Breakdown of Dynamical Scale Invariance in the Coarsening of Fractal Clusters

TL;DR

This paper investigates the breakdown of dynamical scale invariance during the coarsening of fractal clusters, revealing multiple dynamical length scales and exponents, supported by theoretical estimates and numerical agreement.

Contribution

It introduces a new analysis of multiple dynamical length scales and exponents in fractal cluster coarsening, extending previous models and providing theoretical predictions validated numerically.

Findings

Existence of a second dynamical length scale in coarsening.

Measurement of a third dynamical exponent related to solute mass.

Numerical exponents agree with theoretical predictions.

Abstract

We extend a previous analysis [PRL {\bf 80}, 4693 (1998)] of breakdown of dynamical scale invariance in the coarsening of two-dimensional DLAs (diffusion-limited aggregates) as described by the Cahn-Hilliard equation. Existence of a second dynamical length scale, predicted earlier, is established. Having measured the "solute mass" outside the cluster versus time, we obtain a third dynamical exponent. An auxiliary problem of the dynamics of a slender bar (that acquires a dumbbell shape) is considered. A simple scenario of coarsening of fractal clusters with branching structure is suggested that employs the dumbbell dynamics results. This scenario involves two dynamical length scales: the characteristic width and length of the cluster branches. The predicted dynamical exponents depend on the (presumably invariant) fractal dimension of the cluster skeleton. In addition, a robust theoretical estimate for the third dynamical exponent is obtained. Exponents found numerically are in reasonable agreement with these predictions.