Thermal conductivity and viscosity of nanofluids: A review of recent molecular dynamics studies

Chemical Engineering Science 174 (2017) 67–81

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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Review

Thermal conductivity and viscosity of nanofluids: A review of recent molecular dynamics studies Fatemeh Jabbari a,⇑, Ali Rajabpour b,c, Seifollah Saedodin a a

Faculty of Mechanical Engineering, Semnan University, Semnan, Iran Mechanical Engineering Departments, Imam Khomeini International University, Qazvin, Iran c School of Nano-Sciences, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 22 July 2017 Received in revised form 24 August 2017 Accepted 30 August 2017 Available online 1 September 2017 Keywords: Nanofluid Nanoparticles Thermal conductivity Shear viscosity Molecular dynamics simulation Enhanced heat transfer

a b s t r a c t The heat-transfer enhancement of nanofluids has made them attractive and the subject of many theoretical and experimental researches over the last decade. Of the theoretical approaches employed to investigate nanofluid properties, molecular dynamics (MD) simulation is a popular computational technique that is widely used to simulate and investigate thermophysical properties of nanofluids. In this paper, we review and discuss the MD studies conducted on the thermophysical properties of nanofluids, considering the thermal conductivity and shear viscosity as two important factors for the industrial application of nanofluids. In this study, after introducing different MD methods to calculate those parameters, we classify and review various influential effects including the volume fraction of nanoparticles, nanofluid temperature, Brownian motion of the nanoparticles, as well as the nanoparticle shape and size in terms of the thermal conductivity and viscosity of nanofluids. Viscosity has been studied to a lesser extent than the thermal conductivity of nanofluids. In our review, we note the similarities and differences between previous MD reports on nanofluids, and we highlight gaps and potential ideas that may be of interest for future studies. Ó 2017 Elsevier Ltd. All rights reserved.

Contents 1. 2. 3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular dynamics simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Calculation of temperature and pressure in MD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Computation of thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Non-equilibrium molecular dynamics (NEMD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Equilibrium molecular dynamics (EMD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Factors affecting the thermal conductivity of nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Volume fraction of nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Nanofluid temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Brownian motion of the nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. The number and size of nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Aggregation and clustering of nanoparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6. Molecular layer around the nanoparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7. Material and shape of nanoparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.8. Interfacial thermal resistance (Kapitza resistance). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Future ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Computation of shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. E-mail address: [email protected] (F. Jabbari). http://dx.doi.org/10.1016/j.ces.2017.08.034 0009-2509/Ó 2017 Elsevier Ltd. All rights reserved.

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4.1.1. NEMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. EMD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Factors affecting the shear viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Volume fraction of nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Nanoparticle size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Nanofluid temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Nanoparticle aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Future ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Nanofluids are mixtures of two components, namely a base fluid and a small volume fraction (within the range of sub 1% up to 10%) of solid particles with sizes that are usually less than 100 nm (Sergis and Hardalupas, 2014). Fig. 1 represents a molecular dynamics (MD) model of a nanofluid containing carbon nanotube (CNT) as the nanoparticle with water as the base fluid. The concept of nanofluids was first proposed by Das et al. (2008) in the mid-nineties as a way of superseding micron-sized solids that were used inside conventional coolants to enhance their thermal conductivity. Researchers have devoted much effort to further understand the physics governing nanofluids, including calculating the thermal conductivity enhancement of various types of nanofluids using different methods, and to study mechanisms of heat transfer in nanofluids (Choi et al., 2001; Li et al., 2010; Godson et al., 2010; Duangthongsuk and Wongwises, 2009; Esfe et al., 2014). Compared to conventional fluids, a higher thermal conductivity and better convective heat transfer as well as lower pressure drop have made nanofluids a newfound technology that has good potential for heat-transfer applications. Therefore, the use of nanofluids has been considered to meet new challenges in refrigeration techniques and in the thermal handling of high heat flux equipment (Bahiraei, 2016). Researchers are being challenged to detect the many unanticipated thermal characteristics of these fluids, as well as to propose new mechanisms and unusual models to illustrate their behavior. Significant differences have been reported by different researchers in their results obtained for the same problem

78 78 78 78 78 78 79 79 79 79

(Said, 2016; Dalkilic et al., 2017; Huminic et al., 2017; Abu-Nada, 2017). Many theoretical and empirical studies have been performed to determine the thermophysical properties of nanofluids, and different methods and models have been used (Qiao et al., 2017; Satti et al., 2017). For instance, the transient hot wire method is widely used as an experimental technique to measure the thermal conductivity of fluids, where a hot wire is implanted in the fluid, which serves as both a heat source and a thermometer (Kleinstreuer and Feng, 2011; Feng, 2010; Vadasz, 2010). Mathematical models, theoretical analyses, and related computer simulations may provide a better physical perspective that helps clarify presumably anomalous enhancements of the thermophysical properties of nanofluids (Galliero and Volz, 2008; Sun et al., 2011; Gao et al., 2012; Kang et al., 2012). Classical mathematical models such as Maxwell (1873), Hamilton and Crosser (1962), Jeffrey (1973) and Davis (1986) were derived from continuum formulations that generally include only the nanoparticle volume fraction and nanoparticle size/shape, and they assume diffusive heat transfer in both solid and fluid phases. In this paper, we review and focus on MD calculations to determine the thermophysical properties of nanofluids. Owing to the large and growing volume of MD works in the nanofluid area of research, the goal of this review is to find differences and similarities between past MD studies as well as to provide an overview of how well MD simulations can predict the thermophysical properties of the nanofluids. The structure of this paper is organized as follows: First, we briefly describe the method of the MD simulation and to calculate the temperature and pressure, which are two important parameters when predicting the thermophysical properties of nanofluids. Then, we focus on the calculation of the thermal conductivity of nanofluids as the most important thermophysical property of nanofluids in industrial applications, and we present factors that affect the thermal conductivity, as reported in previous MD works. Finally, we describe how MD can be used to calculate the viscosity of nanofluids, which is one of the most important challenges of using nanofluids in industrial applications, and we review previous MD studies. 2. Molecular dynamics simulation

Fig. 1. Nanofluid molecular dynamic model: CNT nanoparticle (red-colored) and surrounding water (blue-colored).

There are two main approaches that are discussed in literature for theoretical studies. The first method uses kinetic theories and continuum models, which are macroscopic methods that include the Boltzmann transport equation, while the second method is based on the first-principle atomistic simulations or quantum mechanical models, which are essential microscopic methods that generally include MD simulations (Zhenhua et al., 2005) (Fig. 2). MD is a computer simulation method that computes the physical movements of all atoms and molecules, and it is therefore a type of N-body simulation. System components (atoms and molecules) are allowed to interact for a fixed period of time until the system reaches a dynamic evolution. In this method, the Newton’s

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Fig. 2. The position of MD in different modeling and simulation approaches: Reynolds-averaged Navier-Stokes (RANS, large eddy simulation (LES), direct numerical simulation (DNS), lattice Boltzmann method (LBM), dissipative particle dynamics (DPD), direct simulation Monte Carlo (DSMC), and MD (Luo et al., 2009).

equations of motion for a system of interacting particles are solved, and the trajectories of atoms and molecules are specified, where forces between the particles and their potential energies are calculated using interatomic potentials or molecular mechanics force fields. The interatomic potentials used in molecular dynamics simulations include two bonded or non-bonded modes that the nonbonded mode shows the formed non-bonded energy that is a sum over interactions between the system particles. The simplest non-bonded potential is the ‘‘pair potential” that is calculated by sum of available energy between pairs of atoms (Allen, 2004; Sachdeva, 2009). There are a variety of potential functions, for instance: Morse potential, Embedded atom model (EAM), etc., while the simplest and most commonly used pair potential in many studies is Lennard-Jones(LJ) Jones, 1924 which the particles are absorbed by each other when they are far apart and are repulsed from each other when they are very close together. For two atoms i and j the LJ potential is represented as:


12 r

uLJ ¼ 4e

r




r6  r

ð1Þ

where e is the depth of the potential well, r is the distance between the particles, and r is the distance at which the potential reaches its minimum. The MD method was first presented by Alder and Wainwright at the end of 1950 (Alder and Wainwright, 1957, 1959). Initially, it was extended within the field of theoretical physics, but it is currently employed mostly in materials science, chemical physics, and the modeling of biomolecules (Rajabpour et al., 2014; Faraji et al., 2016; Bazrafshan and Rajabpour, 2017). Computer simulations are used in an attempt to increase our understanding of the properties of molecules in terms of their structure and the microscopic interactions between them. In addition to conventional experiments, this tool enables us to understand the phenomena that cannot be realized using other methods. The two main groups of simulation techniques are MD and the Monte Carlo (MC) method; in addition, there is a wide range of combined techniques that combine features of both methods. In this study, we focus on MD. The obvious advantage of MD over MC is that it can compute dynamic properties of the system, such as time-dependent responses to perturbations, transfer coefficients, and rheological properties (Allen, 2004; Rajabpour et al., 2011). The main advantage of the MD method is its ability to rapidly review non-equilibrium processes at the atomic level.

Therefore, the MD method is a unique approach to solve a variety of problems (Luo et al., 2009; Allen, 2004; Sachdeva, 2009; Jones, 1924). 2.1. Calculation of temperature and pressure in MD In equilibrium statistical mechanics, the absolute temperature is proportional to the average kinetic energy. In MD, the temperature can be defined according to Eq. (2) when the momenta Pi appear as squared terms in the Hamiltonian for an system that consists of N-atoms (Liu et al., 2012):

2 T¼ 3kB

*

N 1X jPi j2 N i¼1 2mi

+ ð2Þ

where h i represents the ensemble average, kB is the Boltzmann constant, and mi is the mass of atom i. Lion and Allen (2012) studied the computation of the local pressure using MD simulations. They expressed that for a homogeneous system, the pressure is calculated by taking the moment of the average pressure function according to Eq. (3):

* + N1 X ! ! NkB T 1 X P ¼ hPi ¼ r ij : f ij þ V 3V i¼1 j>i !

ð3Þ

!

where V is the volume, ri and rj are the positions of particles i and j, !

!

!

!

respectively; r ij ¼ r i  rj , and f ij indicate the force applied on particle i by particle j. To prevent double counting, the double sum spans all pairs of particles. The first and second terms in Eq. (3) originate from the kinetic energy of the particles and from particle interactions, respectively. Eq. (3) can be achieved in several steps starting from the Clausius virial relation:

* N X jPi j2 i¼1

mi

N X ! ! þ ri : F i

+ ¼0

ð4Þ

i¼1

!

where P i and mi are the momentum and mass of particle i, respec!

tively, and F i is the total force that is applied on particle i from other particles and the walls of the container. For inhomogeneous systems, it can be measured directly within the simulation box via the ‘‘method of planes” (MOP), which was first proposed by Irving and Kirkwood (1950), and later proposed by Todd et al. (1995).

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Fig. 3. (a) Schematic model for non-equilibrium MD simulation to calculate thermal conductivity. The water molecules a have hexagonal pattern. (b) The heat flux runs from the heat source to the heat sink, creating a steady temperature gradient. (c) The calculated thermal conductivity at various temperatures (Cheh et al., 2013).

3. Thermal conductivity 3.1. Computation of thermal conductivity Thermal conductivity is one of the properties of a material and indicates its ability to conduct heat, and it is investigated for heat conduction in Fourier’s Law. Heat transfer occurs at a higher rate across materials having high thermal conductivity compared to materials having a low thermal conductivity. Accordingly, materials with a low thermal conductivity are used as thermal insulation, while materials with a high thermal conductivity are widely used in heat-sink applications. The thermal conductivity of a material may depend on parameters such as the temperature. Because thermal conductivity is the most important factor responsible for enhanced heat transfer, many works have been carried out in this regard (Babaei et al., 2012, 2013a, 2013b). The values of the thermophysical properties of nanofluids are found to be increased to a very large degree relative to the base fluids in which nanoparticles are added, even when the concentration of the nanoparticles is very low. According to Fourier’s law, the thermal conductivity relates the heat flux to the temperature gradient as

J ¼ k

@T @x

ð5Þ

where J is a component of the thermal current, k is a member of the is the gradient of temperature T. thermal conductivity tensor, and @T @x Experimentally, k is generally found by measuring the temperature gradient that results from the heat current. In MD simulations, the thermal conductivity can be calculated using the non-equilibrium molecular dynamics (NEMD) method as well as the equilibrium molecular dynamics (EMD) method. The most commonly used methods for computing thermal conductivity are the ‘‘direct method” (Xue et al., 2004; Liang and Tsai, 2011; Lee et al., 2015) and the

Fig. 4. Heat flux autocorrelation functions (HFACF) of water at 300–550 K (Lee, 2014).

Green-Kubo (Sun et al., 2011; Sarkar and Selvam, 2007; Lu and Fan, 2008; Cu et al., 2011; Cui et al., 2015; Lee et al., 2015) method. The first method is an NEMD method in which a temperature gradient across the simulation cell is imposed, and it is similar to measurement mechanisms in experimental methods. On the contrary, the Green-Kubo method is an EMD method that calculates the thermal conductivity using the fluctuation-dissipation theorem. 3.1.1. Non-equilibrium molecular dynamics (NEMD) There are three general methods to compute the thermal conductivity using non-equilibrium molecular dynamics simulations.

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The first involves launching two regions in different situations of a simulation box. By keeping the two regions at different temperatures, the energy added to the hot zone is equal to the energy removed from the cold zone, and is proportional to the transferred heat flux between the two regions (Ikeshoji and Hafskjold, 1994). The second method involves applying a fixed amount of thermal energy to an area, and it is taken from another area, as schematically shown in Fig. 3, where the temperature gradient between the two regions can be calculated that is used to calculate the thermal conductivity (Teng et al., 2007). The third method is to perform a reverse non-equilibrium MD simulation that uses the RNEMD algorithm proposed by MüllerPlathe (1997). In this approach, the temperature gradient between the two layers is caused by the kinetic energy displacement between the atoms in different layers of the simulation box. 3.1.2. Equilibrium molecular dynamics (EMD) The equilibrium molecular dynamics (EMD) method refers to a system that is checked in the equilibrium state, and it is based on the linear response theory (Hansen et al., 1986). The transfer coefficients (e.g., thermal conductivity and shear viscosity) can be achieved using equilibrium simulations, while all of the transfer processes are non-equilibrium ones, because they are characteristics of the system. According to linear response theory, different transfer coefficients can be stated as a general form of the Green–Kubo formula (Lu and Fan, 2008). In fact, the Green–Kubo formula presents the exact mathematical amount of transfer coefficients S in terms of the integrals of time-correlation functions:

Z

S¼

1

hMðtÞMð0Þidt

ð6Þ

0

Specifically, the Green–Kubo formula for thermal conductivity is

k¼

Z

V 3kB T

2

1

hJðtÞJð0Þidt

ð7Þ

0

where k is the thermal conductivity, V is the volume of the simulation box, T is the system temperature, kB is the Boltzmann’s constant, J is the temporary microscopic heat flux vector, and hJðtÞJð0Þi is the heat-current autocorrelation function (HCACF). HCACF must decay to zero within the integral time length to generate convergence of data. Therefore, to ensure that there is an adequate time for the autocorrelation function to decay to zero, one of the important parameters that is used is the correlation length. If the correlation length is diverted from the right amount, the results contain errors (Jones et al., 2012). The heat-current autocorrelation functions decay monotonically to zero for liquids. In contrast, they have an oscillatory manner for nanoparticles when they decay to zero and display negative values (Fig. 4). In addition, the oscillatory behavior of the HCACF increases with an increasing number of loading nanoparticles (Sarkar and Selvam, 2007; Lee et al., 2015; Keblinski et al., 2002). The microscopic heat flux in a material described with a twobody potential function (u) is defined as follows:

#  "X   X 1 1 1 J¼ ðrmn F mn Þv n en v n þ V 2 2 m–n n

ð8Þ

where Fmn demonstrates the interaction between atom m and atom n, which is ruled by the interaction potential function, r mn demonstrates the distance between atom m and atom n (r mn = r n  rm ), and en demonstrates the per-atom energy of particle n, which is calculated by

en ¼

 X X1 1 u mn v 2n þ 2 2 m–n mn n

ð9Þ

71

where mn demonstrates the mass of the n-th particle. Schelling et al. (2002) compared these two methods for bulk silicon, giving full details of their approach, and their results show that both methods are in generally good agreement with the experimental results. However, the use of the Green-Kubo method for nanofluids needs to be investigated by performing widespread studies and comparisons because nanofluids are inhomogeneous systems (Teng et al., 2007). 3.2. Factors affecting the thermal conductivity of nanofluids 3.2.1. Volume fraction of nanoparticles Most of the investigations on nanofluids showed that the thermal conductivity of nanofluids (knf) will significantly increase with just small nanoparticle volume fractions when compared to the base fluid, and it is enhanced with an increased nanoparticle volume fraction (Teng et al., 2010; Maheshwary et al., 2017). Sarkar and Selvam (2007) computed the thermal conductivity of base fluids and nanofluids (a single copper nanoparticle in argon base fluid) using the Green-Kubo method by performing EMD simulations in a 3D-computing domain for different volume fractions of nanoparticle loading. In their simulation, NVT ensemble has been employed with Nose-Hoover (Hoover, 1985) thermostat and velocity Verlet algorithm as the integration pattern by applying periodic boundary condition in all directions. In addition they used pairwise LJ potential for modelling all the atomic interactions. The thermal conductivity was 0:145 W=m K at a very low nanoparticle concentration (0.4%). The thermal conductivities were 0:156 W=m K and 0:165 W=m K for 1% and 2% nanofluids, respectively. Furthermore, they found that the thermal conductivity enhancement is 20% and 52% compared to the base fluid with 1% and 8% nanoparticles concentration, respectively. The thermal conductivity enhancement was slower at high nanoparticle loading compared to very small loadings. They believed that for higher nanoparticle loadings, the other mechanism can play a considerable role in enhancing the thermal conductivity such as enhanced conduction that originated from a larger aggregation of highly conductive nanoparticles. In addition, in another similar report, Sarkar and Selvam (2007) reported that the thermal conductivity enhancement was not linear. Teng et al. (2007) calculated the variations in the thermal conductivities with an increased volume fraction of suspended copper nanoparticles in liquid argon as the base fluid for a fixed-sized box of nanofluid. They used pairwise Lennard Jones (LJ) potential for modelling the atomic interactions of argon-argon and argoncopper atoms and Morse potential between copper–copper atoms. Also they used Verlet integration algorithm in all simulations. Compared with other predictions, their results show that the predicted values are in relatively good agreement with those in the model proposed by Jang and Choi (2004), but the values are much higher than those anticipated by the Hamilton-Crosser theory. In another study, the nanofluid system including a solid spherical nanoparticle of Al2O3 in water and ethylene glycol molecules by using LJ potential of non-polar molecules and the periodical boundary condition was considered (Lu and Fan, 2008), and the results show that the thermal conductivity can be enhanced after inputting nanoparticles, and it increases clearly with an increase in the volume fraction. Lu and Fan reported that this is because heat-transfer processes occur mainly on the surface of nanoparticles, and nanoparticles have a larger surface area-to-volume ratio. Therefore, the heat-swapping area between fluids and nanoparticles is much larger than for general particles, which may increase the effective heat transfer of nanofluids. Sankar et al. (2008) studied the effective thermal conductivity of Pt nanoparticles in water as a base fluid, while the potential functions employed in the system for water–water interaction,

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water–Pt interaction and Pt–Pt interaction are LJ potential, the potential provided by Spohr and Heinzinger (1988) and two potentials of the Morse and FENE (finitely extendable nonlinear elastic) respectively. Their molecular dynamics simulations are done with velocity rescaling and periodic boundary conditions. Finally they reported that it increases proportionally with the volume fraction. Results for Cu-Ar nanofluids in a volume concentration of 1%, including one spherical nanoparticle with a diameter of 1.8 nm, is 0.147 W=m K (Cu et al., 2011), which is close to the previous simulation result of 0.144 W=m Kin Ref. (Li et al., 2010) for a similar simulation model with an error of 2.1%. Sun et al. (2011) also calculated the thermal conductivity of CuAr nanofluid including a single copper nanoparticle in argon base fluid limited between two parallel walls using an EMD simulation by employing the Green-Kubo formula and periodic boundary conditions in the x and y direction of the simulation system. In their simulation the LJ potential has been modeled all of the interactions between the atoms and they used an effective Velocity-Verlet integration algorithm in all simulations. The results show that the shudder of the nanoparticle in the z-direction (channel altitude direction) is inhibited because there are solid boundaries, and accordingly, the thermal conductivity of the limited nanofluid is clearly anisotropic. An extraordinary enhancement in the thermal conductivity component in the z-direction (k ) reveals strong coupling interactions between the fluid atoms, mainly the nanoparticle atoms and the wall atoms. The effect of nanoparticle concentration on the thermal conductivity of the limited nanofluid is more obvious than that in the macroscale. Mohebbi (2012) investigated 8.75% and 3.1% Silicon nitrideArgon nanofluid systems contains a single nanoparticle at a temperature of 130 K in a non-periodic boundary conditions and using the CHARMM22 force field, and he achieved a 47.9% and 19.7%

increase in thermal conductivity enhancement, respectively. Furthermore, he obtained thermal conductivity value of 0.08 W=m K and 0.085 W=m K for 1.77% and 3.1% nanofluids, respectively. MD-based prediction results show that with an increasing volume fraction of nanoparticles, the rate of thermal conductivity enhancement decreases in MD simulations (Jia et al., 2012). This study investigated nanofluid system including one copper nanoparticle in argon fluid atoms with periodic boundary condition in all directions that the potential employed for modelling interaction is Stoddard–Ford potential (Stoddard and Ford, 1973). The thermal conductivity enhancements for nanoparticle volume fractions of 2, 2.5, 3, 3.5, and 4% are 3, 7, 10, 11, and 12%, respectively. In addition, Javanmardi and Jafarpur (Javanmardi and Jafarpur, 2013) surveyed the effective thermal conductivity of the singlewalled carbon nanotube (SWCNT)-water nanofluids using the Green-Kubo method. The potential functions employed in the nanofluid system for liquid water, carbon nanotube and carbon– water interaction are TIP4P potential, simplified form of Brenner’s potential and LJ potential, respectively. Furthermore nanotube employed is an armchair CNT with chiral vector (12, 0). One of the important properties of nanofluids is the nonlinearity of the thermal conductivity, which they were able to prove. They also reported that the volume fraction of nanoparticles affects the nonlinear behavior of the thermal conductivity of nanofluids. Sedighi and Mohebbi (2014) investigated 1.5%, 3%, and 4.5% SiO2-water nanofluid systems, while Langevin algorithm is employed to fix the system temperature during the simulation, also they used a Velocity-Verlet integration algorithm and CHARMM22 force field potential function in all simulations. Finally they achieved a 9.8%, 11.3%, and 14.9% increase in the thermal conductivity enhancement, respectively. They also confirmed the

Fig. 5. The trend of thermal conductivity with volume fraction in different studies, which show that the thermal conductivity of nanofluids is enhanced with an increased nanoparticle volume fraction.

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F. Jabbari et al. / Chemical Engineering Science 174 (2017) 67–81 Table 1 The increase in the nanofluid thermal conductivity at varying temperatures. Authors

Nanofluid

Nanoparticle concentration (%)

Temperature range (K)

Thermal conductivity enhancement (%)

Sarkar and Selvam (2007) Mohebbi (2012) Javanmardi and Jafarpur (2013) Cui et al. (2015)

Ar-Cu Ar-Silicon nitride Water-CNT Water-Cu

2 4.15 0.1 wt 2

85–103 107–140.132 293–303 293–343

37–68 50–15 2–20 39–24

results by Sarkar and Selvam (2007), which show that the conductivity enhancement was steeper for low volume fractions of nanoparticles compared to higher loadings. This is also evident from studies of other researchers in that thermal conductivity increases with an increased nanoparticle volume fraction (Lee et al., 2015a, 2015b; Babaei et al., 2013; Cui et al., 2014; Bushehri et al., 2016). In order to enable a clear comparison of different studies, in Fig. 5, we present the thermal conductivity enhancement for different nanofluid types versus the nanoparticle volume fraction. From the figure, we see that the range of enhancement for most studies is less than 40% for volume fractions that are less than 5%. We also find that there are different enhancements for the same nanofluid type (Ar-Cu), which may be due to other influencing factors such as the temperature of the system or the shape of nanoparticles. 3.2.2. Nanofluid temperature One of the important and effective factors related to the thermal conductivity is the system temperature. Generally, as the temperature increases, corresponding increases in thermal conductivity are clear for all nanoparticle volume fractions, especially for high concentrations. The thermal conductivity of a nanofluid depends largely on temperature because it increases significantly with respect to the increased temperature. The enhancement of the thermal conductivity process due to temperature changes is shown by Sarkar and Selvam (2007) for three copper nanoparticle volume fractions in argon, i.e., 0.2%, 1.0%, and 2.0%. They found that with increasing temperature, the thermal conductivity increases nonlinearly. They investigated the 0.2% nanofluid system at 85 K and 103 K and they achieved 11% and 31% increases, respectively, in the thermal conductivity, while for the 2.0% nanofluid, the enhancement was from 37% to 68% for the same increase in temperature. In addition, they confirmed the results reported by Sankar et al. (2008) with respect to the significant effects of the temperature on the thermal conductivity at higher temperatures. Mohebbi (2012) calculated the thermal conductivity using a new approach that is based on the combination of two general methods involving MD simulations (EMD and NEMD) in nonperiodic boundary conditions. He found that for a nanofluid system at temperatures of 140 K and 107 K, there is a thermal conductivity enhancement of 15% and 50%, respectively. Therefore, the thermal conductivity enhancement was smaller at high temperatures compared to lower temperatures. Further, the results reported by Javanmardi and Jafarpur (2013) indicate that both factors, namely the concentration and temperature, affect the nonlinear variation of the effective thermal conductivity. Therefore, this nonlinearity is proportional to the temperature; however, it is inversely proportional to the volume fraction of nanoparticles. Actually, the nonlinear response of the thermal conductivity is more evident at higher temperatures. However, evidence of this feature is more easily seen for low nanoparticle volume fractions at fixed temperature. For instance, the thermal conductivity increases to 20% for a 0.1% weight fraction (wt.) of CNT- water nanofluid at 303 K, while this increase is about 2% at 293 K for the same fraction of nanoparticles. Table 1 presents a summary of the effect of temperature on the thermal

Fig. 6. Contribution of each part of thermal conductivity k* of nanofluid: kKK , kPP , and kcc , which represent the Brownian motion, the vibrational or phonon modes, and the interacting potential, respectively (Sun et al., 2011).

conductivity of nanofluids (Cui et al., 2015; Sedighi and Mohebbi, 2014). 3.2.3. Brownian motion of the nanoparticles The Brownian movement of nanoparticles is one of the basic mechanisms employed to realize heat-transfer enhancement in nanofluids. Many studies have been done to consider the effect of Brownian movement on the thermal conductivity of nanofluids (Michaelides et al., 2014), and most of them show that the thermal conductivity will not be affected by the Brownian motion of nanoparticles. In the first studies conducted on the nanofluids using MD simulations, Keblinski et al. (2002) proposed four possible factors that may lead to extraordinary increases in the thermal conductivity, including the Brownian motion of the nanoparticles. To demonstrate the role of Brownian motion in heat transfer, they compared the heat-flux autocorrelation functions for two different simulations of the same system. In the first simulation, they allowed all atoms to move pursuant to Newton’s equation of motion, while in the second simulation, they fixed the center of mass of the nanoparticle to make their position fixed. Their results show that the heat-flux autocorrelation functions for both cases are the same. Therefore, the Brownian motion of nanoparticles has no effect on the thermal conductivity or even a detailed microscopic mechanism of heat transport. In addition, in their study, they showed that thermal diffusion is much quicker than Brownian diffusion, even when the particle size is too small. Evans et al. (2006) presented their results for MD simulations and a kinetic theory reasoning. In their simulation the LJ potential has been modeled all of the interactions between the atoms and they used a Verlet integration algorithm in all simulations. They expressed the thermal conductivity of nanofluids with a good dispersion of nanoparticles in a very good compliance with effective medium theory, and there were no visible increases due to Brownian motion resulting from hydrodynamic effects.

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Fig. 7. Comparison of translational and angular velocity components of nanoparticles under shear velocities of 50 and 0 m/s (Cui et al., 2015a).

Sun et al. (2011) investigated the heat current including the kinetic part (K), the potential part (P), and the collision part (C). Therefore, the thermal conductivity can be considered as a total of nine terms that include the relationship between them (KK, PP, and CC), which interact with each other (PC, PK, etc.). Their results show that kKK and kCC do not play the main role, while kPP is a very important factor, where k represents the thermal conductivity perpendicular to the z-direction (channel height direction). Fig. 6 expresses each of the expressions KK, CC, and PP. Finally, they also confirmed Keblinski’s results about the very small effect of the Brownian motion of nanoparticles on the abnormal enhancement of the thermal conductivity of nanofluids. In another study, Babaei et al. (2013) investigated heat-flux terms, namely the interaction (virial) (kvv) and convective (kcc) terms, and their effect on the thermal conductivity of nanofluids. In their simulation the LJ potential has been modeled all of the atomic interactions between the atoms and they used a velocity Verlet integration algorithm in all simulations. The results of their study demonstrate that the virial contribution (kvv) plays an important role in the thermal conductivity of nanofluids. The convective term (kcc) is directly related to the Brownian motion of nanoparticles, which increases slightly with the volume fraction of nanoparticles, while it is much smaller compared to the virial term. They confirmed the previous results about the impact of the Brownian motion of nanoparticles on the thermal conductivity. Cui et al. (2015a) simulated a nanofluid system, including liquid argon (Ar) as a basic fluid and copper (Cu) nanoparticles that the interatomic interaction between Cu-Cu atoms is modeled by EAM potential and the interactions between Ar-Ar atoms and Cu-Ar atoms are modeled by LJ potential. They compared the translational and angular velocities of nanoparticles during simulations in two different modes: Applying a shear velocity of 50 m/s and no shearing velocity (v = 0 m/s). According to the results shown in Fig. 7, the main reason for the chaotic movements of nanoparticles is the Brownian motion. They compared the required time by heat transfer and the nanoparticle movements in nanofluids. They found that the nanoparticle movement is effective on nanofluid heat-transfer process. The perception of the magnitude of the nanoparticle Brownian motion and the thermal conductivity enhancement of nanofluids by Brownian motion was studied by Cui et al. (2015b) using several

Table 2 Variations in thermal conductivity with increasing nanoparticle size. Author

Nanofluid

Nanoparticle size Range (nm)

Thermal conductivity enhancement (knf/kf)

Teng et al. (2007) Lu and Fan (2008) Sun et al. (2011) Cui et al. (2014)

Ar-Cu

0.075–1.25

1.13–90

WaterAl2O3 Ar-Cu

5–24

1.9–1.15

0.15–0.75

1.65–777.5

Ar-Ag

1–3

316–205

MD simulations. They considered the different effective factors involving nanoparticle diameters, materials, and shapes. They proved the impact of the rotation and migration of nanoparticles for heat-transfer enhancement by defining the ‘‘translational” Peclet number and ‘‘rotational” Peclet number. Lee et al. (2015) also investigated an Ar-Cu nanofluid system and used LJ potential for atomic interaction between Ar-Ar and Cu-Cu atoms and the Lorentz–Berthelot mixing rule for specify the interaction between Cu-Ar atoms. They observed that the volume fraction of nanoparticles does not affect the average kinetic energy. Moreover, the addition of copper nanoparticles does not significantly change the average kinetic energy of the base fluid. Therefore, the Brownian motion and its effects are negligible to the increase in thermal conductivity. The effect of the nanofluid temperature on nanoparticle micromotion was investigated by Cui et al. (2015) in another related study. In their study the nanoparticle was spherical and they applied the x and y axis periodic boundary condition. The interaction between Cu-Cu atoms is modeled by EAM potential and the interatomic interactions between Ar-Ar, Cu-Ar and H2O-Cu are modeled by LJ potential, also interactions between water molecules is SPC/E potential. The micro-motion of nanoparticles was reported to be an important reason for the enhancement of thermal conductivity in nanofluids. The Random Brownian motion of nanoparticles results in the translational movements of nanoparticles because the translational velocity of nanoparticles changes between negative and positive values according to the MD simulation results.

F. Jabbari et al. / Chemical Engineering Science 174 (2017) 67–81

3.2.4. The number and size of nanoparticles Teng et al. (2007) reported that with the increase in the size of copper nanoparticles in the liquid argon, the base fluid thermal conductivity increases, while the nanoparticle volume fraction was fixed equal to 0.688%. They compared the thermal conductivity value for nanofluids with a particle size of 1.86 nm with a base fluid (0.13 W=m K) showing an increase by a factor of 300. Moreover, they investigated the effect of the number of nanoparticles on the nanofluid thermal conductivity. They found that for a system including 27 nanoparticles of radius 0.51 nm, and 8 nanoparticles of radius 0.61 nm, the computed thermal conductivity increased by 75% and 33%, respectively. Simulations on Al2O3-water and Al2O3-ethylene glycol nanofluid systems done by Lu and Fan (2008) showed that the thermal conductivity decreases as the nanoparticle size increases, and it is nearly constant when the nanoparticle diameter is greater than 30 nm. Finally, the greatest increase in the calculated thermal conductivity of nanofluids was reported as 41% in Al2O3- ethylene glycol nanofluid. Sun et al. (2011) found that the thermal conductivity increases significantly with an increased ratio d/h, where h is the channel height. Cui et al. (2014) also concluded that the thermal conductivity of nanofluids decreases with an increased nanoparticle diameter, and in another study (Cui et al., 2015b) for Ar-Cu nanofluid systems, they reported that the rotational and translational speeds of nanoparticles decrease with an increasing nanoparticle diameter. Table 2 presents a summary of the effect of nanoparticle size on the thermal conductivity enhancement of nanofluids. It shows that the thermal conductivity varies for different nanoparticle diameters. The results indicate that the thermal conductivity increases with an increase in the nanoparticle diameter in an ArCu nanofluid system, while in other nanofluid systems, e.g., ArAg, it is decreased. Therefore, the effect of the nanoparticle diameter on the thermal conductivity enhancement for different types of nanofluid is not the same. 3.2.5. Aggregation and clustering of nanoparticles Another factor that affects the increased thermal conductivity is the effect of nanoparticle clustering (Keblinski et al., 2002), which is classified as follows. To verify whether the nanoparticle aggregation occurs in simulation systems, Kang et al. (2012) increased the simulation time to 40 ns for only the case involving two nanoparticles, while more time is required for a larger number of nanoparticles. Because the purpose of their study was to evaluate the effects of nanoparticle aggregation on the thermal conductivity of nanofluids, the nanoparticles were placed close together in the system from the beginning of the simulation time, while the wellknown LJ potential explains interaction between Ar-Ar and Ar-Cu atoms and the EAM potential explains the interaction of Cu-Cu atoms. The results clearly showed that the thermal conductivity of nanofluids increases considerably when nanoparticles are close together. Sedighi and Mohebbi (2014) performed MD simulations to investigate changes in the thermal conductivity for water-silicon dioxide nanofluid while nanoparticle aggregation occurred in the base fluid. Their results showed that the specific heat decreases by about 3%, and the diffusivity increases by about 3.5% while aggregation takes place by adding one nanoparticle in each step; in addition, the thermal conductivity increases with previous trends. By the addition of two and three aggregated nanoparticles, the thermal conductivity enhancement was less than that for the addition of one nanoparticle. The thermal properties of nanofluids at constant volume fraction of nanoparticles were investigated in two cases, i.e., with suspended nanoparticles and aggregated nanoparticles. The results showed that the diffusivity and thermal conductivity of nanofluids increased by about 2% for the case of the

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aggregated nanoparticles, while in comparison, the specific heat of the nanofluid remains almost constant for the suspended nanoparticles. Their results showed that the thermal properties of nanofluids are not affected significantly by the aggregation of nanoparticles. Actually, nanoparticle doping in the nanofluid increases until aggregates become large so that they separate from each other when the nanoparticles become aggregated during the simulation time, and settlement occurs. Furthermore, the thermal properties of the nanofluid are improved by nanoparticle aggregation, but this is temporary. Lee et al. (2015) used the EMD method to compute the thermal conductivity of an Ar-Cu nanofluid system in two cases, namely aggregated and non-aggregated nanoparticles. Their results clearly showed that the thermal conductivity enhancement is better when nanoparticles are aggregated, especially at higher volume concentrations of up to 35%. They believed that a higher number of collisions between nanoparticles in the aggregated state results in thermal conductivity enhancement at low volume concentrations of 2.59% and 3.89%, and it increases the potential energy of copper nanoparticles. This phenomenon enables efficient heat conduction along the aggregation. This verifies the main mechanism governing the increased thermal conductivity by nanoparticle aggregation based only on the conduction. 3.2.6. Molecular layer around the nanoparticle The molecular layer at the interface between the nanoparticles and the base fluid is one of four factors that influence the thermal conductivity value (Xue et al., 2004; Keblinski et al., 2002). The reason for the creation of the molecular layer is that the interactions between homogeneous base fluid atoms were much weaker than those of the base fluid and nanoparticle atoms. In addition, the atoms of the base fluid near the solid walls and near the nanoparticles are absorbed by them. The impact of the molecular layer structure on the thermal conductivity for a multi-shell sphere was studied by Teng et al. (2007). They showed that for different sizes of nanoparticles, the enhancement ratio of the thermal conductivity due to the formation of molecular layer ranges from 1.6 to 3.0. Lv et al. (2011) compared their results with those obtained by Li et al. (2008), and they found that the diameter of the nanoparticle does not affect the thickness of the molecular layer. The molecular layer may be important during heat transfer, even at the distribution layer, and because its thermal conductivity is very high compared to the base fluid, it can act as a heat-transfer ‘‘bridge” between the nanoparticle and the base fluid. In fact, atoms of argon-like solid in a molecular layer have a much higher thermal conductivity than those with liquid argon as the base fluid. Thus, the effective diameter of the nanoparticles is increased in the presence of the molecular layer, and the nanoparticles become heavier. The nanoparticles and the surrounding molecular layer artificially increase the thermal conductivity of the base fluid. Liang and Tsai (2011) calculated the thickness and thermal conductivity of the interfacial molecular layer by performing MD simulations, by applying the periodic boundary conditions in three directions. In their simulation, Berendsen algorithm has been employed (Berendsen et al., 1984) to fix the system temperature. In addition they used LJ potential for modelling all the atomic interactions, and they used a velocity Verlet integration algorithm in all simulations. Their results indicated that increasing the density of the molecular layer causes the thermal conductivity in the region to significantly increase. The effect of the molecular layer presence on the increase in the thermal conductivity of nanofluids in the case of aggregated nanoparticles is much higher than for well-dispersed nanoparticles when the thickness of the molecular layer is 1 nm and its enhancement ratio is 1.6  2.5 times that of the base-fluid thermal conductivity, and it therefore cannot be

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ignored. Mohebbie’s studies (Mohebbi, 2012) also show that argon atoms are attracted by nanoparticles, which leads to an increase in the density near the interface, and the formation of the molecular layer on the interface between the nanoparticles and the liquid. Cui et al. (2015) believed that one of the factors that are responsible for the enhanced thermal conductivity is the absorption layer around nanoparticles. They understood that the effect of temperature on the molecular layer could be seen in the magnitude of the first peak in the number density graph. For example, the magnitude of the first peak in the number density graph is the largest and smallest at 353 K and 293 K, respectively. This phenomenon indicates that the absorbed liquid atoms are more coherent at the higher temperature because the random motion of atoms is more severe. In addition, in another study, Milanese et al. investigated how the water molecules attract the surrounding Cu and CuO nanoparticles with different nanoparticle diameters (Milanese et al., 2016). Their simulations considered a nanoparticle of Cu or CuO in water molecules, while for modelling all the interactions is used ReaxFF potential, that it is suitable potential for hydrocarbon reaction simulations; also they used the Berendsen thermostat to fix the system temperature and periodic boundary condition in all simulation. They studied the density distribution of hydrogen and oxygen around the nanoparticles, and they found that the molecular layer is more closely related to the metal particles. The numerical results show that the molecular layer around a Cu nanoparticle is formed, while the molecular layer around the CuO nanoparticle is not formed significantly. In addition, they investigated the impact of different sizes and volume fractions of metal nanoparticles on the structure of the molecular layer. Their results showed that these parameters do not influence the structure of the base fluid molecules. Finally, they reported that the formation of the molecular layer around the metal nanoparticles play a major role in explaining the experimental results pertaining to the thermal conductivity. In addition to the mentioned results, they reported that atoms of the base fluid in the molecular layer affect the enhanced mean-square displacement of nanofluids (Cu et al., 2016). 3.2.7. Material and shape of nanoparticle In addition to affecting factors related to the thermal conductivity, which we discussed in previous sections, the material and

shape of nanoparticles also play an effective role in the thermal conductivity of nanofluids. Cu et al. (2011) investigated the effect of the nanoparticle shape on the increasing thermal conductivity of nanofluid, while the LJ potential explains atomic interaction between all atoms and they controlled the system temperature by a Nose-Hoover thermostat. They found that spherical copper nanoparticles have fewer energetic atoms compared to long cylindrical copper nanoparticles because spherical nanoparticles have smaller specific surface area than that of cylindrical nanoparticles, and the potential energy of atoms on the surface is less than that of internal atoms. Their results showed that the thermal conductivity of Cu-Ar nanofluids in the presence of a spherical nanoparticle and a long cylindrical nanoparticle increase by 14.84% and 20.31%, respectively, while the volume concentration of the nanoparticles is 1%. They found that at a constant nanoparticle volume fraction, by choosing materials that have a higher thermal conductivity than the nanoparticle, the thermal conductivity of nanofluids can also be increased. In another study done by Cui et al. (2014), they used a periodic boundary condition in MD simulation and employed LJ and EAM potential to describe interactions between argon and copper atoms respectively. Finally, by comparing nanoparticles that have different shapes, they found that the thermal conductivity enhancement is greater if the surface-to-volume (S/V) value is higher. Thus, one of the ways of increasing the thermal conductivity of nanofluid is to choose nanoparticles with a higher S/V (Fig. 8). The effect of the rotational and translational velocities of nanoparticles was also studied by Cui et al. (2015b). The boundary conditions and potential functions that used in this study are similar to their previous study (Cui et al., 2014). They stated that the S/ V ratio changes the rotational and translational velocities of nanoparticles, and consequently, it causes heat-transfer augmentation. Furthermore, they studied the impact of these two factors on the microstructure of the molecular layer at the interface between the nanoparticle and base fluid (Cu et al., 2016). 3.2.8. Interfacial thermal resistance (Kapitza resistance) The interfacial thermal resistance, which is also known as the thermal boundary resistance, or Kapitza resistance, is a measure of the resistance to heat flow. Another factor that affects the thermal conductivity is the thermal resistance (Serebryakov et al.,

Fig. 8. Nanoparticles of different shapes with equal volumes (Cui et al., 2014).

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2017). Vladkov and Barrat (2006) illustrated that the thermal resistance (Kapitza) can be improved by setting either the liquid–solid interaction coefficient or the solid mass density or a combination of them. The results show that the effective thermal conductivity is in good agreement with the Maxwell-Garnett expression. Evans et al. (2006) used three different values for the interactions strength between solid and liquid: esf ¼ 0:25e; 1:25e; and 2:25e. These selections include the different modes of wettability, and correspond to the states of nonwetting, weakly-wetting, and wetting particle, respectively. In addition, they investigated the difference in the wettability effect on the thermal resistance for the solid and liquid as well as the effect of the thermal conductivity of nanofluids (Putnam et al., 2006). Further, a sharp drop in temperature at the interface between the nanoparticles and base fluid can be observed, and it is significant for the nonwetting particle case (Xue et al., 2003). Lee et al. (2015) also calculated the Kapitza resistance and the nanofluid thermal conductivity for several cases for CNT-based nanofluids with applying Nose-Hoover thermostats on the system.

Fig. 10. Autocorrelation function (ACF) and viscosity (l) of water (solid line) and Al2O3 nanofluid (dot line) at 300 K Lou and Yang, 2015.

3.3. Future ideas

nanofluids and the effects of various parameters on the various nanofluids systems.

In spite of the numerous studies that have been carried out with respect to the calculation of the thermal conductivity of nanofluids and the assessment of effective factors using MD simulations, there remain a gap in studies pertaining to some of the effective parameters on nanofluids thermal conductivity, and more research is needed. To prepare many of the nanofluids using experiments, we need to use surfactants to avoid the agglomeration of nanoparticles. It has been experimentally shown that surfactants significantly affect the thermal conductivity of nanofluids (Salehi et al., 2013; Gómez-Villarejo et al., 2017; Sánchez-Coronilla et al., 2017). Thus, this is an important area of research that could be pursued using MD. The effect of the acidity level on the thermal conductivity of nanofluids is another important factor that has not been yet studied via MD. In addition to the above-mentioned issues, most of the studies that have been performed till date examined Ar-Cu nanofluid systems with the Lennard-Jones potential. Owing to the extent of different types of nanofluids systems and their unique features, e.g., the use of CNTs in the base fluid, it is essential to study the trends in terms of the enhancement of the thermal conductivity of

4. Shear viscosity 4.1. Computation of shear viscosity Viscosity is the property of a fluid that opposes the relative motion between two liquid layers, or in a fluid that moves at different velocities. When a fluid is flowing in a tube, its constituent particles move faster near the center of the tube and slower near the tube walls. Therefore, the forces (e.g., the pressure difference between the two ends of the pipe) should overcome the friction between the layers of the liquid to maintain the flow in the tube. According to Newton’s law of viscosity, the value of the tension is proportional to the viscosity of a fluid. Therefore, the dynamic viscosity of a homogeneous fluid is described as follows:

l¼

s

ð@u=@yÞ

¼

s c

ð10Þ

where s represents the local shear stress, ð@u=@yÞ represents the local velocity gradient, and c is called the rate of shear. Similar to

Fig. 9. (a) Schematic model of non-equilibrium MD simulations for water molecules that are placed between two solid walls of silicon. (b) Velocity profile for water inside a nano-channel. (c) The values of the shear viscosity as a function of the temperature for the different water models (Markesteijn et al., 2012).

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the calculation of thermal conductivity using MD simulations, there are two general methods to calculate the shear viscosity. 4.1.1. NEMD The application of shear stress on a simulation box can be referred to as the first non-equilibrium MD simulation method. In the second method, by moving one or two walls, shear stress is generated between layers of fluids. In this method, a type of thermal thermostat is used to prevent the fluid from heating. The third method is a reverse non-equilibrium MD simulation proposed by Muller-Plathe. In this approach, the velocity gradient between two layers is caused by the momentum displacement between the atoms in different layers of the simulation box (Fig. 9). 4.1.2. EMD In this method, the Green-Kubo formula is used, which relates the ensemble mediocre of the autocorrelation of the stress function.

l¼

V kB T

Z

1 0



sxy ðtÞsxy ð0Þ dt

ð11Þ

Where l, V, T, kB, and sxy are the shear viscosity, volume, temperature, Boltzmann constant, and components of the stress tensor, respectively, where the stress tensor is calculated from the following equation (Bushehri et al., 2016):

sxy

" # 1 X 1X ¼ mj v jx v jy þ r ijx f ijy V j 2 i–j

ð12Þ

Results presented by researchers show that at the beginning, the autocorrelation function of the stress rapidly decays with fluctuations, and it is then followed by a slower decay to zero (Fig. 10) Lou and Yang, 2015, which is similar to that of the thermal conductivity (Loya and Ren, 2015). Generally, there are fewer studies related to shear viscosity of nanofluids compared to studies on their thermal conductivity, especially in MD simulations (Bashirnezhad et al., 2016). In the following section, we classify the results of studies to calculate the shear viscosity of nanofluids.

4.2. Factors affecting the shear viscosity 4.2.1. Volume fraction of nanoparticles Lu and Fan (Lu and Fan, 2008) studied the effect of the volume fraction of Al2O3 nanoparticles on the shear viscosity of ethylene glycol and water nanofluids. The results showed that the calculated viscosity of the mixture is larger than that of pure ethylene glycol and water. Upon the addition of larger amounts of nanoparticles to the pure fluid, the shear viscosity increases. These results were also confirmed in another study (Gui et al., 2014) that is related to the shear viscosity of gold–water nanofluid. Liquid argon with aluminum and lithium nanoparticles was investigated by Rudyak and Krasnolutskii (2014, 2015). They used periodic boundary conditions and LJ potential to describe Interaction between the molecules of the base fluid while for modelling interaction between base fluid and nanoparticle atoms is used Rudyak–Krasnolutskii (RK) (Rudyak and Krasnolutskii, 1999, 2001) potential. The results show that with an increase in the nanoparticle volume fraction, the viscosity of the nanofluid increases. The effects of different factors on the shear viscosity of Al2O3 nanofluids were studied by Lou and Yang (2015), and they obtained similar results as in the case of previous studies related to the impact of the nanoparticle volume fraction on shear viscosity. In their simulations, periodic boundary conditions were applied in all the three directions and Nose–Hoover thermostat was used to set the temperature system. The TIP4P/2005 potential is used to describe the interactions among all water molecules and the CLAYFF force field (Cygan et al., 2004) is applied for the particle–particle and water–particle interactions. For example, when the nanoparticle volume fraction increased from 1.24% to 3.72%, the shear viscosity increased from 1.21 mPa s to 3.68 mPa s at 300 K, and the viscosity increased to 3.59 mPa s at 280 K. Therefore, the effect of the nanoparticle volume fraction was more significant at lower temperature. In order to have a good comparison between different studies, the viscosity of different nanofluid types versus the nanoparticle volume fraction is summarized in Fig. 11. It can be deduced that in all studies, the viscosity of the nanofluid is enhanced with an increasing solid volume fraction. The discrepancies between reports on Al2O3 nanofluid may be due to the different nanoparticle diameters (100–450 Ǻ in Lu and Fan (2008) and 10.85 Ǻ in Lou and Yang (2015)). 4.2.2. Nanoparticle size Another important factor investigated by Lu and Fan (2008) pertaining to the viscosity of nanofluids was the size of nanoparticles. The results show that the shear viscosity decreased in the same manner as the thermal conductivity when the particle size increased. The same result was also achieved in a previous study by Rudyak and Krasnolutskii (2014, 2015). In another study, the Al2O3-water nanofluid system was examined by Lou and Yang (2015), and the result was similar to the result reported by Lu and Fan (2008). This phenomenon occurred as a result of increasing the particle-base fluid interaction energy (Eint), because when the nanoparticles become smaller, more molecules of the base fluids surround them, while the volume fraction of nanoparticles is fixed during the simulation time. Actually, by summing the Van der Waals and coulomb terms between nanoparticle and base fluid molecules, we can obtain the particle–base fluid interaction energy. Thus, for systems with greater Eint, the constraint between the base fluid and nanoparticles is stronger, which leads to an increased viscosity.

Fig. 11. Variation of the shear viscosity with the volume fraction based on different studies.

4.2.3. Nanofluid temperature As previously mentioned, the nanofluid temperature is an effective parameter that affects the thermal conductivity, and it can therefore affect the shear viscosity. Lou and Yang (2015) showed

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that with an increasing temperature, the viscosity of nanofluid decreased. Thus, at high temperature, the constraints between nanoparticles and base fluid are easily removed, and for this reason, the viscosity decreases at high temperature. In systems with a large number of nanoparticles, when the temperature increases, the shear viscosity decreases quickly because for systems with higher energy, the effect of the temperature is more substantial. The results obtained showed that interaction in Al2O3–water is stronger than in water molecules. Loya and Ren (2015) also achieved the same result as Lou and Yang (2015) with respect to the effect of nanofluid temperature on viscosity, while in their study COMB(Charge-Optimized Many-Body) potential was used to describe interaction between CuO and water with periodic boundary conditions. 4.2.4. Nanoparticle aggregation In this section, we discuss the aggregation of nanoparticles as another factor that affects the shear viscosity. The effect of nanoparticle aggregation on nanofluid viscosity was investigated by Kang et al. (2012). The results clearly showed that by clustering the nanoparticles, the shear viscosity of the nanofluid increased, while the trends in the increment is more gradual compared with the thermal conductivity. In addition, different nanoparticleclustering shapes lead to different amounts of increase in the shear viscosity. 4.3. Future ideas As previously mentioned, there are fewer studies on shear viscosity of nanofluids compared to the thermal conductivity of nanofluids. With respect to the importance of shear viscosity as a thermophysical property of nanofluids, there are many challenges regarding the viscosity and its influencing factors on the atomic and molecular level that can be examined by performing MD simulations. By considering the MD studies in this research area, we found that some of the effective parameters on shear viscosity of nanofluids, e.g., the Brownian motion of the nanoparticles, molecular layer at the interface between the nanoparticles and the base fluid, material and shape of nanoparticles, surfactants, and the effect of the acidity level have not yet been studied, and may be considered as future research ideas in this field. 5. Summary In this paper, we reviewed the thermophysical properties of nanofluids including the thermal conductivity and shear viscosity, which are calculated by MD simulations. We classified and reviewed the effects of parameters such as the nanoparticle volume fraction, nanoparticle size, nanoparticle shape, nanofluid temperature, nanoparticle materials, Brownian motion of nanoparticles, and the effects of clustering and the molecular layer on the thermal conductivity and shear viscosity. In most studies, the authors showed that their results agree well with the experimental data and mathematical models. Most of the reports on the nanofluid thermal conductivity and shear viscosity showed that these thermomechanical properties increase with increasing nanoparticle volume fraction and by decreasing the nanoparticle diameter. Several studies that focused on the effect of the Brownian motion of nanoparticles on the thermal conductivity of nanofluids using different methods indicate that this parameter does not significantly affect the increase in the thermal conductivity of nanofluids. The results show that the formation of a molecular layer at the interface between the nanoparticle and base fluid, as well as the clustering and aggregation of nanoparticles leads to an increase in the thermal conductivity of nanofluids. The mate-

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rial and shape of nanoparticles also affect the thermal conductivity of nanofluids, and by comparing nanoparticles with different shapes, we found that the thermal conductivity enhancement is higher if the surface-to-volume (S/V) value is higher.

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