Oscillatory solutions to the system of Allen-Cahn and Cahn-Hilliard equations: A spinodal decomposition model

The boundary-value problem for the system of Cahn-Novick-Cohen equations is analyzed. This problem is a quasi-continuum model of the corresponding lattice model for an Fe-Al alloy and simultaneously describes the processes of phase separation (spinodal decomposition) and atomic ordering in sublattices. It is demonstrated that the evolution of the system can occur according to the following three scenarios. (i) Against the background of a disordered state v = 0, spatially nonuniform distributions of the concentration u with respect to a stationary distribution u = u _m that is dependent on the mean mass m are developed at long times t → ∞. (ii) Against the background of a stationary distribution of the concentration u = u _m , spatially nonuniform distributions of the order parameter are developed at t → ∞. (iii) The first and second scenarios can proceed simultaneously for a specific set of parameters (for example, with the dimensionless temperature ? = T/T _c , where T _c is the critical temperature). The results of the calculations performed with so-called asymmetric boundary conditions of wetting in a constant magnetic field for a thin quasi-one-dimensional film consisting of a binary mixture are compared with data obtained from numerical and real experiments.