Homogenization of boundary conditions for surfaces with regular arrays of traps

Abstract

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 radiationtype boundary condition with a properly chosen trapping rate see e.g., Ref. 16 and references therein . We have demonstrated how this procedure works in the case of randomly distributed traps in Refs. 17 and 18. Here we consider the problem with traps regularly distributed over the surface. Our aim is to predict the dependence of on the trap concentration and parameters of the traps and diffusing particles over the entire concentration range. At low concentrations is equal to the product of the concentration and the trapping rate constant of an isolated trap. It turns out that the low-concentration linear dependence of on the trap concentration fails very early, and grows with the concentration much faster even when only a small fraction of the surface is covered by the traps. This happens because “interaction” between traps decays very slowly, as 1 /L, where L is the intertrap distance. As a consequence, collective effects due to this interaction, which lead to the enhancement of the trapping rate, manifest themselves already at low concentrations. In Ref. 17 we reported a boundary homogenization approach for surfaces randomly covered by nonoverlapping circular traps. To describe the enhancement of the trapping rate compared to the linear regime, we introduced the function F of the trap surface fraction and suggested an approximate formula for this function. In Ref. 18 we found that the enhancement due to the collective effects was insensitive to whether the traps were identical or polydisperse and their radii are allowed to fluctuate. This suggests that the enhancement depends only on and is weakly sensitive to the details of the trap arrangement on the surface. To check this hypothesis here we study homogenization of boundaries with regular arrangements of identical traps. Our results support this hypothesis. We find that the values of for the three different arrangements and for random distribution of traps are close to each other the difference is within 20% . In addition, here we study homogenization of periodic nonuniform boundaries formed by alternating absorbing and reflecting stripes. Such boundaries are special because the conventional ideology based on the existence of a trivial limiting behavior of when →0 fails in this case since a stationary flux to an isolated absorbing strip does not exist. Nevertheless, we are able to overcome this difficulty and find the effective trapping rate for such a boundary. In our analysis we use the computer-assisted boundary homogenization approach suggested in Ref. 17: First, based on the limiting behavior and dimensional arguments we express in terms of an unknown dimensionless function of the dimensionless trap surface fraction ,F . We then determine this function using the dependence , which is found by Brownian dynamics simulations as described in Ref. 17 or by solving THE JOURNAL OF CHEMICAL PHYSICS 124, 1 2006