Introduction

Nanofluids as suspensions of nanometer-sized particles are complex fluids widely used in electronic-cooling, lubrication, and biomedical systems (Choi 1995, 2009; Cheng and Cao 2011; Michaelides 2013). The suspended nanoparticles significantly modify the thermophysical properties of the base fluids so that the nanofluids exhibit attractive properties, such as high thermal conductivity, tunable surface tension, viscosity, and rheology (Chakraborty and Padhy 2008; Trisaksri and Wongwises 2007; Branson et al. 2013). These properties strongly affect the heat transfer, flow, and dynamic wetting behavior of nanofluids in microfluidic systems (Wu et al. 2013; Murshed et al. 2008a; Li et al. 2009); hence, nanofluids can be used as “smart materials” in many applications.

Most previous studies of 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. There are a large number of studies on the thermal conductivity (Choi 1995, 2009; Cheng and Cao 2011; Michaelides 2013; Wu et al. 2013; Murshed et al. 2008b; Li et al. 2009), but only a few studies on the nanofluid surface tension, viscosity, and rheology. As shown in Table 1, the surface tensions of nanofluids have been reported to be increased (Zhu et al. 2010; Moosavi et al. 2010; Tanvir and Li 2012), unchanged (Kumar and Milanova 2009; Chen et al. 2011), or decreased (Das et al. 2003; Vafaei et al. 2009) from that of the base fluids,. Even for the same nanofluid, such as Al2O3/H2O nanofluids, the surface tension was reported to be increased by Zhu et al. (2010), but unchanged by Das et al. (2003). Some studies have tried to qualitatively explain the mechanisms for these changes in the nanofluid surface tension (Tanvir and Li 2012; Murshed et al. 2008c; Radiom et al. 2010; Liu and Kai 2012). Tanvir and Li (2012) suggested that the attractive forces between the particles at the liquid–vapor interface increase as the particle concentration increases which increases the surface tension. However, Murshed et al. (2008a), Radiom et al. (2010), and Liu and Kai (2012) proposed that the surface tension reduction may be attributed to the reduction of the cohesive energy at the liquid–vapor interface, since nano-sized particles, when functioning as surfactant molecules, are brought to the lowest interfacial energy location by the Brownian motion. However, all these analyses were only suggestions without providing any direct evidence and some were even contradictory.

Table 1 Studies of the surface tension of nanofluids
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Micro/milli size particles in suspension (colloidal suspensions) increase the base fluid viscosity with several classical models proposed to predict the effective viscosity of colloid suspensions (Einstein 1956; Krieger and Dougherty 1959; Nielsen 1970; Batchelor 1977). However, the viscosity increase for nanofluids is quite different from that of colloidal suspensions with the effective viscosity of nanofluids being quite different from that given by the classical models (Chandrasekar et al. 2010; Nguyen et al. 2008; Lee et al. 2008). The studies of nanofluid viscosity have been reviewed by Eastman et al. (2004), Keblinski et al. (2005), and Mahbubul et al. (2012). Most of these models were only based on hypotheses or empirical correlations between the effective viscosity and macroscopic parameters, such as the loading concentration and temperature. Some correlations considered the effects of particle size as it relates to the Brownian motion (Masoumi et al. 2009; Hosseini et al. 2010); 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 were reviewed by Chen et al. (2007b). According to the summary in Table 2, there is still debate about whether nanofluids exhibit Newtonian (Das et al. 2003; Prasher et al. 2006; Chen et al. 2007a; Susan-Resiga et al. 2012) or non-Newtonian (Wang et al. 1999; He et al. 2007; Murshed et al. 2008b; Chen et al. 2009; Ding et al. 2006; Kole and Dey 2011; Yu et al. 2001, 2009; Kim et al. 2011; Abareshi et al. 2011) behavior in experiments. Even for the same nanofluid, such as Al2O3/H2O nanofluids, Newtonian rheological behavior was reported by Das et al. (2003) and Prasher et al. (2006), while non-Newtonian rheological behavior was reported by Wang et al. (1999), Murshed et al. (2008c), Kole and Dey (2011), and Kim et al. (2011). Most of these studies only reported experimental data, with few focusing on the mechanisms for the rheological transformation from Newtonian to non-Newtonian fluids by adding nanoparticles. Yu et al. (2009) stated that Newtonian or non-Newtonian behavior of nanofluids depends strongly on many factors, such as the kind and shape of the nanoparticles, the volume fractions, and temperatures. Chen et al. (2007b) proposed an aggregation mechanism to interpret the rheological behavior of nanofluids and categorized the rheological behavior of nanofluids into four groups as dilute, semi-dilute, semi-concentrated, and concentrated. However, no direct evidence was provided to support their assumptions.

Table 2 Studies of the rheology of nanofluids
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Most previous studies of nanofluids have focused on properties data measurements. The macroscopic measurements were then used to build models to explain the mechanisms; however, these explanations have not been supported by direct evidence. MD simulations can show the microscopic mechanisms for the molecular interactions in much greater detail to provide a microscopic understanding of the thermophysical properties. MD simulations have been extensively used to investigate the surface tension, viscosity, and rheology of pure liquids (Shi et al. 2006; Mountain 2009; Zhu and Yang 2011; Rutkevych et al. 2010; Hou et al. 2009; Sunda and Venkatnathan 2013; Medina et al. 2011; Thomas and Rowley 2011) or binary mixture solutions (Li et al. 2011; D’Auria and Tobias 2009; Ge et al. 2011; Chen et al. 2010; Kumar et al. 2013). Recently, the MD approach has also been used to model the thermal conductivity of nanofluids (Mohebbi 2012; Cui et al. 2011; Li et al. 2008, 2010; Teng et al. 2008; Sarkara and Selvam 2007) with some results verifying the theories based on macroscopic measurements. However, no MD studies exist that simulate the surface tension, viscosity, and rheology of nanofluids.

The objective of this work is to study how nanoparticles alter the surface tension, viscosity, and rheology of nanofluids from microscopic viewpoints using MD simulations. The simulations used gold–water nanofluids with the interaction potential parameters varied to represent various types of nanofluids. The microscopic pictures of the nanoparticle–water molecule interactions are used to identify the microscopic mechanisms that explain the modified nanofluid surface tension, viscosity, and rheology.

Simulation methods

Simulation systems

The simulated systems had four nanofluid loadings as shown in Fig. 1. Bulk water films (4,500 water molecules) were simulated at 300 K using molecular dynamics (MD) simulations with four gold nanoparticle loadings (particle volume fractions φ = 0, 3.43, 6.77, and 9.81 %). The gold nanoparticles (0.8 × 0.8 × 0.8 nm3 with 32 gold atoms per particle) were equilibrated at 300 K and then absorbed spontaneously by the water film until the nanofluid system reached its minimum energy where the nanoparticles were randomly distributed inside the liquid films, as shown in Fig. 1. The water–water interactions were calculated using the TIP4P-Ew water model (Horn et al. 2004) with the long-range Coulombic forces calculated using the particle–particle particle-mesh technique with the water molecules kept rigid by the SHAKE algorithm (Ryckaert et al. 1977). The gold–gold interactions were calculated using the embedded-atom method (Daw et al. 1993). The 12-6 LJ potential was used to describe the water–gold interactions. The gold nanoparticle wettability was modified by changing the gold–water interaction parameter ε (ε = 0.0070, 0.02714, 0.05427, and 0.08141 eV, with σ = 3.1 Å and a cutoff distance of 9 Å). The simulations were performed using LAMMPS (Plimpton 1995) with NVT ensembles at T = 300 K with a time step of 1 fs in all cases.

Fig. 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 %)

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Surface tension calculation method

In most MD simulations, the surface tension is calculated based on the Young–Laplace equation (Ismail et al. 2006).

$$\Delta p = \frac{{\gamma_{lv} }}{R},$$
(1)

where Δp is the pressure difference between the inside and outside of the droplet, γ 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, the new method was used to calculate the surface tension of the liquid vapor system. Another NVT system was simulated with a nanofluid film with two free surfaces. The time-averaged excess energy of the two systems with and without free surfaces was then used to calculate the surface tension for pure water and for the nanofluids with different particle loadings as

$$\gamma_{lv} = \frac{{\left\langle {E_{\text{w} } } \right\rangle - \left\langle {E_{{\text{w} /o}} } \right\rangle }}{{A_{\text{interface}} }},$$
(2)

where 〈E〉 is the time-averaged total energy, the subscripts w and w/o denote with and without liquid–vapor interfaces, and A interface is the area of the free interface.

Viscosity and rheology calculation method

The reverse non-equilibrium MD (rNEMD) method (Muller-Plathe 1999) was used to calculate the nanofluid viscosity since it is more robust than the equilibrium MD method. The method is based on Onsager linear response theory,

$$j_{\alpha } = - \sum\limits_{\beta } {L_{\alpha \beta } } X_{\beta } ,$$
(3)

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

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)

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

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. A good linear velocity profile was obtained along the z direction that was fit with a straight line with the momentum flux then calculated using Eq. (4).

The velocity swapping period, t, in Eq. (4) was varied to create various momentum fluxes and velocities to relate the viscosity to the shear rate (rheology).

The MD results agree well with experimental data for pure water at various temperatures as shown in Fig. 2 which indicates the accuracy of the present water model and the rNEMD method.

Fig. 2
figure 2

Verification of the rNEMD method: viscosity of pure water at various temperatures

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Results and discussion

Surface tension of gold–water nanofluids

For pure water, the calculated surface tension from the MD simulations was γ = 0.0679 N/m at T = 300 K which differs by 5.6 % from the experimental value of γ = 0.072 N/m (Gittens 1969). The addition of nanoparticles increased the surface tension of the nanofluid with φ = 3.43 % for ε water–gold = 0.05427 eV, but reduced it for 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, acting as surfactant-like particles, as shown in Fig. 3a. 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. 3b.

Fig. 3
figure 3

Snapshots of the 3.43 % nanofluid with a hydrophobic gold nanoparticles (ε = 0.007 eV); b hydrophilic gold nanoparticles (ε = 0.05427 eV)

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The surface tension indicates the unbalanced forces acting on the liquid molecules on the interface due to the van der Waals force, as shown in Fig. 4a. The liquid molecules at the surface do not have an equal number of molecules on the vapor side so they are pulled inwards by the internal molecules which results in the surface tension which forces 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. 4b. 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 inwards due to the presence of the hydrophilic nanoparticles with stronger gold–water interaction forces than for the water–water interactions, which increase the surface tension, as shown in Fig. 4c.

Fig. 4
figure 4

Schematic of the van der Waals force between water molecules and nanoparticle–water molecule near the interface 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

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Figure 5 shows the evidence for the hydrophobic/hydrophilic nanoparticles changing the intermolecular spacing in the interface 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. 5
figure 5

Water density profiles perpendicular to the free surface for pure water and nanofluids with hydrophobic or hydrophilic nanoparticles (φ = 3.43 %)

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Viscosity of gold–water nanofluids

The viscosities of pure water and low loading nanofluids (φ = 0 %, φ = 3.43 % and φ = 6.77 %) are shown in Fig. 6. The viscosity increases with increasing volume concentration, which is consistent with experimental data (Hilsenrath 1995). The increasing viscosity with loading is illustrated by the microscopic picture of one randomly selected gold nanoparticle shown in the insert in Fig. 7, in which the hydrogen atoms are hidden. An absorbed water layer has formed around the gold nanoparticle which can be seen by the water density plotted along the radial direction in Fig. 7. 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 (Einstein 1905),

Fig. 6
figure 6

Nanofluid viscosity for various NP loadings

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Fig. 7
figure 7

Water density near a gold nanoparticle (inset illustration of the water molecule layer)

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$$D_{\text{NP}} = \frac{{k_{\text{B} } T}}{6\pi \eta r},$$
(5)

where D NP is the nanoparticle diffusion coefficient, k B is Boltzmann’s constant, T is the absolute temperature, η 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. (5). Consequently, the increased nanofluid viscosity can be explained by the decreased nanoparticle diffusion coefficient. The absorbed layer is also the basis for the nanofluid thermal conductivity enhancement mechanism (Li et al. 2008, 2010). The interactions between the nanoparticles and the water molecules were changed by modifying ε. As shown in Fig. 8a, 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 of the gold–water interaction parameter, as can be seen in Fig. 8b. The number of gold nanoparticles in the solution affects the viscosity, as shown in Fig. 8b, but does not affect the intensity of the absorbed water layer for a single nanoparticle if the nanoparticle loading is low enough.

Fig. 8
figure 8

Effects of gold nanoparticle–water molecule interactions on a atom number density; b viscosity

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Rheology of gold–water nanofluids

The viscosity versus shear rate variation is shown in Fig. 9. 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 (2006).

Fig. 9
figure 9

Viscosity versus shear rate (rheology) for nanofluids for various loadings (ε = 0.05427 eV)

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The nanoparticles move randomly in the base liquid due to Brownian motion. Comparison of the particle distribution with the high loading in the insert in Fig. 10 with the distribution for a lower loading seen in the insert in Fig. 7 shows that there are more chances for nanoparticles to move closer together for high loadings. A typical microscopic structure of two nanoparticle–water molecule cluster is shown in the insert in Fig. 10. 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. 10, indicating that solidification of the water molecules also occurs in this structure, which 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 rheological behavior.

Fig. 10
figure 10

Solidification effect at high loadings (inset illustration of the molecular spacing)

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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. 11. 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. 11
figure 11

Effects of gold nanoparticle–water molecule interactions on the rheology for φ = 3.43 % and φ = 6.77 % nanoparticle loadings

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Conclusions

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

  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 nanoparticle–nanoparticle interactions. These solidification structures are then disrupted for shear rates exceeding a critical value, which leads to shear-thinning non-Newtonian behavior.