The viscosity of SiO2 nanofluid was simulated at constant temperatures for the systems with different volume concentrations, which are defined as the volume fraction of SiO2 particles. As shown in Fig 2a, the viscosity increases with SiO2 concentration at a given temperature. Moreover, the increase is more significant at lower temperatures. When the concentration increases from 1.2% to 4.8%, for example, the viscosity increases by 1.47 mPa∙s at 280 K and 0.32 mPa∙s at 340 K, respectively. For these simulations, all the SiO2 particles are fixed at 11.2 Å in diameter. A large volume concentration means more particles in the system. The increasing SiO2 particle number increases the particle–water contacting area, leading to increasing contribution from particle-water interaction to the viscosity. As we will show below, the particle–water interaction, which is stronger than water–water interaction, tends to increase the viscosity. The volume concentration dependence of nanofluids has been studied experimentally for quartz, copper oxide, titanium dioxide systems [4, 13, 42, 43]. In Namburu’s measurements , the high concentrations of quartz nanoparticles in ethylene glycol and water mixture leads to great viscosity, and the viscosity variations at low temperature is more significant than that at high temperature. Our calculations produced similar results with the observations.
The temperature dependence of fluids has been well addressed by many authors [4, 11, 12, 42-44]. Increased molecular kinetic energy at high temperature usually makes the fluid viscosity small. This is true for SiO2 nanofluid. Fig 2a also shows the temperature dependence of viscosity for the nanofluid at a given SiO2 concentration. Similar to water, the viscosity of SiO2 nanofluid decreases with temperature. However, the temperature sensitivity of viscosity is different for the systems. The viscosity of systems with higher SiO2 concentration drops more rapidly with temperature. In Fig 2a, the slope of viscosity–temperature curves increases with SiO2 concentration, indicating that the SiO2–water interaction is more important in viscosity contribution at lower temperature. This holds only when the interaction between SiO2-water interfaces is stronger than that between water molecules.
For SiO2 nanofluids with a given volume concentration, their viscosity depends on the particle size. In Fig 3 we compare the viscosity of SiO2 nanofluid with different particle diameters. The large particle, about 18.4 Å in diameter, is larger than the small one (11.2 Å) by 170% in surface area and by 343% in volume. Since the total volume fraction of SiO2 particles is fixed, the system with smaller particle size has greater particle number. As a result, the total surface area increases by a ratio of 18.4:11.2. Therefore, SiO2–water interface interaction plays a more significant role in the system with smaller particle size. Our calculations reveal that at every temperature the system with smaller particles has larger viscosity. Moreover, the difference becomes more remarkable at lower temperature. For example, the small SiO2 particles result in a viscosity of 0.78 mPa∙s at 340 K, only 0.15 mPa∙s larger than that by the large particles. At 280 K, however, their viscosity difference becomes 1.03 mPa∙s. Namburu  measured the viscosity of SiO2 nanoparticles with various diameters of 20, 50 and 100 nm suspended in a 60:40 (by weight) ethylene glycol and water mixture in a wide temperature range from –35 to 50°C, revealing that at same volume fraction the nanofluids with large particle diameters have low viscosity. Our calculated results are consistent with the experimental observations for both the particle size dependence and its variation with temperature.
It is interesting to look into the relative viscosity, which was often used to measure the viscosity of nanofluids. The relative viscosity is defined as the viscosity ratio of nanofluid with respect to pure solvent, μr = μs/μw, where μw is the viscosity of water. The computed μr values for SiO2 nanofluids are shown in Fig 2b. Remarkable concentration dependence can be noted. The ratios are about 1.1, 1.3, 1.6, and 2.2 for the four concentrations, and nearly unchanged within the temperature range except that for the highest concentration of 4.8%. Such terraced increase of μr with respect to concentration confirms that the viscosity increase mainly comes from SiO2–water interaction rather than water–water or SiO2–SiO2 interaction. It has been observed that the relative viscosity of copper oxide nanofluid has very small changes (less than 0.3) over –35 to 50°C at low concentrations . The μr decay at high temperature for the samples with high concentrations was also noted. Similar results were also reported by Prasher for alumina particles suspended in propylene glycol with a volume concentration of 0.5%, 2%, and 3% at 30–50°C . Our calculations reveal that the μr of SiO2 nanofluid is concentration dependent instead of temperature dependent at low SiO2 concentrations. In addition, we also noted the decay of μr at high temperature (above 300 K) for systems with relatively high volume concentrations of 3.6% and 4.8%.
Above calculations demonstrate the important role of SiO2-water interaction. It is therefore interesting to inspect such kind of interaction further. Two computational models were then designed to evaluate the interacting patterns between SiO2 and water by means of DFT calculations, as shown in Fig 4. One is the cluster model in which a water molecule adsorbs onto a (SiO2)6 cluster. The structure of (SiO2)6 cluster was taken from Ref. []. The water H atom binds with one of the O atoms of the cluster via a hydrogen bond. The interaction energy, which is defined as the energy difference between the systems before and after water adsorption, is about 1.43 eV. A similar cluster model, a water molecule adsorbing onto a (H2O)6 cluster, which was taken from Ref. [], gives the interaction energy of 0.65 eV between the water molecule and the water cluster. In the second model, periodical DFT calculations were performed to compute the interaction of a water molecule on the SiO2 (001) and ice (001) surfaces, which were sliced respectively from α-quartz and cube-ice crystal structures. Under this slab model, the computed interaction energy is 1.78 eV for a water molecule on the SiO2 surface and 0.94 eV on the ice surface. Larger interaction energies between SiO2 and water were predicted by both the cluster and the slab models, confirming above speculations from MD computations at the force-field level.
Several expressions have been proposed by Bicerano,  Brinkman,  Duangthongsuk,  Kulkarni  and Namburu  to fit the measured viscosity data of nanofluids, providing an estimation for viscosity variation with particle concentration and/or temperature. Most of these correlations are similar in nature, though different parameters were used to adjust the values for high concentration systems. The effect of particle size, however, is ignored in these correlations. As we found above, the nanofluid systems with different particle sizes may have quite different viscosities even though they have the same volume concentration. Our MD and DFT calculations revealed the decisive role of SiO2-water interaction in the rheological behavior of SiO2 nanofluids. We would explore below the correlation of the viscosity of SiO2 nanofluids with particle-water interaction strength.
Starting from the data in Fig 2, an exponential correlation,
can be fitted. A good correlation with R2 > 0.99 is obtained, as shown in Fig 5. The systems with different SiO2 concentrations have similar slopes (lnA). The concentration effect on the viscosity is then represented only in parameter B. From (2) one has
ln μr = A' - ∆B⁄T (3)
where A´ is a constant and ΔB = B – Bw. Bw is fitted from the viscosity of pure water. Eq. (3) has the similar form with the correlations proposed by Kulkarni  and Namburu  in which both A and B were functions of particle concentrations. From our MD simulations, the SiO2–water interaction energy (Eint) can be obtained by summing up the coulomb and van der Waals terms between SiO2 particles and water molecules. Fig 6 presents the correlation of ΔB with Eint. It is interesting to note that with increasing SiO2-water interaction energy ΔB increases, leading to increasing μr. Therefore, the concentration dependent parameters in previously observed correlations [12, 43] for nanofluids can be further understood as quantities relating to particle-solute interaction, and can be expressed as functions of interaction energy.