Abstract
The paper presents a comprehensive analysis of physical, mathematical, and computational approaches used to describe the transformation and propagation of crystallization and recrystallization fronts. The discussion begins with classical sharp-interface models, where the Stefan’s model serves as the fundamental description of phase-front transformation governed by heat conduction and solutal diffusion. Extensions of this model, via the Gibbs–Thomson relation, incorporate interfacial curvature and interfacial velocity effects, which are necessary for accurate modelling of interfacial-transformation processes. The Mullins–Sekerka theory examined in this work describes the morphological instability of the crystallization front, identifying the conditions, under which a planar interface loses stability and evolves into cellular or dendritic morphology. This theory accounts for impurity diffusion, undercooling, and surface energy, enabling the determination of critical growth rates and characteristic perturbation wavelengths. The Jackson–Hunt theory is considered as classical model of eutectic transformation, describing lamellar or rod-like structures and determining the optimal spacing between lamellae based on the minimal undercooling condition. The work also discusses diffuse interface (phase field) models, emphasizing their advantages for describing complex morphologies and topological changes. The thin-interface analysis is presented as a mathematical framework ensuring quantitative consistency between phase-field simulations and sharp-interface models. Corrections such as antitrapping fluxes guarantee modelling accuracy, even when the interfacial thickness and curvature are increased. The chapters describe finite-difference and finite-element schemes, stability conditions, adaptive meshing, and the role of multilevel and GPU-accelerated solvers in achieving high-accuracy simulations of crystallization and recrystallization processes. Classical theories including interface-motion kinetics description, the Mullins–Sekerka stability limits, and Jackson–Hunt law reproduction are regarded as essential tools for validating quantitative models. Finally, the work offers practical recommendations for selecting between sharp-interface models, phase-field models, multiphase approaches, cellular automata, and hybrid approaches depending on material system complexity, required accuracy, and available computational resources. The presented general concept integrates theory, asymptotic analysis, and modern computational methods to support predictive modelling of interfacial transformations under contemporary materials-processing conditions.