A micromechanical derivation of Fick's law for interfacial diffusion of surfactant molecules

Abstract

A theory is presented for the diffusion of surfactant molecules, modeled as noninteracting Brownian spheres in proximity to, or straddling, the interface between two immiscible fluids. The spheres are assumed to be physicochemically inert except for the existence of short-range forces, which may be attractive and/or repulsive with respect to the interface. The number density of such particles is assumed small. Two alternative views of the diffusion process are presented; one, termed microscopic, in which variations in concentration are resolvable down to the scale l imposed by the short-range attractive or repulsive forces between the particles and the interface, and the other, termed macroscopic, in which variations are only discernible at the much larger scale, L, of gradients in the bulk surfactant concentration distribution, which are assumed to exist parallel to the interface. It is demonstrated that a rigorous theory can be developed, at the microscale, for prediction of the concentration profiles and diffusive flux of surfactant molecules parallel to the interface using only the well-established Stokes-Einstein theory of bulk-phase Brownian diffusion, and low Reynolds number hydrodynamics for the motion of a torque-free particle in close proximity, or straddling, a fluid-fluid interface. On the other hand, a macroscopic description of the same phenomena requires the introduction of new concepts, such as “adsorption” and “surface diffusion,” which are specifically associated with the interface, as seen from the macroscale of O( L). In the present paper, constitutive relationships for these macrosurface processes are derived in a rigorous manner from the more fundamental and complete microscopic description of the system. It is shown that Fick's law is applicable to surface diffusion, with the driving gradient based on a surface-excess concentration for surfactant particles. Furthermore, the surface diffusion coefficient is found to depend upon the Stokesian hydrodynamic resistance of a torque-free sphere translating parallel to the interface, as a function of the distance of its center from the interface, and upon the “adsorptive-potential” energy function which tends to cause the particles to accumulate there. Numerical values await the solution of the low Reynolds number hydrodynamics problem thereby posed, as well as the acquisition of knowledge relating to the potential energy function. Two examples of such potential energy functions are discussed for Brownian particles, one based upon a difference in area-specific, surface-free energies (i.e., solid/liquid interfacial tensions) for the two fluids, and the other on a simple density differential between the particle and the two fluids. Simple geometric models of surfactant molecules are also discussed in the context of potential energy functions for interaction between the molecules and the interface. The possibility of using such models in the analysis of other equilibrium and surface transport phenomena is pointed out.