The asymptotic behavior for a binary alloy in energy and material science: The unified method and its applications

The Cahn–Hilliard system was proposed to the first time by Chan and Hilliard in 1958. This model (or system of equations) has intrinsic participation energy and materials sciences and depicts significant characteristics of two phase systems relating to the procedures of phase separation when the temperature is constant. For instance, it can be noticed when a binary alloy (“Aluminum + Zinc” or “Iron + Chromium”) is cooled down adequately. In this case, partially or totally nucleation (nucleation means the appearance of nuclides in the material) is observed: the homogeneous material in the initial state gradually turns into inhomogeneous, giving rise to a very accurate dispersive microstructure. Next, when the time scale is slower the microstructure becomes coarse. In this work, to the first time, the unified method is presented to investigate some physical interpretations for the solutions of the Cahn–Hilliard system when its coefficients varying with time, and to show how phase separation of one or two components and their concentrations occurs dynamically in the system. Finally, 2D and 3D plots are introduced to add more comprehensive study which help to understand the physical phenomena of this model. The technique applied in this analysis is powerful and efficient, as evidenced by the computational work and results. This technique can also solve a large number of higher-order evolution equations.