Reaction Kinetics of Cluster Impurities,
E. Ben-Naim
We study the kinetics of
clustered immobile reactants in diffusion-controlled single-species
annihilation. We consider the initial conditions where the immobile
reactants occupy a subspace of dimension $d_i$, while the rest of
the $d$-dimensional space is occupied by identical mobile particles.
The Smoluchowski rate theory suggests that the immobile reactants
concentration, $s(t)$, exhibits interesting behavior as a function of
the codimension, $\bar d\equiv d-d_i$. This survival probability
undergoes a survival-to-extinction transition at $\bar d_c=2$. For
$\bar d<\bar d_c$, a finite fraction of the immobile reactants
survives, while for $\bar d\ge \bar d_c$, $s(t)$ decays indefinitely.
The corresponding asymptotic properties of the concentration are
discussed. The theoretical predictions are verified by numerical
simulations in 2D and 3D
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