Imperfect Diffusion-Controlled Reactions

Abstract

This chapter aims at emphasizing the crucial role of partial reactivity of a catalytic surface or a target molecule in diffusion-controlled reactions. We discuss various microscopic mechanisms that lead to imperfect reactions, the Robin boundary condition accounting for eventual failed reaction events, and the construction of the underlying stochastic process, the so-called partially reflected Brownian motion. We show that the random path to the reaction event can naturally be separated into the transport step toward the target, and the exploration step near the target surface until reaction. While most studies are focused exclusively on the transport step (describing perfect reactions), the exploration step, consisting is an intricate combination of diffusion-mediated jumps between boundary points, and its consequences for chemical reactions remain poorly understood. We discuss the related mathematical difficulties and recent achievements. In particular, we derive a general representation of the propagator, show its relation to the Dirichlet-to-Neumann operator, and illustrate its properties in the case of a flat surface.

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 = ∞).…”

“…We note that the notation G q (x, t|x 0 ) is different from that of Refs. [28,43], in which Neumann and Dirichlet propagators were denoted as G κ=0 and G 0 , respectively.…”

“…One sees that the boundary local time characterizes the dynamics of a diffusing particle near the boundary and thus plays a crucial role in the description of various diffusion-mediated phenomena in cellular biology, heterogeneous catalysis, nuclear magnetic resonance, etc. [1][2][3][4][5][6][7][17][18][19][20][21][22][23][24][25][26][27][28]. In these phenomena, a diffusing particle approaching the boundary can change its state due to, e.g., permeation through a pore, chemical reaction on a catalytic germ, or surface relaxation on a paramagnetic impurity [29][30][31].…”

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

“…The reactivity is thus related to the probability of reaction event at the encounter [12][13][14]. Robin boundary condition with homogeneous (constant) reactivity κ was often employed to describe many chemical and biochemical reactions and permeation processes [15][16][17][18][19][20][21][22][23][24][25][26], to model stochastic gating [27][28][29][30], or to approximate the effect of microscopic heterogeneities in a random distribution of reactive sites [31][32][33] (see a recent overview in [34]). In spite of its practical importance, diffusion-controlled reactions with heterogeneous surface reactivity κ(s) remain much less studied.…”

Reversible reactions controlled by surface diffusion on a sphere

“…It states that the net diffusive flux density of particles B toward the target (the left-hand side) is equal to the reactive flux density, κc |∂Ω , minus the flux density related to the dissociation of the metastable complex AB (the last term). The reactivity κ characterizes the difficulty for a particle B to overcome an energy activation barrier for binding to the target [41][42][43] (see also a recent overview in [44,45]). The limit κ = ∞ describes an immediate reaction upon the first encounter (no activation barrier), whereas κ = 0 corresponds to no reaction (infinite activation barrier) on reflecting boundary.…”