A stochastic theoretical approach to study the size-dependent catalytic activity of a metal nanoparticle at the single molecule level.

The catalytic activity of metal nanoparticles is intrinsically heterogeneous due to the heterogeneous distribution of surface catalytic sites and surface restructuring dynamics. Recent advances in single-molecule fluorescence spectroscopy reveal that the rates of product formation and dissociation exhibit size-dependent activities. Here we present a theoretical method to study the size-dependent catalytic activity of a metal nanoparticle using the stochastic approach based on the superposition of renewal processes. We observe that for a single nanoparticle with fewer surface-active catalytic sites, temporal fluctuations in the reaction rate, a phenomenon commonly known as dynamic disorder, are present in both the product formation and product dissociation events. The increase in the number of surface catalytic sites suppresses the effect of dynamic restructuring of the surface, thereby leading to a decrease in dynamic disorder. The proposed formalism provides a theoretical foundation to understand the size-dependent catalytic activity of metal nanoparticles at the single molecule level.