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Continuum-statistical dynamics of colloidal suspensions under kinematic reversibility

Jerome Burelbach

Abstract

We develop a continuum-statistical framework for colloidal dynamics to first order in the applied forces, based on the Lorentz reciprocal theorem and a reassessment of the conditions under which kinematic reversibility emerges in continuum mechanics. The theory recovers known results for buoyancy, phoretic motion, and active swimming, and extends them to include the effect of advective transport and hydrostatic many-body interactions through the colloidal osmotic pressure. The resulting formulation is consistent with Onsager's theory of non-equilibrium thermodynamics, thus providing a unified perspective on reciprocal relations in colloidal transport.

Continuum-statistical dynamics of colloidal suspensions under kinematic reversibility

Abstract

We develop a continuum-statistical framework for colloidal dynamics to first order in the applied forces, based on the Lorentz reciprocal theorem and a reassessment of the conditions under which kinematic reversibility emerges in continuum mechanics. The theory recovers known results for buoyancy, phoretic motion, and active swimming, and extends them to include the effect of advective transport and hydrostatic many-body interactions through the colloidal osmotic pressure. The resulting formulation is consistent with Onsager's theory of non-equilibrium thermodynamics, thus providing a unified perspective on reciprocal relations in colloidal transport.
Paper Structure (22 sections, 171 equations, 3 figures)

This paper contains 22 sections, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Microscopic continuity equations
  3. Colloidal Suspensions: Definitions and Properties
  4. Microscopic conservation of energy and momentum
  5. Microscopic conservation of particle number
  6. Thermodynamic forces in colloidal suspensions
  7. Translational motion of a single colloid
  8. Interfacial decoupling and thermodynamic adsorption equations
  9. Application of the Lorentz reciprocal theorem to induced colloidal transport
  10. Colloidal transport due to bulk fluid forces
  11. Continuum-statistical colloidal dynamics
  12. Homogenization
  13. Macroscopic continuity equations
  14. Colloidal diffusion
  15. Application of the Lorentz reciprocal theorem to colloidal diffusion
  16. ...and 7 more sections

Figures (3)

  • Figure 1: Schematic representation of a suspension on the microscale. The dark and light gray spheres represent different species of microparticles, suspended in a viscous microphase. The viscous microphase consists of small solutes (black dots) dispersed in a solvent reservoir. The dashed lines represent thermodynamic reservoir boundaries.
  • Figure 2: Schematic representation of the reciprocal approach to colloidal motion. a) A single colloid subjected to applied forces. The continuum fields that these forces derive from are depicted by a blue background gradient. The dashed circle around the colloid represents the effective boundary between its interfacial region and the bulk fluid. Instead of resolving the actual microscopic dynamics of a), the reciprocal approach considers two separate auxiliary problems: b) An equilibrium-statistical problem to determine local component distributions, and a continuum-mechanical problem to independently compute c) the applied forces and d) the induced Stokes flow field. A stronger background gradient is used in c) to show that the colloid may modify the fields in its vicinity.
  • Figure 3: A representative volume element of a homogeneous, isotropic suspension at MLTE. The blue background gradient shows that the volume element is subjected to forces due to non-equilibrium boundary conditions.