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
