Interface pinning and slow ordering kinetics on infinitely ramified fractal structures

Abstract

We investigate the time-dependent Ginzburg-Landau (TDGL) equation for a nonconserved order parameter on an infinitely ramified (deterministic) fractal lattice employing two alternative methods: the auxiliary field approach and a numerical method of integration of the equations of evolution. In the first case the domain size evolves with time as $L(t)\ensuremath{\sim}{t}^{{1/d}_{w}}$, where ${d}_{w}$ is the anomalous random-walk exponent associated with the fractal and differs from the normal value $2$, which characterizes all Euclidean lattices. Such a power-law growth is identical to the one observed in the study of the spherical model on the same lattice, but fails to describe the asymptotic behavior of the numerical solutions of the TDGL equation for a scalar order parameter. In fact, the simulations performed on a two dimensional Sierpinski carpet indicate that, after an initial stage dominated by a curvature reduction mechanism in the manner of Allen and Cahn [Acta. Metall. 27, 1085 (1979)], the system enters in a regime where the domain walls between competing phases are pinned by lattice defects. The lack of translational invariance determines a rough free-energy landscape, the existence of many metastable minima, and the suppression of the marginally stable modes, which in translationally invariant systems lead to power-law growth and self-similar patterns. On fractal structures, as the temperature vanishes the evolution is frozen since only thermally activated processes can sustain the growth of pinned domains.