Transition-Metal Nanocluster Size vs Formation Time and the Catalytically Effective Nucleus Number: A Mechanism-Based Treatment

A mechanism-based equation for the size of a forming transition-metal nanocluster vs time has been derived based on the Finke−Watzky two-step mechanism for transition-metal nanocluster nucleation (A → B, rate constant k1) and autocatalytic growth (A + B → 2B, rate constant k2), where A is the nanocluster precursor and B is the growing nanocluster. The resultant equation expresses nanocluster diameter as a function of time, Dt, in terms of k1, k2, the initial concentration of the nanocluster precursor complex, [A]0, and the number of catalytically effective nuclei derived from either (i) the final nanocluster size, Df, or (ii) the number of atoms in the average catalytically effective nucleus, N*, and the induction period time, tind (N* being by definition the number of atoms present in the average size nucleus at the end of the induction period and when observable catalysis begins). By fitting experimentally determined nanocluster size vs time data using this equation, evidence for the validity of the equ...