Influence of nanoparticle properties on the thermal conductivity of nanofluids by molecular dynamics simulation

Abstract

The properties of nanoparticles (including shape, size, material, and volume concentration) may significantly influence the thermal properties of nanofluids. Through molecular dynamics simulations, the aim of this study is to investigate the influence of nanoparticle properties on the thermal conductivity of nanofluids and find an effective criterion for predicting thermal conductivity enhancement. By establishing a series of simulation models, thermal conductivities of nanofluids were calculated on the basis of the Green–Kubo formula. It was found that all the nanoparticle properties that have been considered in this work influence the thermal conductivity of nanofluids, and the influence rules were discussed. Furthermore, there is a positive correlation between the distribution of atomic potential energy and the thermal conductivity of nanofluids. Therefore, the ratio of energetic atoms in nanoparticles is proposed to be the criterion for predicting enhancement of the apparent thermal conductivity of nanofluids.

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1. Introduction

Nanofluids have received considerable attention in recent years due to their strong heat transfer properties, which possess important applications in heat transfer. The concept of nanofluids was first proposed by Choi in 1995 (ref. 1) to describe the fluids containing nanometer-sized particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid.^2,3 Numerous experimental works have discovered that nanofluids exhibit thermal properties superior to those of the base fluid or conventional solid-liquid suspensions.^4–7 Most of the measured thermal properties of nanofluids significantly exceed the values predicted by classical macroscopic theories and models.^8–10 Nanofluids exhibit novel properties, including significantly higher thermal conductivity and enhanced convective heat transfer coefficients, which make them potentially useful in many applications in heat transfer, including engine cooling/vehicle thermal management, hybrid-powered engines, and fuel cells.¹¹ Much research in recent years has been focused on the explanation of the significantly high thermal properties of nanofluids. However, although there is a substantial number of mechanisms proposed and modeling work related to their enhanced thermal conductivity,^12–14 systematic mechanisms or models that are suitable for nanofluids are still lacking. Furthermore, conventional analysis, such as computational fluid dynamics (CFD), failed to reveal the specific microscopic mechanisms in the flow and heat transfer of nanofluids.

One effective way to investigate the enhanced heat transfer in nanofluids, especially the strengthening mechanisms at the microscopic level, is molecular dynamics (MD) simulations.¹⁵ Shukla et al. established an MD simulation model of nanofluids with ideal particles, and their results showed that the liquid adsorption layer at the surface of nanoparticles possesses a higher thermal conductivity than that of the single-phase base fluid.¹⁶ Sobhan et al. investigated the thermal conductivity of Pt–H₂O nanofluids by MD simulation and discussed the influence rule of temperature and volume concentration for the thermal conductivity of nanofluids.¹⁷ Sarkar et al. studied the thermal conductivity and diffusion coefficient of Cu–Ar nanofluids by MD method. Through analyzing the mean square displacement of nanofluid components, they found that the range of fluidic molecular motion is increased due to nanoparticles, and the heat conduction is therefore strengthened.¹⁸ Eapen et al. investigated the heat conduction mechanisms in Xe–Pt nanofluids through MD simulations and revealed that the interfacial effect between the liquid and solid is the main reason for explaining the thermal conductivity enhancement.¹⁹ Selvam et al. simulated the thermal conductivity of Cu–Ar nanofluids with different volume concentrations by MD method and investigated the possible heat conducting mechanisms in nanofluids.²⁰ Liu et al. simulated the thermal conductivities of Al–Ar nanofluids and investigated the influence of volume concentration for thermal conductivity.²¹ Their simulation work revealed that the role of interactions between liquid molecules and solid nanoparticles is stronger than that of irregular Brownian motion of nanoparticles for the thermal conductivity enhancement. Li et al. simulated the thermal conductivities of Cu–Ar nanofluids by MD simulations.^22,23 They focused on the effect of the absorption layer for heat conduction in nanofluids, and proposed the absorption layer at the surface of solid nanoparticle as an important mechanism for explaining the significant increase in thermal conductivity. Liang et al. proved that the thermal conductivity of the interface layer between solid nanoparticles and liquid is 1.6–2.5 times larger than that of the base fluid by MD simulations.²⁴ They have also proposed that the cluster of nanoparticles plays an important role in thermal conduction. Kang et al. simulated thermal conductivity of Cu–Ar nanofluids with various potential functions to examine the influence.²⁵ Their simulations have proved that the absorption layer is one important reason for explaining thermal conductivity enhancement in nanofluids. Lin simulated the thermal conductivity of EG–Cu nanofluids by MD method and studied the mechanisms of heat conduction in nanofluids.²⁶ They have proposed a Layer–Maxwell model for predicting the effective thermal conductivity. Sun et al. simulated the effective thermal conductivity of Cu–Ar nanofluids confined between plates.^27,28 They have proved that nanoparticles rotate fast in the base fluid, and the micro convection effect due to the rotation of nanoparticles is the main reason for explaining the effective thermal conductivity enhancement. According to the existing research, adding nanoparticles could effectively increase the thermal conductivity of nanofluids. However, few studies report systematic MD simulations on the influence of nanoparticle properties on the thermal conductivity of nanofluids and give effective criterion for the thermal conductivity enhancement.

The present paper carries out a series of MD simulations to summarize the influence rules of nanoparticle properties for thermal conductivity of nanofluids, and attempts to find a criterion for estimating the enhancement degree of effective thermal conductivity. Various MD simulation models of nanofluids containing nanoparticles of different shapes, sizes, materials, and volume concentrations are established. Through calculating the thermal conductivity of nanofluids on the basis of Green–Kubo theory, the influence rules of nanoparticle properties for the thermal conductivity are discussed. In addition, the ratio of energetic atoms in nanoparticles is proposed as the criterion for predicting the ability of nanoparticles to enhance the apparent thermal conductivity of nanofluids.

2. Methodology

MD simulation calculates thermal conductivity via integrating the autocorrelation function of microscopic heat flow, and the Green–Kubo formula is written as:²⁹where k is the thermal conductivity, k_B is the Boltzmann constant, V is the volume, T is the temperature, J⃑_q is the instantaneous microscopic heat current vector, and the angular brackets denote the ensemble average. 〈J⃑_q(0)J⃑_q(t)〉 is the autocorrelation function of microscopic heat flow, and the heat flow vector is calculated by:where J⃑_q is the instantaneous microscopic heat current vector, r_i is the position vector of the particle i; E_i represents the excessive energy; h stands for the enthalpy per particle which is calculated as the sum of the average kinetic energy, potential energy, and average virial terms per particle of each species;^18,23 r_i is the position vector of particle i; m_i is the mass of particle i; U(r_ij) is the pair potential between particles i and j.

With regard to the biphasic system in this work, the heat flow vector is calculated by:^30–32

where α and β denote the species of particles, and i and j are the number of particles; N_α and N_β are the numbers of particles α and β; J⃑_q is the heat current vector; m_α is the mass of the particle i; [small nu, Greek, vector]_ja denotes the velocity of a particle j of kind α; I⃑ is the unit vector; h_α represents the average enthalpy of particle α. In the equation, three modes including the kinetic (K), the potential (P), and collision (C) constitute the heat flux.

In the simulation, heat flux is calculated at each simulation time step. As the time steps are not continuous, eqn (1) is actually a form of summation, which can be rewritten as:

where N represents the total calculation steps after the simulation system reaches equilibrium; M represents the calculation steps used for calculating average value, J⃑(m + n) is the heat flux at the (m + n) time step.

Currently, most MD simulations employ empirical or semi-empirical potentials to describe interactions between atoms.^17,18,20,21 Among these potential functions, Lennard-Jones (LJ) is the most commonly used one to describe interactions between liquid atoms. Previous studies have verified that LJ potential can successfully depict interactions between atoms in nanofluids. In this work, LJ potential is employed to describe interactions between Ar atoms. For interactions between Ar and Cu atoms, the Lorentz–Berthelot mixing rule is employed to calculate the LJ parameters. The LJ parameters used in this work are given in Table 1.^29,33 For interactions between Cu atoms, EAM potential is used.²⁹ In EAM potential, the potential energy of an atom, i, is given by.

where r_ij is the distance between atoms i and j, U_αβ is a pair-wise potential function, ρ_β is the contribution to the electron charge density from atom j of type β at the location of atom i, and F is an embedding function that represents the energy required to place atom i of type α into the electron cloud.

Table 1 LJ potential function parameters

Atom pair	σ (Å)	ε (gÅ^2/fs^2)
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
Au–Au	2.637	7.0731314 × 10^−27
Ar–Cu	2.872	1.0415300 × 10^−27
Ar–Fe	2.863	1.1810369 × 10^−27
Ar–Ag	3.025	9.5586860 × 10^−28
Ar–Au	3.021	1.0816230 × 10^−27

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3. Simulation model and verification

3.1 Simulation model

Currently, Ar is chosen as the base fluid in most MD simulation studies on nanofluids, and the accuracy of simulation results is verified to be accurate and reliable.^17,18,20,21 Furthermore, using liquid Ar as the base fluid could sharply reduce the computing time. Therefore, in this work liquid Ar is chosen to be the base fluid.

To investigate the influence of nanoparticle properties on the thermal conductivity of nanofluids, the following factors have been considered: nanoparticle concentration, nanoparticle diameter, nanoparticle material, and nanoparticle shape. In this work, the nanoparticle concentrations include 0.5%, 1%, 2%, 3%; the nanoparticle diameters include 2 nm, 4 nm, 6 nm; the nanoparticle materials include Cu, Ag, Fe, Au; the nanoparticle shapes include spherical, bar-like, and planar. For the Ar-based nanofluids with spherical nanoparticles, taking into account the different materials of Cu, Ag, Fe, and Au, and the volume concentrations of 0.5%, 1%, 2%, and 3%, the MD simulations are 48 times. Moreover, each simulation consumes about 20 hours. By fixing the nanoparticle material as Cu, and changing the nanoparticle shapes under the same reference diameter of 2 nm, the influence of nanoparticle shapes on the thermal conductivity is investigated. For volume concentrations of 0.5%, 1%, 2%, and 3%, the simulations are 8 times. In the model establishment, to ensure the total volumes of nanoparticles in different shapes are the same, the basal diameters of bar-like and planar nanoparticles are 1 nm and 4 nm, respectively, as shown in Table 2.

Table 2 Nanoparticles in different shapes with equal volumes

Nanoparticles in different shapes	Bar-like nanoparticle	Spherical	Planar
Images
Dimension	1 nm × 5.33 nm	2 nm	4 nm × 0.33 nm
Surface area to volume ratio (S/V)	4.375	3	7.061

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3.2 Simulation details and result verification

In this simulation work, canonical ensemble NVT is used. The temperature is fixed at 86 K. The size of the simulation model is specified based on the nanoparticle volume fraction and fluid density, and the entire model particles (including Ar and metallic atoms) were arranged in a face centered cubic (FCC) lattice. The initial velocity was determined by the random sampling of Maxwell–Boltzmann distribution with a given random velocity direction. A periodic boundary condition was used in MD simulation with a cut-off radius of 2.5σ_Ar for the cutting-off of potential energy. The constraint algorithm called “SHAKE” algorithm, which was first proposed by Ryckaert et al., was selected to fix the geometric structure of the nanoparticle,³⁴ and 5^th order Gear algorithm was used to solve the motion equations of the particle. In MD simulations, longer simulation time and more simulation time steps ensure more accurate statistical calculation results.²³ In this work, the total simulation time of each case was set as 4200 ps (about 2.1 million time steps) and 2 fs for each time step.²³ The first 200 ps is the relaxation process to ensure the system reaching equilibrium, and the last 4000 ps simulation is the main process for the thermal conductivity calculation.

In MD simulations, the model should be as small as possible to reduce the time consumption of the simulation. On the other hand, the model should be big enough to avoid large dynamical perturbations in the statistical post-processing. Therefore, there exists a conflict between the result accuracy and the simulation capability of the computer. In order to determine the simulation result independence from the total particle amount, the influence of particle number for the thermal conductivity calculation result is studied first. The simulations take different number of Ar atoms as the object of study, and the thermal conductivities are calculated, as shown in Fig. 1. It could be found in the figure that when the particle number is larger than 500, the influence of particle number on the thermal conductivity calculation results becomes inconspicuous. This result coincides with the result of Sarkar et al.¹⁸ Moreover, in their study, the simulated result agrees with the experimental values when the particle number is larger than 1372 for the nanofluid simulation model. In this work, larger simulation models were used. For instance, in the case of 2 nm Cu nanofluids of 1% volume fraction, the total particle number is 9070. With 4631 Ar atoms, the thermal conductivity of liquid Ar at 86 K is calculated to be 0.12796 W mK⁻¹. This result is consistent with 0.127 W mK⁻¹ by Sarkar et al.¹⁸ and 0.126 W mK⁻¹ by Li et al.,²³ and comparable to the experimental result of 0.132 W mK⁻¹.¹⁸ The relative error between the present simulation result and experimental value is 3.1%, which indicates that the current model is acceptable for the calculation. Fig. 2 shows the influence of simulation time on the calculation of thermal conductivity. It could be found in the figure that after 2 ps, the simulation result begins to stabilize. Fig. 3 illustrates the convergence of heat flux autocorrelation function (HACF) over time for the simulation system of liquid Ar. It could be found that the system is fully converged within 10 ps, which shows that the established system is stable and reliable. Fig. 4 shows the comparison between the present MD simulation results and the existing literature,^18,23,35 and it could be found that the present results are reliable.

Fig. 1 Influence of particle number.

Fig. 2 Influence of simulation time.

Fig. 3 Relationship between HACF and simulation time.

Fig. 4 Comparison between MD simulation results for Cu–Ar nanofluids.

4. Results and discussion

4.1 Simulation results of thermal conductivity

The MD simulations in this work have confirmed that adding nanoparticles can effectively improve the thermal conductivity of nanofluids. All the calculated thermal conductivities of nanofluids are higher than that of the base fluid. In addition, the contributions of several nanoparticle properties, including volume fraction, material, nanoparticle diameter and nanoparticle shape, for the thermal conductivity of nanofluids are different. The simulated results are shown in Table 3.

Table 3 Ratios of thermal conductivity enhancement

Diameters (nm)	Volume fractions (%)	Cu	Ag	Au	Fe
2	0.5	1.112	1.326	1.062	1.036
	1	1.149	1.405	1.112	1.059
	2	1.359	1.498	1.205	1.079
	3	1.414	1.570	1.322	1.096
4	0.5	1.095	1.287	1.055	1.041
	1	1.134	1.325	1.096	1.065
	2	1.303	1.396	1.175	1.085
	3	1.376	1.462	1.202	1.112
6	0.5	1.065	1.244	1.049	1.045
	1	1.097	1.265	1.086	1.079
	2	1.266	1.326	1.152	1.098
	3	1.336	1.369	1.256	1.132

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Fig. 5 shows the MD simulation results of thermal conductivities for nanofluids containing spherical nanoparticles. In this case, the nanoparticle volume fractions, nanoparticle diameter, and thermal conductivity of nanoparticles are considered. The influence rules of these nanoparticle properties for the thermal conductivity of nanofluids could be concluded from the figure. It is obvious that thermal conductivity increases with an increased volume fraction of nanoparticles. Take Cu nanofluids as an example; the ratios of thermal conductivity enhancement are 1.11, 1.15, 1.36 and 1.42 when the nanoparticle volume fractions are 0.5%, 1%, 2% and 3%, respectively. It could also be concluded from the figure that the thermal conductivity of nanofluids increases with a decreased nanoparticle diameter. With the same nanoparticle material and volume fraction, smaller nanoparticles (2 nm) lead to a higher thermal conductivity of nanofluids. For example, for Cu nanofluids with a volume fraction of 1%, the thermal conductivity enhancement ratios for nanofluids containing nanoparticles with diameters of 2 nm, 4 nm and 6 nm are 1.15, 1.135 and 1.10, respectively. Furthermore, the influence of nanoparticle materials on the thermal conductivity could be roughly concluded; the ratio of thermal conductivity enhancement follows the increments of the nanoparticle bulk material. For instance, the thermal conductivities of Ag, Cu, Fe and Au bulk materials at 293 K are 427 W mK⁻¹, 398 W mK⁻¹, 315 W mK⁻¹ and 81.1 W mK⁻¹, respectively. Moreover, the ratios of thermal conductivity enhancement for Ag, Cu, Fe and Au nanofluids are 1.41, 1.15, 1.11 and 1.08 sequentially, when the nanoparticle volume fraction is 1% and the nanoparticle diameter is 2 nm.

Fig. 5 MD simulation results for thermal conductivity of nanofluids.

In order to further examine the impact of nanoparticle properties on the thermal conductivity of nanofluids, a new parameter, the ratio of thermal conductivity enhancement by nanoparticle volume fraction (denoted by K), is introduced. The physical significance of this parameter is to evaluate the specific value of the thermal conductivity increase ratio and nanoparticle volume fraction. The definition of the ratio of thermal conductivity enhancement by nanoparticle volume fraction, K, is written as:

where k and k_f are the thermal conductivities of nanofluids and base fluid, respectively; V_np is the total volume of nanoparticles; and V represents the volume of nanofluids.

Through analyzing the values of K, the impact of nanoparticle properties on thermal conductivity can be further evaluated. Fig. 6 and 7 show the impact of the volume fraction of nanoparticles and thermal conductivity of nanoparticles on the ratio of thermal conductivity enhancement. In the figures, it could be found that with various volume fractions and nanoparticle diameters, the contribution of Ag nanoparticles to the ratio of thermal conductivity enhancement is the highest, followed by Cu, Au, and Fe nanoparticles. However, on comparing K values, it is found that this influence is weakened when the volume fraction is increased, as shown in Fig. 6, and the influence is also weakened by the increased nanoparticle size, as shown in Fig. 7.

Fig. 6 Comparison of K values against nanoparticle volume fraction for various nanofluids.

Fig. 7 Comparison of K values against nanoparticle size for various nanofluids.

Fig. 8–10 show the ratios of thermal conductivity enhancement for nanofluids by nanoparticles with different thermal conductivities. With the same nanoparticle size, the impact of nanoparticles with a larger thermal conductivity on the increase ratio of thermal conductivity is greater. Moreover, on comparing the situations of three kinds of diameters, this influence is found to be more obvious for smaller nanoparticles, which manifests in the value of K.

Fig. 8 Comparison of K values for 2 nm nanoparticles.

Fig. 9 Comparison of K values for 4 nm nanoparticles.

Fig. 10 Comparison of K values for 6 nm nanoparticles.

Fig. 11 shows the impact of the surface area to volume ratio (S/V) or nanoparticle shape on the thermal conductivity of nanofluids. In the present work, three kinds of nanoparticle shapes, including spherical (S/V = 3), planar (S/V = 7.061) and bar-shape (S/V = 4.375), have been considered. It is found that the ratio of thermal conductivity enhancement of nanofluids increases with an increasing nanoparticle S/V value. For the cases of S/V = 3, 4.375, and 7.061, the increase ratios are 1.23, 1.21 and 1.15, respectively.

Fig. 11 Enhancement ratios of thermal conductivity with nanoparticles in different shapes.

Fig. 12 shows the relationship between K and S/V, and it could be found that the contribution of nanoparticles with a larger S/V value is greater for the thermal conductivity enhancement. When the nanoparticle is planar in shape, the K value is greater, which indicates that this type of nanoparticle is more conducive to thermal conductivity enhancement.

Fig. 12 Comparison of K values for nanoparticles with different S/V values.

In Fig. 13, the thermal conductivities of Cu nanofluids are compared to that of Ag nanofluid with spherical nanoparticles, under the condition that the reference diameter is the same at 2 nm. It is found that the nanoparticles with a larger S/V value contribute more for the thermal conductivity improvement. For instance, the thermal conductivity ratio of nanofluid with Cu nanoparticles in planar shape (S/V = 7.061) is higher than that of Cu nanofluids containing bar-like (S/V = 4.375) or spherical nanoparticles (S/V = 3). Furthermore, the ratio of thermal conductivity enhancement by Cu nanoparticles in planar shape (S/V = 7.061) is close to that of Ag nanofluids with nanoparticles in spherical shape.

Fig. 13 Enhancement ratios of thermal conductivity with nanoparticles in different materials.

Fig. 14 compares the K values of Cu nanofluids and Ag nanofluids, containing nanoparticles with different S/V values. It is found that when the volume fraction is low (0.5%), the contribution of nanoparticles with a larger thermal conductivity is more obvious. On increasing the S/V value of nanoparticles, the contribution of nanoparticles for thermal conductivity improvement can be increased. Under the condition that the volume fraction is relatively large (3%), the impact of nanoparticle properties on the thermal conductivity is reduced. In this situation, for nanoparticles with different S/V values and thermal conductivities, the influences on the thermal conductivity enhancement of nanofluids come close.

Fig. 14 Comparison of K values for nanoparticles with different materials.

4.2 Criterion for the enhancement of apparent thermal conductivity

Through MD simulations, it was found that the nanoparticle properties, including the bulk material property and nanoparticle shapes, directly affect the degree of thermal conductivity enhancement. By analyzing the MD simulation results for various nanoparticles, it is found that the distributions of atomic potential energy of these nanoparticles are different. Moreover, there is a positive correlation between the distribution of atomic potential energy and the thermal conductivity of nanofluids. The nanoparticles that are better for thermal conductivity enhancement contain more atoms that possess high atomic potential energy (energetic atoms). Therefore, the ratio of energetic atoms in a nanoparticle could be proposed to be a criterion for estimating the degree of strengthened thermal conductivity in nanofluids. The ratio of energetic atoms in a nanoparticle is defined as:where N_E is the quantity of energetic atoms in a nanoparticle, and N represents the total quantity of atoms in a nanoparticle.

In MD simulations, the values of atomic potential energy that all the atoms possess can be the output at any simulation time. Then, the data are solved by a statistical method, and the distributions of atomic potential energy for nanoparticles can be obtained. With a given judgment standard for energetic atoms, the ratio of energetic atoms in a nanoparticle, E, can be acquired.

Fig. 15 illustrates the distributions of atomic potential energy of nanoparticles in different materials. The results in the figure are the average value of atomic potential energies at the simulation times of 3000 ps, 3500 ps and 4000 ps. It could be found in the figure that the atomic potential energy distributions of different nanoparticles are significantly different. For instance, the general distribution of atomic potential energy of Ag nanoparticles is the largest, and that of Fe nanoparticles is the lowest. Furthermore, this indicates that in Ag nanoparticles, the ratio of atoms with a high potential energy is larger than that of Fe nanoparticles under the condition of the same nanoparticle size and shape. One can nominate the standard for delimiting the energetic atoms in a nanoparticle. For instance, if an atom that possesses an atomic potential energy larger than −8e–19 J is regarded as an energetic atom, the ratio of energetic atoms, E, in a nanoparticle can be then calculated according to eqn (7). In this case, the ratios of energetic atoms E of Ag, Cu, Au, Fe nanoparticles in spherical shape are 0.782, 0.455, 0.374, 0.133, respectively. It can be found that the ratio of energetic atoms in a nanoparticle that possesses a large thermal conductivity is higher than that with a low thermal conductivity under the same condition. Conversely, nanoparticles with a larger E value are better for thermal conductivity enhancement in nanofluids.

Fig. 15 Atomic potential energy distributions of various nanoparticles.

Fig. 16 shows the influence of surface area to volume ratio S/V for the distribution of atomic potential energy. In the figure, three types of Cu nanoparticles with different S/V values (including 3, 4.375 and 7.061) have been considered. It could be concluded that nanoparticles that possess different S/V values show disparate atomic potential energy distributions. If the same standard for delimiting an energetic atoms −8e–19 J is used, then the ratios of energetic atoms in the nanoparticles with S/V values of 3, 4.375, and 7.061 are 0.455, 0.508, and 0.566, respectively. Moreover, this result indicates that nanoparticles with a larger S/V value possess a larger E value, which is better for thermal conductivity enhancement in nanofluids. Fig. 17 illustrates the comparison of atomic potential energy distributions between spherical Ag nanoparticles (S/V = 3) and non-spherical Cu nanoparticles (S/V = 4.375 and 7.061), under the condition that the reference diameter is the same at 2 nm. It could be found that the E value of spherical Ag nanoparticles (S/V = 3) is 0.782, which is still larger than that of Cu nanoparticles with a larger S/V value. Furthermore, this illustrates that the ratio of energetic atoms E in a nanoparticle with a large thermal conductivity is larger.

Fig. 16 Comparison of atomic potential energy distributions for nanoparticles with different shapes.

Fig. 17 Comparison of atomic potential energy distributions for nanoparticles with different materials.

5. Conclusions

In this paper, the influence rules of nanoparticle properties for the thermal conductivity of nanofluids are investigated by MD simulation, and an effective criterion for predicting the enhancement of apparent thermal conductivity is proposed.

1. On the basis of the Green–Kubo theory, thermal conductivities of nanofluids with various nanoparticles have been calculated through MD simulations. Thermal conductivities of nanofluids are found to increase with an increased volume fraction of nanoparticles or decreased nanoparticle diameter; the ratio of thermal conductivity enhancement follows the increment in nanoparticle bulk thermal conductivity and an increased nanoparticle S/V value.

2. Through defining the ratio of thermal conductivity enhancement by nanoparticle volume fraction, K, the impacts of nanoparticle properties on thermal conductivity are further evaluated. It was found that with various volume fractions and nanoparticle diameters, the contribution of Ag nanoparticles for the ratio of thermal conductivity enhancement is the highest, followed by Cu, Au, and Fe nanoparticles. However, on comparing K values, it is found that this influence is weakened when the volume fractions or the nanoparticle size is increased. With the same nanoparticle size, the impact of nanoparticles with a larger thermal conductivity on the increase ratio of thermal conductivity is greater. In addition, this influence is found to be more obvious for smaller nanoparticles, which manifests in the value of K. It could also be found that the contribution of nanoparticles with larger S/V values is greater for the thermal conductivity enhancement. On increasing the S/V value of nanoparticles, the contribution of nanoparticles on thermal conductivity improvement can be increased. However, under the condition that the volume fraction is relatively large (3%), the impact of nanoparticle properties for the thermal conductivity is reduced. In this situation, for nanoparticles with different S/V values and thermal conductivities, the influences on the thermal conductivity enhancement of nanofluids come close.

3. The ratio of energetic atoms in nanoparticles, E, is proposed to be an effective criterion for judging the impact of nanoparticles on the thermal conductivity of nanofluids. It is found that the ratio of energetic atoms in a nanoparticle that possesses a large thermal conductivity is higher than that with a low thermal conductivity, under the same condition. Conversely, nanoparticles with a larger E value are better for thermal conductivity enhancement in nanofluids. Nanoparticles with a larger S/V value possess larger E values and are better for enhancing the thermal conductivity of nanofluids. Moreover, the ratio of energetic atoms, E, in a nanoparticle with a large thermal conductivity is larger.

Acknowledgements

The authors are grateful to the China Postdoctoral Science Foundation funded project (Grant no. 2014T70330, 2013M540284), the Project (HIT.NSRIF.2015116), supported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology, and the Project (HIT(WH)201301), supported by Scientific Research Foundation of Harbin Institute of Technology at Weihai. We acknowledge the reviewers' comments and suggestions very much, which are valuable in improving the quality of our manuscript.

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