SERPEnT

Surface Extrapolation by Reverse-Plotting of Energy Trajectories

I. Intro to Equilibrium Crystal Shapes and Wulff Constructions

     Wulff’s classical theory is a (static) energy minimization algorithm which yields the lowest energy crystal shape defined relative to an outer surface energy shape which is governed by reconstructed atomic lattice surface energies and unique to a given chemical composition. The surface energy at angle θ , γ (θ ) , is related to the expected crystal shape by the following equation:
Geometric construction of the Wulff shape based on anisotropic surface energy shape.
(The Wulff Shape)
where θu is a unit vector in the θ direction, Rd is the real domain of d dimensions containing all vectors x, and Sd−1 refers to a surface in polar (d=2) or spherical (d=3) coordinates. Wulff shapes represent the minimal surface energy orientation for a crystal of a given volume, or equilibrium crystal shape (ECS). The Wulff shape is the convex inner shape bounded by all tangents to an outer surface energy shape (SES). While a single, convex inner ECS is implied by a given SES, there are an infinite number of SES shapes that can correspond to any ECS shape, i.e. the Wulff construction represents an irreflexive geometric set relation.

To the right, a 3D mapping of the ECS of a cube and a corresponding SES shape. The outer SES shape shape is ‘false-colored’ according to the facet orientation energy (red being higher energy, and blue being lower energy facet orientations), and displayed with transparency so that the inner ECS shape can be seen within. The blue funnel shapes pointing inwards towards the facets indicate minima in the SES corresponding to facets in the ECS, and the red regions indicate high-energy orientations in the SES where facets are excluded from forming in the ECS.
Cube ECS (left) and SES (right)
This SES demonstrates just one of an infinite number of possible SES shape mappings to the given cube ECS shape, designed for clarity in demonstrating the geometric relationship between SES/ECS and for visual appeal.

See "Characterization, Simulation, and Modeling of Multiscale Directed-Assembly Systems" (p20-23) for more detail on metering of crystal shapes.



I. Intro to Equilibrium Crystal Shapes and Wulff Constructions
II. Functionalization and Application of the Wulff Constructions
III. SERPEnT plots and Applications
IV. Gallery





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