Kinetic theory of colloid thermodiffusion

The Kramers’ equation of Brownian motion is applied to investigate the motion of a colloidal particle in a medium subjected to a temperature gradient. The equation is generalized in two ways. First, a chemical force is included in order to account for the non-ideality of the colloidal solution, in the thermodynamic sense. Second, the local disequilibrium of the medium gives rise to a force proportional to the temperature gradient, known as the thermophoretic force in the physics of gases. It is found that the latter force dominates in a rarefied gas, while the chemical force is a good candidate in liquid solutions. The description of the cross-over regime is still unsatisfactory. Next, given the force undergone by a colloidal particle, regardless of its physicochemical origin(s), we determine the velocity response. It is demonstrated that the velocity is not proportional to the applied force, in variance with the Stokes’ law of viscous drag invoked in many works and valid in thermally homogeneous media. An additional effective force tends to drive the particle toward places of higher mobility; that effective force is also proportional to the temperature gradient and can be of the same order of magnitude as the applied force. This conclusion is reached in two different ways, using either a transport equation or statistical-dynamical relations akin to the Ehrenfest theorem in quantum dynamics. Finally, the theoretical formula for the Soret coefficient shows that it is neither proportional to the velocity nor to the force.