Understanding Equilibrium Size Distribution of Metal Single Atoms, Clusters, and Particles from Thermodynamics

Dynamic size evolution of supported metal single atoms, clusters and nanoparticles widely observed through in-situ characterization techniques plays a crucial role on the catalytic activity and stability. However, the underlying atomic-level mechanism remains elusive. To gain insights into this dynamic evolution phenomenon, we have derived a universal equilibrium size distribution equation, through a variational method from a thermodynamic perspective (see below). This equation holds under both canonical and grand canonical ensemble, and can be used to predict how particle size distributions change with the environment and the total metal loading ratio, thereby revealing the trends of size evolution without a priori knowledge on kinetics. By integrating machine learning force filed (MLFF) with Monte Carlo (MC) and molecular dynamics (MD) simulations, the size evolution is modelled and the results are consistent with the physical insights derived from our equilibrium size distribution equation.