Growth Theory for an Ensemble of Ellipsoidal Particles

Abstract—When the growth of an ensemble of crystals from metastable melts is modeled, it is important to take into account the shape of growing particles. As experimental data show, the shape of evolving crystals can often be considered ellipsoidal, since it allows us to describe the deviations in the shape of particles from spherical geometry in the first approximation. In this work, we theoretically study the evolution of a polydisperse ensemble of elongated and oblate ellipsoidal crystals in a supercooled single-component melt. The volume growth rates of elongated and oblate ellipsoids with the same supercooling of the melt are analytically found and compared. Elongated crystals are shown to evolve faster than oblate ones, and the difference in their growth rates increases with the supercooling of the melt. These volume growth rates are taken into account to formulate a model describing the evolution of an ensemble of elongated/oblate ellipsoidal particles. An analytical solution to this integro-differential model has been found for two particle nucleation mechanisms in a parametric form for elongated and oblate ellipsoids using the saddle point method. A particle volume distribution function and the time and supercooling of the system are determined depending on the maximum crystal volume, which plays the role of a solution parameter. The constructed solution shows that an ensemble of elongated particles grows and removes the supercooling of the melt faster than an ensemble of oblate particles. As a result, the particle volume distribution function of elongated crystals shifts toward larger crystal sizes than the same distribution for oblate crystals. Considering this behavior, we can conclude that the crystal shape plays a crucial role in the melt supercooling removal dynamics and the volume distribution of particles during crystallization.