Numerical investigation into the dependence of the Allen–Cahn equation on the free energy

Abstract

Phase-field modeling is strongly influenced by the shape of a free energy functional. In the theory of thermodynamics, it is a logarithmic type potential that is legitimate for modeling and simulating binary systems. Nevertheless, a tremendous amount of works have been dedicated to phase-field equations driven by 4-th double-well potentials as a polynomial approximation to the logarithmic type, which is valid only in the critical temperature regime. For comprehensive understanding of polynomial and logarithmic potentials and their relationship in the context of phase-field modeling, we provide and analyze a numerical framework for the Allen–Cahn equation derived from the Ginzburg–Landau functional with a logarithmic potential and its polynomial approximations. We prove that our numerical schemes guarantee the boundedness and energy dissipation properties of the solutions. As for the logarithmic free energy, we characterize different morphological changes of numerical solutions under various atomic binding energy configurations. Comparison of the logarithmic potential with its 2n-th order polynomial approximations reveals difference in the dynamics of spinodal decomposition. In particular, unlike the 6-th order or higher polynomials, the most studied solution by the 4-th order polynomial approximation using the 4-th double-well potential turns out to violate the logarithmic energy dissipation law. The geometric aspect of the Allen–Cahn equation with the logarithmic potential is also confirmed numerically. In summary, this study demonstrates the validity and applicability of the numerical framework for logarithmic and polynomial potentials and supports the need for further mathematical analysis on the logarithmic model.