Prediction of Spherule Size in Gas Phase Nanoparticle Synthesis

Abstract

Surprisingly, there is still no rational yet practical method to reliably predict absolute ‘primary’ nanospherule sizes and, hence, specific surface areas, in gas phase flame nanoparticle synthesis. The present paper summarizes our approach to this important problem, using a plausible and tractable coagulation–coalescence (two-rate process) model, but with an important modification to the rate of nanoparticle coalescence. The Smoluchowski equation is used to describe the particle Brownian coagulation rate process (free-molecule regime), together with the assumption that the particle population follows a self-preserving size distribution. The decisive coalescence process, driven by the minimization of surface energy of the coalescing nanoparticles, is presumed to occur via the mechanism of surface diffusion. However, a curvature-dependent energy barrier for surface-diffusion is proposed, taking into account the extended ‘surface-melting’ behavior of nanoparticles. This is shown here to have the effect of accelerating the coalescence rate of touching nanoparticles, leading to absolute sizes (at the predicted onset of aggregate formation) in encouraging agreement with available experiments. It was found that the coalescence rate, especially with a curvature-augmented surface diffusivity, is far more sensitive to particle size than is the Brownian coagulation rate. As a result, when cast in terms of characteristic process times, a distinct crossover generally exists, allowing the determination of observed ‘primary’ spherule sizes within larger aggregates. This approach is successfully applied here to several published synthesis examples of vapor-derived nanosized alumina and titania. Its broader implications for nanoparticle synthesis in non-isothermal reactors, including our own counterflow diffusion flame reactor, are also briefly summarized.