Critical viscoelastic behavior of colloids

Abstract

The linear and nonlinear frequency-dependent viscoelastic response of a suspension of spherical colloids in the vicinity of the gas-liquid critical point is analyzed in the mean-field region. Explicit expressions for the shear rate and frequency dependence of the static structure factor are derived, starting from the N-particle Smoluchowski equation, which is the fundamental equation of motion for the probability density function of the position coordinates of the spherical colloids. Microscopic expressions for the anomalous parts of the linear and nonlinear response functions are derived, which are then expressed as wave-vector integrals weighted with the static structure factor. These integrals are evaluated in part numerically, leading to explicit results for the viscoelastic response functions. The critical enhancement of both the linear and nonlinear viscoelastic response functions is found to be far more pronounced than for molecular systems as a result of long-ranged hydrodynamic interactions between the colloidal particles. Viscoelastic response functions are found to diverge with the same exponent as the correlation length of the quiescent, unsheared suspension. The frequency spectrum of the linear response functions is found to be extremely broad, while nonlinearity affects only the low-frequency behavior of the lowest-order response functions. The lowest-order response functions attain their linear response values at higher frequencies even far into the nonlinear regime. Nonlinear effects are thus absent at higher frequencies. For these higher frequencies the lowest-order response functions are found to vary with the frequency $\ensuremath{\omega}$ as ${\ensuremath{\omega}}^{\ensuremath{-}1/4}$ close to the critical point and cross over to a ${\ensuremath{\omega}}^{\ensuremath{-}1/2}$ dependence further away from the critical point. In addition to the viscoelastic response of an otherwise quiescent suspension, the viscoelastic response of a stationary sheared suspension is discussed. The response of such a stationary sheared system to a superimposed oscillatory shear flow probes the dynamics of the partially distorted microstructure by the stationary shear flow. The frequency spectrum of the linear viscoelastic response functions is found to be strongly affected by the microstructure distortion due to the stationary shear flow.