Concentrated Dispersions of Spherical Particles

Many semi-empirical expressions are available for describing the shear viscosity of concentrated (Aspersions of hard spheres. Probably the most widely used is the fiuictional form suggested by Krieger and Dougherty  

Both of the above equations reduce to the Einstein limit [eqn. (5.1)] at low concentrations ( f — 0) and to oo as K K The crowding factor K can be treated either as an adjustable parameter for fitting experimental data or as a theoretical parameter equal to the reciprocal of the volume fraction at which diverges to infinity. For random close packing of monodisperse hard spheres, we have 0niax = 0.64 and K = 1.56. An alternative theoretical route to K is to expand 

Measured values of the low shear viscosity for concentrated dispersions of hard spheres may be described reasonably well by the equation -  

This equation gives the correct dilute limiting behaviour to order p, and rj, diverges to infinity at p, = 0.64. Another empirical equation fitting many experimental data sets is the formula 2  

Non-Newtonian behavioitr can be allowed for in the Krieger-Dougherty formalism by allowing K to vary with the shear-rate. The justification for this is the formation at higher shear stresses of a more ordered microstructure, involving strings or layers of particles, characterized by a different value of p. The effect of shear stress r on the crowding factor can be represented by an equation of die form 

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