Viscosity of colloidal suspensions

Simple expressions are given for the Newtonian viscosity ${\mathrm{\eta }}_{\mathrm{N}}$(φ) as well as the viscoelastic behavior of the viscosity η(φ,ω) of neutral monodisperse hard-sphere colloidal suspensions as a function of volume fraction φ and frequency ω over the entire fluid range, i.e., for volume fractions 0<φ<0.55. These expressions are based on an approximate theory that considers the viscosity as composed as the sum of two relevant physical processes: η(φ,ω)=${\mathrm{\eta }}_{\mathrm{\infty }}$(φ)+${\mathrm{\eta }}_{\mathrm{cd}}$(φ,ω), where ${\mathrm{\eta }}_{\mathrm{\infty }}$(φ)=${\mathrm{\eta }}_0$χ(φ) is the infinite frequency (or very short time) viscosity, with ${\mathrm{\eta }}_0$ the solvent viscosity, χ(φ) the equilibrium hard-sphere radial distribution function at contact, and ${\mathrm{\eta }}_{\mathrm{cd}}$(φ,ω) the contribution due to the diffusion of the colloidal particles out of cages formed by their neighbors, on the Péclet time scale ${\mathrm{\tau }}_{\mathrm{P}}$, the dominant physical process in concentrated colloidal suspensions. The Newtonian viscosity ${\mathrm{\eta }}_{\mathrm{N}}$(φ)=η(φ,ω=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 ω is of the form ${\mathrm{\eta }}_{\mathrm{\infty }}$(φ)+${\mathrm{\eta }}_0$A(φ)(ω${\mathrm{\tau }}_{\mathrm{P}}$ ${)}^{\mathrm{-}1\mathrm{/}2}$, in agreement with these experiments, but the theoretical coefficient A(φ) differs by a constant factor 2/χ(φ) from the exact coefficient, computed from the Green-Kubo formula for η(φ,ω). 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.