Thixotropy Hysteresis Loop

There are plenty of papers with theoretical treatments of thixotropy hysteresis loops. I've chosen the approach from a team in Barcelona¹ as it is mathematically tractable and theoretically clear. This starts with the Shear Viscosity app, using the Cross model to create the "equilibrium" line which is what you would measure if your thixotropic fluid had been infinitely well-sheared and had no time to recover. The plot goes up to your chosen maximum shear rate.

The Up line is what you would expect from your thixotropic liquid which starts at near-zero shear (0.1/s) with a viscosity of η_start.The rate at which it approaches the equilibrium line depends on three factors.

1. How fast you scan your shear rate - the slower you scan, the longer it takes to get to your maximum shear rate and, therefore, the sooner (visually, in terms of shear rate) you reach the equilibrium line.
2. The timescale for breaking up your structure, τ_break. The longer this is, the more thixotropic your system will be.
3. This is combined with how much τ_break changes with shear rate, expressed as γ̇ power. If this is 1 then the time decreases proportionally to shear rate

In principle, your "down" curve will be very different from both your up curve (of course) and your equilibrium curve. This depends on τ_build, the timescale for re-building your original structure. This is assumed to be shear-rate independent. You only see significant effects (i.e. different from the equilibrium line) with fast scans and if τ_build is quite large.

You may be surprised that the curves don't look too much like the ones we always see in simple explanations. The reason is that there are lots of ways of looking at the data, and lots of "equilibrium" curves, so you have to first get you equilibrium curve right, which might need one sort of scale (e.g. log-linear viscosity), then view the thixotropy via a log-log plot with shear stress instead of viscosity. When testing the app against published papers I often thought the results were wrong, but then found I was using the wrong scale in the app. Effects that look really interesting in log-log can look quite dull in log-linear.

Thixotropic Area

The area S between the up curve and down curve (though here it's between up and equilibrium) is some sort of indication of the thixotropy of the system. However, the area has no absolute meaning because it depends on scan rate and maximum shear rate.The relative area S_rel which is S divided by the area under the up curve (S_max) is said to be rather more independent of the experimental variables. Both values are calculated for you.

The theory

The title of the paper is "A Simple Theory", so it is easily described - though implementing it was tricky. The core assumption is that the rate of change of viscosity η towards the equilibrium value η_e at any given shear rate is given by:

`(δη)/(δt) = (η-η_e)^2/τ_s`

τ_s is the "structure breaking" or "structure making" timescale and for breaking it is given by t_break/(γ̇ )^p, where p is a power dependence.