On the Influencing Factors and Strengthening Mechanism for Thermal

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On the Influencing Factors and Strengthening Mechanism for Thermal Conductivity of Nanofluids by Molecular Dynamics Simulation Wenzheng Cui,*,† Minli Bai,† Jizu Lv,*,† Guojie Li,† and Xiaojie Li‡ † ‡

School of Energy and Power Engineering, Dalian University of Technology, Dalian, China The State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, China

bS Supporting Information ABSTRACT: Compared with conventional single-phase working fluids, nanofluids possess significantly increased thermal conducting properties, but the mechanism for the increase still lacks perfect theory to explain. The aim of this study was to investigate effects of various influencing factors and determine the strengthening mechanism for thermal conductivity of nanofluids by Molecular Dynamics simulations. Thermal conductivities of nanofluids containing nanoparticles with different materials, volume concentrations, and shapes were calculated based on the Green
Kubo formula. Besides, Radical Distribution Function was first applied to nanofluids to analyze the microstructure. In addition, micromotions of nanoparticles were investigated thoroughly by statistical analysis. It was found for the first time that the effects of influencing factors for thermal conductivity of nanofluids can be explained and forecasted by comparing the proportion of energetic atoms containing in different nanoparticles. The changed microstructure of nanofluids is a newly discovered mechanism, and combined with the micromotions of nanoparticles they are proposed to be the main mechanism for strengthening thermal conductivity of nanofluids. The present work indicates that analyzing the proportion of energetic atoms can be an effective way for predicting thermal conducting properties of nanofluids and suggests the main mechanism for thermal conductivity increase which is the basis of developing new heat transfer theory for nanofluids.

1. INTRODUCTION Nanofluids1 technology is an advantageous method for strengthening heat transfer which denotes the evenly distributed, stably suspended, and high heat conducting binary mixtures composed of a small amount of metal or nonmetal nanoparticles and conventional heat transfer media such as water, alcohol, or oil, etc. Nanofluids show fantastic heat transfer properties that are much better than those of conventional fluids due to the small size effect of nanoparticles. For instance, adding a small amount of 0.5
2% solid nanoparticles in base fluids brings about 30
150% increase in thermal conductivity.2
6 Besides, the technique for nanofluids preparation has determined that precipitation of nanoparticles, or blocking and abrasion of equipment, are less likely to occur. Therefore, nanofluids have a promising future as substitutes for conventional heat transfer media that possess poor thermal conducting properties to meet the increasing heat transfer load in heat exchanging systems. However, the indefinite mechanism for enhancement of thermal conductivity has become an obstacle to the further development of nanofluids. One effective way to study the mechanism of significantly increased thermal conducting properties of nanofluids is with the Molecular Dynamics (MD) simulation.7,8 Because the conventional heat transfer theory aiming at solid
liquid binary fluid containing millimeter- or micrometer-sized particles has been demonstrated to be unsuitable for nanofluids,9,10 it is essential to investigate the influencing factors, the influence extent of these factors, and the strengthening mechanism for thermal conductivity, based on which the new theory for nanofluids could be proposed. Therefore, there have been a lot of references on this issue to discuss the influencing factors11
13 and propose possible r 2011 American Chemical Society

strengthening mechanisms.14
17 However, although various influencing factors were investigated experimentally or by MD simulation, little attention has been paid to making comparison of these influencing factors for some instructive conclusions. And few researchers have addressed the problem of the explanation for the different effects of these influencing factors. Besides, a lot of previous MD simulation studies on nanofluids have focused only on some superficial phenomena that might be explanation for the strengthening mechanisms, but lack further statistical analysis of these phenomena. The present work presents a series of MD simulations on thermal conductivity of nanofluids, examines effects of various influencing factors (including concentration, material, and shape of nanoparticles), and discusses the strengthening mechanism for thermal conductivity of nanofluids. The calculation of thermal conductivity is based on the Green
Kubo formula. Through statistical analysis, it is found for the first time that the effects of influencing factors for thermal conductivity of nanofluids can be explained by analyzing the proportion of energetic atoms contained in different nanoparticles. Besides, Radical Distribution Function is applied to nanofluids also for the first time, with which method the microstructure of nanofluids is statistically analyzed. In addition, the micromotions of nanoparticles are examined by tracing the trajectories, and statistical computation of instantaneous translational and rotational velocities. The combination of microstructural change and micromotions of Received: June 19, 2011 Accepted: October 21, 2011 Revised: October 18, 2011 Published: October 21, 2011 13568

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nanoparticles has been proposed to be the main mechanism for explaining the significantly increased thermal conductivity of nanofluids.

Table 1. LJ Parameters for Each Atom Pair atom pair

) σ (Å

2/fs2) ε (g Å

Ar
Ar

3.405

1.6540190  10
28

Cu
Cu

2.338

6.5581445  10
27

Fe
Fe

2.321

8.4330835  10
27

Ag
Ag

2.644

5.5240287  10
27

Ar
Cu

2.872

1.0415300  10
27

Ar
Fe

2.863

1.1810369  10
27

Ar
Ag

3.0245

9.5586860  10
28

2. SIMULATION METHOD AND MODELS 2.1. Simulation Method. There have been a few research

works reported on thermal conductivity of nanofluids utilizing MD method.7,8,15 In MD method, the thermal conductivity of nanofluids is obtained from the Green
Kubo formula18 which describes thermal conductivity through a time correlation function in the following form: k¼

1 Z 3V kB T 2

∞ 0

ÆJð0ÞJðtÞædt

ð1Þ

where k is the thermal conductivity, V is the volume, T is the temperature, kB is the Boltzmann constant, and the angular brackets denote the ensemble average. The microscopic heat flux J is given by   1 ð2Þ v j εi þ rij Fij vi JðtÞ ¼ 2 i i, j, i6¼ j

∑

∑

where vi is the velocity of particle i, rij is the distance between particles i and j, and Fij is the force experienced by a particle when it interacts with a neighboring particle. The site energy εi is given by 1 1 εi ¼ mi jvi j2 þ 2 2

∑j uij

ð3Þ

where mi is the mass of atom i, and uij is the potential function which will be described below. 2.2. Potential Function. To choose a suitable potential function is a crucial procedure to make sure the results are accurate and reliable in MD simulation.18 At present empirical or semiempirical correlations are adopted in most classic MD simulations. Among them, Lennard
Jones (LJ) potential function is the most commonly known one which is frequently used to describe the interactions between molecules. The LJ potential function (uij) is given by 2 ! !6 3 12 σ σ 5 ð4Þ
uij ¼ 4ε4 rij rij where rij is the interatomic distance between atoms i and j (rij = rj
ri), and ε and σ are parameters describing the bonding energy and bonding distance, respectively. The first term in the above equation on the right side represents the strong repulsion caused by inner electrons or ion overlap and the second one represents electrostatic interaction between dipoles. The models of nanofluids in this work consist of liquid argon and metallic nanoparticles. And the materials of metal nanoparticles involve copper, ferric, and silver atoms. The LJ parameters between homogeneous atoms can be found in literature19 and those between heterogeneous atoms are calculated according to Lorentz
Berthelot law of averages.20 All the LJ parameters concerned in present MD simulations are listed in Table 1. 2.3. Simulation Models. To study the effects of various influencing factors for thermal conductivity of nanofluids, the present work established a variety of nanofluid models for MD simulations. (1) To study the influence of nanoparticle materials,

Figure 1. Models for MD simulation of nanofluids.

three models of nanofluids with argon atoms as base fluid and one spherical nanoparticle comprised of copper, ferric, and silver atoms are established, respectively. Atoms in the metallic nanoparticles are in Face-Center-Cubic (FCC) lattice. The volume concentrations of nanoparticles in these models are approximately 1% and the diameters of nanoparticles are 1.8 nm. The argon atoms are also in FCC lattice, and therefore the side length of cubic simulation box is determined as 6.89 nm according to the lattice constant and the required volume concentration. (2) To study the influence of volume concentration of nanoparticles, aiming at Cu
Ar nanofluids, two models with volume concentrations of approximately 2% and 3% are established, respectively, to compare with Cu
Ar nanofluids with volume concentration of 1%. The lengths of simulation box in these two models are 5.17 and 4.59 nm, respectively. (3) To study the influence of nanoparticle shapes, aiming at Cu
Ar nanofluids with volume concentrations of 1%, 2%, and 3%, models of nanofluids with one long cylindrical nanoparticle are established respectively to compare with those with spherical nanoparticle. The amounts of copper atoms in these models are the same as those models containing spherical nanoparticles. The length of cylindrical nanoparticle is 7.1 nm and the diameter of underside is 0.92 nm. The nanofluid models in volume concentration of 1% with spherical and long cylindrical nanoparticles are shown in Figure 1a and b, respectively.

3. MD SIMULATION DETAILS Canonical ensemble is applied in all MD simulations, and the simulation temperature is fixed at 86 K controlled by a Nose
Hoover thermostat.21 The pressure is 1.013  105 Pa. All the simulation starting configurations are in FCC lattices. The starting orientation of each molecule is random orientation, and starting speed of each molecule is given by random sampling from Maxwell
Boltzmann distribution. Fundamental assumptions of two body effective potential approximation and minimum mirror standards are applied in the simulations. The size of simulation box is decided by density. Spherical truncated method is utilized with cut-off ratio of 2.5σAr, and interactions between 13569

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Figure 2. Thermal conductivities of nanofluids with different volume concentrations.

Figure 3. Thermal conductivities of nanofluids with different materials of nanoparticles.

molecules without truncation distance are adjusted by long-range correction method. Long-range electrostatic interactions are calculated by Ewald summation method.22 The geometric configuration is fixed by SHAKE algorithm and the kinetic equation is solved by 5-order Gear predictor-corrector method.23 The length of each time step is 2 fs in the simulations. For each simulation, a total simulation time of 4200 ps is performed with the first 200 ps as relaxation period. Data of the last 4000 ps are used for statistically calculating physical properties.

4. RESULTS AND DISCUSSION 4.1. Simulation Results for Thermal Conductivity of Nanofluids. To validate the effectiveness of algorithm and model,

before simulating thermal conductivity of nanofluids, the thermal conductivity of liquid argon at 86 K is simulated and calculated by MD method first. According to the report of Sarkar et al.,15 for argon when the total amount of atoms is more than 500, the simulation result will be in good agreement with experimental results; and for nanofluids when the total amount of atoms is more than 1372 the simulation result will be reliable. For more accurate data, the present work has established a relatively large simulation model containing 4631 liquid argon atoms. The result for thermal conductivity of argon at 86 K is 0.12796 W/m 3 K, which is in good agreement with the previous simulation result of 0.127 W/m 3 K15 and experimental result of 0.132 W/m 3 K.24 Compared with the experimental result, the error is 3.1% which is

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Figure 4. Thermal conductivities of nanofluids with different shapes of nanoparticles.

within acceptable limits, thus the present algorithm is reliable and can be used for further investigation of nanofluids. The thermal conductivities of nanofluids containing nanoparticles of the same material but different volume concentrations, of the same volume concentration but different materials, and of the same material and volume concentration but different shapes are calculated and compared by MD method. At first, the thermal conductivities of Cu
Ar nanofluids containing one spherical nanoparticle with volume concentrations of 1%, 2%, and 3% are calculated. Then the thermal conductivities of Fe- and Ag-Ar nanofluids containing one spherical nanoparticle with volume concentrations of 1% are calculated. And finally the thermal conductivities of Cu
Ar nanofluids containing one long cylindrical nanoparticle with volume concentrations of 1%, 2%, and 3% are calculated. The MD simulation results for thermal conductivities of nanofluids are shown in Figures 2, 3, and 4. The result for Cu
Ar nanofluids in volume concentration of 1% containing one spherical nanoparticle with diameter of 1.8 nm is 0.147 W/m 3 K. This result is close to previous simulation result of 0.144 W/m 3 K in literature7 for a similar simulation model with error of 2.1%, by which the effectiveness of the present algorithm and model are further verified. Through the analysis for simulation results, it could be found that all types of nanofluids have an increased thermal conductivity compared with that of base fluid. Among them the nanofluids containing long cylindrical nanoparticles have the maximum increase. For instance, the increase of 3% Cu
Ar nanofluids containing one long cylindrical nanoparticle is 57.03%. Figure 2 shows that in the condition of the same material and shape of nanoparticles, with the increase of volume concentration of nanoparticles, the thermal conductivity of nanofluids increases. Figure 3 shows that in the condition of the same volume concentration of 1%, when the material of nanoparticles possesses higher capability of thermal conducting, the thermal conductivity is greatly increased. For example, when the material of nanoparticles is ferric, the thermal conductivity is increased by 7.81%, whereas the increase is 40.63% when the material is silver. Figure 4 shows that with the same volume concentration and material, a long cylindrical nanoparticle makes more contribution than a spherical nanoparticle. For the same volume concentration of 1%, the increase of Cu
Ar nanofluids containing a spherical nanoparticle is 14.84% while the percentage of that containing a long cylindrical nanoparticle is 20.31%. 4.2. Effects of Influencing Factors for Thermal Conductivity of Nanofluids. It is found that statistically analyzing the 13570

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Figure 5. Proportions of energetic atoms in different nanoparticles.

proportion of energetic atoms contained in each type of nanoparticle is an effective way to explain the differences in simulation results for thermal conductivities of different types of nanofluids. The potential energies of atoms in cylindrical copper, spherical copper, spherical silver, and spherical ferric nanoparticles at simulation times of 3000, 3500, and 4000 ps are calculated and averaged respectively, and the distributions of atomic potential energy are statistically shown in Figure 5a
d. If a metallic atom is regarded as an energetic atom when its potential energy is greater than
7e-19 J, the proportions of energetic atoms in these four types of nanoparticles are 38.96%, 32.79%, 61.78%, and 1.4%, respectively. It can be found that a long cylindrical copper nanoparticle contains more energetic atoms than a spherical copper nanoparticle. The reason is because a cylindrical nanoparticle has a greater specific surface area than that of a spherical one and the potential energy of atoms on the surface is higher than that of internal atoms. In addition, higher thermal conducting materials possess a greater proportion of energetic atoms. For instance, the proportion of energetic atoms in a spherical silver nanoparticle is significantly greater than that of a spherical ferric nanoparticle. Furthermore, changing the shapes of nanoparticles to enlarge the specific surface area will increase the proportion of energetic atoms, however, the proportion of energetic atoms in a nanoparticle made of high thermal conducting material is still significantly greater than that made of low thermal conducting material, even though the shape of the latter has been changed to have a greater specific surface area. For example, the proportion of energetic atoms in a spherical silver nanoparticle is still significantly greater than that of a cylindrical copper nanoparticle. Therefore the proportion of energetic atoms in nanoparticles is a real usable criterion to help guide the understanding of the differences in simulation results and to explain the effects of influencing factors for thermal conductivities of different types of nanofluids.

Nanoparticles with a higher proportion of energetic atoms will contribute more for increasing thermal conductivity of nanofluids. The results of MD simulations have reminded that, during the preparation of nanofluids, adopting high thermal conducting material and in the meanwhile preparing the shape of nanoparticles as elongated linear or needle-like that has greater specific surface area will maximize the thermal conductivity of nanofluids. 4.3. Mechanism for Strengthening Thermal Conductivity of Nanofluids. 4.3.1. Microstructure of Nanofluids. Radical Distribution Function (RDF) is used to characterize the degree of structure disorder for fluids or noncrystalline solids. The physical significance of this function is the average number of particles in the distance between r and r + Δr from a certain particle at the center. It is written as gðrÞ ¼

V nðr, ΔrÞ N 4πr 2

ð5Þ

where V is the volume, N is the number of atoms, r is the distance from a certain particle, and n(r, Δr) is the number of particles in the distance between r and r + Δr. For a multicomponent system (containing more than one kind of particle), the RDF between two kinds of particles or called Pair Correlation Function could be written as  Nα  niβ ðr, ΔrÞ V gαβðrÞ ¼ ð6Þ Nα Nβ i ¼ 1 4πr 2 Δr

∑

where the subscripts α and β identify the type of particles. niβ(r, Δr) is the number of particles β in the distance between r and r + Δr with a certain particle α at the center. For nanofluid containing a spherical copper nanoparticle, the RDF for atom pairs of Ar
Ar, Cu
Cu, Cu
Ar, and the overall RDF for nanofluid are shown in Figure 6a
d. In Figure 6a, the 13571

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Figure 6. RDF of Cu
Ar nanofluids.

first peak in the RDF of Ar
Ar atom pair appears at 3.85, and the first trough appears at 5.2, then the curve oscillates and converges to 1 which demonstrates the typical short-range order structure feature of liquids. In Figure 6b, the ordered RDF of Cu
Cu atom pair shows the typical long-range order structure of metal. Figure 6c shows the RDF for Cu
Ar atom pair, compared with that of Ar
Ar atom pair shown in Figure 6a, the first peak reduces and in the position of the first trough in Figure 6a there appears a new peak which has a phenomenon of split. Figure 6d shows the overall RDF of Cu
Ar nanofluid which demonstrates the feature of a fluid doped by metal. Compared with Figure 6a, at 2.4 before the first peak in Figure 6a there appears a new peak; the primary peak at 3.85 has increased slightly; between the first peak and the first trough there appears a new peak; at the position of the primary trough in Figure 6a there appears another new peak; and at the position of original second peak there appears a new peak which makes the peak value increase obviously. All the positions of new peaks appeared in Figure 6d are corresponding

to the peaks in Figure 6b. These phenomena indicate that due to the adding of nanoparticles, nanofluids show a long-range order structure feature which is similar to that of a solid. Because nanofluids have microstructures similar to solids the thermal conductivity increases. It is generally acknowledged that the mechanism of heat conducting through liquids is analogous to that of gases in qualitative which is mainly attributed to irregular thermal movements of molecules, while the mechanism of heat conducting in solid depends largely on lattice vibration which refers to the vibration of atoms or molecules near their balance positions. The differences in mechanism of heat conducting have determined that the thermal conductivity of solid is generally greater than that of liquid with several orders of magnitude. Through analysis of RDF for nanofluids, it could be concluded that due to the adding of nanoparticles, the base fluid has been transformed to a fluid that is similar to solid in microstructures which makes its capability of heat conducting dramatically increased. 13572

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Industrial & Engineering Chemistry Research The MD simulation results by Yu et al.25 and Xue et al.26 have proved that the liquid atoms of base fluid near a nanoparticle will be absorbed to the solid surface of the nanoparticle and form an absorption layer with a thickness of several atomic distances. Through continuous observation in the process of MD simulations liquid atoms in this layer are found to be evenly distributed and will move together with the nanoparticles. In real nanofluids, due to the small scale of nanoparticles, a low particle concentration will make the base fluid crawl with nanoparticles. These nanoparticles absorb liquid atoms around them and change them from the state of random and chaotic into uniform arrangement which is similar to solid. When there are plenty of nanoparticles at work, the microstructure of holistic nanofluids will present as a long-range order feature which has been proved above. The RDF

Figure 7. Trajectories of nanoparticles.

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of nanofluids has shown that the modification function of nanoparticles for base fluid is one of the key factors for the significantly increased thermal conductivity of nanofluids. 4.3.2. Micromotions of Nanoparticles. By tracking copper nanoparticles in MD simulations, it is found that the solid nanoparticles move and rotate due to the bombardment of fluid molecules. In Figure 7 the trajectories of nanoparticles in different shapes from 200 ps (t1) to 4200 ps (t2) are shown and the trajectories vividly describe the micromotions of nanoparticles. The initial position at t1, trajectory during 4000 ps, and final position at t2 of nanoparticles are indicated in red, green, and ice blue. It is found that the phenomenon of micromotions of nanoparticles is obvious, and the rotation of cylindrical nanoparticle is especially clear for its geometrical structure. Furthermore, by practical measuring it is found that the cylindrical nanoparticle adjusts its size due to the obviously larger interatomic forces in the height direction. The bottom diameter of cylindrical nanoparticle increases to 1.11 nm and the height shortens to 5.05 nm, while the diameter of spherical nanoparticle has no evident change. For clearer understanding of the micromotions of nanoparticles, aiming at nanoparticles in different shapes, the instantaneous translational velocities along, and rotational linear velocities around, three directions are statistically analyzed, as shown in Figures 8 and 9. From the figures it can be seen that the micromotions of nanoparticles are random and high-speed. By comparison, the

Figure 8. Translational velocities of nanoparticles.

Figure 9. Rotational linear velocities of nanoparticles. 13573

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translational velocity components of spherical nanoparticles are larger than those of cylindrical nanoparticle, which is mainly because of the smooth shape. And by comparison of the rotational linear velocity components of nanoparticles, it is found that the cylindrical shape of nanoparticle contributes to rotating. Compared with displacement, the rotation of nanoparticles is believed to be more conducive to accelerating micro convection in base fluid due to the stirring effect. The micromotions of nanoparticles will enhance relative motion between nanoparticles and base fluid, which trigger micro convection in local region of fluid. The effect of micro convection will strengthen energy transfer between nanoparticles and liquid, and will increase the thermal conductivity of nanofluids. Therefore, the micromotion of nanoparticles is another key factor for the significantly increased thermal conductivity of nanofluids.

transfer theory, the new found mechanism is proposed to be one of the key factors for the significantly increased thermal conductivity of nanofluids. In addition, by tracing the trajectories, and statistically analyzing instantaneous translational and rotation velocities, the micromotion behaviors of nanoparticles have been thoroughly studied. The micromotions of nanoparticles trigger micro convection in base fluid and will strengthen energy transfer between nanoparticles and liquid. Therefore, it is another key factor for the significantly increased thermal conductivity of nanofluids. In generally, the combination of changed microstructure of nanofluids and micromotions of nanoparticles is the main mechanism for explaining the significantly increased thermal conductivity of nanofluids.

’ ASSOCIATED CONTENT

5. CONCLUSIONS On the basis of MD simulations on influencing factors (including concentration, material, and shape of nanoparticles) for thermal conductivity of nanofluids, we have examined the effects of these influencing factors and presented for the first time that the effects can be forecasted by statistically analyzing the proportion of energetic atoms containing in disparate nanoparticles. Through applying RDF analysis for this new kind of fluids we have first found and proposed the changed microstructure of nanofluids as a new mechanism for explaining the significantly increased thermal conductivity. And by statistically analyzing the micromotions of nanoparticles during MD simulation, we have further developed the microscopic mechanism of the increased thermal conductivity of nanofluids. The combination of changed microstructure of nanofluids and micromotions of nanoparticles has been demonstrated to be the main mechanism for explaining the significantly increased thermal conductivity of nanofluids. The main conclusions of this paper are as follows: 1 All kinds of nanofluids will contribute to thermal conducting properties of base fluid. With the same material of nanoparticles, the thermal conductivity of nanofluids increases with increasing the volume concentration of nanoparticles. With the same concentration, adopting high thermal conducting material will greatly improve thermal conductivity of nanofluids. With the same material and concentration, nanoparticles in shapes with greater specific surface area will further contribute to increase of thermal conductivity. 2 It is found that the effect of various influencing factors can be examined and compared by statistically analyzing the proportion of energetic atoms in disparate nanoparticles. With this method the material of nanoparticles is found to be the main influencing factor for thermal conductivity. In addition, changing the shape of nanoparticles from sphere to long cylinder, which is actually increasing the specific surface area, will also increase the percentage of energetic atoms. Therefore, the present work has proposed that with appropriate high concentration of nanoparticles, adopting high thermal conducting material combining with manufacturing nanoparticles into shapes with high specific surface area will maximize the thermal conductivity of nanofluids. 3 Through applying RDF analysis of nanofluids, it is found for the first time that due to the adding of nanoparticles the microstructure of nanofluids changes and shows long-range order structure which is similar to solid. Based on heat

bS

Supporting Information. Parameters of LJ potential for various metals listed in Table S1; MD simulation data of translational and angular velocity components of nanoparticles with different shapes in Tables S2 and S3, respectively. This material is available free of charge via the Internet at http://pubs. acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel/Fax: +86-411-84706305. E-mail: [email protected]. edu.cn or [email protected].

’ ACKNOWLEDGMENT The support of the National Natural Science Foundation of China (50576008, 50876016, and 51006015) and China Postdoctoral Science Foundation (20100470070) is gratefully acknowledged. ’ REFERENCES (1) Choi, U. S. Enhancing Thermal Conductivity of Fluids with Nanoparticles. In Developments and Applications of Non-Newtonian Flows, San Francisco, CA, Nov 12
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