Growth of Domains and Scaling in the Late Stages of Phase Separation and Diffusion-Controlled Ordering Phenomena

Abstract

These lectures consider the kinetics of phase changes, induced by a sudden change of external thermodynamic parameters. E.g., we treat a system with a second-order transition at a critical temperature T_c (Fig. 1, left part). For T₀ > T_c the system is disordered, while for T < T_c there is an order parameter ± ψ (implying one-component orderings, e.g., an Ising model; later we discuss generalizations). We consider a “quenching experiment”: The system is brought from an initially disordered state at T₀ to a state at T where in equilibrium the system should be orderedl. Since no sign of ψ is preferred, the system starts forming locally ordered regions of either sign, separated by domain walls. Due to the unfavorable interface free energy cost, this situation is not thermodynamically stable — there is a driving force to reduce this free energy. Thus the random motion of walls, induced by statistical fluctuations, leads to a growth of a characteristic length L(t) of the ordered regions with the time t after the sudden quench performed at t = 0. Typically, one expects L(t)^∝t^a for large t, and the excess internal energy ΔE(t)^∝t-a a’ : a, a’ are the characteristic exponents of interest here. Sometimes even slower growth [L(t)∝lnt] might occur, see below.