Diffusion within nanometric and micrometric spherical-type domains limited by nanometric ring or pore active interfaces. Part 1: conformal mapping approach

A theoretical approach is developed for investigation of diffusion within nanometric or micrometric spherical-type domains limited by two types of active interfaces. Typical representative examples of either system consist in: (a) insoluble droplets (defined by spherical cupolas) of electroactive liquids or redox dendrimers adsorbed onto an electrode surface placed in an electrolyte or (b) biological vesicles during exocytotic release occurring in living cells. In situation (a), the active interface is an infinitely small ring defined along the triple-phase interface at the junction between the droplet surface, the electrode surface, and the electrolyte. In situation (b), the interface is a nanometric pore connecting the inside of the vesicle to the outer medium. Either situation imposes extremely difficult conditions for modeling the diffusional behavior of these two systems because of: (a) the conflicting symmetries between the spherical domains shapes vis-à-vis the pore or ring active interface which acts as a diffusional sink or a source and (b) the near-infinite and heterogeneous concentration gradients which develop at the active interface. Despite these intrinsic severe difficulties, introducing an original conformal mapping transform leads to a great simplification of the numerical solution of diffusional problems. This allows the development of fast and extremely precise simulations of the problem at hand. The efficiency of the simulations is tested in each case by imposing diffusion-controlled conditions so as to examine situations which enhance the intrinsic difficulties of the problems. In this first part, only diffusion within the inner domain defined by the vesicle or the droplet is considered in simulations since this is the domain in which the most difficult diffusional situations occur, but the general theory developed encompasses the most global circumstances, viz., when diffusion in both the inner and outer domains controls the system's behavior and when generalized Butler-Volmer-type kinetics apply at the active interface. It is also shown that this conformal mapping approach leads to the design of a one-dimensional approximation which provides results accurate within a few percent compared to the full treatment of the system. Such equivalence, which is reminiscent of the hemisphere-disk or hemicylinder-band equivalence at ultramicroelectrodes, provides efficient and accurate simulations of the system at hand and may prove extremely valuable for semi-quantitative predictions.