Analytically predicting the viscosity of hard sphere suspensions from the particle size distribution

Abstract

Suspension rheology is of widespread importance to industry and research. Hard spheres represent a benchmark by which to compare other particle suspensions, and there are a variety of analytical and numerical models available to describe their rheology. However, it is experimentally challenging to produce ideal hard spheres, where surface forces are negligible between particles, and where phase volume is precisely defined. Beyond the dilute regime, the model by Maron and Pierce [1] and Quemada [2], which we refer to as the MPQ model, is commonly used analytically to describe the relative viscosity of hard sphere suspensions as a function of phase volume and a maximum packing fraction (ϕm). We show that obtaining ϕm from empirical fits can lead to misinterpretation of experimental data. We reveal that reasonable prediction of the viscosity is obtained using the MPQ model when ϕm is set to the geometric random close packing fraction ϕrcp, which is independently defined from the particle size distribution using the packing model of Farr and Groot [3]. This ‘theoretical’ approach is tested using a wide variety of experimental data on colloidal and non-colloidal hard spheres without need for any fitting parameters or empiricisms. In addition, plotting the inverse of the square-root of viscosity as a function of phase volume, which linearises the MPQ model, provides a convenient means by which to clearly see where suspensions deviate from the model due to such effects as particle aggregation, particle softness and measurement errors. We also demonstrate the necessity of this approach by accurately predicting the viscosity of microgel suspensions up to ϕrcp; empirical fits across the full data set are erroneous because particle deformation and viscoelasticity lead to values of ϕ>ϕrcp. This approach provides a suitable unambiguous theoretical baseline for comparison to experimental studies on suspension rheology involving polydisperse size distributions.