Diffusion to nonoverlapping or spatially correlated traps

Abstract

The survival probability S(t) is computed for a particle diffusing in a medium containing a distribution of static spherical traps whose centers may be correlated in an arbitrary way. This extends the author’s previous work on overlapping traps, which correspond to complete lack of correlation. The correlation is accounted for by introducing a trap-center pair correlation function in a second-cumulant approximation for averages over the distribution of traps. Otherwise the methods are the same as those used previously wherein the average number of different sites sampled in a random walk is used in a first-cumulant approximation for averages over all possible walks. Specific results are obtained for nonoverlapping traps (hard-sphere correlation) which occupy an arbitrary fraction f of the volume and for point traps which interact via attractive or repulsive potentials. For nonoverlapping traps S(t) has the same functional dependence exp(-t-$t^{1/2}$) found for overlapping traps, but the parameters are increased so that the average trapping rate $R_{\mathrm{in}}$ is greatly enhanced, even after account is taken of the reduced trapping volume caused by overlap. $R_{\mathrm{in}}$ is not greatly affected by having centers repel one another, but strong attraction produces clustering which greatly reduces $R_{\mathrm{in}}$ in a manner consistent with a simple physical picture. The characteristic inverse length κ which appears in the $e^{\mathrm{-}\kappa r}$/r spatial dependence of the distribution of filled traps is calculated for nonoverlapping traps. Agreement between theory and simulations is very good for κ. Simulation data for $R_{\mathrm{in}}$ are strongly dependent on discrete lattice effects (much more so than for κ); but extrapolation to the continuum limit gives good agreement with theory. The effective diffusion coefficient is calculated from its relation to $R_{\mathrm{in}}$ and κ.