Viscosity of colloidal suspensions

Abstract

Simple expressions are given for the Newtonian viscosity ${\mathrm{\ensuremath{\eta}}}_{\mathrm{N}}$(\ensuremath{\varphi}) as well as the viscoelastic behavior of the viscosity \ensuremath{\eta}(\ensuremath{\varphi},\ensuremath{\omega}) of neutral monodisperse hard-sphere colloidal suspensions as a function of volume fraction \ensuremath{\varphi} and frequency \ensuremath{\omega} over the entire fluid range, i.e., for volume fractions 00.55. These expressions are based on an approximate theory that considers the viscosity as composed as the sum of two relevant physical processes: \ensuremath{\eta}(\ensuremath{\varphi},\ensuremath{\omega})=${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\varphi})+${\mathrm{\ensuremath{\eta}}}_{\mathrm{cd}}$(\ensuremath{\varphi},\ensuremath{\omega}), where ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\varphi})=${\mathrm{\ensuremath{\eta}}}_{0}$\ensuremath{\chi}(\ensuremath{\varphi}) is the infinite frequency (or very short time) viscosity, with ${\mathrm{\ensuremath{\eta}}}_{0}$ the solvent viscosity, \ensuremath{\chi}(\ensuremath{\varphi}) the equilibrium hard-sphere radial distribution function at contact, and ${\mathrm{\ensuremath{\eta}}}_{\mathrm{cd}}$(\ensuremath{\varphi},\ensuremath{\omega}) the contribution due to the diffusion of the colloidal particles out of cages formed by their neighbors, on the P\'eclet time scale ${\mathrm{\ensuremath{\tau}}}_{\mathrm{P}}$, the dominant physical process in concentrated colloidal suspensions. The Newtonian viscosity ${\mathrm{\ensuremath{\eta}}}_{\mathrm{N}}$(\ensuremath{\varphi})=\ensuremath{\eta}(\ensuremath{\varphi},\ensuremath{\omega}=0) agrees very well with the extensive experiments of van der Werff et al., [Phys. Rev. A 39, 795 (1989); J. Rheol. 33, 421 (1989)] and others. Also, the asymptotic behavior for large \ensuremath{\omega} is of the form ${\mathrm{\ensuremath{\eta}}}_{\mathrm{\ensuremath{\infty}}}$(\ensuremath{\varphi})+${\mathrm{\ensuremath{\eta}}}_{0}$A(\ensuremath{\varphi})(\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathrm{P}}$${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}2}$, in agreement with these experiments, but the theoretical coefficient A(\ensuremath{\varphi}) differs by a constant factor 2/\ensuremath{\chi}(\ensuremath{\varphi}) from the exact coefficient, computed from the Green-Kubo formula for \ensuremath{\eta}(\ensuremath{\varphi},\ensuremath{\omega}). This still enables us to predict for practical purposes the viscoelastic behavior of monodisperse spherical colloidal suspensions for all volume fractions by a simple time rescaling.