On the Modified Eguchi-Oki-Matsumura System

TL;DR

This work studies joint order-disorder transformation and phase separation within the Eguchi-Oki-Matsumura (EOM) framework by replacing the conserved Cahn-Hilliard (CH) dynamics with a convective-viscous CH equation and seeking exact traveling-wave solutions. The authors derive an exact traveling-wave solution using a linear u-v coupling $u=\gamma v+\beta$ after nondimensionalization, yielding a tanh-profile for the non-conserved field $v(z)$ and a corresponding linear relation for $u(z)$, with front speed $\sigma$ determined by the model parameters. Key results include explicit expressions for $v(z)$ and the roots $v_{1,2}$, and a set of algebraic constraints that link the parameters (notably $y=\gamma^2+1=\frac{D}{\varepsilon}$ and $\eta(y)=\frac{2y-3}{(y-1)y}$) to ensure reality and positivity, together with a velocity formula $\sigma^2=2D\frac{(y-1)}{(4-y)}[Dy^2+y(a-D)+b]$. The findings delineate the admissible coupling regimes and show how viscosity and external convective terms influence the propagation of coupled ordering and phase-separation fronts, contributing to the understanding of type-$C$ order-parameter dynamics in phase-field models.

Abstract

To describe the simultaneous order-disorder transformation and phase separation Eguchi, Oki and Matsumura [\doi{10.1557/proc-21-589}] introduced the system of two equations: one equation, governing the evolution of a conserved order parameter, and the second equation for the non-conserved order parameter. The key feature of their model is the free energy functional, which contains the square gradient terms of the both order parameters and a fourth power polynomial depending on both order parameters. According to the general Hohenberg-Halperin classification it is the type C model. We show that if the dynamics of the conserved order parameter is governed by the convective-viscous Cahn-Hilliard equation, this system allows exact traveling wave solution.