## Predicting the viscosity of mixtures with RheoCube

11 March 2021

**Mihai Gherghe, MSc, and Christophe Vaillant, PhD**

The viscosity of fluid mixtures is arguably one of the most commonly investigated properties. This especially rings true from the point of view of creating new formulations for products. Viscosity itself plays a key role in a wide range of phenomena. For example, the diffusion of a cleaning agent towards a target surface, or particle sedimentation in a coating formulation. From a technological point of view, the way viscosity changes is crucial to understanding fluid behavior during manufacturing operations. Imagine this for mixing, mass transport, and energy transfer calculations.

Predicting the viscosity of any chemical combination, however, turns out to be pretty complicated. For example when trying to find the right balance between keeping the calculations simple and having the best accuracy. There is no simple, universal equation that can predict the viscosity for an arbitrary mixture. Models of viscosity can vary from fairly simple, straight-line approximations based on the relative volume or mass fractions of the components, right up to full atomistic molecular dynamics simulations. Unfortunately, the simplest types of model typically fail to capture even the simplest trends.

**Simple models of viscosity**

Any microscopic model of viscosity can be broken down into a few basic parts. Typically this involved the composition (how much stuff there is) and how viscous each component is. More complicated models can also include a relative energy contribution to try capturing some of the interactions in more detail. These models also usually perform better. Calculating this quickly gets complicated due to the number of effects that can contribute to the overall free energy. For instance, some systems can arrange themselves into complicated networks, where each molecule is stuck to another by hydrogen bonds. Capturing this effect quantitatively with a simple model is exceedingly difficult.

In RheoCube, we make use of the SPH method to simulate complex fluids on the mesoscale. However, for each point in this simulation we need to have a good prediction of the microscale result. As such, we have tried to find the simplest microscopic model that can still give a good level of accuracy. The model we use_{1 }essentially uses a ratio of the component’s cohesive energies (energy required to keep a fluid held together and not turn into a gas) to the total cohesive energy. These cohesive energies are based on the readily-available Hansen Solubility Parameters (HSPs), which we use in other parts of our simulations to quantify the solubility of fluids_{2}.

**A difficult case: water-glycerol mixtures**

The bigger the difference between the viscosities of different components of a mixture, the more difficult it is to simulate. This due to the relative time scales involved. Molecules in a fluid move at different speeds depending on the viscosity. We can always adapt our simulation to suit the right time scale. However, if we have big differences in these scales then we must adapt our simulations to suit both our largest and smallest time scales. Simulations here need to run longer to properly capture the longest and shortest time scales of the mixture.

This situation namely happens in the water-glycerol mixture. Here the viscosity of glycerol is about 1500x that of water. In RheoCube, as we are simulating this mixture on the mesoscale, we sidestep the issue of the long time scales. This is done by scaling down the viscosity of glycerol, effectively making the time scale shorter. Afterwards, we calculate the viscosity from the simulation and multiply by the same factor. Of course, this introduces a small error, but we can extrapolate to the exact result by repeating the same simulation with different scaling factors.

As you can see in the graph, the naïve model based purely on the relative amount of glycerol (blue dots, dashed line) is bad at capturing the viscosity values, with the viscosities being way too high compared to the experimental values^{3}. The RheoCube predictions (yellow dots, dashed line), however, are much closer over the whole range. It is important to note that for quantities like viscosity, predictions will never give perfect agreement with experiment. This is because the slightest error in anything gets blown up exponentially.

Getting the prediction of viscosity right is very important in RheoCube’s mesoscale simulations and for experimentalists creating new fluid mixtures. This is why RheoCube’s model for viscosity is such a powerful tool for determining viscosities in complex mixtures. Our simple model that takes into account various complicated effects arising from molecular interactions, all using readily-available data, to bring simple viscosity calculations quickly to our clients’ doorstep.