Crystal nucleation and growth from the undercooled liquid: A nonclassical piecewise parabolic free‐energy model

An undercooled liquid exhibits crystalline fluctuations, some of which grow into crystal of macroscopic dimension, while smaller fluctuations disappear. We present a model which allows for exact analytic characterization of the inhomogeneous critical nucleus, the smallest fluctuation which will give rise to crystal growth, in terms of a single spatially varying order parameter for the degree of crystallinity. The model is built around the square‐gradient approximation for the free energy with a simple double‐parabolic form for the homogeneous component. We study the radius, free energy of formation, and profile of the critical nucleus as functions of the liquid undercooling and compare these with results from an earlier nonclassical theory and from the classical capillarity approximation. The time evolution of the order parameter is described by a phase‐field equation which is easily solved numerically for growth dynamics of initially supercritical fluctuations or for the regression of subcritical profiles.

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