A phenomenological description of transition from an unstable to a (meta)stable phase state, including microscopic and mesoscopic scales, is presented. It is based on the introduction of specific memory functions which take into account contributions to the driving force of transformation from the past. A region of applicability for phase-field crystals and Swift-Hohenberg-type models is extended by inclusion of inertia effects into the equations of motion through a memory function of an exponential form. The inertia allows us to predict fast degrees of freedom in the form of damping perturbations with finite relaxation time in the instability of homogeneous and periodic model solutions.
The phase-field model of Echebarria, Folch, Karma, and Plapp [Phys. Rev. E 70, 061604 (2004)] is extended to the case of rapid solidification in which local nonequilibrium phenomena occur in the bulk phases and within the diffuse solid-liquid interface. Such an extension leads to the fully hyperbolic system of equations given by the atomic diffusion equation and the phase-field equation of motion. This model is applied to the problem of solute trapping, which is accompanied by the entrapment of solute atoms beyond chemical equilibrium by a rapidly moving interface. The model predicts the beginning of complete solute trapping and diffusionless solidification at a finite solidification velocity equal to the diffusion speed in bulk liquid.
Equations of fast phase transitions, in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation, are analyzed. These equations have a singular perturbation due to the second derivative of the order parameter with respect to time, which appears due to phenomenologically introduced local nonequilibrium. To develop unconditionally stable computational schemes, the Eyre theorem [D. J. Eyre, unpublished] proved for the classical equations, based on hypotheses of local equilibrium, is used. An extension of the Eyre theorem for the case of equations for fast phase transitions is given. It is shown that the expansion of the free energy on contractive and expansive parts, suggested by Eyre for the classical equations of Cahn-Hilliard and Allen-Cahn, is also true for the equations of fast phase transitions. Grid approximations of these equations lead to gradient-stable algorithms with an arbitrary time step for numerical modeling, ensuring monotonic nonincrease of the free energy. Special examples demonstrating the extended Eyre theorem for fast phase transitions are considered.
A model for diffusion and phase separation which takes into account hyperbolic relaxation of the solute diffusion flux is developed. Such a 'hyperbolic model' provides analysis of 'hyperbolic evolution' of patterns in spinodal decomposition of binary systems. Analytical results for the dispersion relation and critical parameters (such as wavelength and amplification rate of decomposition) are analyzed in comparison with outcomes of classic CahnHilliard theory. It is shown that the hyperbolic model predicts the amplification rate behaviour that is typically observed in experiments on spinodal decomposition.
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