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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Off-Lattice Monte Carlo Simulation ofHeat Transfer through Carbon NanotubeMultiphase Systems Taking into AccountThermal Boundary ResistancesFeng Gong a , Dimitrios V. Papavassiliou b & Hai M. Duong aa Department of Mechanical Engineering , National University ofSingapore , Singaporeb School of Chemical, Biological, and Materials Engineering ,University of Oklahoma , Norman , Oklahoma , USAPublished online: 17 Mar 2014.

To cite this article: Feng Gong , Dimitrios V. Papavassiliou & Hai M. Duong (2014) Off-Lattice MonteCarlo Simulation of Heat Transfer through Carbon Nanotube Multiphase Systems Taking into AccountThermal Boundary Resistances, Numerical Heat Transfer, Part A: Applications: An International Journalof Computation and Methodology, 65:11, 1023-1043, DOI: 10.1080/10407782.2013.850972

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OFF-LATTICE MONTE CARLO SIMULATION OF HEATTRANSFER THROUGH CARBON NANOTUBEMULTIPHASE SYSTEMS TAKING INTO ACCOUNTTHERMAL BOUNDARY RESISTANCES

Feng Gong1, Dimitrios V. Papavassiliou2, and Hai M. Duong11Department of Mechanical Engineering, National University of Singapore,Singapore2School of Chemical, Biological, and Materials Engineering, University ofOklahoma, Norman, Oklahoma, USA

An off-lattice Monte Carlo model was developed to study the heat transfer in three-phase

systems containing carbon nanotubes (CNTs) randomly orientated and distributed at the

interface of another two media. Thermal energy was simulated by a large number of discrete

thermal walkers with a random movement in the CNTs and a Brownian motion in the other

two media. Thermal boundary resistances (TBRs) were calculated according to the

Acoustic Mismatch theory and were applied in the simulation using the probabilities of the

thermal walkers to travel across the interfaces. The numerical models were validated by com-

paring the simulation results with the theoretically calculated values for a spherical heat

source. Using the current model, heat transfer properties with complex morphology of CNTs

(diameter, orientations, and aspect ratio), as well as different heat sources, can be quantified.

1. INTRODUCTION

Due to their excellent electrical and mechanical properties, carbon nanotubes(CNTs) have been widely studied for chemical, biomedical, material science applica-tions, and in other diverse fields [1–3]. For instance, carbon nanotubes can be appliedas the stabilizers in Pickering emulsions [4, 5]. Wherein, CNTs are well absorbed anddistributed on the surface of droplets, functioning as an interfacial barrier to hinderdroplet coalescence [6]. In the biomedical field, CNTs have been attracting tremen-dous attention in studies of cancer cell hyperthermia and tumor ablation due to theirstrong optical absorbance of near infrared radiation (NIR: wavelength from 700 to1100 nm), while healthy tissue is highly transparent to NIR [7]. After being functio-nalized with proper anti-drugs or DNA, CNTs can selectively attach to the cancercells in biological solution containing both normal cells and cancer cells. Following

Received 12 August 2013; accepted 13 September 2013.

Address correspondence to Hai M. Duong, Department of Mechanical Engineering, National

University of Singapore, Singapore 117576. E-mail: mpedhm@nus.edu.sg

Color versions of one or more of the figures in the article can be found online at

www.tandfonline.com/unht.

Numerical Heat Transfer, Part A, 65: 1023–1043, 2014

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2013.850972

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NIR, selective cancer cell destruction can be achieved [8, 9]. In this process, CNTsmay selectively attach to the interface between cancer cells and surrounding media(biological solution or normal cells) during their travel in the biological systems.

In material science, multi-phase polymer nanocomposites have attracted con-siderable interest, owing to their outstanding electrical and mechanical propertieswith the incorporation of nanotubes, nanolayers or nanoparticles into the polymermatrix [10–13]. Raja et al. studied nanocomposites with metal decorated multi-walled carbon nanotubes (MWNTs) in thermoplastic polyurethane (PU) [12]. Theeffective thermal conductivity was increased by 300% compared with pristine PUby dispersing 5wt% CNTs. Higher concentration of CNTs resulted in higher effec-tive thermal conductivity of the composite. Zhang et al. synthesized nylon-6 basednanocomposites incorporating CNTs and nanoclays as the nanofillers [11]. Themechanical properties of the composites, such as the tensile stress and yield stress,as well as the thermal stability, were enhanced dramatically compared to neat nylondue to the incorporation of CNTs and nanoclays. Liao et al. fabricated aMWNTs=carbon fibre (CF)=vinyl ester (VE) nanocomposite by dispersing MWNTsinto the CF=VE composite [13]. With 1.0 phr MWNTs (phr: parts per hundred partsof VE resin by weight), the MWNT=CF=VE nanocomposite displayed a 29.8%increase in the flexural strength and 19.9�C increase in glass transition temperaturecompared with the neat CF=VE composite.

Considering the compositions of the above CNT application systems, it isnoted that they are all multi-phase systems (more than two phases). Thermal proper-ties of CNT double-phase systems, such as CNT-polymer nanocomposites and

NOMENCLATURE

Acnt CNT surface area

C specific heat capacity

Cpm1 specific heat capacity of the

medium 1

Cf thermal equilibrium factor

Cf1 thermal equilibrium factor between

CNTs and medium 1

Cf2 thermal equilibrium factor between

CNTs and medium 2

Dm thermal diffusivity of the medium

fm1�cnt probability for a thermal walker

jumping from medium 1 to a CNT

fcnt�m1 probability for a thermal walker

jumping from a CNT to medium 1

fm2�cnt probability for a thermal walker

jumping from medium 2 to a CNT

fcnt�m2 probability for a thermal walker

jumping from a CNT to medium 2

fm1�m2 probability for a thermal walker

jumping from medium 1 to medium 2

fm2�m1 probability for a thermal walker

jumping from medium 2 to medium 1

K thermal conductivity

N number of walkers in a computational

cell

NIR near infrared radiation

q energy power of the continuous heat

source

Q energy released from the instantaneous

point source

r distance to the center of the heat source

r0 radius of the spherical heat source

Rbd thermal boundary resistance

t time

Dt time increment

T0 initial temperature

DT temperature increase

V volume of a bin

Vm1 sound velocity in medium 1

Vcnt volume of a CNT

x0 position of the plain source in x direction

x distance to the plain source

a thermal diffusivity of the bulk medium

q density of the medium

qm1 density of the medium 1

e energy of a thermal walker

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CNT-based nanofluids have been studied by both experiments and computations[14, 15]. However, while some experiments have been carried out to investigate thethermal properties of CNTs multi-phase systems [10, 11], few computational studiesfocus on their thermal properties. CNT multi-phase systems are much more compli-cated than the two-phase systems, due to the interactions between any pair of phases.New models are necessary to predict and optimize the thermal properties of CNTmulti-phase systems.

In CNT-polymer composites, the measured thermal conductivity is alwaysmuch lower than the values calculated by the classical Maxwell theory. This is dueto the thermal resistance at the CNT-polymer interface, known as Kapitza thermalboundary resistance [16]. The thermal boundary resistance at the interface plays acrucial role in the transfer of heat in CNT-based composites and this resistance con-trols the thermal conductivity of the composites. Molecular dynamics simulation(MD) can be applied to explore the thermal boundary resistance between CNTand different ambient media in the micro-scale. Through MD simulation,Maruyama et al. estimated the thermal boundary resistance between single-walledcarbon nanotubes (SWNT) and water to be 2.0� 10�7m2 �K=W [17]. Huxtableet al. also conducted MD simulations to study the heat flow from a(5, 5) SWNTto surrounding octane liquid and calculated the thermal boundary resistancebetween the SWNT and octane to be 0.4� 10�7m2 �K=W [18]. Inspired by theseMD simulations, Duong et al. investigated the effects of thermal boundary resistanceon the effective thermal conductivity of CNT-based nanofluids and nanocompositesthrough the use of mesoscopic Monte Carlo methods [19, 20]. The calculatedthermal conductivities showed good agreement with experimental results.

In the CNT-stabilized oil=water emulsions, CNTs are located at the interfacebetween water and oil. The different thermal boundary resistances at the interfacesof CNT-water and CNT-oil both affect the transfer of heat in the emulsions. Thiscould be one of the reasons why Maxwell theory was not applicable in predictingthe effective thermal conductivity of the emulsions [21]. Patel et al. found that ther-mal boundary resistances also existed at liquid-liquid interfaces [22] and they calcu-lated the thermal boundary resistance between water and organic liquid to be in therange of 0.27–1.5� 10�8m2 �K=W through MD simulation. In this case, the thermalboundary resistance between water and oil is also required to be considered whenstudying the thermal properties of CNT-stabilized oil=water emulsions. The modelsof CNT two-phase systems developed by Duong et al. can successfully predict theeffective thermal conductivities of CNT-based composites and nanofluids [14, 23,24]. However, taking into account the thermal boundary resistances between anypair of phases, those models cannot be used to study the thermal properties of CNTsthree-phase systems.

In this study, a 3-D model was developed to investigate the heat transfer inCNT three-phase systems based on an off-lattice Monte Carlo method. In thismodel, thermal energy was simulated by a large number of discrete thermal walkerstravelling passively with a Brownian motion in the three-phase system. Thermalboundary resistances between any two phases were taken into account by introdu-cing a travelling probability to a thermal walker jumping across the interfaces,and the probability was calculated according to the Acoustic Mismatch theory.The thermal equilibrium factors, which depend on CNT geometry and interfacial

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area, were determined numerically by a modified method based on our previouswork [20]. The model was validated by comparing the simulation results with thetheoretical calculation of temperature distribution of a spherical heat source in aninfinite bulk medium. Finally, potential applications of the model are also discussedin this article.

2. METHODOLOGY

2.1. Computational Algorithm

A 3-D model is developed with two different media and CNTs that are distrib-uted at the interface between the two media. As shown in Figure 1, medium 1 (m1 inFigure 1) is built as a cube, and medium 2 (m2 in Figure 1) is built as a sphere locatedat the centre of medium 1. In the simulations presented here, the side of the cuberepresenting medium 1 is 2 mm and the diameter of the sphere representing medium2 is 1 mm. The CNTs are considered as solid cylinders and are randomly distributedover the surface of medium 2. The CNT radius for the simulations presented here ischosen to be 3 nm. The model, as presented in Figure 1, is representative of a cancercell in a biological medium or of a CNT-stabilized droplet in a Pickering emulsion.

CNTs are considered as heat sources in the developed model. Heat energy ismodeled by releasing a large number of thermal walkers (10,000 in this study) withthe same energy from a CNT in each small time step. Thermal walkers in CNTs areassumed to travel randomly in the ballistic phonon transport regime with an infinitespeed due to the ultrahigh intrinsic thermal conductivity of CNTs [25]. In both media1 and 2, thermal walkers move following a Brownian motion [26]. The Brownianmotion can be described by changes in the position of each thermal walker in eachtime step. These changes of position take values from a normal distribution with azero mean and a standard deviation, r, that depends on the thermal diffusivity of

Figure 1. Schematic plot of computational model. The pink cubic box is medium 1 (m1) and the red sphere

in the center is medium 2 (m2). Black cylinders distributed on the medium 2 surface are CNTs. Red dots in

different mediums are thermal walkers, and the white dots represent the thermal boundary resistances at

interfaces. The CNTs are enlarged for easy comprehension.

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the medium and can be expressed as follows.

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DmDt

pð1Þ

Where Dt is the time increment, and Dm is either the thermal diffusivity of medium 1or the thermal diffusivity of medium 2 (in which a thermal walker is travelling) thatcan be calculated as follows.

Dm ¼ K

qCpð2Þ

Where K, q, and Cp are the thermal conductivity, density, and specific heat capacityof the medium, respectively.

The computational box is divided by 200 mesh points on each side (total of200� 200� 200 computational cells) and the temperature distribution is calculatedfrom the number of walkers in each computational cell. As the initial condition,the same temperature is assumed everywhere in the model. The temperature in eachcell can be described as follows.

T ¼ T0 þ DT ¼ T0 þNe

qVCpð3Þ

Where T0 and DT are the initial and increased temperature. N, e, and V are thenumber of walkers, the energy of one thermal walker and the volume of the cell,respectively. The other parameters are the same as expressed in Eq. (2).

To model the transfer of heat in the computational system, the followingassumptions are made:

1. The transfer of thermal energy is passive.2. The interactions between thermal walkers are ignored.3. The thermal properties of all media (e.g., density, thermal conductivity, and spe-

cific heat capacity) do not change with the temperature over the modeled range.4. The thermal boundary resistances are identical at the same interface for the

walkers entering or exiting a medium. For example, the thermal boundary resist-ance for a thermal walker travelling from medium 1 to a CNT is equal to thatfrom a CNT to medium 1.

5. Infinite boundary condition (neglecting the thermal walker when it steps out ofthe cube) and periodic boundary condition were both applied. The whole systemhad the same initial temperature.

Due to the different movements of walkers in CNTs and other media, jumpingacross the interface is also different for thermal walkers in CNTs and in either medium1 or 2. Once a thermal walker in medium 1 reaches the interface between a CNT andmedium 1, it jumps into the CNT with a probability fm1�CNT, which is related to thethermal resistance at the interface, or it stays in medium 1 with a probability of 1-fm1�CNT. A similar rule governs the motion of a walker in medium 1 reaching theinterface between medium 1 and medium 2. The walkers in medium 2 travelsimilarly to the walkers in medium 1. Since the walkers in the CNTs have an infinite

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speed, it is assumed that they may reach the interfaces between CNTs and medium 1or 2 in a single time step. In this case, the thermal walkers in CNTs will travel into themedium 1 with a probability of fCNT�m1, jump into medium 2 with a probability offCNT�m2, and stay in the original CNT with a probability of 1- fCNT�m1� fCNT�m2.

The Acoustic Mismatch theory is based on an interpretation of the thermalboundary resistance (Kapitza resistance) as an average probability of the phononstransmitting across the interface [27]. According to this theory, the probability,fm1�CNT, can be given by the following.

fm1�CNT ¼ 4

qm1Cpm1Vm1Rbdð4Þ

Where qm1, Cpm1, Vm1, and Rbd are the density, specific heat capacity of medium 1,the sound velocity in medium 1, and the thermal boundary resistance between CNTsand medium 1, respectively.

According to assumption (4), at the interface between medium 1 and CNTs,the thermal boundary resistances are the same when a thermal walker enters or exitsa CNT, however, this does not indicate that fm1�cnt equals fcnt�m1. In the thermalequilibrium, the average walker density (i.e., the number of walkers per CNT vol-ume) in CNTs does not change with time in order to keep a constant temperature.This means that within each time step, the heat flux exiting a CNT should be equalto the heat flux entering the CNT. All the thermal walkers inside CNTs may travelinto surrounding medium 1 in a time step, owing to their infinite speed in CNTs.However, for the walkers in medium 1, only those around the CNTs’ surfacesmay jump into CNTs due to the random Brownian motion in medium 1. Therefore,the two probabilities, fm1�CNT and fCNT�m1 are related as follows.

VCNT fCNT�m1 ¼ CfrACNT fm1�CNT ð5Þ

Where VCNT and ACNT are the volume and surface area of CNTs, respectively,and Cf is a thermal equilibrium factor, which depends on the interfacial area andthe geometry of the CNTs. The relationship between fCNT�m2 and fm2�CNT is similar,but the thermal equilibrium factor is different due to the different interface area.At the interface between medium 1 and medium 2, it is reasonable to assumethat fm1�m2 is equals to fm2�m1, owing to the large interfacial area and the similarityof the random Brownian motion in media 1 and 2.

2.2. Determination of the Thermal Equilibrium Factor ofm1-CNT(Cf1) and m2-CNT (Cf2)

To numerically determine the thermal equilibrium factors, a smaller system isbuilt with 10 CNTs and both medium 1 and medium 2. The CNTs are kept at thesame size as in the model described in the previous section. A cube of medium 1is built with a length of 0.5 mm and a sphere of medium 2 is built with a diameterof 0.13 mm (maintaining the same CNT=matrix volume ratio as in the previousmodel) and is located in the centre of the cube. The smaller computational systemwas divided into 50 grids on each side (50� 50� 50 computational cells) and one

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thermal walker was released from the center of each cell (instead of releasing themfrom CNTs). The thermal walkers followed the same rules of motion as describedin the previous section. A homogeneous initial temperature is thus utilized as theinitial condition and periodic boundary conditions are applied in x, y, and z direc-tions. Walker densities in the CNTs and media 1 and 2 are calculated and recordedevery 500 time steps and the simulation is stopped once the walker densities nolonger change with time (the thermal equilibrium state). In an adiabatic system,the temperature is identical everywhere at the thermal equilibrium state accordingto the second law of thermodynamics. Since the whole system has a homogeneousinitial temperature, the temperature in the CNTs, medium 1 and medium 2 are theidentical when reaching the thermal equilibrium state, so we have the following.

Nc � eVc � qc � Cpc

¼ Nm1 � eVm1 � qm1 � Cpm1

¼ Nm2 � eVm2 � qm2 � Cpm2

ð6Þ

Where Nx, Vx, qx, and Cpx are, respectively, the number of the thermal walkers, thevolume, density, and specific heat capacity of a medium (subscript x designatesCNTs, medium 1, and medium 2). The average walker densities in CNTs, medium1, and medium 2 can be related as follows.

Nc

Vc:Nm1

Vm1:Nm2

Vm2¼ qcCpc : qm1Cpm1 : qm2Cpm2 ð7Þ

Table 1. Thermal properties and parameters used in the simulation

Medium 1 [43]

Density (kg=m3) 1325

Specific heat capacity (J=(Kg K)) 3750

Thermal conductivity (W=(m K)) 0.52

Sound velocity (m=s) 1603

Medium 2 [44]

Density (kg=m3) 1025

Specific heat (J=(Kg K)) 3400

Thermal conductivity (W=(m K)) 0.45

Sound velocity (m=s) 1500

Carbon nanotubes

Density (kg=m3) [45] 1357

Thermal conductivity (W=(m K)) [25] �3500

Specific heat (J=(Kg K)) [46] 841(at 300K)

Other simulation parameters

Computational box size (bins) 200� 200� 200

CNT diameter (nm) 6

Aspect ratio of CNT, L=D 5� 80

Number of thermal walkers released from each CNT 10,000

Time increment (ns) 0.02

Thermal equilibrium factor Cf 1 0.25

Thermal equilibrium factor Cf 2 0.045

To quantify all the parameters in our model, a real case of the cancer treatment is applied.

We consider a cancer cell in a cubic tissue box and the CNTs are randomly orientated and

distributed on the cancer cell surface, so the parameters of medium 1 are taken from the

human tissue and those of the medium 2 are taken from cancer cell.

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The parameters used in the simulations can be found in Table 1. According toassumption (3), the thermal properties of media do not change over the modeledtemperature range, so the average walker density in CNTs, medium 1 and medium2 have a constant ratio in the thermal equilibrium state. Based on our previous work[20, 28, 29], the values of the thermal equilibrium factors were reached when the ratioof the walker densities calculated in Eq. (7) was obtained (Cf1¼ 0.25 and Cf2¼0.045). The obtained thermal equilibrium factors were used in the later simulationsdescribed from here on.

3. RESULTS AND DISCUSSION

3.1. Random Walker Algorithm Validation

In order to validate the random walker algorithm, the normalized temperaturedistribution is calculated from simulation results and compared with the values cal-culated from the theories. Simulations are carried out to model different heat sourcesin three dimensions.

3.1.1. Instantaneous point heat source. An instantaneous point heatsource was built and fixed in the center of a cube made of medium 1. In the first timestep, 90,000 thermal walkers are released from the heat source and the walkers jumprandomly as described in the methodology section. According to the theoreticalresults given by Carslaw and Jaeger [30], the transient temperature distributionfrom an instantaneous point heat source to the surrounding medium is expressedas follows.

Tðr; tÞ ¼ T0 þQexpð�r2

4atÞqCpð4patÞ

32

ð8Þ

Where a, q, and Cp are the thermal diffusivity, density, and specific heat capacity ofthe medium, respectively. T0, Q, t, and r are, respectively, the initial temperature, thereleased energy from the point source, the time, and the distance from the pointsource. Normalized temperature distribution from the point source to the surround-ing medium is calculated to compare with the theoretical values, as shown in Figure 2.

3.1.2. Continuous point heat source. Similarly, a continuous point heatsource is built and fixed in the center of a cube of medium 1. In each time step,90,000 thermal walkers are released continuously from the heat source and all thewalkers travel randomly as described in the methodology. According to Carslawand Jaeger [30], the transient temperature distribution with time from a continuouspoint heat source to the surrounding is given as follows.

Tðr; tÞ ¼ T0 þq

4pqCparerfcð rffiffiffiffiffiffiffi

4atp Þ ð9Þ

Where q is the energy power of the continuous point heat source and the other para-meters are the same as described in Eq. (8). Normalized temperature distributionfrom the continuous point source is calculated to compare with the theoretical valuesand the fitting results are shown in Figure 3.

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Figure 2. Fitting results of the simulation under instantaneous point source at different times. The theor-

etical values are plotted in lines, and the simulation results are plotted with symbols. The unit of time is

microseconds (us).

Figure 3. Fitting results of the simulation under a continuous point source at different times. The theor-

etical values are plotted in lines, and the simulation results are plotted with symbols. The unit of time is

milliseconds (ms).

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3.1.3. Continuous plane heat source. A continuous plane heat source ismodeled by releasing thermal walkers continuously from one surface of the cubicmodel, for example, from the plane of x¼ 0. Periodic boundary conditions areapplied in the other two directions, which can be viewed as a constant heat fluxcoming from the x¼ 0 plane and transferring along the x direction. The transienttemperature distribution with time from the plane source can be calculated asfollows [30].

Tðx; tÞ ¼ qt

pa

� �12

e�ðx�x0Þ2=4at � q x� x0j j2a

erfcx� x0j j2

ffiffiffiffiffiat

p ð10Þ

Where x0 is the position of the plane heat source and x is the distance from theplane in x direction. The other symbols are as in Eqs. (8) and (9). Similarly,normalized temperature distribution from the plane heat source along the x directionis calculated to compare with the theoretically calculated results, as shown inFigure 4.

3.1.4. Continuous spherical heat source. A continuous spherical heatsource is simulated by releasing thermal walkers continuously from the surface ofa sphere. The sphere has a diameter of 1 mm and is in the center of a cube with alength of 2 mm. The transient temperature distribution can be calculated from thenumber of walkers inside and outside the sphere in a spherical coordinate systemfrom the sphere centre. Theoretically, the temperature distribution from the

Figure 4. Fitting results of the simulation under continuous plane source at different times. The theoretical

values are plotted in lines, and the simulation results are plotted with symbols. Here, the grid unit is used

for brevity and one grid unit equals 10 nm. The unit of time is microseconds (us).

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spherical heat source at different time is given as follows [30].

Tðr; tÞ ¼ q

8parr0

2ðat=pÞ12 e�

ðr�r0 Þ24at � e�

ðrþr0Þ24at

� �� r� r0j jerfc r� r0j j

2ðatÞ12

þ rþ r0j jerfc rþ r0j j2ðatÞ

12

( ) ð11Þ

Where r0 is the radius of the spherical heat source, and the other symbols are thesame as described in Eqs. (8) and (9). We present in Figure 5 the comparisonsbetween the simulation results and the theoretical values.

As shown in Figures 2–5, excellent agreement between the simulation resultsand the theoretical values are obtained in the models for different heat sources. Itcan be concluded that the random walker algorithm is successfully validated, andthe algorithm is accurate enough to predict the heat conduction in a medium. It isnoted that the agreements between simulations results and theoretical values atshorter times are better than those at longer times. In real cases, the thermal energytends to diffuse randomly from the positions with higher temperature to thepositions with lower temperature. However, it is still possible for some energy totransfer back from the lower temperature positions to those with higher tempera-tures. This means the thermal walkers are possible to jump back when they stepout of the computational box. In the model, the walkers that step out of the compu-tational box are neglected due to the applied boundary condition (infinite boundary

Figure 5. Fitting results of the simulation under a continuous spherical source at different times. The

theoretical values are plotted in lines, and the simulation results are plotted with symbols. Here, a grid unit

is used for brevity and one grid unit equals 10 nm. The surface of the sphere is at the grid 50 (500 nm

radius). The unit of time is microseconds (us).

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condition). This may explain the difference between the simulation results and thetheoretical values at the longer times.

3.2. CNT Three-Phase Model Validation

In order to validate the newly developed CNT three-phase model, simulationsare conducted by setting CNTs on the sphere as heat sources. Thermal walkers arereleased continuously from CNTs in each time step. Figure 6 is the distribution atdifferent times of the walkers released from CNTs in the first time step. As shownin Figure 6a, all the walkers are released from the randomly distributed CNTsinitially. Then the walkers randomly move out from CNTs to medium 1 or medium2 (Figures 6b and 6c). Finally, the walkers released in the first time step can travel

Figure 6. The distribution of the thermal walkers released in the first time step at the end of different time

steps. (a) Time step 1; (b) time step 1000; (c) time step 10,000; and (d) time step 100,000. The CNTs are not

plotted in the figures for brevity.

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everywhere in the computational box (Figure 6d). The walkers released in thefollowing time steps have similar movements to those released in the first time stepdue to the large quantity of the thermal walkers released from each CNT (10,000in each time step).

As CNTs are randomly orientated and distributed on the surface of the sphere,it can be imagined that a large quantity of CNTs with tiny lengths can uniformlycover the surface of the whole sphere. Under this circumstance, the sphere withuniformly distributed CNTs can be viewed as a spherical heat source, and thetemperature distribution of these two cases should be similar. Based on this, theCNT three-phase model is validated by comparing the simulation results withthe theoretical values of the spherical heat source as described in Eq. (11). Figure 7is a presentation of the validation results of the simulations using differentnumbers of CNTs with different aspect ratios.

As shown in Figure 7, the simulation results and the theoretical values arein excellent agreement with different numbers and aspect ratios of CNTs, indicatingthe successful validation of the CNT three-phase model. The simulation results

Figure 7. Validation results of the CNT multiphase model with different numbers and aspect ratios of the

CNTs. (a) 400 CNTs, L=D¼ 20; (b) 800 CNTs, L=D¼ 10; (c) 1600 CNTs, L=D¼ 5. Here, one grid equals

10 nm and grid 50 is the position of the sphere surface. The unit of time is microseconds (us).

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obtained with smaller aspect ratio have a better agreement with the theoretical valuesthan those using bigger aspect ratios. This is so because CNTs with smalleraspect ratio have smaller lengths, which are more similar to point heat sourcecompared to the large sphere (1,000-nm diameter). The smaller CNTs have a moreuniform distribution on the sphere surface, making the model even more similarto the case of a spherical heat source. It should be noted that thermal boundaryresistances are not included in the validation simulations. This is because in thetheoretical result, the medium is homogeneous without any interfaces, so there areno thermal boundary resistances. In order to more closely approach the theoreticalcase, thermal boundary resistances are reasonably ignored in the validationsimulations.

3.3. Determination of the Number of Walkers Releasedfrom a Single CNT

In order to determine the number of walkers released from a CNT in the cur-rent model, values ranging from 10,000 to 100,000 (with an increment of 10,000)were adopted to conduct the simulation and compare the results with the theoreticalvalues [19]. The simulation results agreed well with the theoretical values at all caseswith different number of thermal walkers. In Figure 8 we present the simulationresults using 10,000 and 100,000 walkers together with the theoretical values. Meansquare errors were calculated to characterize the agreement between simulationresults and theoretical values. As shown in Table 2, the larger the number of walkersapplied, the smaller the mean square errors, indicating better agreement with thetheoretical values. However, larger number of walkers required much longer time to

Figure 8. Fitting results of the simulation using 10,000 and 100,000 walkers with the theoretical values. t1

is the simulation result using 10,000 walkers; t2 is the simulation result using 100,000 walkers; and T is the

theoretical value. The unit of time is microseconds (us).

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run the simulations. Considering the balance between time efficiency and simulationaccuracy, the number of walkers released from a CNT was assumed to be 10,000,which would be used in the results presented next.

3.4. Effects of Thermal Boundary Resistance on the TemperatureRise of Media 1 and 2

Simulations were carried out to study the effects of thermal boundary resist-ance between each two of medium 1, medium 2, and CNTs on the temperature riseof medium 1 and medium 2. CNTs randomly located on the surface of medium 2 asthe heat source, thereby releasing thermal walkers continuously. Temperature increaseof medium 1 and medium 2 were calculated according to Eq. (3). Since thermalboundary resistance in nano-scale usually falls into the magnitude of 10�8m2 �K=W[17, 18, 22], the values of 0.1, 1.0 and 10� 10�8m2 �K=W were chosen as the valuesof thermal boundary resistance among medium 1, medium 2, and CNTs. As shown inFigure 9a, the temperature of medium 1 increased with the rise of thermal boundaryresistance between medium 2 and CNTs (Rbd m2�CNT ). On the contrary, the tempera-ture of medium 2 decreased when increasing the value of Rbd m2�CNT . Higher thermalresistance between medium 2 and CNTs more greatly hinders thermal energytransferring from CNTs to medium 2, hence fewer walkers jump to medium 2 fromthe CNTs. The smaller number of walkers in medium 2 accounts for the lowertemperature of medium 2. However, owing to the large number of walkers in CNTs,more walkers may travel to medium 1, leading to a high temperature of medium 1.Figure 9b is a presentation of the effects of thermal boundary resistance betweenmedium 1 and CNTs on the temperature of medium 1 and medium 2. Higherthermal boundary resistance between medium 1 and CNTs (Rbd m1�CNT ) generatesa higher temperature of medium 2 but a lower temperature of medium 1. Within highRbd m1�CNT , thermal walkers are highly impeded jumping from CNTs to medium 1,which induces a lower temperature of medium 1. Due to large number of walkers inCNTs, larger quantity of walkers will travel to medium 2, thereby leading to a highertemperature of medium 2.

Thermal boundary resistance between medium 1 and medium 2 (Rbd m1�m2)does not exhibit an apparent effect on the temperature of medium 1 and medium2 in the current work, as shown in Figure 9c. This can be ascribed to the similarthermal diffusivities of medium 1 and medium 2 (1.05� 10�7 versus 1.29� 10�7m2=s).). Since medium 1 and medium 2 have similar thermal diffusivity, thermal walkershave similar traveling speed in medium 1 and medium 2. Due to the large interfacialarea between medium 1 and medium 2, the numbers of walkers travelling from

Table 2. Mean square errors (MSE) of simulations using 10,000 and

100,000 walkers

Time (us) 10,000 walkers 100,000 walkers

0.2 0.013799 0.012956

0.4 0.013494 0.012015

1 0.017498 0.016828

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medium 1 to medium 2 and those from medium 2 to medium 1 should probably beequal and finally reach equilibrium. This equilibrium could be achieved in the case ofboth low and high thermal boundary resistance. When medium 2 has a much biggerthermal diffusivity than medium 1 (e.g., 5.0� 10�7 versus 1.05� 10�7m2=s), thewalkers in medium 2 are more possible to jump into medium 1. High thermal bound-ary resistance will greatly block walkers in medium 2, rather than letting them jumpinto medium 1, which will induce a high temperature of medium 2. As shown inFigure 10a, simulation results also confirmed this prediction. Similarly, when ther-mal diffusivity of medium 1 is much larger than that of medium 2 (e.g., 5.0� 10�7

versus 1.29� 10�7m2=s), the walkers in medium 1 have higher probabilities to travelinto medium 2. Higher thermal resistance will more significantly hinder walkers tra-veling into medium 2 from medium 1, leading to a higher temperature of medium 1as well as a lower temperature of medium 2. Simulation results also exhibit sametrend as the prediction, as shown in Figure 10b.

Figure 9. Effects of thermal boundary resistances on the temperature of medium 1 and medium 2. (a)

Effect of Rbd m2�CNT ; (b) effect of Rbd m1�CNT ; and (c) effect of Rbd m1�m2. Normalized temperatures by

the lowest temperature are presented for simplicity.

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3.5. Potential Applications of the CNT Multiphase Model

Both SWNTs [31, 32]and MWNTs [33, 34] have been investigated widely as thelocal heating agents in cancer photothermal therapies. CNTs functionalized withfolic acid [8], anti-drugs [35] and DNA proteins [36] have shown no cytotoxicityto the biological systems both in vitro and in vivo, indicating potential clinical appli-cations of CNTs as local heating agents for cancer hyperthermia. The functionalizedCNTs can selectively attach to the cancer cells due to over expressed receptors on thecancer cell membranes [8], inducing selective destruction of the cancer cells under theillumination of near-infrared laser. Due to the high transparency of the biological

Figure 10. Effects of Rbd m1�m2 on the temperature of medium 1 and medium 2 using different thermal

diffusivity of medium 1 and medium 2. (a) 5.0� 10�7 versus 1.29� 10�7m2=s; and (b) 1.05� 10�7 versus

5.0� 10�7m2=s. Normalized temperatures by the lowest temperature are presented for simplicity.

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system (biofluids and cells) to near-infrared irradiation, it can be assumed that onlyCNTs absorb the main laser energy in a short irradiation time when CNTs areembedded in biological systems. In this case, the newly developed model can beapplied to investigate the cancer treatment using CNTs and near-infrared irradia-tions. The sphere depicting medium 2 can be viewed as a cancer cell, while the cubecan be viewed as biological fluid. CNTs can absorb laser energy and release thermalheat to the surrounding biological system (cancer cell and biofluids). Using thedeveloped CNT three-phase model, the effects of various parameters (CNT concen-trations, irradiation time and thermal boundary resistances) on the temperatures ofthe cancer cell and surrounding biological fluid can be quantified. The effective ther-mal conductivities of the biological systems containing CNTs also can be calculated.The CNTs three-phase model developed in this work can be applied to optimize theconditions of cancer treatment using CNTs and near-infrared lasers for the practicalclinical applications.

The CNT-stabilized Pickering emulsions have attracted much attention in thepast few years, due to their applications in food science, cosmetics, biomedicine andoil industry [37–39]. The physical properties of the emulsions need to be character-ized in detail for their large scale applications. As CNTs locate at the interfaces of theother two materials, obtaining same geometry with the CNT three-phase model, themodels may be applied to study the thermal properties of the solid-stabilized emul-sions, especially the effective thermal conductivity. By modifying the geometry of theCNTs cylinders, the model can also be used to study the thermal properties of othersolid-stabilized emulsions using nanoparticles. Moreover, multiphase polymerblends have similar structure to emulsions [40, 41].CNTs or nanoparticles localizeat the interfaces between two polymers to stabilize polymer droplets. The obtainedpolymer blends can take advantage of the attractive features of both polymers.The effects of different thermal properties of each component on the properties ofpolymer blends can be investigated by the newly developed model. With modifyingthe geometries of the sphere and CNTs, the model may be applied to predict thethermal properties of multiphase nanocomposites, for example, the nanocompositesmade of nanofibers, nanoparticles and polymer matrix [42].

4. CONCLUSION

An off-lattice Monte Carlo method is successfully developed to model the heattransfer mechanism in multiphase systems containing CNTs. Thermal boundaryresistances at the interfaces between any two mediums have been taken into accountby introducing a probability for random walkers travelling across the interfaces. Thevalidation results show that the random walker algorithm can successfully model theheat transfer of different heat sources in bulk mediums and the CNT multiphasemodel can be applied to predict the temperature distribution in CNT multiphase sys-tems. Thermal boundary resistances have different effects on the temperature of thecomponents. Higher Rbd m1�CNT induces lower temperature of medium 1 but highertemperature of medium 2. On the contrary, higher Rbd m2�CNT results in higher tem-perature of medium 1 but lower temperature of medium 2. Rbd m1�m2 does not exhi-bit apparent effect on the temperature of components due to the similar thermalproperties of the major components. The CNT multiphase model may be potentially

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applied to optimize the conditions of cancer treatment using CNTs and near-infraredirradiations and predict the effective thermal conductivities of emulsions, polymerblends and multiphase nanocomposites.

FUNDING

The authors would like to thank MOE Tier 1, R-265-000-361-133 for financialsupport. Feng Gong acknowledges the supercomputer calculation support fromHPC at the National University of Singapore.

REFERENCES

1. W. Yang, K. R. Ratinac, S. P. Ringer, P. Thordarson, J. J. Gooding, and F. Braet,Carbon Nanomaterials in Biosensors: Should You use Nanotubes or Graphene?,Angew. Chem. Int. Ed. Engl., vol. 49, no. 12, pp. 2114–2138, 2010.

2. K. Kostarelos, A. Bianco, and M. Prato, Promises, Facts and Challenges for CarbonNanotubes in Imaging and Therapeutics, Nat. Nanotechnol., vol. 4, no. 10, pp. 627–633, 2009.

3. R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Carbon Nanotubes — the RouteToward Applications, Science, vol. 297, no. 5582, pp. 787–792, 2002.

4. W. Wang, E. D. Laird, Y. Gogotsi, and C. Y. Li, Bending Single-Walled Carbon Nano-tubes Into Nanorings using a Pickering Emulsion-Based Process, Carbon, vol. 50, no. 5,pp. 1769–1775, 2012.

5. W. C. Chen, R. K. Wang, and K. J. Ziegler, Coating Individual Single-Walled CarbonNanotubes with Nylon 6,10 through Emulsion Polymerization, ACS Applied Materials& Interfaces, vol. 1, no. 8, pp. 1821–1826, 2009.

6. M. Shen and D. E. Resasco, Emulsions Stabilized by Carbon Nanotube-Silica Nano-hybrids, Langmuir, vol. 25, no. 18, pp. 10843–10851, 2009.

7. Y. Xu, W. E. Heberlein, M. Mahmood, A. I. Orza, A. Karmakar, T. Mustafa, A. R. Biris,D. Casciano, and A. S. Biris, Progress in Materials for Thermal Ablation of Cancer Cells,J. Mater. Chem., vol. 22, no. 38, pp. 20128, 2012.

8. N. W. S. Kam, M. O’Connell, J. A. Wisdom, and H. J. Dai, Carbon Nanotubes asMultifunctional Biological Transporters and Near-Infrared Agents for Selective CancerCell Destruction, Proc. Natl. Acad. Sci. U. S. A., vol. 102, no. 33, pp. 11600–11605, 2005.

9. C. Iancu and L. Mocan, Advances in Cancer Therapy through the Use of CarbonNanotube-Mediated Targeted Hyperthermia, Int. J. of Nanomedicine, vol. 6, no. pp. 1675–1684, 2011.

10. R. Scaffaro and A. Maio, Enhancing the Mechanical Performance of Polymer BasedNanocomposites by Plasma-Modification of Nanoparticles, Polym. Test., vol. 31, no. 7,pp. 889–894, 2012.

11. C. Zhang, W. W. Tjiu, T. X. Liu, W. Y. Lui, I. Y. Phang, and W. D. Zhang, DramaticallyEnhanced Mechanical Performance of Nylon-6 Magnetic Composites with Nanostruc-tured Hybrid One-Dimensional Carbon Nanotube-Two-Dimensional Clay NanoplateletHeterostructures, J. Phys. Chem. B, vol. 115, no. 13, pp. 3392–3399, 2011.

12. M. Raja, A. M. Shanmugharaj, S. H. Ryu, and J. Subha, Influence of Metal NanoparticleDecorated CNTs on Polyurethane Based Electro Active Shape Memory NanocompositeActuators, Mater. Chem. Phys., vol. 129, no. 3, pp. 925–931, 2011.

13. W.-H. Liao, H.-W. Tien, S.-T. Hsiao, S.-M. Li, Y.-S. Wang, Y.-L. Huang, S.-Y. Yang,C.-C. M. Ma, and Y.-F. Wu, Effects of Multiwalled Carbon Nanotubes Functionalization

HEAT TRANSFER THROUGH NANOTUBE MULTIPHASE SYSTEMS 1041

Dow

nloa

ded

by [

NU

S N

atio

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4

on the Morphology and Mechanical and Thermal Properties of Carbon Fiber=Vinyl EsterComposites, ACS Appl. Mater. Interfaces, vol. 5, no. 9, pp. 3975–3982, 2013.

14. H. M. Duong, N. Yamamoto, K. Bui, D. V. Papavassiliou, S. Maruyama, and B. L.Wardle, Morphology Effects on Nonisotropic Thermal Conduction of Aligned Single-Walled and Multi-Walled Carbon Nanotubes in Polymer Nanocomposites, J. Phys.Chem. C, vol. 114, no. 19, pp. 8851–8860, 2010.

15. N. Yamamoto, A. J. Hart, E. J. Garcia, S. S. Wicks, H. M. Duong, A. H. Slocum,and B. L. Wardle, High-Yield Growth and Morphology Control of Aligned CarbonNanotubes on Ceramic Fibers for Multifunctional Enhancement of Structural Composites,Carbon, vol. 47, no. 3, pp. 551–560, 2009.

16. K. Bui, B. P. Grady, and D. V. Papavassiliou, Heat Transfer in High Volume FractionCNT Nanocomposites: Effects of Inter-Nanotube Thermal Resistance, Chemical PhysicsLetters, vol. 508, no. 4–6, pp. 248–251, 2011.

17. S.Maruyama, Y. Igarashi, Y. Taniguchi, and J. Shiomi, AnisotropicHeat Transfer of Single-Walled Carbon Nanotubes, J. of Therm. Sci. and Tech., vol. 1, no. 2, pp. 138–148, 2006.

18. S. T. Huxtable, D. G. Cahill, S. Shenogin, L. P. Xue, R. Ozisik, P. Barone, M. Usrey,M. S. Strano, G. Siddons, M. Shim, and P. Keblinski, Interfacial Heat Flow in CarbonNanotube Suspensions, Nat. Mater., vol. 2, no. 11, pp. 731–734, 2003.

19. H. M. Duong, D. V. Papavassiliou, K. J. Mullen, B. L. Wardle, and S. Maruyama,Calculated Thermal Properties of Single-Walled Carbon Nanotube Suspensions, J. Phys.Chem. C, vol. 112, no. 50, pp. 19860–19865, 2008.

20. H. M. Duong, D. V. Papavassiliou, K. J. Mullen, and S. Maruyama, ComputationalModeling of the Thermal Conductivity of Single-Walled Carbon Nanotube–PolymerComposites, Nanotechnology, vol. 19, no. 6, art. no. 065702, 2008.

21. H. Y. Sul, H. Hong, and Y. T. Kang, Thermal Conductivity Measurement of Oil in Water(O=W) Nanoemulsion and Oil in H2O=LiBr (O=S) Binary Nanoemulsion for AbsorptionApplication, Int. J. of Refrigeration-Revue Internationale Du Froid, vol. 34, no. 6, pp. 1360–1365, 2011.

22. H. A. Patel, S. Garde, and P. Keblinski, Thermal Resistance of Nanoscopic Liquid-LiquidInterfaces: Dependence on Chemistry and Molecular Architecture, Nano Lett., vol. 5,no. 11, pp. 2225–2231, 2005.

23. H. M. Duong, D. V. Papavassiliou, K. J. Mullen, B. L. Wardle, and S. Maruyama,A Numerical Study on the Effective Thermal Conductivity of Biological Fluids ContainingSingle-Walled Carbon Nanotubes, Int. J. Heat Mass Transfer, vol. 52, no. 23–24,

pp. 5591–5597, 2009.24. H. M. Duong, N. Yamamoto, D. V. Papavassiliou, S. Maruyama, and B. L. Wardle,

Inter-Carbon Nanotube Contact in Thermal Transport of Controlled-MorphologyPolymer Nanocomposites, Nanotechnology, vol. 20, no. 15, art. no. 155702, 2009.

25. A. A. Balandin, Thermal Properties of Graphene and Nanostructured Carbon Materials,Nat. Mater., vol. 10, no. 8, pp. 569–581, 2011.

26. A. Einstein, The Electrodynamics of Moving Bodies, Annals of Physics, vol. 17, no. 549,pp. 891–921, 1905.

27. E. T. Swartz and R. O. Pohl, Thermal-Boundary Resistance, Reviews of Modern Physics,vol. 61, no. 3, pp. 605–668, 1989.

28. K. Bui, H. M. Duong, A. Striolo, and D. V. Papavassiliou, Effective Heat TransferProperties of Graphene Sheet Nanocomposites and Comparison to Carbon NanotubeNanocomposites, The J. of Physical Chem. C, vol. 115, no. 10, pp. 3872–3880, 2011.

29. K. Bui and D. V. Papavassiliou, Numerical Calculation of the Effective Thermal Conduc-tivity of Nanocomposites, Numer. Heat Transfer A, vol. 63, no. 8, pp. 590–603, 2013.

30. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford, ClarendonPress, New York, 1959.

1042 F. GONG ET AL.

Dow

nloa

ded

by [

NU

S N

atio

nal U

nive

rsity

of

Sing

apor

e] a

t 18:

54 1

7 M

arch

201

4

31. F. Zhou, D. Xing, Z. Ou, B. Wu, D. E. Resasco, and W. R. Chen, Cancer PhotothermalTherapy in the Near-Infrared Region by using Single-Walled Carbon Nanotubes,J. Biomed. Opt., vol. 14, no. 2, pp. 021009, 2009.

32. H. K. Moon, S. H. Lee, and H. C. Choi, In Vivo Near-Infrared Mediated TumorDestruction by Photothermal Effect of Carbon Nanotubes, ACS Nano, vol. 3, no. 11,pp. 3707–3713, 2009.

33. C. Iancu, L. Mocan, C. Bele, A. I. Orza, F. A. Tabaran, C. Catoi, R. Stiufiuc, A. Stir, C.Matea, D. Iancu, L. Agoston-Coldea, F. Zaharie, and T. Mocan, Enhanced LaserThermal Ablation for the In Vitro Treatment of Liver Cancer by Specific Delivery ofMultiwalled Carbon Nanotubes Functionalized with Human Serum Albumin, Int. J.Nanomedicine, vol. 6, pp. 129–141, 2011.

34. J. W. Fisher, S. Sarkar, C. F. Buchanan, C. S. Szot, J. Whitney, H. C. Hatcher, S. V.Torti, C. G. Rylander, and M. N. Rylander, Photothermal Response of Human andMurine Cancer Cells to Multiwalled Carbon Nanotubes after Laser Irradiation, CancerRes., vol. 70, no. 23, pp. 9855–9864, 2010.

35. N. Shao, S. Lu, E. Wickstrom, and B. Panchapakesan, Integrated Molecular Targeting ofIGF1R and HER2 Surface Receptors and Destruction of Breast Cancer Cells using SingleWall Carbon Nanotubes, Nanotechnology, vol. 18, no. 31, art. no. 315101, 2007.

36. S. Ghosh, S. Dutta, E. Gomes, D. Carroll, R. D’Agostino, J. Olson, M. Guthold, andW. H. Gmeiner, Increased Heating Efficiency and Selective Thermal Ablation of Malig-nant Tissue with DNA-Encased Multiwalled Carbon Nanotubes, ACS Nano, vol. 3,

no. 9, pp. 2667–2673, 2009.37. E. Dickinson, Food Emulsions and Foams: Stabilization by Particles, Curr. Opin. Colloid

Interface Sci., vol. 15, nos. 1–2, pp. 40–49, 2010.38. J. Frelichowska, M. A. Bolzinger, J. Pelletier, J. P. Valour, and Y. Chevalier, Topical

Delivery of Lipophilic Drugs from o=w Pickering Emulsions, Int. J. Pharm., vol. 371,nos. 1–2, pp. 56–63, 2009.

39. M. Chiesa, J. Garg, Y. T. Kang, and G. Chen, Thermal Conductivity and Viscosity ofWater-in-Oil Nanoemulsions, Colloids and Surfaces A—Physicochemical and Eng.Aspects, vol. 326, no. 1–2, pp. 67–72, 2008.

40. A. C. Baudouin, D. Auhl, F. F. Tao, J. Devaux, and C. Bailly, Polymer Blend EmulsionStabilization using Carbon Nanotubes Interfacial Confinement, Polymer, vol. 52, no. 1,pp. 149–156, 2011.

41. H. K. F. Cheng, T. Basu, N. G. Sahoo, L. Li, and S. H. Chan, Current Advances in theCarbon Nanotube=Thermotropic Main-Chain Liquid Crystalline Polymer Nanocompo-sites and Their Blends, Polymers, vol. 4, no. 2, pp. 889–912, 2012.

42. H. S. Xia, G. H. Qiu, and Q. Wang, Polymer=Carbon Nanotube Composite EmulsionPrepared through Ultrasonically Assisted In Situ Emulsion Polymerization, J. Appl.Polym. Sci., vol. 100, no. 4, pp. 3123–3130, 2006.

43. C. Sturesson and S. AnderssonEngels, A Mathematical Model for Predicting the Tem-perature Distribution in Laser-Induced Hyperthermia, Experimental Evaluation andApplications, Physics in Medicine and Biology, vol. 40, no. 12, pp. 2037–2052, 1995.

44. X. A. Xu, A. Meade, and Y. Bayazitoglu, Numerical Investigation of Nanoparticle-Assisted Laser-Induced Interstitial Thermotherapy Toward Tumor and Cancer Treat-ments, Lasers Med. Sci., vol. 26, no. 2, pp. 213–222, 2011.

45. G. H. Gao, T. Cagin, and W. A. Goddard, Energetics, Structure, Mechanical andVibrational Properties of Single-Walled Carbon Nanotubes, Nanotechnology, vol. 9,no. 3, pp. 184–191, 1998.

46. Y. Xiao, X. H. Yan, J. X. Cao, J. W. Ding, Y. L. Mao, and J. Xiang, Specific Heat andQuantized Thermal Conductance of Single-Walled Boron Nitride Nanotubes, Phys. Rev.B, vol. 69, no. 20, pp. 205415, 2004.

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