Spinodal Decomposition in Binary Mixtures

We study the early stage of the phase separation of a binary mixture far from its critical point of demixing. Whenever the mixture of two mutually repulsive species is quenched to a temperature below its critical point of miscibility, the effect of the enthalpic repulsive force prevails upon the entropic tendency to mix, so that the system eventually separates into two coexisting phases. We have developed a highly non linear model, in close analogy with the linear theory of Cahn and Hilliard, where a generalized free energy is defined in terms of two parameters, psi and a, the first describing the equilibrium composition of the two phases and the second denoting a characteristic length scale that is inversely proportional to the equilibrium surface tension. The linear stability analysis predicts that any perturbation of the initial mixture composition with wave number k smaller than sqrt(2 psi / a) will grow exponentially in time, with a maximum growth corresponding to k_max = sqrt (psi/a). A numerical solution of the equation shows that nonlinear effects saturate the exponential growth, and that the concentration distribution tends to a steady state, periodic profile with wavelength lambda = 2 pi a / sqrt(psi) corresponding to the fastest growing mode of the linear regime. The main result of our theoretical model is that this steady state does not depend on the form of thye initial perturbation to the homogeneous composition profile