Theory of unstable growth

Abstract

A long-time universal fixed point for domain growth in a general nonconserved time-dependent Ginzburg-Landau model is established. Both the scaling function F(x) and the growth law or scaling length L(t) associated with this fixed point are shown to have universal features. The scaling function depends only on the spatial dimensionality, not on the form of the degenerate double-well potential, the lattice or continuum spatial structure, or on the initial conditions. The growth law, measured in units of the equilibrium interfacial width ξ, is found to have a universal amplitude multiplying the expected ${\mathit{t}}^{1/2}$ curvature drive time dependence. The universal amplitude in L(t) is connected to the large-distance behavior of the scaling function through a nonlinear eigenvalue problem. For intermediate distances, ξ≤R≤L, the scaling law obeys Porod’s law, F=1-α‖R‖/L, with α= √2/π(d-1) , where d is the dimensionality of the system. The theory developed here is accurate for all times after an initial quench from a completely disordered state to a temperature well below the critical temperature. Comparisons with direct numerical simulations show excellent agreement at early and intermediate times. For later times various features predicted by the theory are in very good agreement with the simulations. It appears, however, that the selection process determining the amplitude of L(t) is rather sensitive to finite-size effects, and a direct comparison between theory and simulation on this point requires simulations on much larger systems.