Conservation and constitutive equations for adsorbed species undergoing surface diffusion and convection at a fluid-fluid interface

Abstract A rigorous mathematical theory is presented for mass transfer of a solute by diffusion and convection at and near an interface between two immiscible solvent liquids. The objective of this study is to determine, for a simple model system, the conditions under which a detailed resolution of the concentration and mass flux profiles near the interface can be replaced (insofar as macroscopically observable profiles are concerned) with jump conditions that relate the bulk or macroscopic concentration and flux values on the two sides of the interface. The “statistical—mechanical” model underlying this theoretical development consists of spherical Brownian particles, either wholly immersed in one of the two contiguous fluids, or else straddling the interface. Adsorption forces tending to cause accumulation of the Brownian “surfactant” particles at the interface are regarded as deriving from a position-dependent potential energy function. This system is characterized by three independent length scales: a macroscale L , which is relevant to solute concentration gradients in the bulk fluids away from the interface; and two smaller scales: I , which is characteristic of the potential energy gradients normal to the interface, and I 1 , which provides a proper scale for the hydrodynamic “wall effects” of the fluid interface upon the particle motion. It is the macroscale L which is characteristic of the normal continuum description of solute mass transfer. Relative to L , the microscales l and l 1 are vanishingly small. Singular perturbation techniques are employed to provide a complete spatial resolution of solute concentration and flux profiles, right down to the scales l 1 and/or l , with δ = l/L and δϵ = l 1 /L is independent small parameters. Three distinct situations emerge for δ ⪡ 1, depending upon the value of ϵ and the rate of decrease of any hydrodynamic wall effects with increasing distance from the undeformed interface. First, when ϵ ⪡ 1 and/or the wall effects fall off faster than linearly with increasing distance, a set of local interface jump conditions can be derived which includes both the effects of the potential energy of attraction/repulsion and hydrodynamic wall effects, while transport in the contiguous bulk-phase fluids occurs at a rate that is consistent with the advection velocity and diffusivity in an infinite, unbounded fluid. Second, when δ ⪡ 1, but ϵδ = O(1), and “wall effects” again fall off faster than linearly, a set of local interface jump conditions can still be derived, but these must be supplemented by “hindered” transport corrections for advection and diffusion in the bulk-phase fluids. Finally, in other circumstances, we show that a local theory is not possible; here, one cannot ignore the detailed features near the interface. In those cases where local jump conditions can be derived rigorously, the qualitative features of these interfacial jump conditions are discussed in order to obtain a physical understanding of interfacial transport processes for our model system. It is shown that rigorously derived interface conditions for these cases resemble the normally assumed macroscopic jump conditions for interfacial mass transfer. We consider particularly those conditions for which interface transport effects are macroscopically significant.