A General Kinetic-Optical Model for Solid-State Reactions Involving the Nano Kirkendall Effect. The Case of Copper Nanoparticle Oxidation

Oxidation of copper nanoparticles (NPs) and other solid-state reactions have been shown recently to involve the nanoscale Kirkendall effect (NKE). The process, resulting in the formation of internal voids, has been rarely considered in most of the simple reaction kinetic models currently available. Here we present a general solid-state reaction kinetic model based on the assumption of steady-state diffusion profiles for rationalizing the evolution of the optical behavior observed in plasmonic Cu NPs undergoing solid-state oxidation to Cu2O. While the model is applied here to an oxidation process, it is applicable in principle to any system showing Kirkendall voiding. An analytical expression of the rate law for spherical NPs is derived, and implications of the model are discussed. By combining the general kinetic model with Mie scattering solutions under the quasi-static approximation, the extinction cross section of Cu NPs as a function of the Cu-to-Cu2O conversion fraction is modeled, for NPs of different initial sizes and for different degrees of shrinkage upon formation of a Cu2O shell layer. The experimental observation of an initial increase in the surface plasmon (SP) band extinction intensity, followed by a decrease, is qualitatively reproduced using the combined model. This characteristic behavior of Cu nanoplasmonic systems is not expected theoretically in sufficiently small NPs; however, NPs of >similar to 6 nm in diameter are expected to exhibit this type of behavior. Our results suggest that the NKE may be important in describing the optical behavior of plasmonic NPs subject to solid-state oxidation.