Generalized Stokes laws for active colloids and their applications

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.

Direct verification of the fluctuation-dissipation relation in viscously coupled oscillators

“…We emphasize that this expression is not limited to the translationally invariant Green's function of unbounded flow, 8πη G(r) = (∇ 2 I − ∇∇)r, but holds generally for any Green's function and is both symmetric and positivedefinite [19,20]. The mutual friction tensors decay inversely with distance in an unbounded fluid and more rapidly in the proximity of boundaries.…”

“…The bold-face notation, with Cartesian indices suppressed, is used for both vectors and tensors. In the limit of slow viscous flow in the fluid, the friction tensors can be calculated from the Stokes equation using a variety of methods [16][17][18][19][20]. To leading order the result is…”

Filament actuation by an active colloid at low Reynolds number

“…n ls n la n lt -of rank l,l 1 andl 2; they correspond to symmetric traceless, antisymmetric and trace combinations of the reducible indices. We denote the three irreducible parts by s ( ) V l and the suffixes s = s a t , , are self-explanatory [30,31]. For convenience, we introduce special notations for the s = l a 1 and s = l a 2 coefficients.…”

“…By invoking the linearity of the Stokes equations, it can be shown that the contact forces and torques are linear functions of the boundary condition [31]. They are related to linear velocity V m , angular velocity W m and irreducible modes of active slip s ( ) V m l as, In the equation above, we have used the summation convention for repeated particle (n, m) and mode s ( ) l indices.…”

“…This reflects the constant consumption of energy that is necessary to maintain the active slip, independent of the state of motion of the sphere. A method for computing the g matrices in terms of the Green's function has been proposed recently [31]. From there it follows that in an unbounded fluid, the g s…”