Abstract
This paper seeks a vertex based approach to faceted anisotropy in phase field modelling of crystal growth. We examine Wulff shapes and the connection they have with phase field formulations. On inspecting current approaches to modelling facets within phase field we observe that there are two distinct approaches: one implements the faceting purely in the kinetic parameter thus avoiding the complications of taking gradients of discontinuous functions; while the second is based upon regularisation of the anisotropy function within the free energy function. Armed with our new insight into the operation of anisotropy within phase field we refine the second of these and advocate a vertex based approach to facet anisotropy modelling. Results include regular and irregular morphologies and hill-valley growth. We also present high undercooling effects on faceting, which can cause breaks in facets and, in the case of irregular shapes, distortion of the underlying Wulff shape.
Highlights
Since the establishment of a thermodynamic basis [1] and the pio neering efforts of [2], the application of phase field to crystal growth has become almost mandatory over the last decades
We have presented a very general method for the phase-field simu lation of arbitrary faceted crystals in both 2- and 3D, illustrated for a range of both regular and irregular morphologies
The method lends itself both to a relatively straightforward geometrical interpretation of the simulated morphology, via the Wulff shape, and is of comparable ease to implement for arbitrary vertices as, for example, [25,26], but offers a more canonical approach
Summary
Since the establishment of a thermodynamic basis [1] and the pio neering efforts of [2], the application of phase field to crystal growth has become almost mandatory over the last decades. Developing phase-field models for sharp corners and facets is challenging because sharp corners are necessarily associated with negative stiffness and facets with a Wulff shape with missing orienta tions (A′ (n) undefined) This leads to modifications of the model by rounding vertices, [17], and convexification, [20], and by adding higher order terms, [21]. It follows that a ca nonical formulation is the most natural starting point of the different formulations, leading to the most straightforward geometric interpre tation It must be modified for direct implementation, the end result being a formally convex Wulff shape with approximately sharp corners and near flat edges (in 2D). The following subsections illustrate and explicate the concept of a Wulff shape and its relation to the anisotropy - Section 2.1, and how this mathematical object relates to a phase field formulation - Section 2.2
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