Emergent Parabolic Scaling of Nano-Faceting Crystal Growth

The dynamics of slightly undercooled crystal-melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment-detachment limited kinetics finds expression through a certain singularly perturbed, hyperbolic-parabolic, geometric partial differential equation.

Among its solutions, we discover a remarkable family of 1D convex- and concave- translating fronts whose fixed asymptotic angles deviate from the thermodynamically expected Wulff angles in direct proportion to the degree of undercooling: a non equilibrium (thermokinetic) effect.

We also present a novel geometric matched-asymptotic analysis that demonstrates that the slow evolution of the large-scale features of 1D solutions $\mathcal{I}$ are captured by a Wulff-faceted interface $\mathcal{A}$ evolving under an intrinsic facet dynamics. This emergent dynamics possesses a Peclet length $L_\text{p}$ below which a spatio-temporal symmetry of parabolic type appears. We thereby theoretically predict, and numerically verify, that within the sub-Peclet regime the universal scaling law $\mathcal{L} \sim t^{1/2} $ governs the time $t$ evolution of the characteristic length $\mathcal{L}$ of the interface $\mathcal{I}$.

Related Article: Stephen J. Watson, "Emergent Parabolic Scaling of Nano-Faceting Crystal Growth", Proceedings of the Royal Society A, Vol. 471 (Issue 2174) , DOI: 10.1098/rspa.2014.0560