Flow in Hard Sphere Systems

In Chapter 3 (Section 3.5.2) the viscosity of a hard sphere model system was developed as a function of concentration. It was developed using an exact hydrodynamic solution developed by Einstein for the viscosity of dilute colloidal hard spheres dispersed in a solvent with a viscosity rj0. By using a mean field argument it is possible to show that the viscosity of a dispersion of hard spheres is given by 

Intuitively we might associate the low shear limit with the order-disorder transition at pm(0) = 0.495. However literature data for the packing fraction in this limit is more widely scattered. We must remember that the approach to equilibrium in these systems can take a while to progress. So it is feasible that some systems have been measured away from the equilibrium state when the samples have been transferred and placed in the measuring geometry on an instrument. We could 

If we extrapolate this line until it equates with the value of rjn= 1 where r) a) = rj(0) then 

A plot of rj(o) versus the log of reduced stress shows a linear slope between the high and low shear limits. We can use this feature with our master curve to define another value of b. This slope is given by 

The slope between the low and high shear rate limits has a constantly changing gradient. In order to define the point of inflection on this slope and its tangent the second derivative is required  

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