Low-Shear Particle/Dispersion Viscosity
The formulae for low-shear viscosities of particles/dispersions are described here. The equations cover everything from emulsion particles through soft matter particles to fully solid particles.
If we have an emulsion or fillers or pigments, how does the viscosity depend on the volume fraction? Actually, the question is about "low shear" viscosity as the high-shear viscosity decreases rapidly with shear as shown in the High Shear Particles app.
The overall viscosity depends on the initial viscosity of the bulk phase, η0 and the volume fraction of the dispersed component, φ. Many equations are used to describe this dependence and three are shown here. The Einstein equation obviously is the standard for solid particles.
However, for this is the limiting factor when k, which is the ratio of the viscosity of the dispersed material to that of the bulk phase, ηdispersed/η0 reaches a very high value. For more modest k values, the Taylor equation is used:
The Dougherty-Krieger formula is popular but tends to overshoot at high φ values. It incorporates a "close packing" fraction φm at which the viscosity becomes infinite. In the app φm is assumed to be 0.74. The factor of 2.5 is the default value for the "intrinsic viscosity" term. Because the viscosity rises exceptionally above φ=0.61 (near the random close packing limit) the calculation is not performed above that value.
Reading some early Krieger literature it seems that this curve is for viscosities at reasonably high shear stress. At low stress the curve behaves as if φm is ~0.58, so viscosities take off much faster as φ increases.
The most realistic, at least for emulsions, is also the most monstrous - the Yaron, Gal-Or. This uses the k from the Taylor model and instead of φ uses λ=φ0.333.
`η=η_0(1+5.5φ[4λ^7+10-(84/11)λ^2+(4/k)(1-λ^7 )]/(10(1-λ^10)-25λ^3(1-λ^4)+(10/k)(1-λ^3 )(1-λ^7)))`
Another choice, based on an extensive evaluation of real emulsion viscosities (rather than dispersions) with very large ranges of k is from Pal1:
where ηr is given by the following with φm=0.637 (this is a general value which in principle can be fitted. For simplicity, it is fixed in the app and the plot stops at φ=0.6 before the curve gets too large):
None of the equations has any dependence on dispersed particle size. This is surprising but supported by the data as long as the particles are nicely dispersed.
All you have to do is choose a range over which to plot (limited by φmax), enter η0 and, for Taylor and for Yaron, Gal-Or, ηdispersed. You can read out viscosity values with the mouse. For those with solid dispersed particles, set ηdispersed to its maximum value. As you can easily see, above a few 1000 cP the exact values doesn't matter.
As a quick check, the Wt% at your maximum φ is calculated from the density of your particle relative to the medium.
The most important point from all this is that apparently small changes in the base viscosity can make a huge difference to the dispersion's viscosity. Suppose the base viscosity is 1cP and at your chosen φ the viscosity is 100cP. Now increase the base viscosity by 4cP. This is trivial. But not for the dispersion which now has a viscosity of 500cP, not 104cP! My own experience out in the world of coating machines is that this effect of apparently trivial changes in the base viscosity is largely unknown.