Abstract
The force per unit area on the surface of a colloidal particle is a fundamental dynamical quantity in the mechanics and statistical mechanics of colloidal suspensions. Here we compute it in the limit of slow viscous flow for a suspension of N spherical active colloids in which activity is represented by surface slip. Our result is best expressed as a set of linear relations, the ‘generalized Stokes laws’, between the coefficients of a tensorial spherical harmonic expansion of the force per unit area and the surface slip. The generalized friction tensors in these laws are many-body functions of the colloidal configuration and can be obtained to any desired accuracy by solving a system of linear equations. Quantities derived from the force per unit area—forces, torques and stresslets on the colloids and flow, pressure and entropy production in the fluid—have succinct expressions in terms of the generalized Stokes laws. Most notably, the active forces and torques have a dissipative, long-ranged, many-body character that can cause phase separation, crystallization, synchronization and a variety of other effects observed in active suspensions. We use the results above to derive the Langevin and Smoluchowski equations for Brownian active suspensions, to compute active contributions to the suspension stress and fluid pressure, and to relate the synchrony in a lattice of harmonically trapped active colloids to entropy production. Our results provide the basis for a microscopic theory of active Brownian suspensions that consistently accounts for momentum conservation in the bulk fluid and at fluid-solid boundaries.
Highlights
Einstein used these laws in his phenomenological theory of Brownian motion and obtained a relation between the diffusion coefficient of a spherical colloid and the friction constant in Stokes law, the so-called Stokes-Einstein relation, the first example of a fluctuation-dissipation relation [1–3]
Smoluchowski, in 1911, presented an iterative method to calculate the force per unit area on a moving sphere in the presence of another and thereby initiated the study of hydrodynamic interactions between colloidal particles [4]
Through the course of these studies it became apparent that the force per unit area on the surface of the colloid is the key dynamical quantity necessary for developing both the mechanics and the statistical mechanics of suspensions
Summary
In all of the above examples, the matching condition at the edge of the boundary layer is the continuity of the fluid velocity, which contains, in addition to the rigid body motion of the colloid, the contribution from the active flow vA This “slip” velocity is a general, possibly time-dependent, vector field on the colloid surface, subject only to the constraint of mass conservation vA · n dS = 0. The generalized Stokes laws immediately provide the quantities of interest to suspension mechanics: the force, the torque, and the stresslet on the colloids Through their use, succinct expressions are obtained for the fluid flow, the fluid pressure, and the entropy production. We foresee many other instances where FIPS should provide an accurate representation of the physical forces that drive active aggregation
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