Theory of thermophoresis. I. General considerations and mode-coupling analysis

The phenomenon of the motion of suspended particles induced by temperature gradients in fluids, known as thermophoresis, is considered. We show that thermophoresis is related to the Soret effect. The thermophoretic force acting on a particle immersed in a fluid which has a temperature gradient, $\overrightarrow{\nabla }T$, is given phenomenologically by $-{\eta }_T\overrightarrow{\nabla }T$. ${\eta }_T$ is shown to be related to the thermal-diffusion ratio by ${\eta }_T=k_B(k_T-1)$. In the limit of a dilute suspension we find a molecular expression for $k_T$ in terms of an equilibrium time-correlation function of the momentum of a suspended particle and the dissipative heat current. As a by-product, we also obtain the force acting on a such a particle due to a pressure gradient. We further show, using time-scale arguments, that a local-equilibrium calculation of the thermophoretic force cannot be adequate. Finally, we show, using a mode-coupling approach, that the hydrodynamic contribution to the thermophoretic force coefficient is negligible, unlike in the case of Stokes friction. We conclude that although thermophoresis is a macroscopic effect (not a fluctuating one) observed in fluids, it is not given by standard hydrodynamics. In a subsequent paper, with the use of kinetic theory, we identify the microscopic processes responsible for thermophoresis, calculate the thermophoretic force coefficient for moderately dense gases, and compare our results to experiment.