The steady shear viscosity of filled polymeric liquids described by a linear superposition of two relaxation mechanisms

Filled polymeric liquids often exhibit apparent yielding and shear thinning in steady shear flow. Yielding results from non-hydrodynamic particle—particle interactions, while shear thinning results from the non-Newtonian behavior of the polymer melt. A simple equation, based on the linear superposition of two relaxation mechanisms, is proposed to describe the viscosity of filled polymer melts over a wide range of shear rates and filler volume fraction.The viscosity is written as the sum of two generalized Newtonian liquid models. The resulting equation can describe a wide range of shear-thinning viscosity curves, and a hierarchy of equations is obtained by simplifying the general case. Some of the parameters in the equation can be related to the properties of the unfilled liquid and the solid volume fraction. One adjustable parameter, a yield stress, is necessary to describe the viscosity at low rates where non-hydrodynamic particle—particle interaction dominate. At high shear rates, where particle—particle interactions are dominated by interparticle hydrodynamics, no adjustable parameters are necessary. A single equation describes both the high and low shear rate regimes. Predictions of the equation closely fit published viscosity data of filled polymer melts.n power-law index - n₁,n₂ power-law index of first (second) term - shear rate - steady shear viscosity - ₀ zero-shear rate viscosity - _{0, 1},_{0, 2} zero-shear rate viscosity of first (second) term - time constant - ₁,₂ time constant of first (second) term - µ_r relative viscosity of filled Newtonian liquid - ₀ yield stress - ø solid volume fraction - ø_m maximum solid volume fraction