Effective thermal conductivity in nanofluids of nonspherical particles withinterfacial thermal resistance: Differential effective medium theoryXiao Feng Zhou and Lei Gao Citation: J. Appl. Phys. 100, 024913 (2006); doi: 10.1063/1.2216874 View online: http://dx.doi.org/10.1063/1.2216874 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v100/i2 Published by the American Institute of Physics. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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JOURNAL OF APPLIED PHYSICS 100, 024913 �2006�

Effective thermal conductivity in nanofluids of nonspherical particleswith interfacial thermal resistance: Differential effective medium theory

Xiao Feng ZhouDepartment of Physics, Suzhou University, Suzhou 215006, China

Lei Gaoa�

CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China and Department of Physics,Suzhou University, Suzhou 215006, China

�Received 24 November 2005; accepted 18 May 2006; published online 27 July 2006�

By taking into account the interfacial thermal resistance across the solid particles and the hostliquids, we present differential effective medium theory to estimate the effective thermalconductivity in nanofluids of nonspherical solid particles. It is found that high enhancement ofeffective thermal conductivity can be achieved when the nanoparticles’ shape is deviated much fromthe spherical one. On the other hand, increasing the interfacial thermal resistance results in anappreciable degradation in the thermal conductivity enhancement. To one’s interest, our theoreticalresults are in good agreement with recent experimental data on nanofluids. In particular, ourtheoretical predictions successfully show the nonlinear dependence of effective thermal conductivityon the volume fractions of nanotubes. © 2006 American Institute of Physics.�DOI: 10.1063/1.2216874�

I. INTRODUCTION

Nanofluids, a class of solid/liquid suspensions, are likelyto be the future heat transfer media in many industrial sectorssuch as power generation, micromanufacturing, and chemi-cal and metallurgical industries. Nanofluids are also impor-tant for the production of nanostructured materials,1,2 for en-gineering of complex fluids,3 as well as for enhancement ofwetting and spreading behavior.4 Especially, the develop-ment of nanofluids based on carbon nanotubes is a promisingdirection in nanotechnology, including nanoprobes,5 fieldelectron emitters,6 nanotweezers,7 nanobearings,8 and so on.

Nanofluids have attracted great interest recently becauseof their enhanced thermal properties.9–16 Actually, the addi-tion of very small amount of nanoparticles in liquids substan-tially results in large enhancement of conductivity with no orlittle effect on the mechanical �hydrodynamic� properties ofliquid suspensions. Nanofluids, such as carbon nanotube dis-persed in oil �decene� �Refs. 10 and 15� and Al2O3 nanopar-ticles in water �ethylene glycol �EG��,16 were reported topossess large enhancement of the inherent poor thermal con-ductivity of liquids. In addition, Choi et al.10 found a non-linear dependence of the effective thermal conductivity onthe nanotube volume fraction.

Following those experimental studies, many modelshave been proposed for studying the anomalous enhance-ment of effective thermal conductivity. The conventionalmodel of effective thermal conductivity of two-phase com-posites was briefly summarized by Choi et al.10 However, theexperimental predictions for the enhanced thermal conduc-tivities of nanofluids are much higher than those predicted bythe traditional models. To account for the anomalous en-hancement, the convection caused by the Brownian motion

a�

Electronic mail: lgaophys@pub.sz.jsinfo.net

0021-8979/2006/100�2�/024913/6/$23.00 100, 0249

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of nanoparticles was proposed as one of the physicalmechanisms.17–19 On the other hand, other models such asthe stationary and moving particle model20 and layering ofliquid molecules at the particle-liquid interface21–23 were alsoproposed to investigate the thermal properties of nanofluids.

It is known that, as dispersed in nanofluids, nanoparticleshave unique solid/liquid interface. Without taking into ac-count the interfacial effects and the particles’ shape, tradi-tional models valid for spherical inclusions cannot be used topredict the effective thermal conductivity of nanofluids. Inthis connection, Nan et al. generalized the Maxwell-Garnettapproximation to derive a quite simple formula for the effec-tive thermal conductivity of carbon nanotube-based compos-ites. The magnitude of large thermal conductivity enhance-ment observed in experiments was well predicted within themodel of Nan et al.24,25 However, the model cannot explainthe interesting phenomenon that the effective conductivity ofnanotube suspensions is nonlinear with nanotube volumefractions. In fact, in nanofluids, both the geometric aniso-tropy and the interfacial effects play important roles in en-hancing the thermal properties of nanofluids. The geometricanisotropy results from the large aspect ratio of granular in-clusions, while the physical anisotropy originates from theinterfacial thermal resistance.25 In this paper, by taking intoaccount both the geometric anisotropy and the interfacial ef-fects, we present differential effective medium theory to in-vestigate the effective thermal conductivity of nanofluids.For randomly isotropic spherical inclusions without interfa-cial resistance, our formula degenerates to the well-knowndifferential one proposed by Bruggeman.26 We aim at study-ing the dependence of thermal behavior on the nanoparticles’shape and interfacial resistance. To verify the validity of ourformula, we shall show that our theoretical results are ingood agreement with those of experimental reports. To one’s

interest, even for extremely low volume fractions, our differ-

© 2006 American Institute of Physics13-1

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024913-2 X. F. Zhou and L. Gao J. Appl. Phys. 100, 024913 �2006�

ential model predicts the nonlinear relation between the ef-fective thermal conductivity and the volume fraction suc-cessfully.

II. THEORETICAL CONSIDERATIONS

Now let us consider the composites in which randomlyoriented ellipsoidal nanoparticles with low volume fractionsf and the thermal conductivity Kp are embedded in a liquidwith the thermal conductivity Km. In order to take into ac-count the interfacial effects, we assume that the ellipsoidalnanoparticles are coated with layers of thickness � and con-ductivity Ks. Without loss of generality, the interfacial ther-mal resistance is assumed to be concentrated on a surface ofzero thickness defined as

RBd = lim�→0,Ks→0

��/Ks� .

Due to the interfacial thermal barrier, the thermal conductiv-ity of coated ellipsoidal particles along the i �i=x ,y ,z� sym-metric axis must be anisotropic and are written as27

Ki =Kp

1 + QRBdLiKp=

Kp

1 + ULiKp, �1�

where Q= �ab+ac+bc� / �abc� and U=QRBd, a, b, and c, arethe radii of an ellipsoid particle along the x, y, and z axes,respectively, and the depolarization factors Li are dependenton the particles shape given by

Lx =abc

2�

0

+� ds

�s + a2�R�s�, Ly =

abc

2�

0

+� ds

�s + b2�R�s�,

�2�

Lz =abc

2�

0

+� ds

�s + c2�R�s�,

where R�s����s+a2��s+b2��s+c2�.Since the embedded ellipsoidal particles are randomly

oriented, the whole nanofluids will be isotropic.28,29 Accord-ing to the traditional Maxwell-Garnett theory,31 the effectivethermal conductivity Ke of composites can be expressed as

Ke = Km�1 +f

3��x + �y + �z� , �3�

where

�i =Kp − Km − ULiKpKm

Km + Li�UKpKm + Kp − Km� − ULi2KpKm

,

�i = x,y,z� .

Starting with a homogeneous host material, we calculatethe change in Ke from Ke=Km at f =0 to Ke+�Ke at �f ,

�Ke = Km�x + �y + �z

3�f . �4�

To carry out further iterations, we simply reply Km by Ke ofthe homogenized composites and �f with �f / �1− f� due tothe overlap effect.32 We arrive at the final differential equa-

tion

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dKe

df=

Ke

3�1 − f�

��i

Kp − Ke − ULiKpKe

Ke + Li�UKpKe + Kp − Ke� − ULi2KpKe

.

�5�

Integrating the above equation and imposing the initial con-dition that Ke=Km at f =0, we obtain

1 − f = �Km

Ke�3b/a�Km + m

Ke + m�3c/a�Km + n

Ke + n�3d/a�Km + p

Ke + p�3e/a

,

�6�

where parameter a is given by

a = − 3�1 + LxLy + LyLz + LzLx� i=x

z

�1 + UKpLi� ,

while parameters m, n, and p are three solutions of the fol-lowing nonlinear equation for Ke:

i�j�k,i=x

z

��Kp − Ke − ULiKpKe�

��Ke + Lj�UKpKe + Kp − Ke� − ULj2KpKe�

� �Ke + Lk�UKpKe + Kp − Ke� − ULk2KpKe�� = 0.

As a result, b, c, d, and e are determined by

b + c + d + e = i

�1 − Li��1 + UKpLi� ,

bmnp = Kp3LxLyLz,

mn�b + e� + mp�b + d� + np�b + c�

= Kp2�1

2 i�j�k,i=x

z

�LiLj + Li2LjLk� + �3U − 3�

i

Li ,

m�b + d + e� + n�c + e + b� + p�b + c + d�

= Kp�1 − i�j

LiLj + 3��U − 1�2 − U2Kp2�

i

Li�+ UKp

2 i�j�k,i=x

z

LiLj�1 + LiLk − Li − UKpLiLk� .

Equation �6� is our differential effective medium theory,which is rather complicated to investigate the thermal behav-ior in nanofluids containing spherical and nonspherical inclu-sions. In what follows, for simplicity, we assume the granularinclusions to be spheroidal in shape. Such assumption isquite reasonable for carbon nanotubes composites.21,24 Forspheroidal particles with a=b and c �for P=c /a�1 or��1�, we have prolate or �oblate� spheroid�, then according

to Eq. �2�, one yields

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024913-3 X. F. Zhou and L. Gao J. Appl. Phys. 100, 024913 �2006�

Lx = Ly =�P

4�P2 − 1�3/2�2P�P2 − 1 + lnP − �P2 − 1

P + �P2 − 1� for P � 1

P

4�1 − P2�3/2�� − 2P�1 − P2 − 2 arctanP

�1 − P2� for P � 1,�Lz = 1 − 2Lx. �7�

Substituting Eq. �7� into Eq. �6�, we can simplify Eq. �6� as

1 − f = �Km

Ke�3A/D�Km + S+

Ke + S+�3C+�Km + S−

Ke + S−�3C−

, �8�

where

D = �1 + 3Lx��− 1 + Kp�Lx − 1�U + Kp2Lx�2Lx − 1�U2� ,

S ± =Kp�− 1 − 3KpLx

2U + 2Lx�3 + KpU�� ± E

2�1 + 3Lx��Kp2Lx�2Lx − 1�U2 − 1 + Kp�Lx − 1�U�

,

A =Kp

2�2Lx − 1�Lx

�S+S−�,

C± = ±2LxS± − 2Lx

2S± + AS ± F�S±�D�S+ − S−�

,

with

E = �9 + 2Kp�4 − 9Lx2�U + Kp

2Lx�8 − 72Lx2 + 81Lx

3�U2,

F�x� = ± Kp2LxU�− 1 + 2Lx��1 + 2Lx − 2LxxU + 2Lx

2xU�

± Kp�− 1 + 2Lx2xU − 4Lx

2�1 + xU� + Lx�3 + 2xU�� .

For numerical calculations, we would like to introduce adimensionless thermal barrier resistance factor , defined as

= �RBd

aKm for P � 1

RBd

cKm for P � 1,�

then U�ARBd in Eq. �8� is replaced with

U = ��2 + 1/P� for P � 1

�1 + 2P� for P � 1�

Here we would like to mention that =0 corresponds to theperfect interface at which no interfacial thermal resistanceexists and =� corresponds to the limit of completely insu-lating interface. In particular, for spherical inclusions with nointerfacial thermal resistance, i.e., P=1 and =0, one ob-tains

1 − f =Kp − Ke

Kp − Km�Km

Ke�1/3

. �9�

Equation �9� is the traditional differential effective mediumtheory.26 In comparison, the work of Nan et al.24,27 reduces

30

to the Maxwell-Garnett formula

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Ke − Km

Ke + 2Km= f

Kp − Km

Kp + 2Km. �10�

III. NUMERICAL RESULTS

In Fig. 1, we apply our theory to analyze the dependenceof effective thermal conductivity on the nanoparticles’ aspectratio. The enhancement Ke /Km is plotted as a function of theaspect ratio P at f =0.05 and Kp /Km=500 with perfect inter-face between the matrix and inclusions �=0�. It is evidentthat for both oblate and prolate particles, Ke /Km increasessignificantly with increasing the anisotropy of the inclusionshape. However, for prolate granules �P�1�, the tendency ofthe enhancement Ke /Km gradually reaches a calm value withfurther increasing the aspect ratio. This can be well under-stood that when the inclusions possess large aspect ratio�prolate� or small aspect ratio �oblate�, they can more easilyform a path for heat flow through the composites, which isnot provided by the spherical inclusions �P=1�. Therefore,nonspherical shape especially disk shape of the inclusions isreally helpful to achieve appreciable enhancement of thethermal conductivity.

In Fig. 2, we investigate the effective thermal conductiv-ity of nanofluids of prolate spheroidal particles with aspectratio P=100 �see Fig. 2�a�� and of oblate spheroidal particleswith P=0.01 �see Fig. 2�b��. Again, for nonspherical inclu-sions, small volume fraction of inclusions results in largeenhancement of effective thermal conductivity. Moreover,the thermal conductivity enhancement induced by the prolate�oblate� nanoparticles without thermal interface resistancewould be much larger than the one with interface thermal

FIG. 1. The effective thermal conductivity enhancement Ke /Km as a func-

tion of aspect ratio for the volume fraction f =0.05, Kp /Km=500, and =0.

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024913-4 X. F. Zhou and L. Gao J. Appl. Phys. 100, 024913 �2006�

resistance. Actually, large thermal resistance results in an en-ergy loss and subsequently leads to a degradation in the ef-fective thermal conductivity.

When the nanoparticles are spherical in shape �P=1�, westudy the effect of interfacial thermal barrier resistance onthe effective thermal conductivity of nanofluids, as shown inFig. 3. We find that the interfacial thermal barrier resistancealways diminishes the enhancement of the effective thermalconductivity. As �1 �=0 corresponds to no thermal re-sistance�, the enhancement of Ke /Km obviously increaseswith increasing the volume fraction of the inclusions. Inter-estingly, the effective thermal conductivity of the nanofluids

FIG. 2. Ke /Km as a function of f for various thermal resistance factor . Theaspect ratio P is chosen to be P=100 in �a� for prolate particles and P=0.01 for oblate particles in �b�.

FIG. 3. The same as in Fig. 2 but for spherical nanoparticles.

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holds the constant for =1, which indicates that the contri-bution of the high thermal conductivity is entirely balancedby the effect of the interfacial resistance. On the contrary, for�1, the effective thermal conductivity became lower thanthat of the matrix, and the effective thermal conductivitydecreased with increasing the volume fraction of the spheri-cal inclusions.

In Figs. 4 and 5, we would like to compare our theoret-ical results given by Eq. �8� with recent experimental resultson nanotube/oil10 and nanotube/decene15 nanofluids, respec-tively. For numerical calculations, the thermal conductivityof oil, decene, and carbon nanotubes are taken to be 0.1448,0.14, and 1000 W/m K, respectively, as adopted in otherworks.10,15 In addition, for carbon nanotubes, since the aspectratio P is always large, Lx is estimated as 0.4986. It is foundthat our theoretical results are in good agreement with

FIG. 4. The enhancement of effective thermal conductivity Ke /Km as afunction of the volume fraction f in nanotube/oil nanofluids with Kp

=1000 W/m K �Ref. 24� and Km=0.1448 W/m K �Ref. 10�.

FIG. 5. The same as in Fig. 4 but for nanotube/decene nanofluids with Kp

=1000 W/m K �Ref. 24� and Km=0.14 W/m K �Ref. 15�.

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024913-5 X. F. Zhou and L. Gao J. Appl. Phys. 100, 024913 �2006�

nanotube/oil experimental data for =5.792�10−2, corre-sponding to the interfacial thermal resistance RBd=1�10−8 m2 K/W. This estimated RBd value is of the sameorder as those reported in carbon nanotubes matrix.33 Toone’s interest, in comparison with other theoreticalworks,23,25 our model is able to predict the nonlinear depen-dence of effective thermal conductivity on the volume frac-tion of nanotube/oil nanofluids. For nanotube/decene nanof-luids �see Fig. 5�, the theoretical predictions provide the bestfit for nanotube/decene experimental data when we choosethe dimensionless thermal resistance =2.352�10−2 �orRBd=4.2�10−7 m2 K/W�. We note that the thermal conduc-tivity enhancement for nanotube/oil nanofluids �Km

=0.1448 W/m K� with volume fraction f =0.01 is 160%,much higher than the 19.6% enhancement in nanotube/decene nanofluids �Km=0.14 W/m K� with the same volumefraction, although the thermal conductivity of host medium isalmost the same. Actually, the interfacial thermal resistanceis strongly dependent on the type of bonding between thesolid and the liquid.34 Therefore, a large discrepancy in thethermal conductivity enhancement in nanotube/oil andnanotube/decene systems may result from the different inter-facial thermal resistances.

Furthermore, we compare the predictions of our differ-ential effective medium theory with the experimental data ofnanofluids consisting of spherical Al2O3 nanoparticles in wa-ter and ethylene glycol �EG�.16 Numerical results are shownin Fig. 6. For comparison, the results for single-sphereBrownian model19 �BM� are also shown. In contrast tosingle-sphere Brownian model, we find that our differentialeffective medium theory predicts the tendency of the en-hancement in thermal conductivity relatively well. To fit theexperimental data we choose RBd=0.77�10−8 m2 K/W �=0.121� �for Al2O3/water� and RBd=1�10−9 m2 K/W �=0.067� �for Al2O3/EG�, which have the same magnitude asreported in Refs. 19 and 35.

IV. CONCLUSIONS

In summary, by taking into account the geometric aniso-

FIG. 6. The same as in Fig. 4 but for Al2O3/water and Al2O3/EG nanofluids;parameters are Kp=46 W/m K, Km=0.604 W/m K for water, and Km

=0.258 W/m K for EG.

tropy and the interfacial effects, we present a differential

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effective medium approximation to predict the effective ther-mal conductivity of nanofluids. We numerically investigatethe effects of aspect ratio P and the effects of interfacialthermal resistance on the effective thermal conductivity ofthe system. It is found that the adjustment of the particleshape is really helpful to achieve appreciable enhancement ofeffective thermal conductivity. On the other hand, the inter-facial thermal resistance always diminishes the enhancementof effective thermal conductivity. Furthermore, our theoreti-cal results on the effective thermal conductivity of nanotube/oil, nanotube/decene nanofluids, and Al2O3/water �or EG�nanofluids are in good agreement with the experimental data.To one’s interest, our model is able to predict the nonlineardependence of effective thermal conductivity on the volumefraction of nanotube/oil nanofluids.

Some comments are in order. In nanotube/oil nanofluids,a significant nonlinear behavior was observed at very lowvolume fractions, with the conductivity-volume fractioncurve exhibiting an upward-concave dependence.10 By con-trast, in nanotube/water nanofluids,36 a negative curvaturewas predicted. The interfacial resistance and uncertainty inthe effective conductivity of carbon nanotubes are proposedto be the main reason. In our present paper, by taking intoaccount the interfacial resistance, we successfully explain thepositive curvature within our differential effective mediummodel. However, the negative curvature needs our furthercareful examination. In addition, experimental studies ofelectrical conductivity37 reveal a small percolation thresholdfor composites and no thermal percolation threshold exists.Since the thermal conductivity and electric conductivity aredescribed by the same continuum equation. It would be ofgreat interest to develop the general theory to explain suchcontrasting phenomena.38 Our initial numerical results showthat the Bruggeman effective medium theory can be gener-alized to explain both the existence of percolation thresholdin the electric conductivity and the lack of thermal percola-tion threshold. Furthermore, to get deep understanding ofnanofluids, the effects of Brownian motion, networks, andthe interactions induced by the fillings may be taken intoaccount simultaneously especially for nanoparticles withsmall sizes.

ACKNOWLEDGMENTS

This work was supported by the National Natural Sci-ence Foundation of China under Grant No. 10204017 and theNatural Science of Jiangsu Province under Grant No.BK2002038 �for one of the authors �L.G.��.

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