Particle Rheology

The app is based on a very full analysis of particulate systems by Campbell et al ¹. Amazingly, the effects of interest here can be described via just two parameters (E and r are used for Yield Stress, see below):

1. φ_c, the critical percolation threshold which is the volume fraction where "percolation" sets in - i.e. the particles form coherent clusters that extend across the whole volume.
2. φ_m which is the maximum packing fraction.

Up to φ_c, all that happens is that the low shear viscosity (relative to the continuous phase viscosity) increases according to one of the typical laws discussed in the viscosity app which all involve φ_m. In this app we use a more appropriate law based on our two inputs:

η_rel=exp[(φ_m-φ_c)/(φ_m-φ)]-1

Above φ_c we start to get shear thinning because the particles behave more and more as a solid mass, which means there is no dissipation. This follows a power law depending on the shear rate, γ̇, and the exponent n which is less than 1:

η = γ̇^n-1

Happily, n depends, again, only on φ and our two inputs:

n=[(φ/φ_m)^-0.667-1]/[(φ_c/φ_m)^-0.667-1]

Finally, the viscosity falls to an infinite shear value, η_∞ which depends on a "dissipation ratio" ε which, fortunately, is related to n via:

n=1-4/3ε+1/3ε⁴

As ε's meaning is rather obscure, the authors point out that ν=ε² is the "volume fraction of percolating clusters" which, of course, approaches 1 as φ increases. The ratio of η_∞ to the original low shear viscosity η₀ is given by:

η_∞/η₀=(1-ε)²[1-8ε/3+2ε²-ε⁴/3]

Plotting the data

There are many ways to look at the results of these formulae. As most of us are interested in viscosity versus shear rate, that's the default. It needs a log plot to show the large range of properties (you can look at the non-log to confirm this). Because viscosities get super-high close to φ_m, the plots cover a range from φ_c/2, φ_c, then some unequal steps to 0.9φ_m.

The η_∞ don't show because they are down in the very small range. However, if you choose to plot stress rather than shear rate, you see a sudden increase in an otherwise smooth curve when the η_∞ becomes the limit.

φ_c and φ_m

For spherical particles above ~1μm, φ_c should be around 0.28 (see below for discussions on estimating this and the Percolation app). For smaller particles, Brownian motion intervenes and values might rise somewhat. For elliptical particles, φ_c decreases as the ellipse gets more pronounced.

φ_m is trickier. A value around 0.63 should be normal, but self-associating particles (i.e. with poor dispersants) show values closer to 0.5 and poly-disperse systems can get up to 0.8 as the smaller particles can pack into the close-packed spaces of the larger ones. An effect currently not taken into account is that for spheres at high shear, φ_m is ~0.71. So entering 0.71 will give you wrong values at low shear but better values at high shear.

It is well-known that multi-modal distributions can give a higher φ_m. Although it is possible to provide some complex algorithms, I've chosen the Qi and Tanner model based on the chosen φ_m, size ratio, λ, and the fraction, k, of smaller particles. Although there is brave talk about getting packing fractions >0.8, for any normal size ratio of ~10:1, the gain (automatically added to the viscosity calculations), at 33% small particles, is modest (0.63→0.70), though the impact on viscosity for high φ values can be large.

Shape and agglomeration

If the shape starts to change its fractal dimension d from 3 (sphere) to 2 (platelets) and starts to form agglomerates of N particles, then φ_m gets replaced by φ* given by equations from a review by Bicerano et al²:

φ*=φ_mN^1-3/d

As soon as d becomes less than 3 and N greater than 1, φ* decreases rather rapidly so the viscosity at a given φ can increase rapidly. When you play with N and d nothing much seems to happen in the graph, but look at the shown φ=xx values - they decrease rapidly, which means that at any given φ, viscosity is increasing sharply. I tried keeping the lines at φ values fixed with respect to φ_m, but the graphs quickly got out of hand.

Because spheres have d=3 I was confused about how flocculating spheres (N>1) cause increases in viscosity, because nothing happens when d=3. But the fractal dimension of flocculating spheres ~2. It takes only a moment playing with mild flocculation (N=3, d=2.5) to see why we need really good dispersants if we want low viscosity!

The percolation threshold, φ_c also changes with shape and can be estimated from the Aspect Ratio, A_f via:

φ_c=(9.875A_f+A_f²)/(7.742+14.61A_f+12.33A_f^1/5+1.763A_f²+1.658A_f³)

High Shear Corrections

At high shear, two things happen:

1. φ_m increases because we go from a random packing limit to a close packing limit.
2. Flocculated particles (i.e. a combination of N and d) become ripped apart by the flow and behave closer to unflocculated particles.

The science of both these effects is well-known, but I don't have a reliable way to add the effects. Instead I've created an ad-hoc correction (which you can turn on and off): φ_m increases by 0.07 (the case for spherical particles) up to a shear of 1000 which is here considered to be "infinite"; N decreases by 1 over this shear-rate range. Both changes are conservative, but the reductions in viscosity are clearly seen. If some more accurate way to simulate these effects is provided, I'll be happy to update the code.

Shear Thickening

In a well-protected particle dispersion, shear thinning continues to an "infinite shear" plateau. However, many systems, at higher loadings, show a large increase of viscosity at high shear rates. This is a transition from "hydrodynamic flow" where the particles are like cars driving along a multi-lane highway to a state of "lubrication dynamics" where (effectively) the flow around each "car" sucks the neighbouring one into contact. This creates (effectively) a gigantic local pile-up ("hydroclusters") which inhibits the flow along the highway. The theory of this effect is complex, so the Shear Thickening option is just a visual representation of the effect.

The Particle Rheology MCT app says more about the various flow regimes.

Yield Stress

The YODEL model for yield stress from Flatt and Bowen³ is a great combination of simplicity and power. The equation, using the nomenclature of this page is:

Yield Stress = m₁φ(φ-φ_c)²/(φ_c-φ)

The φ values have already been discussed, so all we need is m₁. Unfortunately, at present I don't properly understand how to calculate m₁ from typical values. However, it clearly depends on the size of the repulsion forces between particles, which we can estimate from the DLVO app as being a few kT, and 1/r, the radius of the particle. To provide values that are typical of those found in the original paper I have simply used a fudge factor. So the general trends are correct, the details aren't. Hopefully I will be able to update the calculations at some point. My (poor) understanding is that the value of E to be used is the overall attractive force (which is a negative value in DLVO). If you have sufficient steric or charge repulsion to avoid attraction then, I assume, there is no yield stress. At high volume fractions this seems non-intuitive.

The plot goes up to 0.91φ_m, the same as the maximum value in the viscosity plot.