Lie symmetry analysis, closed-form solutions and dynamics for the kinetics of phase separation in iron (Fe–Cr–X (X=Mo, Cu)) based on ternary alloys

The Cahn–Hilliard equation is a mathematical equation crucial in materials science, capturing significant qualitative aspects of two-phase systems associated with phase separation processes, under the assumptions of isotropy and constant temperature. In this paper, the dynamics of phase separation in iron-based ternary alloys is investigated through the application of the Cahn–Hilliard equation. Solution of the Cahn–Hilliard equation represents the concentration of one of two phases in a system which is undergoing phase separation. Unlike previous research that primarily focused on the intrinsic chemical potential A(u)= \(u^3-u\) $u^3-u$ , our work considers a broader range of intrinsic potentials, including A(u)= \(\varphi u+\psi \) $\phi u+\psi$ , \(\mu u^{2}+\nu u\) $\mu u^2+\nu u$ , \(\theta u^{n}+\delta u\) $\theta u^n+\delta u$ where \(n \ge 3\) $n\ge 3$ , \((\alpha u+\beta )^n\) ${(\alpha u+\beta )}^n$ where \(n \ge 2\) $n\ge 2$ and \(e^{u}\) $e^u$ . The study employs the Lie symmetry approach to reduce the Cahn–Hilliard equation to ordinary differential equations (ODEs), which are then solved using the Kudryashov method, the modified auxiliary equation method, and the power series method. The analytical solutions of the reduced ODEs are utilized to analyze the kinetics of phase separation. This approach has led to the discovery of novel analytical solutions for these cases, which are further analyzed through 3D and contour plots generated by Maple software. Our findings reveal new dark, dark-bright soliton-type visual representations, providing a comprehensive analysis of the kinetics of phase separation. This research not only advances the understanding of phase separation in iron-based ternary alloys but also introduces generalized and novel solutions through the Lie symmetry analysis of various intrinsic potentials.