Modeling the intrinsic viscosity of polydisperse disks

In this work, we model the rheology of dilute colloidal oblate spheroids in their high aspect ratio limit of circular disks. Theoretical models for the intrinsic viscosity, [η], of disks in shear flow are reviewed: The shear-independent, monodisperse Kuhn-Kuhn model, its polydisperse form by van der Kooij, and the shear-dependent models of Stewart and Sorenson, Leal and Hinch, and Brenner. Based on these previous works, three analytical models are introduced to describe the shear response over the entire range of practically accessible rotational Peclet numbers (Pe) and aspect ratios. Using the fact that [η] is linearly additive for sufficiently dilute systems we derive a general expression for polydisperse disks as a function of the two independent variables of particle diameter D and thickness t, that is, assuming D and t to be uncorrelated independent variables. We then argue for continuum modeling being preferable to discrete for using rheological measurements to estimate particle size distribution parameters. Computational results are shown for the continuum model in shear flow and generalized to uniaxial and planar extension, as well as to different particle distributions such as lognormal, normal, and bimodal. Finally, a modified form of [η], which we describe as innate viscosity (η), is suggested as an alternative method of modeling rheology of dilute dispersions.