.9/0.1/99.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9 Experimental data, Naffakh et al. [13] Our model Nan's model [20], Rbd=6.55×10-10m2K/W

Nan's model [20], Rbd=1.0×10-8m2K/W

Eff

ectiv

eTh

erm

al C

ondu

ctiv

ity, K

eff (

W/m

K)

Composite (SWCNTwt%/WS2wt%/PEEKwt%)

Fig. 2 – Validation of the developed model by comparing our

simulation results with measured thermal conductivities of

composites with different compositions 0.1/0.9/99.0, 0.5/0.5/

99.0 and 0.9/0.1/99.0. Keeping the dimensions of the

SWCNTs (2 nm diameter and 500 nm length) and the WS2

nanoparticle (110 nm diameter) constant, the side lengths of

the PEEK cube in 0.1/0.9/99.0, 0.5/0.5/99.0 and 0.9/0.1/99.0

are 760, 925 and 1580 nm, respectively. The slightly lower

measured thermal conductivity of the 0.9/0.1/99.0

composites than that of 0.5/0.5/99.0 composites may be

ascribed to the SWCNT contacts and bundles. The

comparisons between our model and EMTs (MG-EMT and

Nan’s model) are presented in (a), while the results

calculated with different Rbd in Nan’s model are presented in

(b). The error bars in Fig. 2 and in the following Figs. 3–5

represent the standard deviations of the results obtained

from 3 separate simulations with different spatial

distribution of SWCNTs. (A color version of this figure can be

viewed online.)

310 C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6

Keff ¼3Km þ Kf Vf

3� 2Vfð6Þ

wherein Km, Kf and Vf are the thermal conductivity of matrix,

the thermal conductivity of filler and the volume fraction of

filler, respectively. For nearly spherical particles, MG-EMT

can be simplified to be [38,40]

Keff ¼ Kmð1þ 3Vf Þ ð7Þ

Nan et al. [20,41] modified MG-EMT by taking into account

into interfacial thermal resistance and the geometry of car-

bon nanotubes, resulting in a new expression of Keff:

Keff ¼ Km

ð3þ Vf ðb? þ bjjÞÞð3� Vf b?Þ

ð8Þ

with

b? ¼2 d KSWCNT � Kmð Þ � 2RbdKSWCNTKmð Þ

d KSWCNT þ Kmð Þ þ 2RbdKSWCNTKmð9aÞ

bjj ¼L KSNCWT � Kmð Þ � 2RbdKSWCNTKm

LKm þ 2RbdKSWCNTKmð9bÞ

where KSWNT, d, L and Rbd are the thermal conductivity of

SWCNT, the diameter and length of SWCNT, and the interfa-

cial thermal resistance between SWCNT and matrix, respec-

tively. Since EMTs were proposed to predict Keff of double-

phase composite, Keff of SWCNT/WS2/PEEK composite was

calculated in two main steps: (1) calculate Keff of SWCNT/

PEEK composite using Eqs. (6) and (8); (2) considering

SWCNT/PEEEK composite as matrix, calculate Keff of the mul-

tiphase composite using Eq. (7).

With Km = 0.23 W/mK, KSWCNT = 2000 W/mK, d = 2 nm,

L = 500 nm, and Rbd = 1 · 10�8 m2K/W [13,35,42], the Keff values

of different composites were calculated and presented in

Fig. 2(a). As shown in Fig. 2(a), MG-EMT overestimated Keff of

SWCNT/WS2/PEEK composites. This can be ascribed to the

interfacial thermal resistance at SWCNT–PEEK interface,

which greatly impedes heat transfer at interface, but was

not taken into account in this approach. Whereas, Nan’s

model seemed to underestimate Keff of SWCNT/WS2/PEEK

composites with Rbd = 1 · 10�8 m2K/W at SWCNT–PEEK inter-

face. This underestimation may be due to the synergistic

effects of SWCNT and WS2 nanoparticles in multiphase com-

posites [13,14], which were not considered in the separate cal-

culations of Nan’s model. In our work, interface thermal

resistance at SWCNT–PEEK interface (Rbd) was back-calcu-

lated to be 1.425 · 10�8 m2K/W by varying the phonon trans-

mission probability until the calculated Keff matched the

measured value in the case of the 0.5/0.5/99.0 composite.

With the similar way, using Nan’s model, the Rbd was deter-

mined to be 6.55 · 10�10 m2K/W by varying Rbd to match cal-

culated Keff with the measured value. This value of Rbd is

much lower than the widely reported interfacial thermal

resistance at SWCNT–polymer interface (i.e., �10�8 m2W/K

[35]). Moreover, with this Rbd, the calculated Keff of the other

two composites showed larger deviations compared to our

results, as shown in Fig. 2(b).

3.2. Effects of interfacial thermal resistance on theeffective thermal conductivity of multiphase composites

Interfacial thermal resistance at matrix-nanofiller interfaces

accounts for the significant discrepancy between theoreti-

cally expected and experimentally measured thermal con-

ductivity in the multiphase composites [14]. Interfacial

thermal resistance arises from the difference in vibrational

spectra of the atoms in each phase and possible interface

defects such as the formation of a rigidified polymer layer

C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6 311

with finite thickness, interface gaseous products and interfa-

cial voids [43,44]. The effects of the interfacial thermal resis-

tance at the PEEK–SWCNT and PEEK–WS2 interfaces on the

effective thermal conductivity of multiphase composites were

quantitatively investigated by varying the average phonon

transmission probability appearing in Eq. (2). Since the inter-

facial thermal resistance typically falls between 10�9 and

10�6 m2K/W in nanoscale applications at room temperature

[45–48], the resistance at the SWCNT–PEEK (RSWCNT–PEEK)

interface was varied from 0.1158 · 10�8 to 115.8 · 10�8 m2K/

(a)

(b)

0.1 1 10 1000

5

10

15

20

Kef

f/KPE

EK

SWCNT-PEEK Interfacial Thermal ResistanceRSWCNT-PEEK (×10-8m2K/W)

Parallel SWCNTs Random SWCNTs Perpendicular SWCNTs

0.01 0.1 10.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Kef

f/KPE

EK

WS2-PEEK Interfacial Thermal Resistance

RWS2-PEEK (×10-8 m2K/W)

Parallel SWCNTsRandom SWCNTsPerpendicular SWCNTs

Fig. 3 – Effects of interfacial thermal resistance at: (a)

SWCNT–PEEK interface and (b) WS2–PEEK interface on the

effective thermal conductivity of SWCNT/WS2/PEEK

composites. The 0.5/0.5/99.0 composite was utilized for the

generation of these figures. In (a), Rbd at SWCNT–PEEK

interface was varied from 0.1158 to 115.8 · 10�8 m2K/W

while Rbd at WS2–PEEK was kept constant as

0.232 · 10�8 m2K/W. In (b), Rbd of SWCNT–PEEK interface

was kept at 1.158 · 10�8 m2K/W as Rbd of WS2–PEEK

interface was modified from 0.0232 to 2.232 · 10�8 m2K/W.

(A color version of this figure can be viewed online.)

W, corresponding to a transmission probability between

0.001 and 1. Similarly, the thermal resistance at the WS2–PEEK

ðRWS2–PEEKÞ interface was varied from 0.0232 · 10�8 to

2.32 · 10�8 m2K/W, corresponding to a transmission probabil-

ity between 0.005 and 0.5. As presented in Fig. 3(a), at different

SWCNTorientations, the effective thermal conductivity of the

multiphase composite significantly decreases with the rise of

RSWCNT–PEEK [41]. Higher RSWCNT–PEEK will more greatly reduce

heat transfer between the SWCNT and the matrix phase,

expressed in our model as a lower probability of thermal

walkers traveling in SWCNTs, thus weakening the

enhancement of the effective thermal conductivity. At high

RSWCNT–PEEK, differently orientated SWCNTs resulted in

similar Keff, which was close to KPEEK. This is because a high

RSWCNT–PEEK, causes heat to transfer mainly through the PEEK

matrix, rather than through SWCNTs. To obtain SWCNT/WS2/

PEEK composites with high Keff, low RSWCNT–PEEK is desired,

which may be achieved by proper functionalization to bridge

the phonon spectra of SWCNTs and PEEK [49–51].

The effects of RWS2–PEEK on the effective thermal conductiv-

ity of SWCNT/WS2/PEEK composites are presented in Fig. 3(b).

Similar to RSWCNT–PEEK, a higher RWS2–PEEK led to a lower Keff. As

shown in Fig. 3(a) and (b), the influence of RSWNT–PEEK on Keff is

much more significant than that of RWS2–PEEK [13], which may

be explained by the following reasons: (i) ultrahigh thermal

conductivities make SWCNTs dominate in the transfer of

heat through the composite; (ii) much larger interfacial area

of SWCNTs in the representative volume element may induce

a stronger effect of the SWCNT–PEEK interface than that of

0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

Kef

f /KPE

EK

SWCNT Mass Fraction (wt%)

Parallel SWCNTs Random SWCNTs Perpendicular SWCNTs

Fig. 4 – Effects of SWCNTs orientation and mass fraction on

the effective thermal conductivity of SWCNT/WS2/PEEK

composites. The SWCNTs were orientated in parallel,

randomly and perpendicularly to the heat flux. Constant

heat flux was applied along the x-direction, as shown in

Fig. 1(a). The mass fraction of SWCNTs was varied from

0.1 wt% to 0.5 wt% by changing the number of SWCNTs from

317 to 1585, while the mass fraction of the WS2 nanoparticle

was kept at 0.5 wt%. More results can be found in Table 2, in

which each value was obtained by conducting three

separate simulations with different spatial distribution of

SWCNTs. (A color version of this figure can be viewed

online.)

Table 2 – Summary of simulation results to investigate the effects of SWCNTs orientation, SWCNTs mass fraction and interfacial thermal resistance at SWCNT–PEEK andWS2–PEEK Interfaces on the effective thermal conductivity*.

Keff/KPEEK

RPEEK–WS2 (·10�8 m2K/W) (fPEEK–WS2 )

0.0232 (0.50) 0.232 (0.05) 2.32 (0.005)

RPEEK–SWNT (·10�8 m2K/W) (fPEEK–SWNT) RPEEK–SWNT (·10�8 m2K/W) (fPEEK–SWNT) RPEEK–SWNT (·10�8 m2K/W) (fPEEK–SWNT)

wt% (No. of SWCNTs) 0.1158(1.000)

1.158(0.100)

11.58(0.010)

115.8(0.001)

0.1158(1.000)

1.158(0.100)

11.58(0.010)

115.8(0.001)

0.1158(1.000)

1.158(0.100)

11.58(0.010)

115.8(0.001)

Parallel SWCNT0.1 (317) 3.329 1.589 1.010 0.950 3.352 1.599 1.012 0.938 3.402 1.608 1.031 0.9470.3 (951) 10.832 3.312 1.214 0.966 10.699 3.359 1.228 0.965 10.442 3.359 1.218 0.9660.5 (1585) 18.833 5.072 1.413 0.984 19.684 5.185 1.403 0.992 18.639 5.125 1.415 0.987

Random SWCNT0.1 (317) 2.673 1.445 1.096 1.065 2.664 1.431 1.104 1.059 2.689 1.431 1.100 1.0550.3 (951) 8.524 2.624 1.452 1.311 7.863 2.666 1.442 1.310 8.224 2.672 1.436 1.2890.5 (1585) 9.041 3.471 1.710 1.541 9.609 3.500 1.718 1.550 9.612 3.472 1.733 1.536

Perpendicular SWCNT0.1 (317) 0.938 0.946 0.946 0.944 0.933 0.928 0.928 0.942 0.950 0.934 0.932 0.9480.3 (951) 0.942 0.948 0.940 0.942 0.932 0.931 0.946 0.931 0.926 0.934 0.939 0.9350.5 (1585) 0.940 0.941 0.938 0.936 0.940 0.946 0.940 0.937 0.957 0.946 0.944 0.935* The results in each case were obtained by averaging the results of three separate simulations with different spatial distribution of SWCNTs.

31

2C

AR

BO

N7

8(2

01

4)

30

5–

31

6

0 100 200 300 400 500 600 700 800 900 10000

5

10

15

20

25

30

Kef

f /KPE

EK

SWCNT Length (nm)

Parallel SWCNTsRandom SWCNTsPerpendicular SWCNTs

(a)

0.1/0.9/99.0 0.5/0.5/99.0 0.9/0.1/99.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 2 nm SWCNTs4 nm SWCNTs6 nm SWCNTs8 nm SWCNTs

Eff

ectiv

e T

herm

al c

ondu

ctiv

ity, K

eff (W

/mK

)

Composite (SWCNTwt%/WS2wt%/PEEKwt%)

(b)

Fig. 5 – Effects of the morphology of SWCNTs on the effective

thermal conductivity of SWCNT/WS2/PEEK composites: (a)

aspect ratio (length/diameter); (b) diameter. The 0.5/0.5/99.0

composite was used to study the effect of aspect ratio by

varying the length of SWCNTs from 100 to 900 nm (aspect

ratio from 50 to 450) with a constant diameter (2 nm). The

three compositions utilized in (b) were the same as those

used in our model validation. The diameter of SWCNTs was

varied from 2 to 8 nm at each composition. At each

composition, the effective thermal conductivity decreases

with the increase of the SWCNT diameter. (A color version of

this figure can be viewed online.)

C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6 313

the WS2–PEEK interface; (iii) the presence of long cylindrical

SWCNTs at random locations can lead to observing the effect

of SWCNTs throughout the whole computational domain,

while the spherical WS2 nanoparticle has a more localized

effect. This is also consistent with the experimental findings

by Cherkasova et al. [52], where higher enhancement in ther-

mal conductivity was obtained for the cylindrical particles

than the spherical ones.

3.3. Effects of SWCNT orientation and mass fraction onthe effective thermal conductivity of multiphase composites

Models with different SWCNT orientation relative to the

direction of the heat flux were built to investigate the effect

of SWCNT orientation on the effective thermal conductivity

of SWCNT/WS2/PEEK composites. As shown in Fig. 1(a)–(c),

SWCNTs were orientated randomly, in parallel and perpen-

dicularly to the heat flux, respectively. A side view along the

heat flux of the parallel model in Fig. 1(b) was presented in

Fig. 1(d), where isolated SWCNTs were observed. As presented

in Fig. 4, SWCNTs orientated in parallel with the heat flux

induced the highest effective thermal conductivity, compared

with the SWCNTs orientated randomly or perpendicularly to

the heat flux. SWCNTs orientated along the heat flux effec-

tively facilitated heat transfer through the composite, induc-

ing a significant enhancement of the thermal conductivity,

which is consistent with experimental findings [36,53,54]. As

shown in Fig. 4, the effective thermal conductivity of compos-

ites with 0.5 wt% of SWCNTs parallel to the heat flux was cal-

culated to be 1.84 W/mK, which was still much lower than the

value 11.3 W/mK estimated from the Maxwell theory [55].

This is due to interfacial thermal resistance at the SWCNT–

PEEK and WS2–PEEK interfaces, which greatly hinder heat

transfer through SWCNTs and the WS2 nanoparticles. In the

model with perpendicularly orientated SWCNTs, due to the

isolation and low mass fraction of SWCNTs (less than

0.5 wt%), SWCNTs impeded heat transfer along the heat flux

direction, resulting in a thermal conductivity even lower than

that of neat PEEK matrix, which is consistent with the find-

ings of Kang et al. [56] when applying their mass transfer

model to heat transfer.

Since SWCNTs dominated the heat transfer in SWCNT/

WS2/PEEK composites, the mass fraction of the SWCNTs

was modified from 0.1 wt% to 0.5 wt% to study its influence

on the effective thermal conductivity. As shown in Fig. 4, in

the composites with SWCNTs orientated randomly and in

parallel, the effective thermal conductivity increased with

increasing SWCNTs mass fraction [36]. A higher mass fraction

of SWCNTs meant more SWCNTs acting as thermally con-

ducting channels along the heat flux direction, allowing more

heat to transport through SWCNTs, and therefore induced a

higher thermal conductivity of the composite. However, as

isolated SWCNTs were orientated perpendicularly to the heat

flux, mass fraction seemed to show little effect on the effec-

tive thermal conductivity, due to the SWCNTs acting as barri-

ers to the heat flux. More simulation results can be found in

Table 2, where the orientation and mass fraction of SWCNTs,

as well as the interfacial thermal resistance were quantita-

tively investigated in detail. Similar to SWCNTs, higher mass

fraction of WS2 nanoparticles induced higher effective

thermal conductivity of the composite. However, this effect

is not as significant as that of SWCNT, which may be

explained by the aforementioned mechanism in Section 3.2.

3.4. Effects of SWCNT morphology on the effectivethermal conductivity of multiphase composites

A SWCNT, as a one-dimensional nanomaterial, can exhibit an

aspect ratio (the ratio of length over diameter) as high as 1000

[57]. The effect of the SWCNT aspect ratio on the effective

thermal conductivity of multiphase composites was investi-

gated by varying the length from 100 to 900 nm (aspect ratio

314 C A R B O N 7 8 ( 2 0 1 4 ) 3 0 5 – 3 1 6

from 50 to 450), while keeping a constant diameter of 2 nm.

The results using various aspect ratios are presented in

Fig. 5(a). In composites with parallel and randomly orientated

SWCNTs, the effective thermal conductivity increases signif-

icantly with increasing aspect ratio [43,58]. Because of the bal-

listic phonon transport mechanism in SWCNTs, longer

SWCNTs more effectively transport heat through the compos-

ite than short ones, thereby inducing a higher thermal con-

ductivity [57]. On the other hand, molecular dynamics

simulation results have shown that the lower stiffness of

longer SWCNTs may increase the overlap in the vibration

spectra of SWCNTs and polymer matrix, hence leading to a

lower interfacial thermal resistance at the SWCNT–polymer

interface [49]. A lower interfacial thermal resistance can more

greatly enhance the effective thermal transfer properties as

discussed in Section 3.2.

The diameter of SWCNTs was also quantitatively investi-

gated for its influence on the effective thermal conductivity

of multiphase composites. As presented in Fig. 5(b), for all

composites with different composition, SWCNTs with smaller

diameter resulted in a higher effective thermal conductivity.

This can be attributed to the lager surface area of SWCNTs

with smaller diameters at the same mass fraction [37,59].

The larger surface area affords a larger interfacial area for

more effective heat transfer at SWCNT–PEEK interfaces, and

hence results in a higher effective thermal conductivity of

the composites. The diameters utilized here fall into the

diameter ranges of single-walled carbon nanotubes, double-

walled carbon nanotubes (DWCNTs) and multi-walled carbon

nanotubes (MWCNTs) (e.g., 2 nm for SWCNTs, 4 nm for

DWCNTs [60] and 6�8 nm for MWCNTs [61]). It may be antic-

ipated from the present study that SWCNTs are more effec-

tive nanofillers for improving thermal conductivity than

both DWCNTs and MWCNTs.

4. Conclusions

Effective heat transfer properties of multiphase polymer

composites were quantitatively studied by an off-lattice Mote

Carlo model. The interfacial thermal resistance at the

polymer–nanofiller interface was quantified according to the

acoustic mismatch theory. By matching the experimentally

measured thermal conductivity of SWCNT/WS2/PEEK com-

posites, the interfacial thermal resistance at the SWCNT–PEEK

and WS2–PEEK interfaces was estimated to be 1.425 · 10�8 and

0.116 · 10�8 m2K/W, respectively. High interfacial thermal

resistance at both SWCNT–PEEK and WS2–PEEK interfaces

forced heat to mainly transfer inside the PEEK matrix,

inducing a low effective thermal conductivity. It was also

found that the interfacial thermal resistance at the SWCNT–

PEEK interface dominated the transfer of heat in the compos-

ite, because of the large SWCNT–PEEK interfacial area, the

ultrahigh thermal conductivity of the SWCNTs and their long

cylindrical geometry.

SWCNTs with orientation in parallel with the applied heat

flux result in the highest thermal conductivity of the compos-

ite compared with the randomly and perpendicularly orien-

tated SWCNTs. Because of the ballistic heat transfer in

SWCNTs and the high interfacial area, higher effective

thermal conductivity in the composite could also be achieved

by utilizing SWCNTs with larger aspect ratio and smaller

diameter, as well as a higher mass fraction of SWCNTs. These

findings have provided not only a quantitative understanding

of the heat transfer mechanism in multiphase composites,

but also an effective way to propose synthesis goals for mul-

tiphase composites with high effective thermal conductivity.

By proper modification of the geometry and thermal proper-

ties of nanofillers, the developed model may be applied to

investigate the thermal properties of other multiphase sys-

tems having 2+ nanofillers, such as SWCNT-stabilized emul-

sions, polymer blends, nanofluid systems and other diverse

multiphase composites.

Acknowledgements

The authors would like to thank SERC 2011 Public Sector

Research Funding (PSF) Grant – R-265-000-424-305 for the

financial support. This work was done while Dimitrios Papav-

assiliou was serving at the National Science Foundation (NSF).

Any opinion, findings, and conclusions or recommendations

expressed in this material are those of the authors and do

not necessarily reflect the views of the NSF. We would also

like to thank Mr. Yunlong Xie from Nanjing University for

his guidance and discussions. Feng Gong acknowledges the

supercomputer calculation support from HPC in NUS.

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