Covariant description of the colloidal dynamics on curved manifolds

Abstract

Brownian motion is a universal characteristic of colloidal particles embedded in a host medium, and it is the fingerprint of molecular transport or diffusion, a generic feature of relevance not only in physics but also in several branches of science and engineering. Since its discovery, Brownian motion, also known as colloidal dynamics, has been important in elucidating the connection between the molecular details of the diffusing macromolecule and the macroscopic information on the host medium. However, colloidal dynamics is far from being completely understood. For instance, the diffusion of non-spherical colloids and the effects of the underlying geometry of the host medium on the dynamics of either passive or active particles are a few representative cases that are part of the current challenges in soft matter physics. In this contribution, we take a step forward to introduce a covariant description of the colloidal dynamics in curved spaces. Without the loss of generality, we consider the case where hydrodynamic interactions are neglected. This formalism will allow us to understand several phenomena, for instance, the curvature effects on the kinetics during spinodal decomposition and the thermodynamic properties of colloidal dispersion, to mention a few examples. This theoretical framework will also serve as the starting point to highlight the role of geometry on colloidal dynamics, an aspect that is of paramount importance to understanding more complex transport phenomena, such as the diffusive mechanisms of proteins embedded in cell membranes.