Extended dynamical density functional theory for colloidal mixtures with temperature gradients

TL;DR

This paper extends dynamical density functional theory (DDFT) to multi-species colloidal mixtures and includes energy density, enabling the study of colloidal dynamics under temperature gradients with hydrodynamic interactions.

Contribution

The authors develop a generalized DDFT framework for colloidal mixtures and incorporate energy density, allowing analysis of thermodiffusion and hydrodynamic effects.

Findings

Derived explicit cross-coupling terms for mixtures with hydrodynamic interactions.

Formulated an extended DDFT including energy density for temperature gradient scenarios.

Reproduced hydrodynamic limit transport coefficients within the extended DDFT.

Abstract

In the past decade, classical dynamical density functional theory (DDFT) has been developed and widely applied to the Brownian dynamics of interacting colloidal particles. One of the possible derivation routes of DDFT from the microscopic dynamics is via the Mori-Zwanzig-Forster projection operator technique with slowly varying variables such as the one-particle density. Here, we use the projection operator approach to extend DDFT into various directions: first, we generalize DDFT toward mixtures of $n$ different species of spherical colloidal particles. We show that there are in general nontrivial cross-coupling terms between the concentration fields and specify them explicitly for colloidal mixtures with pairwise hydrodynamic interactions. Secondly, we treat the energy density as an additional slow variable and derive formal expressions for an extended DDFT containing also the energy density. The latter approach can in principle be applied to colloidal dynamics in a nonzero temperature gradient. For the case without hydrodynamic interactions the diffusion tensor is diagonal, while thermodiffusion -- the dissipative cross-coupling term between energy density and concentration -- is nonzero in this limit. With finite hydrodynamic interactions also cross-diffusion coefficients assume a finite value. We demonstrate that our results for the extended DDFT contain the transport coefficients in the hydrodynamic limit (long wavelengths, low frequencies) as a special case.