Theoretical Investigation on Surface-mediated Search Dynamics of a Reactant*

The process of a reactant search for a target located on a surface is ubiquitous in Nature, in particular in chemical- and biological processes. We study the dynamics of a reactant search for a small target in a 2D surface from the bulk both in continuum and discrete model. We find that depending on the scanning length $\lambda$ of the reactant on the surface, which is determined by the reactant-surface interactions, the search dynamics shows different behavior: (i) for small $\lambda$, the reactant find the target via 3 dimensional bulk diffusion, (ii) for large $\lambda$, it find the target via surface diffusion, and (iii) for intermediate $\lambda$, the reactant find the target via a combination of 3D and 2D motion which can minimize the search time $T$. The search times $T$ in a continuum- and discrete model are close, but not the same even we take the parameters equivalently. We also study how the search time is dependent on the surface size and the position of the target in the discrete model. Finally, we discuss the relevance of our results with some recent experiments.