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Soft Matter and Biological Physics

Eur. Phys. J. E 1, 159-177

Interfacial reaction kinetics

B. O'Shaughnessy1 - D. Vavylonis2

1 Department of Chemical Engineering, Columbia University, 500 West 120th Street, New York, NY 10027, USA
2 Department of Physics, Columbia University, 538 West 120th Street, New York, NY 10027, USA
bo8@columbia.edu, dvav@phys.columbia.edu

Received 8 June 1998 and Received in final form 10 September 1999

Abstract
We study irreversible A-B reaction kinetics at a fixed interface separating two immiscible bulk phases, A and B. Coupled equations are derived for the hierarchy of many-body correlation functions. Postulating physically motivated bounds, closed equations result without the need for ad hoc decoupling approximations. We consider general dynamical exponent z, where $x_t\sim t^{1/z}$ is the rms diffusion distance after time t. At short times the number of reactions per unit area, ${\cal R}_t$, is 2nd order in the far-field reactant densities $n_{\rm A}^{\infty},n_{\rm B}^{\infty}$. For spatial dimensions dabove a critical value $d_{\rm c}=z-1$, simple mean field (MF) kinetics pertain, ${\cal R}_t\sim Q_b t n_{\rm A}^{\infty} n_{\rm B}^{\infty}$ where Qb is the local reactivity. For low dimensions $d<d_{\rm c}$, this MF regime is followed by 2nd order diffusion controlled (DC) kinetics, ${\cal R}_t
\approx x_t^{d+1} n_{\rm A}^{\infty} n_{\rm B}^{\infty}$, provided $Q_b >
Q_b^* \sim (n_{\rm B}^{\infty})^{[z-(d+1)]/d}$. Logarithmic corrections arise in marginal cases. At long times, a cross-over to 1st order DC kinetics occurs: ${\cal R}_t \approx x_t n_{\rm A}^{\infty}$. A density depletion hole grows on the more dilute A side. In the symmetric case ( $n_{\rm A}^{\infty}=n_{\rm B}^{\infty}$), when $d<d_{\rm c}$ the long time decay of the interfacial reactant density, $n_{\rm A}^{\rm s}$, is determined by fluctuations in the initial reactant distribution, giving $n_{\rm A}^{\rm s} \sim t^{-d/(2z)}$. Correspondingly, A-rich and B-rich regions develop at the interface analogously to the segregation effects established by other authors for the bulk reaction ${\rm A + B} \rightarrow\emptyset$. For $d>d_{\rm c}$ fluctuations are unimportant: local mean field theory applies at the interface (joint density distribution approximating the product of A and B densities) and $n_{\rm A}^{\rm s} \sim t^{(1-z)/(2z)}$. We apply our results to simple molecules (Fickian diffusion, z=2) and to several models of short-time polymer diffusion (z>2).

PACS
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 68.45.Da Adsorption and desorption kinetics; evaporation and condensation - 82.35.+t Polymer reactions and polymerization

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