- Conference date: 5-11 August 2007
- Location: Park City, Utah (USA)
Crystal growth morphology results from an interplay of crystallographic anisotropy and growth kinetics, the latter consisting of interfacial processes as well as long‐range transport. Mathematical modeling of crystal growth shapes is important to our understanding of fundamental crystal growth phenomena as well as to improvement and optimization of practical processes for crystal growth. Such modeling results in a difficult free boundary problem because one must piece together solutions of partial differential equations, via boundary conditions, on a crystal surface whose location and shape are yet to be determined. Moreover, this problem is complicated because the nature of long‐range transport leads to natural instabilities of shape, so‐called morphological instabilities, on the scale of the geometric mean of a transport length and a capillary length. The resulting shapes can be cellular or dendritic but can also exhibit corners and facets related to the underlying crystallographic anisotropy. Growth subsequent to morphological instability can be modeled by means of the phase field model, which is a mesoscopic diffuse interface model that eliminates interface tracking. The phase field is an auxiliary parameter that identifies the phase; it is continuous but makes a transition over a thin region, the diffuse interface, from its constant value in a crystal to some other value in the nutrient phase. Coupled partial differential equations that govern the time evolution of the phase field and accompanying fields (such as temperature and composition) can be formulated on the basis of an entropy functional and irreversible thermodynamics. Anisotropies can also be incorporated. Examples of computed cellular and dendritic morphologies show the transition from shallow to deep cells, liquid encapsulation, dendritic sidebranching, tip splitting, coarsening, solute microsegregation and many other phenomena that have been observed experimentally. This presentation will emphasize the fundamentals of phase field theory and also introduce more recent developments such as crystal phase field theory, which is based on density functional theory and incorporates some aspects of crystallinity, as well as Lattice Boltzmann algorithms that may be used to incorporate convection in fluid phases.
- Crystal growth
- Interface diffusion
- Crystal field theory
- Partial differential equations
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