Adsorption and Permeation Events in Molecular Diffusion

Abstract

How many times can a diffusing molecule permeate across a membrane or be adsorbed on a substrate? We employ an encounter-based approach to find the statistics of adsorption or permeation events for molecular diffusion in a general confining medium. Various features of these statistics are illustrated for two practically relevant cases: a flat boundary and a spherical confinement. Some applications of these fundamental results are discussed.

Keywords: Brownian motion; biochemistry; confinement; diffusion; encounter-based approach; geometric complexity; heterogeneous catalysis; permeation; reversible reactions; surface reaction.

Figure 1

(a) A simulated trajectory of a molecule diffusing inside a disk $\mathsf{\Omega }$ with an adsorbing circular boundary $\mathsf{\Gamma }=\partial \mathsf{\Omega }$. A triangle indicates the starting point ${\mathit{x}}_0$, while three filled squares show three positions, at which the molecule was adsorbed to the boundary. (b) The associated boundary local time ${\ell }_t$ (obtained along the same simulation) crosses three random thresholds ${\widehat{\ell }}_1$, ${\widehat{\ell }}_1+{\widehat{\ell }}_2$, and ${\widehat{\ell }}_1+{\widehat{\ell }}_2+{\widehat{\ell }}_3$ (horizontal lines) at random times ${\mathcal{T}}_1$, ${\mathcal{T}}_2$, and ${\mathcal{T}}_3$ (vertical lines). Each such crossing corresponds to an adsorption on the boundary. Colors distinguish successive time periods between adsorption events.

Figure 2

Statistics of adsorption events at the origin for diffusion on a half-line with a constant reactivity $q=1$. (a) $Q_n\left(t\right|0)$ as a function of n for 64 values of t, chosen equidistantly on a logarithmic scale, from $t={10}^{-1}$ (dark blue) to $t={10}^3$ (dark red), with $D=1$. (b) $Q_n\left(t\right|0)$ as a function of n for $t=10$. Filled circles present the exact solution (50), the solid line shows the approximation (52), whereas the dash-dotted line indicates the asymptotic relation (53).

Figure 3

Statistics of adsorption events at the origin for diffusion on a half-line for the adsorption mechanism with a fixed threshold ${\ell }_0=1$. (a) $Q_n\left(t\right|0)$ as a function of n for 64 values of t, chosen equidistantly on logarithmic scale, from $t={10}^{-1}$ (dark blue) to $t={10}^3$ (dark red), with $D=1$. (b) The effective reactivity $\kappa \left(t\right|x_0)$ defined in Equation (25), with $N_1\left(t\right|x_0)$ given by Equation (56) and $L\left(t\right|x_0)$ given by Equation (54).

Figure 4

Statistics of adsorption events on the adsorbing sphere of radius $R=1$ with a constant reactivity $q=1$ (top panels) and with the Mittag–Leffler adsorption model with $\alpha =0.5$ and ${\ell }_0=1$ (bottom panels). (a,c) $Q_n\left(t\right|R)$ as a function of n for 64 values of t, chosen equidistantly on logarithmic scale, from $t={10}^{-1}$ (dark blue) to $t={10}^2$ (dark red), with $D=1$. (b,d) The same probabilities with n rescaled by the mean number of adsorptions.