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4.1 Introduction

The suspensions of nanoparticles in the nanofluids significantly modify the properties of the base fluids. Therefore, nanofluids exhibit attractive properties, such as high thermal conductivity and tunable surface tension, viscosity, and rheology. Various attempts have been made to understand the mechanisms for these property modifications caused by the additional nanoparticles. However, these mechanisms are still unclear due to the lack of direct nanoscale evidences.

Most previous studies on the thermophysical properties of nanofluids have explored the effects of the nanofluid parameters, such as the nanoparticle loading, diameter, and material and the base fluid type on the thermal conductivity, surface tension, viscosity, and rheology. Among these properties, the surface tension, viscosity, and rheology are related to the dynamic wetting, a process dominated by the surface tension and viscous forces. Therefore, the surface tension, viscosity, and rheology are defined as wettability-related parameters in this book. The modification of surface tension, viscosity, and rheology by adding nanoparticles greatly affects the dynamic wetting behaviors. These effects are regarded as the bulk dissipation here. Nanofluids are reported to have higher thermal conductivity than the base fluids, which were explained by several established models, e.g., Brownian motion, the solid/liquid interface layer around the nanoparticles, or the ballistic phonon transport hypotheses [1–10]. There have been many studies on thermal conductivity, but only a few studies on nanofluid surface tension, viscosity, and rheology. Table 4.1 summarizes the studies on the nanofluid surface tensions [11]. The studies are still controversial because the surface tensions of nanofluids have been reported to be increased [12–14], unchanged [15, 16], or decreased [17, 18] compared with that of the base fluids. Even for the same nanofluids, such as Al2O3/H2O nanofluids, surface tension was reported to be increased by Zhu et al. [12], but unchanged by Das et al. [17]. Some studies have tried to qualitatively explain the mechanisms of these modifications in the nanofluid surface tension [14, 19–21]. Tanvir and Li [14] suggested that the attractive forces between the particles at the liquid–vapor interface increase as the particle concentration increases which increase the surface tension. However, Murshed et al. [19], Radiom et al. [20], and Liu and Kai [21] proposed that the surface tension reduction may be attributed to the reduction of the cohesive energy at the liquid–vapor interface, because nanosized particles are brought to the lowest interfacial energy level by the Brownian motion. They also suggested that the nanoparticles function as surfactant molecules since nanoparticles are absorbed onto the liquid–vapor interface to reduce the surface tension. However, all these analyses are only suggestions without any direct evidences, and some are even contradictory.

Table 4.1 Studies of the surface tension of nanofluids
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The nanofluids are usually regarded as colloidal suspensions. Rigorously speaking, colloidal suspensions are mixture solutions containing micro-/milli-sized particles, while nanofluids containing nanosized particles. Several classical models have been proposed to predict the effective viscosity of colloidal suspensions [22–25]. However, these models failed to predict the effective viscosity of nanofluids [26–28]. The models of nanofluid viscosity have been reviewed by Eastman et al. [29], Keblinski et al. [30], and Mahbubul et al. [31]. Most of these models were only based on hypotheses or empirical correlations between the effective viscosity and macroscopic parameters, such as the loading fractions or temperatures. Some correlations had considered the effects of particle size, which is related to the Brownian motion [32, 33]; however, undetermined constants in these models must be fit from macroscopic experimental measurements. Thus, more studies are needed to identify the enhancement mechanism with microscopic evidence.

The rheological properties of nanofluids had been reviewed by Chen et al. [34]. According to the summary in Table 4.2, there is still debate about whether nanofluids exhibit Newtonian [16, 35–37] or non-Newtonian [38–47] in experiments. Even for the same nanofluid, such as Al2O3/H2O nanofluids, Newtonian rheological behavior was reported by Das et al. [17] and Prasher et al. [35], while non-Newtonian rheological behavior was observed by Wang et al. [38], Murshed et al. [40], Kole and Dey [43], and Kim et al. [46]. Most of these studies mainly reported the experimental data, with few focused on the mechanisms for the rheological transformation from Newtonian to non-Newtonian fluids by adding nanoparticles. Yu et al. [45] stated that the rheological properties of nanofluids depend strongly on many factors, such as nanoparticle material, shape, the loading fractions, and temperatures. Chen et al. [34] proposed an aggregation mechanism to interpret the rheological behavior of nanofluids and categorized the rheological behavior of nanofluids into four groups such as dilute, semi-dilute, semi-concentrated, and concentrated. However, they did not consider the mechanism of rheological transform. In addition, no direct evidence was provided to support their hypotheses.

Table 4.2 Studies of the rheology of nanofluids
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Most previous studies of nanofluid have focused on collecting the thermophysical property data. The macroscopic measurements were then used to establish models to explain the mechanisms; however, these explanations have not been supported by direct evidence. MD simulations can provide microscopic understanding of the thermophysical properties in much greater detail, which have been extensively used to investigate the surface tension, viscosity, and rheology of pure liquids [48–55] or binary mixture solutions [56–60]. Recently, the MD simulations have also been used to model the nanofluid thermal conductivity [61–67] with some promising results that verify the theories based on macroscopic measurements. However, few MD studies have focused on the surface tension, viscosity, and rheology of nanofluids.

The objective of this chapter is to study how nanoparticles alter the surface tension, viscosity, and rheology of nanofluids from microscopic viewpoints using MD simulations. The results reveal the roles of additional nanoparticles on the modification of wettability-related parameters and then provide the guidelines in building nanofluid dynamic wetting models.

4.2 Simulation Methods

4.2.1 Simulation Systems

Figure 4.1 shows the simulated systems had four nanofluid loadings. Bulk water films (4500 water molecules) were simulated at 300 K using the molecular dynamic simulations with four gold nanoparticle loadings (φ = 0, 3.43 %, 6.77 %, and 9.81 %). The gold nanoparticles (0.8 × 0.8 × 0.8 nm3) randomly distributed inside the bulk liquid. The four-point TIP4P-Ew water model, PPPM technique, and SHAKE algorithm were used to describe the water–water interactions [68, 69]. The EAM [70] was used for the gold–gold interactions. The simulation details have been described in Chap. 3. A 12-6 LJ potential with σ = 3.1 Å [71] and a cutoff distance of 9 Å was used to describe the water–gold interactions. The gold nanoparticle wettability was changed with different gold–water interaction parameter ε (ε = 0.0070, 0.02714, 0.05427 and 0.08141 eV). The NVT ensembles at T = 300 K with a time step of 1 fs were used to calculate the thermodynamic properties. The simulations were performed with LAMMPS software packages [72].

Fig. 4.1
figure 1

Illustrations of the molecular structures of pure water and three gold nanoparticle nanofluid loading fractions (the water molecules are transparent to illustrate the nanoparticle positions for φ = 3.43 %). Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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4.2.2 Surface Tension Calculation Method

In most MD simulations, the surface tension is calculated based on Young–Laplace equation [73], as shown in Eq. (4.1).

$$\Delta p = \frac{{\sigma_{\text{LV}} }}{R}$$
(4.1)

where Δp is the pressure difference between the inside and outside of the droplet, σ LV is the liquid–vapor surface tension, and R is the droplet radius. However, the pressure fluctuates greatly in LAMMPS. For example, for a set pressure of 1 bar, the standard deviation of the fluctuations is about 40 bar. Thus, the surface tension cannot be calculated with such large pressure variations even with time averaging. Therefore, a new method, referred to as excess energy method here, was used to calculate the surface tension. The simulation details are shown in Chap. 3.

4.2.3 Viscosity and Rheology Calculation Method

The reverse non-equilibrium MD (rNEMD) method [74] was used to calculate the nanofluid viscosity. The method is based on Onsager linear response theory,

$$j_{\alpha } = - \sum\limits_{\beta } {L_{\alpha \beta } } X_{\beta }$$
(4.2)

where j α is the momentum flux, X β is the driving force (velocity gradient), and L αβ is the diffusion coefficient (viscosity).

As shown in Fig. 4.2, a momentum flux was imposed on the bulk liquid to generate a velocity gradient. The diffusion coefficient, in this case the viscosity, was then obtained using:

$$j_{\alpha } \left\langle {p_{x} } \right\rangle = \frac{{P_{x} }}{2tA}$$
(4.3)

where P x is the pressure in the x direction, t is the velocity swapping period, and A is the area of xy plane.

Fig. 4.2
figure 2

Momentum exchange in the rNEMD simulation method. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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To impose the momentum flux onto the bulk liquid, two water molecules were moved in the bottom and middle plates against the intended current, one with the minimum velocity and the other with the maximum velocity. The velocities were then swapped in another calculation. The momentum flux then generated a velocity gradient, dV x /dz. As shown in Fig. 4.3, a good linear velocity profile was obtained along the z direction that was fit with a straight line with the momentum flux and then calculated using Eq. (4.3).

Fig. 4.3
figure 3

Velocity profile in the rNEMD method. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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The velocity swapping period, t, in Eq. (4.3) was varied to create various momentum fluxes and velocities to relate the viscosity to the shear rate (rheology).

Figure 4.4 compares the present MD results with the experimental data for pure water at various temperatures. The good agreement indicates the accuracy of the present water model and the rNEMD method.

Fig. 4.4
figure 4

Verification of the rNEMD method: viscosity of pure water at various temperatures. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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4.3 Results and Discussion

4.3.1 Surface Tension of Gold–Water Nanofluids

For pure water, the calculated surface tension from the MD simulations was σ = 0.0679 N/m at T = 300 K, close to the experimental value of σ = 0.072 N/m [75]. For the 3.43 % nanofluid, the addition of nanoparticles increased the surface tension for the nanoparticles with ε water–gold = 0.05427 eV, but reduced it for those with ε water–gold = 0.0070 eV. The different tendencies for these two cases are related to the different water–gold interactions (ε water–gold) and water–water interactions (ε water–water = 0.0071 eV for TIP4P-Ew). For ε water–gold < ε water–water, the nanoparticles are hydrophobic, so they tend to stay on the free surface (liquid–vapor interface), acting as surfactant-like particles, as shown in Fig. 4.5a. For ε water–gold > ε water–water, the nanoparticles are hydrophilic, so they tend to submerge into the bulk liquid, acting as non-surfactant particles, as shown in Fig. 4.5b.

Fig. 4.5
figure 5

Snapshots of the 3.43 % nanofluid with a hydrophobic gold nanoparticles (ε = 0.007 eV); b hydrophilic gold nanoparticles (ε = 0.05427 eV). Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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The surface tension indicates the unbalanced forces acting on the liquid molecules at the interface due to the van der Waals force, as shown in Fig. 4.6a. The liquid molecules at the surface do not have an equal number of molecules in the vapor side, so they are pulled inward by the internal molecules, which results in the surface tension, pulling the liquid surface to contract to the minimum area. Thus, the nanoparticle wettability is responsible for the different surface tensions. For hydrophobic nanoparticles, the nanoparticles gather on the free surface. The repulsion force between the nanoparticles and the water molecules increases the intermolecular spacing at the interface and reduces the attraction forces between the water molecules inside the bulk liquid and the ones on the free surface regions, thus reducing the surface tension, as shown in Fig. 4.6b. However, some hydrophilic nanoparticles are transported toward the interfacial region by Brownian motion where the attraction forces between the nanoparticles and the water molecules reduce the intermolecular spacing at the interface. The water molecules at the free surface are more strongly pulled inward due to the presence of the hydrophilic nanoparticles with stronger gold–water interaction forces than those with the water–water interactions, which increase the surface tension, as shown in Fig. 4.6c.

Fig. 4.6
figure 6

Schematic of the van der Waals forces between water molecules and nanoparticle–water molecule near the interfacial region: a water molecular interaction for pure water; b gold nanoparticle–water molecule interactions for ε water–gold < ε water–water; c gold nanoparticle–water molecule interactions for ε water–gold < ε water–water. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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Figure 4.7 shows the evidences for the hydrophobic/hydrophilic nanoparticles changing the intermolecular spacing in the interfacial region. The water density remains constant in the bulk liquid but decreases sharply near the interface. The thickness of the reduced density region, defined as the interface width, d, decreases with increasing nanoparticle wettability, leading to the increased surface tension. In addition, the interface width, d, decreases with increasing φ for the hydrophilic nanoparticles, but increases with increasing φ for the hydrophobic nanoparticles.

Fig. 4.7
figure 7

Water density profiles perpendicular to the free surface for pure water and nanofluids with hydrophobic or hydrophilic nanoparticles (φ = 3.43 %). Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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Therefore, for nanofluids, the dynamic wetting can be facilitated by adding hydrophobic nanoparticles, but inhibited by adding hydrophilic nanoparticles, which is related to the modification of nanofluid surface tension.

4.3.2 Viscosity of Gold–Water Nanofluids

Figure 4.8 shows the viscosities of pure water and low loading nanofluids (φ = 0 %, φ = 3.43 %, and φ = 6.77 %). The viscosity increases with increasing volume concentration, which agrees with experimental data [76]. The increasing viscosity with loading is illustrated by the microscopic picture of one randomly selected gold nanoparticle shown in the inset in Fig. 4.9, in which the hydrogen atoms are hidden. An absorbed water layer has formed around the gold nanoparticle, which can be observed by plotting the water density along the radial direction, as shown in Fig. 4.9. The water density near the nanoparticle at 19 Å is five times that of the bulk liquid, indicating a solid-like absorbed water layer around the gold nanoparticle. The effect can be explained by the Einstein diffusion equation [77],

$$D_{\text{NP}} = \frac{{k_{\text{B} } T}}{6\pi \mu r}$$
(4.4)

where D NP is the nanoparticle diffusion coefficient, k B is Boltzmann’s constant, T is the absolute temperature, μ is the base liquid viscosity, and r is the particle radius. The absorbed water layer increases the equivalent nanoparticle radius which hinders nanoparticle diffusion within the base liquid according to Eq. (4.4). Consequently, the increased nanofluid viscosity can be explained by the decreased nanoparticle diffusion coefficient. The absorbed layer is also reported as the reasons for the nanofluid thermal conductivity enhancement [51, 53]. The interactions between the nanoparticles and the water molecules were changed by modifying ε. As shown in Fig. 4.10a, the density of the absorbed water layer increases with the increasing gold–water interaction parameter. Thus, smaller diffusion coefficients occur for stronger NP–water molecule interactions, and the nanofluid viscosity can be expected to increase with increasing the gold–water interaction parameter, as shown in Fig. 4.10b. The number of gold nanoparticles in the solution affects the viscosity, as shown in Fig. 4.10b, but does not affect the intensity of the absorbed water layer for a single nanoparticle if the nanoparticle loading is low enough.

Fig. 4.8
figure 8

Nanofluid viscosity for various NP loadings. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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Fig. 4.9
figure 9

Water density near a gold nanoparticle (Inset illustration of the water molecule layer). Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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Fig. 4.10
figure 10

Effects of gold nanoparticle–water molecule interactions on a atom number density; b viscosity. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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4.3.3 Rheology of Gold–Water Nanofluids

Figure 4.11 shows the viscosity versus shear rate variation, as known as rheological relations. For pure water and low nanofluid loadings (φ = 3.43 % and φ = 6.77 %), the viscosity remains constant for shear rates from 10−1 to 3 × 102 s−1. However, the viscosity of the φ = 9.81 % nanofluid remains constant only with a narrow shear rate range (from 10−1 to 100 s−1) and decreases as the shear rate increases from 100 to 3 × 102 s−1, indicating shear-thinning non-Newtonian rheological behavior. This rheology transformation at high shear rates was also seen experimentally by Carré and Woehl [78].

Fig. 4.11
figure 11

Viscosity versus shear rate (rheology) for nanofluids for various loadings (ε = 0.05427 eV). Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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The inset in Fig. 4.12 shows a typical microscopic structure of two nanoparticle–water molecule cluster. Comparison of the particle distribution with the high loading in the inset in Fig. 4.12 with the distribution for a lower loadings seen in the inset in Fig. 4.9 shows that there are more chances for nanoparticles to move closer together for high loadings since nanoparticles always move randomly in the base liquid due to Brownian motion. The solid-like absorbed water layers around the nanoparticles are also observed in this structure. Additionally, the water density between these two closed nanoparticles is several times larger than that of the bulk liquid density, as shown in Fig. 4.12, indicating that solidification of the water molecules also occurs in this structure, which is defined here as a solidification structure due to the additional nanoparticle–nanoparticle interactions. This solidification structure further increases the nanofluid viscosity for higher loadings. The solidification structures remain unchanged at low shear rates, where the nanofluids exhibit Newtonian rheological behavior. However, there is a critical shear rate above which the solidification structures are disrupted and the solidified water molecules move more freely, which reduces the nanofluid viscosity, leading to the shear-thinning non-Newtonian rheological behavior.

Fig. 4.12
figure 12

Solidification effect at high loadings (Inset illustration of the molecular spacing). Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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When the gold–water interactions increase for lower loadings (φ = 3.43 % and φ = 6.77 %), the viscosity increases, while the rheology remains unchanged, as shown in Fig. 4.13. Therefore, the mechanism for the solidification effect, which is more likely at high loadings, is related to but differs from the mechanism for the solid-like absorbed water layer that alters the viscosity.

Fig. 4.13
figure 13

Effects of gold nanoparticle–water molecule interactions on rheology for φ = 3.43 % and φ = 6.77 % nanoparticle loadings. Reprinted from Ref. [11], with kind permission from Springer Science+Business Media

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4.4 Conclusions

The surface tension, viscosity, and rheology of gold–water nanofluids were examined using molecular dynamic simulations. The effects of the gold nanoparticle loadings and particle wettability were examined. The main conclusions are as follows:

  1. 1.

    The microscopic mechanism for the surface tension variation is related to the nanoparticle wettability. The repulsion of water molecules by surfactant-like nanoparticles increases the molecular spacing on the free surface which reduces the fluid surface tension. The attraction induced by non-surfactant nanoparticles reduces the molecular spacing on the free surface which increases the surface tension.

  2. 2.

    A solid-like absorbed water layer around the nanoparticles increases the equivalent nanoparticle radius and hinders the nanoparticle mobility within the base fluid which increases the nanofluid viscosity.

  3. 3.

    The nanofluid rheological behavior depends on the nanoparticle loading. For low loadings, the viscosity increases with increasing gold–water interaction forces, but remains unchanged with the shear rate, indicating Newtonian behavior. For high loadings, water molecule solidification is observed between neighboring nanoparticles due to the strong particle–particle interactions. These solidification structures are then disrupted for shear rates exceeding a critical value, which leads to shear-thinning non-Newtonian rheological behavior.