Effect of SWCNT volume fraction on the viscosity of water-based nanofluids

Nanofluids have received a great deal of interest in recent years because of their various unique features. According to the findings, the addition of nanotubes to the base materials can drastically alter their properties. In the present work, the viscosity of a typical water-based nanofluid containing single-walled carbon nanotubes is estimated using the molecular dynamics simulation for different volume fractions ranging between 0.557 and 3% at two temperatures (298 K and 313 K). The temperature of the systems is controlled using a Nose-Hoover thermostat. For calculating viscosity, the Green–Kubo equilibrium method is used. The enthalpy, potential, kinetic, and total energies are calculated to determine the system’s stability. In addition, the influence of molecular mass on these energies is studied. The nanotube under investigation is an armchair(6,6)-type single-walled carbon nanotube. The results highlight the promise of the molecular dynamics simulation technique as a powerful tool in the prediction of nanofluid properties besides the experimental results. The value of viscosity will decrease as the temperature rises, much like the base fluid. Furthermore, it is shown that the viscosity is proportional to the volume fraction of water-SWCNT nanofluid. According to the results, a new viscosity relationship for volume fractions in the range of ϕ ≤ 3% is proposed. The viscosity, temperature, and volume fraction are all linked together in this equation.


Introduction
Scientists have discovered that the existence of solid nanometer-sized particles in the base fluid would have changed its transitional properties such as heat transfer and viscosity significantly. Adding nanoparticles to the base fluid to change its properties is more common than adding larger particles to shrink the suspension instability. Furthermore, nanoparticles' importance and usage, especially nanotubes in different sciences, lead researchers to struggle to expose their manner in specific conditions. In the past, studying molecules was a difficult and exhausting try. Researchers had to do experimental works to find out the phenomena behind them, but it took too much time and expense. When computer simulations started, they brought a revolution to science to save time and money. For instance, molecular dynamic simulation (MDS) is a well-known molecular-level simulation technique [1]. In addition to the MDS, there are some other methods like mathematical modeling to understand the behavior of nanofluids. Tlili [2] studied the nanofluid and hybrid nanofluid flow between two eccentric pipes and model it by mathematic. They chose C 2 H 6 O 2 as a base fluid and Ni and Al 2 O 3 as nanoparticles. By perturbation technique, nonlinear PDEs converted into ODEs. The MDS has been evolving so fast over the last decades. In the present investigation, this well-known method is used. Features at the nanoscale have an obvious contrast with those at the macroscale. Thus, a bit different formula and rules are used in the MD. Nanotubes' invention built a new route for scientists to evolve technology faster. Their desire to apply this new high-tech invention in the industry makes them develop new nanotubes and never look back since the MDS and experiments accompany each other. There are different nanotubes, such as carbon nanotubes (CNTs) and boron-nitride nanotubes (BNNTs). Also, they can be categorized as single-wall or multi-wall nanotubes. Leong et al. [3] desire to know how the weight percentages of nanoparticles in the presence of various surfactants would make the viscosity variation. The results revealed that surfactants impressed nanofluids' viscosity and made their viscosity decrease compared to nanofluids without it. Also, if the weight percentage of nanoparticles is increased, the viscosity will be increased. Halelfadl et al. [4] have studied experimentally dispersed MWCNT in the water. They examined the influence of temperature and volume fraction variation on the nanofluid properties. They expounded that for lower particle content, nanofluids behave in the Newtonian manner. Furthermore, they revealed that for high shear rates, viscosity does not vary sharply with temperature. Tlili et al. [5] investigated the mass/energy transference for Newtonian/non-Newtonian fluids. They revealed that mass/energy transference relies on viscosity variation. Besides, they showed that Casson fluid properties are bigger than the normal fluids. Hemmat et al. [6,7] have studied the thermal conductivity of the Si 2 O-MWCNT and Al 2 O 3 -SWCNT in the based fluid, ethylene glycol. Also, for Si 2 O-MWCNT, they have investigated its cost performance for industrial applications. The MDS was employed by Kuang et al. [8] to investigate selfassembly of different cases of BNNT and CNT immobilization into nanofluids which is influenced by increasing at equivalent nanoparticle radius. Golberg et al. [9] have prepared a review article that obtained various BNNT properties and dependent articles. Different diameters of SWCNT in which water flowed inside it have been investigated by Zhang et al. [10]. They showed viscosity dependence on the temperature and potential energy by using numerical analysis. Their results, at high pressure, have great matching with the experimental viscosity of water. The viscosity ascends nonlinearly according to the diameters increment, as well. In the end, they suggested a semi-empirical formulation to compute the viscosity of water inside SWCNT. Hybrid nanofluid is a novel category of nanofluid and has been chosen and used due to its capability in improving the rate of base-fluid rheology. Shafiq Ahmad et al. [11] analyzed the flow and heat transfer of SWCNT-MWCNT/water. Furthermore, they presented the influence of viscosity, velocity, and heat generation variations in detailed information. They showed that the solid volume fractions decreased velocity but also increased temperature distribution. Vakili-Nezhaad and Dorany [12] have investigated the distributed SWCNT in lube oil cut nanofluids for different temperatures, lengths of the nanotube, and weight fractions. Murshed et al. [13] indicated that if temperature decreases or the volume fraction of the nanoparticle increases, the nanofluid's viscosity will increase. In this research, nanoparticles with spherical and cylindrical shapes were studied. Lu et al. [14] applied MDS to compute the viscosity of gold nanofluids. Immobilization of nanoparticles into the nanofluids could increase its viscosity through its equivalent nanoparticle radius increment. A review about viscosity and thermal conductivity of some nanofluids for experimental has have been one by Ravisankar and Chand [15]. Generally, most of the case studies struggle to explore the connection between the base fluids and nanofluids when investigating the viscosity and thermal conductivity. Moreover, it has revealed that there is not any generalized correlation to use for whole kinds of nanofluids. This study will employ the MDS method to calculate the viscosity of distributed SWCNT in pure water as the base fluid for different volume fractions at two temperatures (298 K and 313 K). The well-known opensource molecular simulation package, LAMMPS, is used [16]. To simulate the interactions between the particles, TIP4P, the model for liquid water molecules, Tersoff potential for CNT atoms, and Lennard-Jones potential for interactions between carbon atoms and water molecules are applied.

Methods
MDS is used for simulating the typical nanofluid. It is a method that links theory and experiment for detecting properties of atoms and molecules such as thermal conductivity, diffusion, and viscosity. Also, progressing in MDS and technology provides ways to simulate systems on a larger scale with much more atmos. For calculating the viscosity of a liquid, MDS offers several methods: the equilibrium molecular dynamics (EMD) and the non-equilibrium molecular dynamics (NEMD) [17]. Sometimes, the EMD approach, in contrast with NEMD, will be able to converge with difficulty. Guo et al. [18] and Chen et al. [19] sought to overcome this flaw by adopting enough statistics with a careful integration time technique.
On the other hand, the EMD method can calculate different properties for the same state point simultaneously in a single simulation. Nevertheless, the NEMD method does not have this option [20]. In this work, the EMD approach is used to compute the viscosity of dispersed carbon nanotube in the water. Green-Kubo formula is applied to estimate nanofluid's viscosity in this article, as well as lots of researches that used this theory for computing thermodynamic properties (see, e.g., [21][22][23]). MDS is based on Newton's second law (F = ∑ Ma) and the interatomic forces are obtained by the atomic potential. Hence, the most sensitive part of each simulation with MD is the atomic potential for interactions between components. Each simulation in MDS begins with the simulating box of the system and its contents. For this purpose, VMD software [24] is applied to make a box of pure water with a TIP4P structure ( Fig. 1-A) and an armchair carbon nanotube, type (6,6) at a length of 3 nm ( Fig. 1-B). Then, CNT is dispersed in pure water ( Fig. 1-C).
The main features of the water TIP4P structure are shown in Fig. 2, and the values of its parameters are tabulated in Table 1 [25]: The 12-6 Lennard-Jones potential describes the interactions between CNT and water molecules.
where σ is the effective diameter of molecules, ε is the potential well depth, and r = |r i − r j | is the distance between molecules i and j. Tersoff potential models the interactions between carbon atoms [26]: where F R is a two-body expression and F A includes threebody interactions. The sums in the formula are over all neighbors J and K of atom I within a cutoff distance = R + D. The  n, β, λ 2 , B, λ 1 , and A parameters are just utilized for two-body interactions. The m, Υ, λ 3 , c, d, and cos θ 0 parameters are just utilized for three-body interactions. The R and D parameters are used for both two-body and three-body interactions. The value of m must be 3 or 1 [16]. The parameters for the 12-6 Lennard-Jones potential for Carbon are tabulated in Table 2 [1,27].
The interaction parameters of water-nanotube are derived by Lorentz and Berthelot mixing law, in which i and j show the kind of particle. As mentioned previously, the TIP4P water model is used for all simulations, so its energy parameters and length scales are [27,28].
A canonical (NVT) ensemble is applied to retain the particles number, N, the system volume, V, and the system temperature, T, as constant values in the simulation process. Finally, all the information that is used as simulation inputs is tabulated in Table 3.
The purpose of this article is to compute nanofluid's viscosity with LAMMPS software, which is an opensource MD package [16]. The nanofluid, whose viscosity is to be assessed, is regarded as a carbon nanotube dispersion in pure water with a particular volume fraction at room temperature. The Green-Kubo approach is used, which is an EMD method. The trend of this theory is written below [22,29]: where the xy component of the stress tensor has the following form for a binary mixture: where subscript α = 1 refers to molecules, subscript α=2 refers to nanoparticles, m i is the mass of a particle, V is the volume of the system, T is the temperature of the medium, τ is the time to reach a plateau value, F is the force acting on a molecule or nanoparticle, x and y are the coordinates of the molecule or nanoparticle, and N 1 and N 2 are the numbers of molecules and nanoparticles, respectively. The angular brackets [22] in Eq. (10) and Eq. (11) mean an ensemble average. These equations were derived [29] for molecular systems. However, some authors [30] believe that they have the same format for dispersed systems, particularly nanofluids [31].

Results and discussion
In the present study, the influence of change in volume fraction is investigated by changing the number of water molecules. As a result, the simulation box size will differ while the size and type of nanotube are constant in all simulations. The percentage of altered volume fractions is ϕ = 0.557, 1, 2, 3%. A typical nanofluid is created with an SWCNT (6,6) added among the water molecules in a confined volume. Simulations  have been done at two temperatures, 298 K, and 313 K, with different volume fractions. The system is equilibrated in the canonical ensemble for 16 ps, which will be more efficient for this system.

Thermodynamic equilibration
To understand that the simulated system is in a stable state and the obtained results are reliable, besides all the other attempts necessary to achieve the accuracy of the  Potential energy. c) Total energy simulation, the equilibration of nanofluid must be investigated. The system kinetic energy, potential energy, temperature, and pressure have been monitored in this study. The stabilization of the system has been done for 16 ps. The main idea behind this stabilization is to keep the temperature at a remarkable degree. Thus, if it comes out of the desired range with its tolerance, the velocities are rescaled, and the temperature goes back to the target temperature [1]. The fluctuation of the total energy will show the stability of the system. The calculated values ought to preserve at a specific range, and their oscillations must be small enough. Figures 3 and 4 indicate the kinetic energy, the potential energy, and the total energy of water-SWCNT nanofluid for different volume fractions at T = 298K and T = 313K. It is crystal clear that the summation of potential and kinetic    that depends on the molecular mass of the system. If the molecular mass is calculated for each volume fraction, it will be revealed that the average molecular mass decreases by the increase in the volume fraction. The average temperature will be set to the desired temperature (298 K or 313 K) by the Nose-Hoover thermostat, although the system temperature is oscillating. It means that the velocity related to temperature is almost the same for all volume fractions. The only parameter that makes changes in the kinetic energy is average molecular mass. Equation (12) is used to calculate the average mass fraction, and the results are tabulated in Table 4.
Figures 5 and 6 presented the temperature and enthalpy fluctuation curve of nanofluid at different volume fractions versus time. Enthalpy is shown at the equilibrium process. Also, temperatures are displayed in the viscosity calculation process. Achieving the equilibrium state in the system is happened by the NVT ensemble. Also, the enthalpy of nanofluid converges within 16 ps, which signifies the stable conditions for the production of thermodynamic and physical properties.

Viscosity calculation
In this section, the amount of viscosity calculated by MDS and is compared with theoretical models. Several theoretical formulations have been developed to correlate the relative viscosity of nanofluids to particle volume fraction. The relative viscosity (μ r ) is described by the ratio of nanofluid's viscosity, μ nf , to base fluid's viscosity, μ bf : These formulations are drawn from the Einstein initiate model [32], which is based on the idea of viscous fluid, including non-interacting hard spheres with particle volume fraction of less than 1%: where μ r is the relative viscosity as defined above and ϕ is the volume fraction of nanoparticles in the base fluid. Later, Brinkman [33] developed Einstein equation for suspensions with medium particle volume fraction, usually less than 4%: As a result of the Brownian motion and interaction among particles, Batchelor [34] proposed the following equation.
In Eq. (16), η is the intrinsic viscosity, and k H is the Huggin coefficient. The values of η and k H are 2.5 and 6.5, respectively, for spherical particles.
Maron and Pierce derived another equation applied in this article considering the Ree-Eyring flow equations [35,36]. Equation (17) was obtained from a minimum principle applied to the energy dissipated by viscous effects.
The ϕ m is the maximum particle volume fraction when the viscosity is infinite. The viscosity certainty is measured by plotting the arithmetic averages (v ave ¼ v xx þv yy þv zz 3 Þ during the time evolution. Figure 7 shows the values of average viscosity for different volume fractions and temperatures. The time evolution has continued until the average viscosity converged to a certain amount. Besides, it is evident that the average viscosity increases by an increment in the volume fraction. This is because by increasing the volume fraction, the fluid will be demanded more energy to move, and the viscosity of the liquid will increase.   Figures 8 and 9 indicate the values of v xx , v yy , v zz , and average viscosity at T = 298K and T = 313K for ϕ = 0.557%, ϕ = 1%, ϕ = 2%, and, ϕ = 3%, respectively. The time evolution has continued until the average viscosity converged to a certain amount. It is apparent that the amount of average viscosity increases on the increase in the volume fraction, and also, it is decreased while the temperature is increased. Because of the increment in volume fraction, the amount of liquid (water) has reduced compared to the amount of solid (CNT). Therefore, the system fluidity has decreased, and the fluid has needed more energy to move. It means that viscosity has increased by the escalation in volume fraction. Figures 10 and 11 indicate the result of MD calculation and the comparison with two theoretical equations, Batchelor and Maron-Pierce equations. MD would be able to estimate the trend of viscosity in different temperatures and volume fractions. Thus, the MD can be chosen as an acceptable method to calculate the viscosity of nanofluids where there is no experimental answer.
The exact amounts of viscosities at both temperatures (298 K and 313 K) have been tabulated in Table 5. Generally, viscosity and volume fraction have a direct relationship, but the temperature does not. Therefore, MD correctly estimates the ascending behavior of viscosity by incrementing the volume fraction and temperature differences.

Correlation
According to the obtained results from MD, a theoretical formula for calculating water-SWCNT viscosity is offered by the authors for ϕ ≤ 3%.
Unfortunately, the water-SWCNT nanofluid lacks sufficient experimental data. As a result, this correlation is compared to results from water-MWCNT nanofluids [4], and the inaccuracy is not impossible to overcome. The main contribution of this study is to demonstrate the use of MDS to calculate the viscosity of water-SWCNT nanofluids at certain temperatures and volume fractions. Figure 12 shows the experimental data and correlation predictions at low-volume fractions. As it has illustrated, the correlation shows acceptable agreement with the experiment. Figure 13 shows the capability of the correlation equation in the prediction volume fraction and the temperature influence on the viscosity variation. It has revealed that viscosity decreased by increasing temperature and increased by volume fraction increment. Both of these have been predicted by the correlation. Also, the results of pure water [37] are shown in this figure, which presented that the correlation predicted a higher amount than pure water, which is entirely reasonable.

Conclusions
Using molecular dynamics modeling, we investigated the effect of volume fraction on the viscosity of a typical water-SWCNT nanofluid at two different temperatures (298 K and 313 K). The volume fraction of nanofluids, including CNT as nanoparticles, has been discovered to play a key influence in their viscosity. The average arithmetic viscosity vectors in various volume fractions are also calculated. Furthermore, the volume fraction increment increases the amount of viscosity. By comparing the MDS result to the two theoretical equations, Batchelor and Maron-Pierce, the correctness of the MDS result can be determined. The veracity of their findings has already been proven by a large body of experimental evidence. It has been observed that raising the effective volume fraction by increasing the volume fraction will increase viscosity. MD can estimate viscosity within an acceptable range; hence, it can be used when experimental data is lacking or conducting research is problematic. In the end, the authors lproposed a theoretical correlation based on the findings. Volume fraction, temperature, and viscosity are all connected in the formula. In the allowed range of volume fractions, it exhibited good agreement with theoretical and experimental results.