Molecular wall effects: are conditions at a boundary "boundary conditions"?

Abstract

This paper addresses and answers "no" to the question of whether the literal molecular-dynamically-derived species velocities prevailing at a solid surface bounding a two-component fluid continuum undergoing molecular diffusion constitute the appropriate species-velocity boundary conditions to be imposed upon the fluid continuum. In a broader context, generic boundary condition issues arising from the presence of different length scales in continuum-mechanical descriptions of physical phenomena are clarified. This is achieved by analyzing a model problem involving the steady-state diffusion of a dilute system of Brownian spheres (the latter envisioned as tractable models of solute "molecules") through a quiescent viscous solvent continuum bounded laterally by solid plane walls. Both physicochemical (potential energy) and hydrodynamic (steric) wall interaction effects experienced by the Brownian spheres are explicitly accounted for in our refined, microscale continuum model of the diffusion process. Inclusion of these "solid-wall-fluid" (s-f ) boundary-generated forces [above and beyond the usual "fluid-fluid" (f-f ) intermolecular forces implicit in the conventional Fick's law macroscale continuum description] serves to simulate the comparable s-f molecular boundary forces modeled in molecular dynamics simulations of the diffusional process. A singular perturbation framework is used to clarify the physical interpretation to be ascribed to "continuum-mechanical boundary conditions." In this same spirit we also clearly identify the origin of the physical concept of a "surface field" as well as of the concomitant surface transport conservation equation for strongly adsorbed species at solid walls. Our analysis of such surface phenomena serves to emphasize the fact that these are asymptotic, surface-excess, macroscale concepts assigned to a surface, rather than representing literal molecular material entities physically confined to the surface. Overall, this paper serves to illustrate the manner in which molecular simulations need to account for these different length scales and corresponding scale-dependent concepts if such analyses are to avoid drawing incorrect inferences regarding the molecular origins of continuum-mechanical boundary conditions.