Homogenization of boundary conditions for surfaces with regular arrays of traps

The problem of trapping of diffusing particles by nonoverlapping absorbing patches randomly or regularly located on a surface arises in numerous settings. Examples include diffusion current to ensembles of microelectrodes, ligand binding to cells, mass transfer to heterogeneous surfaces, ligand accumulation in cell culture assays, etc. (see Refs. 1–15 and references therein). The problem is extremely complicated because the boundary conditions on the surface are nonuniform: absorbing on the patches and reflecting otherwise. There is, however, an approximation that greatly simplifies the analysis when the layer of medium above the surface is sufficiently thick. The approximation is based on the fact that, far from the surface, fluxes and concentrations become uniform in the lateral direction and, therefore, indistinguishable from those in the case of uniformly absorbing surface. Keeping this in mind, one can replace the nonuniform boundary conditions on the surface by a uniform radiation-type boundary condition with a properly...