On the motion of two particles translating with equal velocities through a colloidal dispersion

Abstract

We study the motion of two colloidal (‘probe’) particles translating along their line of centres with fixed, equal velocities in an otherwise quiescent colloidal dispersion. The moving probes drive the microstructure of the dispersion out of equilibrium; resisting this is the Brownian diffusion of the dispersion ‘bath’ particles. As a result of the microstructural deformation, the dispersion exerts an entropic, or thermal, force on the probes. The non-equilibrium microstructure and entropic forces are computed to first order in the volume fraction of bath particles, as a function of the probe separation ( d ) and the Péclet number ( Pe ), neglecting hydrodynamic interactions. Here, Pe is the dimensionless velocity of the probes, which sets the degree of microstructural distortion. For Pe ≪1—the linear-response regime—the microstructural deformation is proportional to Pe . In this limit, for sufficiently large d , the deformation is nearly fore-aft symmetric about each probe; consequently, the entropic forces on them (which are opposite the direction of motion) are almost equal. However, for sufficiently small d , the symmetry is broken and, rather unexpectedly, the entropic force on the trailing probe is now in the direction of motion, whereas the force on the leading probe remains opposite to the direction of motion. Away from equilibrium, Pe >1 (and for all d ), the leading probe acts as a ‘bulldozer’, accumulating bath particles in a thin boundary layer on its upstream side, while leaving a wake of bath-particle free suspending fluid downstream, in which the trailing probe travels. In this (nonlinear) regime, the entropic forces are once more both opposite the direction of motion; however, the force on the leading probe is greater (in magnitude) than that on the trailing probe. Finally, far from equilibrium ( Pe ≫1), the entropic force on the trailing probe vanishes, whereas the force on the leading probe approaches a limiting value, equal to that for a single probe moving through the dispersion.