Boundary Homogenization and Capture Time Distributions of Semipermeable Membranes with Periodic Patterns of Reactive Sites

Abstract

We consider the capture dynamics of a particle undergoing a random walk in a halfspace bounded by a plane with a periodic pattern of absorbing pores. In particular, we numerically measure and asymptotically characterize the distribution of capture times. Numerically we develop a kinetic Monte Carlo (KMC) method that exploits exact solutions to create an efficient particlebased simulation of the capture time that deals with the infinite half-space exactly and has a run time that is independent of how far from the pores one begins. Past researchers have proposed homogenizing the surface boundary conditions, replacing the reflecting (Neumann) and absorbing (Dirichlet) boundary conditions with a mixed (Robin) boundary condition. We extend previous work to asymptotically determine the leakage parameter for the mixed boundary condition for arbitrary periodic pore configurations in the dilute fraction limit. In this asymptotic limit, we pose and solve an optimization problem for the Bravais lattice which maximizes the capture rate of the absorbing pores, finding the hexagonal lattice to be the global maximum.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains

“…When the domain Ω is unbounded, one also needs to impose a regularity condition at infinity: G q (x, t|x 0 ) → 0 as |x| → ∞ (similar condition has to be imposed for the related boundary value problems (12,18,19), see below). The Robin boundary condition (8) appears in a large variety of physical, chemical and biological applications [19][20][21][47][48][49][50][51][52][53][54][55][56][57], as well as the effective boundary condition after homogenization [58][59][60][61][62][63] (see an overview in [28]). The subscript q allows us to distinguish three types of boundary condition: Neumann (q = 0), Robin (0 < q < ∞), and Dirichlet (q = ∞).…”

“…Even though this observation may suggest an approach to a steady-state limit, this is not the case, given that the mean boundary local time slowly grows, see Eq. (63).…”

Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces

“…On one hand, matched asymptotic analysis, dual series technique, and conformal mapping were applied to establish the behavior of the mean first-passage time in both twoand three-dimensional domains [42][43][44][45][46][47][48][49][50]. On the other hand, homogenization techniques were used to substitute piecewise constant reactivity κ(s) by an effective homogeneous reactivity [31][32][33][51][52][53][54][55][56][57][58]. More recent works investigated how the mean reaction time is affected by a finite lifetime of diffusing particles [59][60][61], by partial reactivity and interactions [63,64], by target aspect ratio [65], by reversible target-binding kinetics [66,67] and surface-mediated diffusion [68][69][70][71], by heterogeneous diffusivity [72], and by rapid re-arrangments of the medium [73][74][75].…”

Reversible reactions controlled by surface diffusion on a sphere

“…Berg and Purcell estimated the reaction * Electronic address: denis.grebenkov@polytechnique.edu rate controlled by bulk diffusion toward multiple small patches evenly distributed over the surface of a sphere [14]. This seminal paper and the associated homogenization concept were further extended by many authors [15][16][17][18][19][20][21][22][23]. Shoup et al devised an efficient approximation to compute the reaction rate from the bulk to an active circular region on the surface of a sphere, even in the presence of rotational motion [24].…”