Concentrated Dispersions of Spheres

There are numerous equations in the literature describing the concentration dependence of the viscosity of dispersions. Some are from curve fitting whilst others are based on a model of the flow. A common theme is to start with a dilute dispersion, for which we may define the viscosity from the hydrodynamic analysis, and then to consider what occurs when more particles are added to replace some of the continuous phase. The best analysis of this situation is due to Dougherty and Krieger18 and the analysis presented here, due to Ball and Richmond,19 is particularly transparent and emphasises the problem of excluded volume. The starting point is the differentiation of Equation (3.42) to give the initial rate of change of viscosity with concentration  

Now consider a suspension at a volume fraction (p with a viscosity of rj((p). When a small increase in concentration of this suspension is caused by replacing some of the medium by more particles, the expected increase in viscosity from the value rj is 

Hence to calculate an increase in viscosity, Equation (3.51) must be integrated up to the value of interest so 

This effective medium or mean field assumption is easy to understand if there is a very large size difference between the newly added particles and any there previously, for example if we think of adding particles to a molecular liquid, we merely treat the liquid as a structureless continuum. However when the dimensions become comparable, the finite volume of the particles present prior to each addition must be considered, i.e. new particles can only replace medium and not particles. The consequence of this crowding is that the concentration change is greater than expressed in Equation (3.52) and it must be corrected to the volume available  

Where (pm is the maximum concentration at which flow is possible -above this solid-like behaviour will occur. q /(pm is the volume effectively occupied by particles in unit volume of the suspension and therefore is not just the geometric volume but is the excluded volume. This is an important point that will have increasing relevance later. Now integration of Equation (3.53) with the boundary condition that as 

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