Effective flows across diffusio-phoretic membranes

Flows enabled by phoretic mechanisms are of significant interest in several biological and biomedical processes, such as bacterial motion and targeted drug delivery. Here, we develop a homogenisation-based macroscopic boundary condition that describes the effective flow across a diffusio-phoretic microstructured membrane, where the interaction between the membrane walls and the solute particles is modelled via a potential approach. We consider two cases where potential variations occur (i) at the pore scale and (ii) only in the close vicinity of the boundary, allowing for a simplified version of the macroscopic flow description, in the latter case. Chemical interactions at the microscale are rigorously upscaled to macroscopic phoretic solvent velocity and solute flux contributions, and added to the classical permeability and diffusivity properties of the membrane. These properties stem from the solution of Stokes advection–diffusion problems at the microscale, some of them forced by an interaction potential term. Eventually, we show an application of the macroscopic model to develop minimal phoretic pumps, showcasing its suitability for efficient design and optimisation procedures.