Molecular dynamics simulation of effective thermal conductivity and study of enhanced thermal transport mechanism in nanofluids Suranjan Sarkar and R. Panneer Selvam
Citation: J. Appl. Phys. 102, 074302 (2007); doi: 10.1063/1.2785009 View online: http://dx.doi.org/10.1063/1.2785009 View Table of Contents: http://aip.scitation.org/toc/jap/102/7 Published by the American Institute of Physics
JOURNAL OF APPLIED PHYSICS 102, 074302 共2007兲
Molecular dynamics simulation of effective thermal conductivity and study of enhanced thermal transport mechanism in nanofluids Suranjan Sarkara兲 and R. Panneer Selvam Computational Mechanics and Nanotechnology Modeling Laboratory, Bell 4190, University of Arkansas, Fayetteville, Arkansas 72701, USA
共Received 11 December 2006; accepted 5 August 2007; published online 1 October 2007兲 Nanofluids have been proposed as a route for surpassing the performance of currently available heat transfer liquids in the near future. In this study an equilibrium molecular dynamics simulation was used to model a nanofluid system. The thermal conductivity of the base fluid and nanofluid was computed using the Green-Kubo method for various volume fractions of nanoparticle loadings. This study showed the ability of molecular dynamics to predict the enhanced thermal conductivity of nanofluids. Through molecular dynamics calculation of mean square displacements for liquid phase in base fluid and for liquid and solid phases in nanofluid, this study tried to investigate the mechanisms involved in thermal transport of nanofluids at the atomic level. The result showed that the thermal transport enhancement of nanofluids was mostly due to the increased movement of liquid atoms in the presence of nanoparticle. Diffusion coefficients were also calculated for base fluid and nanofluids. Similarity of enhancement in thermal conductivity and diffusion coefficient for nanofluids indicates similar transport process for mass and heat. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2785009兴 I. INTRODUCTION
Nanofluids are fluids which contain suspended nanometer sized metallic or nonmetallic particles or nanotubes. Compared to ordinary fluids containing microsized particles, nanofluids are more stable and do not clog micro- or nanochannels during flow. They show 30%–150% increase in thermal conductivity for only 0.5%–2% addition of solid particles.1–5 The thermal conductivity 共TC兲 of nanofluid also shows temperature dependency6 and nanoparticle size dependency.7 Recent research has also shown that nanofluids exhibit higher heat transfer coefficients in laminar flow8 and in pool boiling experiments compared to the base fluids.9 Due to their improved heat transfer characteristics, nanofluids have potential applications in many heat transfer areas. Continuum models such as Maxwell10 and HamiltonCrosser 共HC兲 theory11 for predicting the effective thermal conductivity of a matrix that contains dispersion of particles was applied by the researchers to predict the thermal conductivity of nanofluids 共composite of solid nanoparticles dispersed in liquid matrix兲. But HC model considers only static composite material and it not only underpredicts the relative increase in thermal conductivity obtained by the addition of nanoparticles but also is unable to explain the temperature and size dependency of the thermal conductivity of nanofluid suspensions. The HC model11 of thermal conductivity for spherical nanoparticles suspended in nanofluids can be expressed as below. keff k p + 2k f − 2共k f − k p兲 , = kf k p + 2k f + 共k f − k p兲
共1兲
where keff, k p, and k f are the thermal conductivities of nanofluid, solid particles, and fluid, respectively, and is the a兲
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concentration of nanoparticles in nanofluids. This model indicates that thermal conductivity of nanofluid is merely a function of only its component element’s conductivity and their concentration in nanofluids. HC model does not consider the movements of solid and liquid atoms and their possible collisions which can transport heat in nanofluids and may lead to increased thermal conductivity. Temperature and nanoparticle size effects are not considered in HC model. Keblinski et al.12 proposed four possible mechanisms for the anomalous conductivity behavior seen in nanofluids. Several analytical13–15 models had been proposed thereafter with different levels of acceptance in the scientific community. The earliest large scale microscopic simulation was performed16 using Brownian dynamics. They assumed that the nanoparticles were much bigger than the solvent or base fluid particles. Therefore, the solvent particles were omitted and their effects were represented by a combination of random and frictional forces. The solute particles were then allowed to move according to Newton’s second law of motion. The forces on the solute particles were evaluated from an assumed form of a two-body empirical potential. A better alternative was to employ interatomic potentials and perform true molecular dynamics simulations. Molecular dynamics has been successfully used to predict the thermal conductivities of a wide variety of liquids17,18 and solids19,20 with varying degrees of success, and hence it can be extended to calculate thermal conductivity of nanofluids which is a solidliquid composite system. Keblinski et al.12 introduced that the thermal conductivity of nanofluids can be calculated using Green-Kubo methods considering a solid and liquid system with simple Lennard-Jones 共LJ兲 systems. But they did not report any conclusive computed results for nanofluid thermal conductivity. A very recent study21 employed molecular dynamics 共MD兲 simulation to predict the thermal
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conductivity in model nanofluid system of Xe and Pt. They showed a significant increase of thermal conductivity compared to what Maxwell-HC model predicted. Though the result was qualitative, the method showed potential to correctly predict the right trend of thermal conductivity enhancement with increased nanoparticle loading. The mechanism for higher thermal transport in nanofluids is still a much debated discussion among the researchers. Keblinski et al.12 initially suggested that the key factors in understanding higher thermal properties of nanofluids are the ballistic, rather than diffusive, nature of heat transport inside the nanoparticle. They also showed through an order of magnitude analysis that the energy transport due to Brownian diffusion is two orders of magnitude slower than the energy transport due to thermal conduction in the liquid. They concluded based on this analysis that movement of nanoparticles due to Brownian motion is too slow to transport significant amounts of heat through nanofluid. On the other hand, recent studies14,22–24 argued that the enhancement in the effective thermal conductivity of nanofluids is not due to the direct Brownian motion of the nanoparticles, but may be due to the localized convection caused by the Brownian motion of the nanoparticles and demonstrate it through an order of magnitude analysis.14 Keblinski and Thomin25 and Evans et al.26 used MD simulation and kinetic theories to demonstrate that the hydrodynamics effects associated with Brownian motion have only a minor effect on the thermal conductivity of nanofluid. Until now, these competing theories had not been reconciled and another theory based on the aggregation of nanopaticles was proposed by Prasher et al.27,28 to explain the enhancement of nanofluid thermal conductivity. Hence to summarize, different proposed mechanisms exist in the literature for explaining the enhanced thermal conductivity of nanofluids such as 共i兲 ballistic phonon transport of heat through solid nanoparticles, 共ii兲 ordered layering of liquid around the solid, 共iii兲 thermal energy transfer due to increased interatomic interactions arising from interatomic potential, 共iv兲 enhanced thermal energy transfer due to Brownian motion of nanoparticle, 共v兲 localized convection created in fluid due to Brownian motion of nanoparticles, and 共vi兲 agglomeration of highly conductive nanoparticle in nanofluids. In this paper, we have modeled a model nanofluid system of copper and argon of varying nanoparticle concentrations using MD simulation. MD is a computational method that simulates the real behavior of materials and calculates physical properties of these materials by simultaneously solving the equation of motion for a system of atoms interacting with a given potential. This method provides a needed supplement to experimental measurements, which can be extremely difficult at such length scales. As MD simulation only assumes the form of interatomic potentials of a system and it directly and accurately calculates the movements of the particle at the atomic level, the same simulation with statistical mechanics can predict most accurate nanoscale flow and transport phenomenon compared to any classical model based on continuum mechanics. In MD simulations the thermal conductivity can be computed either using nonequilibrium molecular dynamics 共NEMD兲 共or direct method兲
or equilibrium molecular dynamics 共EMD兲 共or Green-Kubo method兲. The NEMD method relies on imposing a temperature gradient across the simulation cell and is therefore analogous to the experimental situation. By contrast, the Green-Kubo approach is an EMD method that uses current fluctuations to compute the thermal conductivity via the fluctuation-dissipation theorem. Using the Green-Kubo formalism in this paper, we have calculated the thermal conductivity of the base fluid and the effective thermal conductivity of the nanofluids. We also calculated the mean square displacement 共MSD兲 of liquid atoms in base fluid, and MSD of solid and liquid atoms in nanofluid through MD simulation to investigate the mechanism for enhanced thermal transport in nanofluids. As diffusion coefficient is easily obtained from MSD calculations, we calculated the diffusion coefficients of base fluid and nanofluids. Finally, we tried to correlate the enhancement of thermal conductivity with the enhancement of diffusion coefficients in nanofluids and the similarity of mass and heat transport phenomena. II. SIMULATION DETAILS A. Numerical procedure
Using equilibrium molecular dynamics technique in a three-dimensional computational domain, the current paper calculated thermal conductivity of copper in argon 共a model nanofluid兲 using Green-Kubo formalism. In this work, we developed a base fluid model of argon and a nanofluid model of copper particles in argon. Although argon is not a real base fluid material used in experiments, it is the best choice for an initial nanofluid thermal conductivity molecular dynamics study. One important reason for this is the availability of a good interatomic potential for argon. The widely accepted LJ potential matches experimental data for bulk fluid argon reasonably well, employs meaningful physical constants as parameters, and posses a simple, two-body form which requires much less computation time than more complex potentials involving other terms 共e.g., simple point charge potential for water兲. In our simulation, the interatomic interactions between solid copper nanoparticles, base liquid argon atoms, and interactions between solid copper 共Cu兲 and liquid argon 共Ar兲 were all modeled by pairwise LennardJones potential29 with appropriate Lennard-Jones parameters,
共rij兲 = 4
冋冉 冊 冉 冊 册 rij
12
−
rij
6
,
共2兲
where and are energy and length scales, respectively, and rij is the intermolecular distance between atoms i and j. Though most accurate potential for modeling copper is embedded atom method 共EAM兲 potential as it can also take care of metallic bonding, but in our present study LJ potential was used to reduce the computational time. To get the most quantitatively accurate results, more accurate EAM potential for that material should be used; but to predict the qualitative trends of thermal conductivity enhancement and study the reason of higher thermal transport, which are the aims of this paper, considering argon as the base fluid and modeling the interactions between copper atoms with LJ potential is a sensible choice. For argon, the LJ parameters and are equal
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TABLE I. LJ parameters used for copper in argon nanofluid in reduced units. ll 1.0
ss
sl
ll
ss
sl
39.676
6.3
1.0
0.738
0.8693
to 3.405 Å and 1.67⫻ 10−21 J, respectively,30 and for copper, the LJ parameters and are equal to 2.3377 Å and 65.625⫻ 10−21 J, respectively.31 By the introduction of a solid in the liquid, the simulation became polyatomic, i.e., a total of six LJ parameters were needed,32 namely, ss, ll, sl, ss, ll, sl, where the subscripts s and l refer to the system type 共solid or liquid, respectively兲. To determine the parameters for the solid-liquid interactions, the common Berthlot mixing rule33 was used,
sl =
1 3kBVT2
共t兲 =
冕
共5兲
where is the thermal conductivity, V the system volume, T the system temperature, kB the Boltzman constant, J the heat current vector, and the angular brackets denote the ensemble average, or, in the case of MD simulation, the average over time and 具J共0兲J共t兲典 is the heat current autocorrelation function 共HCACF兲. The heat current vector was calculated as37,38 N
JQ =
N
冋
N
1 1 d d ri共Ei − h兲 = 兺 ri mivi2 + 兺 共rij兲 兺 2 2 j⫽i dt i=1 dt i=1
册
共6兲
−h ,
N
sl = 冑ssll .
共3兲
In this study the LJ parameters for liquid, i.e., ll共ref兲 and ll共ref兲, were set equal to unity and taken as reference because simulations were performed in nondimensional units. ss, sl, ss, and sl were calculated with respect to the ll and ll with the help of Berthlot mixing rule for cross interaction 关Eq. 共3兲兴. In nondimensional units, the following six LJ parameters were used and shown in Table I for the MD simulation of nanofluid. For a two-component system, the Lennard-Jones potential in reduced form is ij ij共r * 兲 = 4 ref
具J共0兲J共t兲典dt,
0
where vi is the velocity of particle i, ⌽共rij兲 the pair potential between particles i and j, ri the position vector of the particle i, and h is the enthalpy per particle. Taking the time derivative of Eq. 共6兲 gives the following result for the heat flux:
ss + ll 2
and
*
t
冋冉 冊 冉 冊 冉 冊 冉 冊 册 ij ref
12
1 r*
12
ij − ref
6
1 r*
.
共4兲
The simulations were performed at specified temperature and hence NVT ensemble was used where the total number of atoms, the system volume, and temperature were constant throughout the simulation. To employ NVT, the NoseHoover thermostat34 was used. To improve computational efficiency, only the neighbors of an atom within a certain cutoff radius 共rcut兲 equal to 2.8ref were included in the force calculations because far away atoms had a negligible contribution to the total force on a given atom. Verlet neighbor list33 was also constructed at the initial step of the simulation and modified whenever required to reduce the computational time. Velocity Verlet algorithm35 was used as the integration scheme.
B. Green-Kubo theory of calculating thermal conductivity by EMD simulation
An EMD simulation relates the equilibrium heat current autocorrelation function to the thermal conductivity 共TC兲 via the Green-Kubo theory.36
N
共7兲
where Fij is the force on atom i due to its neighbor j from the pair potential and the energy Ei. For a single component system, the last term of Eq. 共6兲 is zero. For the twocomponent system in our simulation we used Eq. 共8兲, an extended form of Eq. 共7兲, to calculate the heat current vector JQ according to37,38 2
JQ =
N␣
1
兺 兺 m␣v2j␣v j␣ ␣=1 j=1 2 2
6
N
N
1 JQ = 兺 viEi + 兺 兺 rij共Fij · vi兲 − 兺 vih, 2 i j⫽i i i
2
N
␣ 1 − 兺兺兺 2 ␣=1 =1 j=1
N
兺
k=1
冋
r j ␣k
册
共r j␣k兲 − 共r j␣k兲I v j␣ r j ␣k
k⫽j 2
−
N␣
v j␣ . 兺 h␣ 兺 j=1
␣=1
共8兲
The subscripts ␣ and  denote two different kinds of particles and j, k count the number of particles. N␣ and N are the number of particles of kinds ␣ and . Thus the heat current JQ in Eq. 共8兲 is composed of a kinetic part, a potential part, and a term containing the partial enthalpies. v j␣ denotes the velocity of a particle j of kind ␣, h␣ denotes the average enthalpy per particle of species ␣, and I is the unit tensor. We calculated the average enthalpy as the sum of the average kinetic energy, potential energy, and average virial terms per particle of each species. Though Eq. 共8兲 is used ideally for a homogeneous system where density gradients are negligible, but we used this in our simulation as we found a good agreement between the heat flux determined from NEMD and EMD simulations within a reasonable 7.6% variation. Since the simulations were performed for discrete MD steps of length ⌬t, including the time averaging, Eq. 共5兲 for calculating TC can be rewritten as39
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N−m
1 ⌬t 共t M 兲 = 兺 J共m + n兲J共n兲, 2兺 3kBVT m=1 N − m n=1
共9兲
where tM is given by M⌬t and J共m + n兲 is the heat current at MD time step 共m + n兲. The average over time of heat current is known as heat current autocorrelation function 共HCACF兲. C. Initial configuration and MD simulation
In our model a single nanoparticle was considered in base fluid atoms. To start with, all the atoms in the nanofluid system were arranged in a regular fcc lattice. The solid nanoparticle was formed by carving spheres out of a fcc lattice of atoms. Periodic boundary condition was applied in all directions of three-dimensional cubic simulation cell. A total of 2048 atoms were considered in the system. This leads to nanoparticle size of nearly 2 nm diameter for 2% nanofluid. The nanoparticle sizes varied from 1.5 nm to 3 mm diameter for different nanoparticle loadings from 1% to 8% as a total of 2048 atoms were used in the simulation cell in all simulations. Then molecular dynamics simulation was started using velocity Verlet integration scheme and continued for 1 000 000 time steps. Each small time step was considered as 4 fs. Initial 100 000 time steps were ignored and the results were not considered for calculating thermal conductivity to allow the system to reach equilibrium. After 100 000 time steps, the heat current J共t兲 was calculated at each time step according to Eq. 共8兲. Then the thermal conductivity was calculated according to Green-Kubo theory with the help of Eq. 共9兲. The HCACF decays very rapidly and 10 000 time steps were sufficient for time averaging of heat current autocorrelation function. III. RESULTS AND DISCUSSIONS A. Validation of model and simulation system size dependency
It is necessary in any simulation to validate the model with some widely known results. Hence we calculated the thermal conductivity of liquid argon at its state point T* = 0.71 and * = 0.844, where T* and * are nondimensional units of temperature and density. We varied the number of argon atoms from 32 to 2048 and found that the thermal conductivity gradually increased from 0.093 to 0.127 W m−1 K−1. The steady convergence of thermal conductivity of liquid argon after 500 atoms was shown in Fig. 1. The final value of 0.127 W m−1 K−1 is in good comparison with the experimental value of 0.132 W m−1 K−1.18 So our result is within 4% of the experimental value. Similar simulations were performed when a nanoparticle was present in the system, and total number of atoms in the system was varied from 32 to 2048 atoms. Thermal conductivity of 1% nanofluid was found to increase gradually from 0.145 to 0.175 W m−1 K−1 共Fig. 1兲. We observed that after 1372 atoms, the thermal conductivity of the nanofluid system converged well as compared to 500 atoms for argon base fluid. This system size dependency study was performed to find the exact simulation system size for which our finite MD simulation cell represents as closely as infinite
FIG. 1. 共Color online兲 Thermal conductivity of liquid argon and 1% copper in argon nanofluid as a function of number of particles in the MD system.
nanofluid bulk domain for calculation of properties such as thermal conductivity accurately. Hence, for all property calculation of nanofluids in this paper, a total of 2048 atoms were used. B. Modeling the nanofluid and calculation of thermal conductivity
Six sets of runs were performed for copper loadings of 0.2%, 0.4%, 1%, 2%, 4%, and 8% by volume. One set was kept with only base fluid atoms for making comparison with nanofluids. The equilibrium structure of nanofluid appeared as in Fig. 2. Thermal conductivity of base fluid and nanofluids was computed using Green-Kubo method by EMD simulation for different volume fractions of nanoparticle loading, and the thermal conductivity values were plotted 共Fig. 3兲 for different volume percent of nanoparticle loading. At very low nanoparticle concentration 共0.4%兲, the thermal conductivity was 0.145 W m−1 K−1. The thermal conductivities were 0.156 and 0.165 W m−1 K−1 for 1% and 2% nanofluids, respectively. Hence for copper in argon nanofluid system, we found thermal conductivity enhancement up to 20% compared to the base fluid with 1% nanoparticle concentration. A recent study by Eapen et al.38 reported up to 35% TC enhancement for 1% Pt in Xe nanofluid through EMD simulation. The conductivity enhancement was steeper at very
FIG. 2. 共Color online兲 Computed equilibrium nanofluid structure for 2% loading of Cu nanoparticle 共dark: copper nanoparticle; light: argon atoms兲.
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FIG. 3. 共Color online兲 Thermal conductivity of nanofluids with different nanoparticle loadings.
FIG. 5. 共Color online兲 Mean square displacement with time for base fluid, liquid phase in 1% nanofluid, and solid phase in 1% nanofluid.
low nanoparticle loading 共up to 0.4%兲 compared to higher loadings, as found in some experimental studies,2 but not predicted by available analytical and computational models.14,15,21 MD simulation of copper-argon nanofluid predicted up to 52% in TC enhancement, whereas the HC theory predicted up to 26% enhancement of thermal conductivity for 8% nanofluids. Hence from our MD simulation we observed that thermal conductivity enhancement at higher loading was two times greater than that predicted by the HC model. MD simulation is a very useful tool and can be used to model nanofluid thermal conductivity using more realistic material and more accurate potential in future. We also plotted the HCACF for different nanofluids and base fluids in Fig. 4. It was observed that HCACF decayed to zero in 1.8 ps for base fluid and it decayed to zero in 2 – 4 ps for nanofluids. HCACF stayed correlated more strongly and for a longer time 共not shown in figure兲 for nanofluids with higher nanoparticle loading. A closer look further revealed that in the base fluid, HCACF decayed to zero monotonically, whereas for nanofluids it decayed to zero in an oscillatory manner 共Fig. 4, HCACFs are shown only up to 0.5 ps兲. With increased nanoparticle loading oscillation of HCACF was found to increase. While decaying to zero, we also observed similar to Keblinski et al.12 that HCACF was oscillating to some negative value. In materials where the fluctuations are long lived the HCACF decays slowly. The
thermal conductivity is related to the integral of the HCACF, and is accordingly large. Hence the thermal conductivity of nanofluid was higher than base fluid and within nanofluids, conductivity was found to increase for higher nanoparticle loading. In liquids such as base fluids, thermal fluctuations are quickly damped, leading to a small integral of the HCACF and a low thermal conductivity.
FIG. 4. 共Color online兲 Decay of heat autocorrelation function for different loadings of nanoparticle in nanofluids with correlation time.
C. Mechanism of thermal transport and high thermal conductivity in nanofluid
In the following paragraphs of this section, we have tried to explain systematically the reasons for enhancement of thermal conductivity in nanofluids through our MD simulation. MD can easily track the displacements or movements of all the atoms in liquid and solid. Their movements at atomic scale are the reason for any observed macroscopic properties. Here we calculated the MSD of solid and liquid atoms in nanofluid and liquid atoms in base fluid to investigate the mechanism of enhanced thermal transport. The MSD is a measure of the average distance an atom travels. It is defined as30 MSD共t兲 = 具⌬ri共t兲2典 = 具共ri共t兲 − ri共0兲兲2典,
共10兲
where ri共t兲 − ri共0兲 is the 共vector兲 distance traveled by atom i over some time interval of length t, and the squared magnitude of this vector is averaged 共as indicated by the angle brackets兲 over many such time intervals. We took the average of this quantity over all the atoms of liquid or solid, summing i from 1 to N and dividing by N, to get the MSD of the solid and liquid atoms in the nanofluid or bulk fluid. The rate of increase of the mean square displacement depends on how often the atoms suffer collisions. In our MD simulation we calculated separately the average MSD of both solid and liquid atoms in nanofluids. The MSD of base fluid was also calculated for comparison. The MSD of all liquid atoms in base fluid and MSD of both solid and liquid atoms in 1% nanofluid were shown in Fig. 5. The time allowed for calculating MSD was 50 ps. The MSD of the liquid phase in the 1% nanofluid was found to be 28 times higher than the nanoparticle 共solid phase兲 in the nanofluid and approximately 1.41 times higher than the MSD of
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FIG. 6. 共Color online兲 Mean square displacement with time for base fluid and liquid phase in different nanofluids.
FIG. 7. 共Color online兲 Diffusion coefficient and thermal conductivity enhancement in nanofluids with varying nanoparticle loading.
the liquid phase in base fluid. Hence in the presence of the nanoparticle in 1% nanofluid, the enhancement in liquid atom movements was quite significant. The MSDs calculated for liquid phase in base fluid and for liquid phase in 0.2%, 0.4%, and 1.0% nanofluids with time were also plotted in Fig. 6. The MSD of liquid phase was found to increase more steeply with time in nanofluids compared to base fluids, and slope was steeper for higher loading of nanoparticle 共up to 1%兲, indicating further increase in movement of fluid atoms in nanofluids. The increased movement of liquid atoms was because of the interatomic interactions due to potential difference from metal to liquid and may be due to the influence of heavier nanoparticle. 40 times higher cohesive energy of solid copper atoms compared to liquid argon atoms 共ss is almost 40 times of ll兲 as well as the heavier mass of copper atoms 共1.6 times higher than mass of argon兲 did not allow the copper atom to move much like argon atoms. The movements of solid copper atoms, i.e., Brownian motion of nanoparticle, are far slow to transport the heat. In comparison, the enhanced movement of much faster liquid atoms in nanofluids creates localized fluid motion surrounding the nanoparticle, which is responsible for enhanced thermal transport in nanofluids and hence enhanced thermal conductivity. This is one of the main mechanisms found from our MD studies for higher thermal transport in nanofluids. The similar type of mechanism was proposed by some recent researchers and described as “localized convective field in fluids.”22,23 Through the MD simulation we confirmed the existence of such enhanced movement of fluid atoms at atomic level. Further study is underway to compute the MSD of liquid atoms at different distances from the nanoparticle cluster and to find the effect of nanoparticle-fluid interaction on the increased liquid motion particularly at the interface. A recent study by Eapen et al.38 showed that a strong cluster fluid attraction through the self-correlation of potential flux was important for enhanced thermal conductivity of nanofluids and highlighted the significance of surface interaction. In our future work we would breakdown the HCACF into components related to liquid alone, solid alone, and the liquidliquid coupling, as seen in Refs. 12 and 38, and try to corre-
late between the enhanced liquid motion in the nanofluid, particularly at the interface, and the effect of different correlations mentioned above. The limiting slope of MSD 共t兲, considered for time intervals sufficiently long for it to be in the linear regime, is related to the self-diffusion constant D by Eq. 共11兲,33 lim t→⬁
d 具⌬ri共t兲2典 = 6D. dt
共11兲
From the above equation the diffusion coefficients of base fluids 共D f 兲 and nanofluids 共Deff兲 were calculated through MD simulation. The diffusion coefficient for base fluid 共argon兲 D f was found 2.10⫻ 10−9 m2 / s at the state points considered for our simulation. The literature value is 2.17⫻ 10−9 m2 / s.30 Hence, our MD simulation can calculate diffusion coefficient quite accurately, and as diffusion coefficient was calculated here from MSD data, it is an indirect validation of our calculated MSD value. The diffusion coefficients of 0.2%, 0.4%, and 1.0% nanofluids 共Deff兲 were also computed, and the diffusion coefficient enhancement 共Deff / Df兲 in nanofluids was reported in Fig. 7. With increasing nanoparticle concentration in base fluid the diffusion coefficients of nanofluids increased. Up to 1% of nanoparticle loading, the diffusion coefficient was found to increase by 41% compared to the diffusion coefficients of base fluid. The thermal conductivity enhancements for corresponding nanofluids were also plotted for comparison in Fig. 7. Interestingly, it was found that both thermal conductivity and diffusion coefficient enhancement behaved similarly with the increasing loading of nanoparticles in base fluid though their slopes were different 共Fig. 7兲, indicating strong correlation between enhanced atom movements and thermal conductivity of nanofluids. For higher nanoparticle loadings the other mechanism such as enhanced conduction due to larger aggregation of highly conductive nanoparticle,27,28 can play a significant role toward enhanced thermal conductivity. Though MD study of the agglomeration of nanoparticles in nanofluid and its effect on thermal conductivity enhancement could be of immense help, but as our model at present only considers single nanoparticle in the simulation cell the aggregation mechanism is not applicable for our present study.
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IV. CONCLUSIONS
In this paper, we have modeled the nanofluid systems of copper and argon of varying nanoparticle 共Cu兲 concentrations, and using the Green-Kubo formalism we have systematically determined the thermal conductivity of the base fluid and the nanofluid. The model was validated with experimentally determined liquid argon conductivity. The thermal conductivity of nanofluid was found to increase linearly with increased volume percent of copper nanoparticle loading. The conductivity enhancement was steeper at very low nanoparticle loading 共up to 0.4%兲 compared to higher loadings. It was also observed that thermal conductivity enhancement at higher loading 共up to 8%兲 was two times greater than that predicted by the HC model for copper in argon nanofluid system. From mean square displacement calculation using MD, it was shown that in nanofluids the movement of liquid atoms increases significantly 共1.41 times in 1% nanofluid兲 compared to the movement of liquid atoms in base fluid. The nanoparticle movement was 28 times slower than that of the liquid phase in 1% nanofluids. So, it is not the slow Brownian motion of nanoparticle, but the highly enhanced and fast enough surrounded liquid atom movement which helps to transport heat quickly in nanofluid and causes higher thermal conductivity of nanofluids. This is the main mechanism for enhanced thermal conductivity of nanofluids in our simulation. The diffusion coefficients were also calculated for nanofluids and base fluids. Similarity of enhancement pattern for diffusion coefficients and thermal conductivity in nanofluids also indicates the similar type of transport mechanism for mass and heat in nanofluids.40 For higher nanoparticle loading, enhanced conduction due to larger aggregation of highly conductive nanoparticle as indicated by recent researchers can also be an important mechanism for enhanced thermal conductivity. As our model at present only considers single nanoparticle in the simulation cell, the aggregation mechanism is not applicable for this study. In near future we will perform MD simulations with multiple nanoparticles to study the effect of agglomeration kinetics toward the enhanced thermal conductivity of nanofluid. ACKNOWLEDGMENTS
The authors acknowledge the useful discussions with Dr. P. Keblinski. The authors acknowledge the support received from the ONR 共Award No. N00014-05-1-0889兲 through the University of Arkansas to perform this work. The primary author 共S.S.兲 would like to thank the support of the Doctoral Academy Fellowship provided by the Graduate School of University of Arkansas. 1
S. Lee, S. U. S. Choi, S. Lu, and J. A. Eastman, J. Heat Transfer 121, 280 共1999兲. J. A. Eastman, S. U. S. Choi, W. Yu, and L. J. Thompson, Appl. Phys. Lett. 78, 718 共2001兲.
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