The irrelevance of phantom nuclei in crystallization kinetics: An integral equation approach

Abstract

Abstract An integral equation for crystallization kinetics that takes into account impingement and prevents the emergence of phantom nuclei is derived. We assume that crystallization proceeds by the growth of crystalline spherical domains, which develop from stable nuclei that emerge in the untransformed phase. In contrast to previous formulations, our approach does not require elaborate geometrical considerations or the need of concepts such as correlation functions or extended volume. Instead, by introducing two crystallization-dependent factors, we assume that nucleation and growth are restricted to the untransformed phase. It is shown that in the special case in which the nucleation rate is made constant, allowing in this way the emergence of unphysical phantom nuclei, our integral expression reduces to the classical Avrami exponential equation. It is found that this latter expression predicts values very close to those corresponding to the case in which phantom nuclei are not allowed to exist, confirming in this way the numerical insignificance of phantom nuclei. Geometric and kinetic arguments are given to shed light on this issue. For comparison purposes, it is illustrated that in the case of two-dimensional crystallization, phantom nuclei are slightly more relevant.