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Emergent Parabolic Scaling of Nano-Faceting Crystal Growth

Presented by: 
S Watson University of Glasgow
Tuesday 11th August 2015 - 11:00 to 12:00
INI Seminar Room 2
Nano-faceting of material interfaces is a paradigmatic, non-equilibrium self-assembly process which arises in a wide variety of physical settings; for example, high-efficiency photo-electrochemical cells yielding solar-energy storage through hydrogen production, and enantiomer-specific heterogeneous catalysts with application to biology. 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 {\em Wulff} angles in direct proportion to the degree of undercooling: a non-equilibrium ({\em 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

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons