A Mean-Field Theory for Coarsening Faceted Surfaces

TL;DR

This paper develops a mean-field theoretical framework for understanding the scale-invariant length distributions during the coarsening of one-dimensional faceted surfaces, drawing parallels with Ostwald ripening and incorporating coalescence effects.

Contribution

It introduces a novel mean-field model that captures the coarsening dynamics of faceted surfaces, extending classical theories with convolution terms for coalescence.

Findings

Exponential distribution solution approximates the scale-invariant state.

Model highlights the role of convolution terms in coarsening.

Agreement with experiments is limited by uncorrelated neighbor assumption.

Abstract

A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems [1-3], but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling the work of Smoluchowski [4] and Schumann [5] on coalescence. The model is solved by the exponential distribution, but agreement with experiment is limited by the assumption that neighboring facet lengths are uncorrelated. However, the method concisely describes the essential processes operating in the scaling state, illuminates a clear path for future refinement, and offers a framework for the investigation of faceted surfaces evolving under arbitrary dynamics. [1] I. Lifshitz, V. Slezov, Soviet Physics JETP 38 (1959) 331-339. [2] I. Lifshitz, V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35-50. [3] C. Wagner, Elektrochemie 65 (1961) 581-591. [4] M. von Smoluchowski, Physikalische Zeitschrift 17 (1916) 557-571. [5] T. Schumann, J. Roy. Met. Soc. 66 (1940) 195-207.