Molecular dynamics simulation of thermal conductivity of Cu Ar nanofluid using EAM potential for Cu Cu interactions

Transcription

1 Appl Phys A (2011) 103: DOI /s z Molecular dynamics simulation of thermal conductivity of Cu Ar nanofluid using EAM potential for Cu Cu interactions Hongbo Kang Yuwen Zhang Mo Yang Received: 17 January 2011 / Accepted: 15 March 2011 / Published online: 6 April 2011 Springer-Verlag 2011 Abstract Mechanism of heat conduction in copper-argon nanofluids is studied by molecular dynamics simulation and the thermal conductivity was obtained using the Green Kubo method. While the interatomic potential between argon atoms is described using the well-known Lennard Jones (L J) potential, a more accurate embedded atom method (EAM) potential is used in describing the interatomic interaction between copper atoms. It is found that the heat current autocorrelation function obtained using L J potential to describe the copper-copper interatomic interaction fluctuates periodically due to periodic oscillation of the instantaneous microscopic heat fluxes. Thermal conductivities of nanofluids using EAM potentials were calculated with different volume fractions but the same nanoparticle size. The results show that thermal conductivity of nanofluids are almost a linear function of the volume fraction and slightly higher than the results predicted by the conventional effective media theory for a well-dispersed solution. A solid-like base fluid liquid layer with a thickness of 0.6 nm was found in the simulation and this layer is believed to account for the small discrepancy between the results of MD simulation and the conventional effective media theory. H. Kang M. Yang College of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai , China Y. Zhang ( ) Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA zhangyu@missouri.edu 1 Introduction Nanofluids are solid-liquid composite materials consisting of solid nanoparticles or nanofibers, with size typically on the order of nm, suspended in the liquids [1]. Nanofluids have attracted wide-spread attentions in the past decade due to reportedly significant enhancement of thermal properties at low nanoparticle volume fractions. Lee et al. measured the thermal conductivity of nanofluids that contained Al 2 O 3 and CuO nanoparticles in water and ethylene glycol [2]. They used volume fractions of 1 5% and obtained a large thermal conductivity enhancement up to 20%. Even though the potential of using nanofluids as heat transfer media was evident from ceramic nanofluids, the emergence of metallic nanoparticle and carbon nanotube based nanofluids was a major step forward. It was reported that a small amount (less than 1% volume fraction) of copper nanoparticles and carbon nanotubes dispersed in ethylene glycol and oil increased the inherently poor thermal conductivity of the liquid by 40 and 150%, respectively [3, 4]. In 2002, Keblinski et al. [5] suggested that there may be four potential mechanisms that could account for the large thermal conductivity enhancement of nanofluids: (1) Brownian motion of nanoparticles, (2) molecular-level layering of the liquid at the nanoparticle surface, (3) nature of heat transport in nanoparticles, and (4) the effects of nanoparticles clustering. Since then, many researchers worked on understanding and verification of these mechanisms. Since nanoparticles suspending in base fluids are much smaller than any other solid solute in the conventional liquid-solid mixtures, the Brownian motion of nanoparticles becomes significant. However, the results of the research worked on this are controversial: some researchers suggested that Brownian motion and microconvection induced by Brownian motion was the main reason for explaining the significant enhancement of thermal conductivity in

2 1002 H. Kang et al. nanofluids [6 8], while some others disagreed and believed thermal conductivity enhancement due to Brownian emotion is not possible [9, 10]. Performing an experiment, Yu et al. [11] found that there existed a solid-like liquid layer at the interface between the solid and liquid. Many researchers suggested that this solid-like liquid layer, which has a more ordered structure and larger thermal conductivity than the bulk base fluid, is the main mechanism that accounts for the significantly increased thermal conductivity in nanofluids [12 14]. On the contrary, Wilson et al. [15] and Nie et al. [16] argued that the liquid layer effect on the enhancement of thermal conductivity in nanofluids is negligible based on experiment study and theoretical analysis, respectively. In the recent years, the importance of the nanoparticle aggregation effect on the enhanced thermal conductivity in nanofluids is recognized by some researchers, who believed that the conventional effective media (EM) theory is still valid on predicting the thermal conductivity in nanofluids if the nanoparticle aggregation effect is taken into account using the Hashin and Shtrikman (H S) model [1, 10]. Some experiments have demonstrated that there does exist nanoparticle clustering [17, 18]. Keblinski et al. [1] summarized the published experiment data and compared them with the H S model. They concluded that these data lay between the well-known H S effective medium bounds. The mechanism of the thermal conductivity enhancement in nanofluids is still in debate and not conclusive. The thermal conductivity enhancement may not be due only to any one of the above mechanisms. Thermal conductivity of an imaginary nanofluid is calculated using the nonequilibrium molecular dynamics (NEMD) method by Evans et al. [9]. The effect of solidliquid interaction strength is studied and the results showed that thermal conductivity did not have a significant enhancement for a nanofluid with well-dispersed nanoparticles. Eapen et al. [10] calculated the thermal conductivity of the Pt Xe nanofluid using the equilibrium molecular dynamics (EMD) method with each nanoparticle consisting of only 10 atoms. Thermal conductivity of the Cu- Ar nanofluid is obtained by Sarkar et al. [19] and Li et al. [20] using the EMD method. Different effects were studied separately. In these two papers, the volume fraction of the nanofluid is changed as the nanoparticle diameter is changed. In this paper, a molecular dynamics (MD) simulation is performed to reveal the mechanism of thermal conductivity enhancement in nanofluids. One nanoparticle with a fixed diameter is placed in the simulation box and the periodic boundary condition is used. This configuration corresponds to the situation that the nanoparticle is well dispersed in the base fluid. 2 Methodology The nanofluids system formed by disperse copper nanoparticles in liquid argon is studied in this work. Argon is chosen as the base fluid because it is an ideal choice for an initial molecular dynamics study on thermal conductivity of nanofluids. The widely accepted Lennard Jones (L J) potential matched experimental data reasonably well for bulk argon fluid. The argon-argon and argon-copper interatomic interactions are modeled by the pairwise L J potential. [ ( ) σ 12 ( ) ] σ 6 Φ(r ij ) = 4ε (1) r ij r ij where the L J potential parameters for copper argon and argon argon are ε Ar Ar = J, σ Ar Ar = nm, ε copper Ar = J, σ copper Ar = nm [19]. For the copper copper interatomic interactions, we choose a more accurate potential the embedded atom method (EAM) potential that takes the metallic bonding into account. The total potential energy E i (for pure metal) of the atom i is given by [21]: ( ) E i = F i ρ i (r ij ) + 1 φ ij (r ij ) (2) 2 j i j i where F i is the embedding energy of atom i and it is a function of the atomic electron density ρ; φ is a short-range pair potential interaction between the atoms i and j. The difference between using EAM potential and L J potential to describe the interatomic interaction among copper atoms will be discussed later in this work. The Green Kubo (G K) method is used to compute the thermal conductivity of the nanofluids. Unlike the nonequilibrium molecular dynamics (NEMD) method, the G K method represents an equilibrium molecular dynamics (EMD) technique. There is no imposed driving force, and hence the system is always in the linear response regime. The thermal conductivity is obtained from [22]: 1 k = J(0) J(τ) dτ (3) 3k B VT 2 0 If the system is homogeneous, the formula can be reduced to k = 1 k B VT 2 0 Jx (0) J x (τ) dτ (4) where J x is an arbitrary component of J, the instantaneous microscopic heat flux vector, and the thermal conductivity k can be determined by averaging over the entire three components. The form of J (0) J (τ) gives the heat current

3 Molecular dynamics simulation of thermal conductivity of Cu Ar nanofluid using EAM potential 1003 autocorrelation function (HCAF). It should be noted that the number 0 in J (0) means the time origin of the HCAF and can be chosen at any time step. J (τ) is the heat flux vector at a time that is τ later than the J (0), in other words, it delayed time τ with respect to J (0). As the total simulation time is finite, the range for the value of τ will change when different time origins are chosen. The angular brackets denote the time average. Since the simulations are performed for discrete MD steps of length t, (4) for calculating thermal conductivity can be rewritten as k(t M ) = t k B VT 2 M m=0 N m 1 J x (m + n)j x (n) (5) N m n=1 where t M is given by M t and J x (m + n) is the heat current in the x-direction at time step (m + n), which means m time steps delayed with respect to J x (n). The instantaneous microscopic heat flux for a binary system is calculated as [23]: N J = v i e i + 1 N N N r ij (f ij v i ) v i h α (6) 2 i i j i where e i denotes the internal energy, i.e., the summation of kinetic energy and potential energy, f ij is the force on atom i due to its neighbor j from the potential, and h α is the mean partial enthalpy which is calculated as the sum of the average kinetic energy, potential energy, and virial item energy for atom type α. It is necessary to note that in multicomponent systems, like nanofluids in this paper, one may obtain a nonphysical result if this partial enthalpy item is not taken into account. In this work, the instantaneous heat flux vector is calculated in every time step and all the time steps were used as time origins of the HCAF. Also, note that the unit of the heat flux vector in (6) isnotw/m 2, rather, it is in the unit of W m. The real heat flux unit of the system is obtained when it is divided by the volume of the total system. 3 Results and discussions The nanofluids system was created initially with an FCC (face centered cubic) lattice structure; and the nanoparticle was formed by putting copper atoms in a spherical region just in the middle of the simulation box where argon atoms were carved out. As the density of the liquid argon and copper are different, the number of atoms occupying the same volume is somewhat different. The time step used in this simulation is 4 fs and an NVE ensemble was used throughout the simulation. The cutoff radius in L J potential was set to be 2.6σ Ar, which is in the range of σ Ar proposed by Vogelsang et al. [24]. In this range, the thermal conductivity is independent of the cutoff radius. To obtain i an equilibrium state, an initial 20,000 time steps were used with atom velocities rescaled in order to make the system equilibrate at the desired temperature. The determination of the total simulation time steps that were used to get the thermal conductivity in nanofluids will be discussed later. To validate the simulation method, we calculated the thermal conductivity of the liquid argon at T = 86 K and ρ = 1418 kg/m 3, which has the experiment data to compare with. Sarkar et al. [19] studied the influence of the atom number on the simulation result. They suggested that the results were in good agreement with the experimental value for pure argon when the number of atom is greater than 500. To be on the safe side, we chose 4,000 atoms in the simulation domain to compute the thermal conductivity of liquid argon. In other words, the simulation box has a size of 10 lattices along three directions. A parallel running was performed using 8 CPUs. Figure 1 shows the result of our simulation for thermal conductivity of the liquid argon. Sixteen independent runs were carried out to get an average. Although the standard deviation of these runs is somewhat large, the flat shaped curve shows that the result is good, i.e., the standard deviation of the independent runs does not affect the final result of the thermal conductivity. It only affects the number of runs that is needed to get an average. That is to say, large standard deviation means more runs are needed to obtain good results. We studied the effect of the total simulation time on the value of the standard deviation and the result is shown in Fig. 2. It is seen from Fig. 2 that longer simulation time can yield smaller standard deviation. Considering the trade-off between the computational time and the accuracy of the result, simulation time steps (besides the initial 20,000 time steps for equilibrium) were chosen in all the simulations in this paper. Under the chosen number of time steps, 16 runs are sufficient to get a good result. To obtain the thermal conductivity, as described in (5), a proper M value is needed. This M value is the upper limit of the integral of the HCAF. Although the HCAF decays to zero with respect to time and fluctuates around zero, the integral will deviate from a reasonable value when M becomes large for a statistical reason. As a result, the M value should be larger than the time steps which the HCAF needs to decay to zero and also should be much smaller than the total simulation time steps to ensure a reasonable statistical averaging. In addition, the integral of the HCAF, i.e., the thermal conductivity, has some fluctuations as well and this makes the M value affect the final result significantly. In this work, multiple-runs are carried out to reduce the fluctuation of the thermal conductivity. Instead of choosing an arbitral M value, we take advantages of M from 4,000 all through to 30,000 and then get an average and the results are shown in Fig. 1. The thermal conductivity of the liquid argon obtained in this simulation is W/m K, which agreed very well with the experiment value of W/m K.

4 1004 H. Kang et al. Fig. 1 Thermal conductivity of pure argon Fig. 2 Standard deviation versus simulation time The copper-argon nanofluids system is studied next. It should be pointed out that in an MD simulation only the lattice thermal conductivity contribution can be obtained for conductors, i.e., metal or alloy. Thus, only the phonon thermal transport contribution is calculated in the copper nanoparticles. Four sets of cases were performed for copper loadings of 2%, 3.18%, 4.1%, and 5.5% by volume and all of them had the same particle diameter of 2.7 nm. As we use the EAM potential, not the usual L J potential, to describe the copper-copper interatomic interaction, it is necessary to compare the differences between the results obtained by these two different potentials. The first comparison was made to the computation times. Since the EAM potential is more complicated than the L J potential, it is widely believed that the former is more time consuming. For this reason, many researchers did not like to use the EAM potential. However, we find that the EAM potential only takes no more than 10% more computational time than the L J potential in our simulation to a system that hasasizeof lattices and total simulation time of 10 6 time steps. The result of different potential is compared for a case of nanofluids having 5.5% volume fraction copper nanoparticles. The thermal conductivities obtained from EAM potential and L J potential is W/m K and W/m K, respectively. In addition, the shapes of the curves also have differences that need to be point out (see Fig. 3). Equation (3) indicates that the thermal conductivity is the product of a constant (if a certain system is chosen) and the integral of the heat current autocorrelation function. Therefore, the thermal conductivity is really determined by the HCAF. It can be obviously seen from Fig. 3 that the thermal conductivity is very different at the beginning due to the different HCAFs of each potential. The normalized HCAF at the beginning is shown in Fig. 4. While both HCAFs oscillate and then decay to zero quickly, the HCAF obtained using L J potential fluctuates more severely and exhibit some periodic behavior. This periodic fluctuation demonstrates that the instantaneous microscopic heat fluxes have some periodic oscillations. We also decomposed the instantaneous microscopic heat flux into two parts (not shown in graph): one is due to the base fluid, i.e., the liquid argon, and the other one is due to the copper nanoparticle. It was found that the periodicity like fluctuation of the total HCAF was caused by the fluctuation of the copper microscopic heat

5 Molecular dynamics simulation of thermal conductivity of Cu Ar nanofluid using EAM potential 1005 Fig. 3 Comparison of thermal conductivities of 5.5% vol. case using different potential Fig. 4 Comparison of HCAFs of 5.5% vol. case using different potential current autocorrelation function. The HCAF of the base fluid part decays to zero monotonically as mentioned in many references [24, 25]. Therefore, the results suggested that using L J potential to describe the interatomic interaction between copper atoms makes the HCAF fluctuate periodicity like. It should be pointed out that the fluctuation of the instantaneous microscopic heat flux of the copper nanoparticle is due to the phonon reflection in the particle; the period of this fluctuation will change when the nanoparticle diameter is changed. This is because the phonons have a strong reflection at the copper-liquid interface and could be constrained inside. There also exists the fluctuations using EAM poten-

6 1006 H. Kang et al. Fig. 5 Thermal conductivities with different volume fractions tial, but compared with the L J potential, the fluctuations decay rapidly and do not appear the periodicity. Figure 5 shows the thermal conductivity enhancement obtained using EAM potential and each data point has an error bar which denotes the standard deviation of the independent runs for each case. The thermal conductivities of the four different cases are 0.142, 0.150, 0.158, and W/m K, respectively. Compared to the thermal conductivity of pure liquid argon obtained above, i.e., W/m K, the thermal conductivity enhancement of these different volume fraction cases are 6%, 12%, 18%, and 28%, respectively. Note that these enhancements are almost linear and do not deviate that much from the results predicted by the Maxwell model at low volume fraction of the nanoparticles. When the volume fraction goes high, this deviation becomes larger. We studied the relative density of the base fluid around the nanoparticle and found that in a shell, which has a thickness of 0.6 nm from the nanoparticle surface, the fluid density is about 20 40% higher, as is shown in Fig. 6. Wealso followed the retention rate of argon atoms in this shell and found that most of the atoms ( 70%) remain in the shell as the time goes by. Thus, the copper nanoparticle with diameter of 2.7 nm has a 0.6 nm-thick solid-like base fluid layer. The thermal conductivity obtained by a renovated Maxwell model [13] is shown in Fig. 7. To obtain thermal conductivity using this model, three parameters are needed: thickness of the solid-like liquid layer, thermal conductivity of the layer, and thermal conductivity of the nanoparticle. The thickness of the solid-like liquid layer is obtained through the simulation, i.e., 0.6 nm, and the thermal conductivity of the layer is 2k f, that is, W/m K, according to the simulation result obtained by McGaughey et al. [25]. Since the diameter of the copper nanoparticle is 2.7 nm in this simulation, which is much smaller than the mean free path of the copper atom, the size effect should be taken into account. The thermal conductivity of the copper nanoparticle can be obtained from [26]: k p k b = Kn where k p is the thermal conductivity of the nanoparticle and k b is the thermal conductivity of the bulk value. Kn is the so-called Knudsen number and is defined as the ratio of the mean free path and the nanoparticle diameter. At 86 K, which was used to perform the MD simulation, the thermal conductivity of the bulk copper is W/m K and the mean free path is about 71 nm. Using these parameters and (7), of the thermal conductivity of the copper nanoparticle can be obtained as 19 W/m K. The thermal conductivity obtained by the renovated Maxwell model based on these parameters is plotted in Fig. 7. As can be seen, when the volume fraction of the nanoparticles is small, the discrepancy between the conventional effective media theory- Maxwell model [27] and the renovated Maxwell model is not significant. However, the discrepancy becomes larger when the volume fraction is higher. Our simulation results lies between these two models. We believe that the small discrepancy between the simulation result and the result predicted by the conventional effective media theory is due to this solid-like liquid layer around the nanoparticle. The effect of the nanoparticle diameter on thermal conductivity of nanofluids is studied next. Many researchers have mentioned that the nanoparticle diameter has an effect on the thermal conductivity of nanofluids. Through experiment, most researchers suggested that the thermal conductivity would increase as the nanoparticle diameter decrease [17, 28]. However, Beck et al. [29] reported a completely contrary result, i.e., the thermal conductivity of nanofluids decreases as the nanoparticle diameter decrease. In MD simulation, if the simulation box size is fixed, changing the diameter of the nanoparticle means the volume fraction of the nanoparticle also changes [19]. So, it is necessary to study (7)

7 Molecular dynamics simulation of thermal conductivity of Cu Ar nanofluid using EAM potential 1007 Fig. 6 Base fluid relative density around the copper nanoparticle and the tracking of base fluid atoms Fig. 7 Comparison of the results obtained by MD simulation, renovated Maxwell model and Maxwell model Fig. 8 Effect of nanoparticle diameter on thermal conductivity whether this particle size effect has an impact on the thermal conductivity of nanofluids. We choose a set of nanoparticle diameters, i.e., 1.9 nm, 2.3 nm, and 2.7 nm, with a fixed 2% volume fraction and the result is shown in Fig. 8. Obviously, the thermal conductivity of these three cases is almost the same. Thus, the nanoparticle size has little effect on the thermal conductivity of nanofluids in such a nanoparticle diameter range. 4 Conclusions Thermal conductivity of a copper-argon nanofluid is studied by a molecular dynamics simulation. To compute the thermal conductivity of nanofluids by the molecular dynamics simulation using the Green Kubo method, several independent runs are needed to obtain an averaged result. The standard deviation of these independent runs is of little effect

8 1008 H. Kang et al. on the averaged result, if sufficient runs are performed. It is found that using EAM potential to describe the interatomic interaction between copper atoms is not that time consuming, but the results obtained by these two different potentials are different. The heat current autocorrelation function obtained using L J potential fluctuates periodicity like, which indicates that the instantaneous microscopic heat fluxes have some periodic oscillation. A solid-like base fluid liquid layer with a thickness of 0.6 nm was found in the simulation and this layer is believed to account for the discrepancy between the simulation result and the result obtained by the conventional effective media theory. The nanoparticle size has little effect on the thermal conductivity of nanofluids in such a nanoparticle diameter range. Acknowledgements Support for this work by the US National Science Foundation under Grant Number CBET and Chinese National Natural Science Foundation under Grants Nos and is gratefully acknowledged. References 1. P. Keblinski, R. Prasher, J. Eapen, J. Nanopart. Res. 10, 1089 (2008) 2. S. Lee, S.U.S. Choi, S. Li, J.A. Eastman, J. Heat Transf. 121, 280 (1999) 3. J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Appl. Phys. Lett. 78, 718 (2001) 4. S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Appl. Phys. Lett. 79, 2252 (2001) 5. P. Keblinski, S.R. Phillpot, S.U.S. Choi, J.A. Eastman, Int. J. Heat Mass Transf. 45, 855 (2002) 6. A. Gupta, R. Kumar, Appl. Phys. Lett. 91, (2007) 7. P. Bhattacharya, S.K. Saha, A. Yadav, P.E. Phelan, R.S. Prasher, J. Appl. Phys. 95, 6492 (2004) 8. R. Prasher, P. Bhattacharya, P.E. Phelan, Phys. Rev. Lett. 94, (2005) 9. W. Evans, Appl. Phys. Lett. 88, (2006) 10. J. Eapen, J. Li, S. Yip, Phys. Rev. Lett. 98, (2007) 11. C.J. Yu, A.G. Richter, J. Kmetko, S.W. Dugan, A. Datta, P. Dutta, Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 63, (2001) 12. K.C. Leong, C. Yang, S.M.S. Murshed, J. Nanopart. Res. 8, 245 (2006) 13. W. Yu, S.U.S. Choi, J. Nanopart. Res. 5, 167 (2003) 14. H.U. Kang, S.H. Kim, J.M. Oh, Exp. Heat Transf. 19, 181 (2006) 15. O.M. Wilson, X.Y. Hu, D.G. Cahill, P.V. Braun, Phys. Rev. B, Condens. Matter Mater. Phys. 66, (2002) 16. C. Nie, W.H. Marlow, Y.A. Hassan, Int. J. Heat Mass Transf. 51, 1342 (2008) 17. S.M.S.Murshed, K.C.Leong, C.Yang, Int. J. Therm. Sci.44, 367 (2005) 18. H.T. Zhu, C.Y. Zhang, S.Q. Liu, Y.M. Tang, Y.S. Yin, Appl. Phys. Lett. 89, (2006) 19. S. Sarkar, R.P. Selvam, J. Appl. Phys. 102, (2007) 20. L. Li, Y.W. Zhang, H.B. Ma, M. Yang, J. Nanopart. Res. 12, 811 (2009) 21. M.S. Daw, M.I. Baskes, Phys. Rev. B, Condens. Matter Mater. Phys. 29, 6443 (1984) 22. D.A. McQuarrie, Statistical Mechanics (University Science Books, Sausalito, 2000) 23. C. Hoheisel, Theoretical Treatment of Liquids and Liquid Mixtures (Elsevier, Amsterdam, 1993) 24. R. Vogelsang, C. Hoheisel, J. Chem. Phys. 86, 6371 (1987) 25. A.J.H. McGaughey, M. Kaviany, Int. J. Heat Mass Transf. 47, 1783 (2004) 26. Z.M. Zhang, Nano/Microscale Heat Transfer (McGraw-Hill,New York, 2007) 27. J.C. Maxwell, Electricity and Magnetism (Clarendon, Oxford, 1873) 28. M. Chopkar, P.K. Das, I. Manna, Scr. Mater. 55, 549 (2006) 29. M.P. Beck, Y.H. Yuan, P. Warrier, A.S. Teja, J. Nanopart. Res. 11, 1129 (2009)