Diffusion-Controlled Reactions: An Overview

Abstract

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski, who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics, and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

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

Figure 1

(Top row) Rescaled concentration $\left[A\right](\mathit{x},t)/{\left[A\right]}_0=1-\frac{R}{\left|\mathit{x}\right|}\mathrm{erfc}\left(\left(\right|\mathit{x}|-R)/\sqrt{4Dt}\right)$ of reactant A near a perfectly reactive catalytic sphere of radius R (in gray) at different time instances (here $\mathrm{erfc}\left(z\right)$ is the complementary error function) [3]. (a) Homogeneous concentration at $t=0$; (b) Formation of a thin depletion zone in a short time $Dt/R^2=0.1$; (c,d) Progressive growth of the depletion zone at longer times $Dt/R^2=1$ and $Dt/R^2=10$; (e) Approach to a steady-state concentration $\left[A\right](\mathit{x},\infty )/{\left[A\right]}_0=1-R/\left|\mathit{x}\right|$ as $t\to \infty$. (Bottom row) Rescaled concentration $\left[A\right](\mathit{x},t)/{\left[A\right]}_0=1-\frac{R-R_{\kappa }}{\left|\mathit{x}\right|}\left\{\mathrm{erfc}\left(\frac{\left|\mathit{x}\right|-R}{\sqrt{4Dt}}\right)+e^{Dt/R_{\kappa }^2+\left(\right|\mathit{x}|-R)/R_{\kappa }}\mathrm{erfc}\left(\frac{\left|\mathit{x}\right|-R}{\sqrt{4Dt}}+\frac{\sqrt{Dt}}{R_{\kappa }}\right)\right\}$ of reactant A near a partially reactive catalytic sphere of radius R [31], with reactivity $\kappa R/D=1$ and $R_{\kappa }=R/(1+\kappa R/D)$), at the same time instants: $t=0$ (f), $Dt/R^2=0.1$ (g), $Dt/R^2=1$ (h), $Dt/R^2=10$ (i), and $t=\infty$ (j).

Figure 2

Various microscopic origins of imperfect surface reactions. (a) When the reactant A arrives onto the catalytic surface C, an activation energy barrier $E_a$ has to be overcome for a chemical transformation of A into B; if failed, the reactant leaves the vicinity of C and, thus, resumes its bulk diffusion. (b) A macromolecule can spontaneously switch its conformational state from “active” (in red) to “passive” (in blue) with the rate $k_a$, and back (with the rate $k_d$), while its reaction on the catalytic surface (in gray) or with another macromolecule (a receptor, an enzyme, a DNA strand, etc.) is only possible in the “active” conformational state. (c) The reactant can be temporarily trapped by a buffer molecule (in green) that makes it inactive for the considered surface reaction; their association/dissociation kinetics is usually described by forward and backward rates $k_a$ and $k_d$. (d) An ion can pass through an open channel, while it is reflected back from a closed channel. (e) The escape of a semi-flexible polymer through a small hole can be described by an entropic barrier that leads to partial reactivity when the first arrival to the hole does not guarantee the passage. (f) An inert (gray) surface is covered by reactive catalytic germs (black spots) so that the reactant may fail to react upon the first arrival, and thus, resumes its bulk diffusion until the next encounter, and so on. Similarly, a protein can search for a specific (target) site on a DNA chain for successful binding.

Figure 3

(a) Encounter-dependent reactivity $\kappa (\ell )$ from the gamma model with $q=1$ and three values of $\nu$. (b) The overall reaction rate $J\left(t\right)$ on a spherical catalyst of radius R, rescaled by Smoluchowski’s rate $J_S=4\pi DR{\left[A\right]}_0$, with $q=1$ and three values of $\nu$. Dotted curve represents Equation (7) for a perfectly reactive sphere (it formally corresponds to $\nu =0$).