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, D(t), 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, D(f), or (ii) the number of atoms in the average catalytically effective nucleus, N*, and the induction period time, t ind (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 equation is obtained for Ir(0) nanoclusters formed from the well-studied system of H2 reduction of the precursor [(1,5-COD)Ir x P2W15Nb3O62](8-). The D(t) equation is then used to determine N* for nine prior Ir(0) nanocluster preparations from five different [(1,5-COD)Ir(+)]n [anion(n-)] precursors. Also given is a relationship allowing one to interconvert between nanocluster size data and nanocluster precursor concentration data, again when the two-step nucleation and growth mechanism has been shown to apply. Some of the key experimental factors that are known to affect the kinetics of nanocluster formation, and therefore nanocluster size, are also summarized. A look ahead to needed future work is also provided.