The interface that separates a crystalline solid phase from another phase usually displays anisotropic properties associated with the underlying crystalline symmetry of the solid phase. This anisotropy is characterized by the dependence of interfacial properties, such as surface tension or attachment kinetic coefficients, on the local orientation of the interface. For the understanding and eventual control of many materials processing techniques, it is frequently necessary to take into account the effects of such interface anisotropy. An example is the preferred growth directions that are observed during dendritic solidification via free growth from a single-component supercooled material, or during the directional solidification of a binary alloy. In general, the crystalline anisotropy affects the shape of the interface that separates the various phases, and so can play a fundamental role in establishing the degree of solute inhomogeneity, grain size, and defect structure that determine whether a given product is accepted or rejected for commercial use. The study of anisotropic interfaces is an area in which NIST materials scientists have made a number of important fundamental contributions.
Diffuse interface models of phase transitions, wherein the region separating distinct phases is represented by a zone of finite thickness rather than a mathematically sharp interface of zero width, have also been an areas of active NIST research recently; models of this type in materials science were developed by Cahn and Hilliard in the late 1960's. Diffuse interface theories that effectively incorporate the effects of interface anisotropy have been developed recently, but to date they have been limited to cases of mild anisotropy. For sharp interface models, it is well known that energy-minimizing shapes can exhibit corners or edges if the surface tension anisotropy is large enough; by forming the edge or corner, the overall surface energy is reduced by omitting certain high-energy orientations all together. A diffuse interface model of a highly-anisotropic system then faces a serious complication: the model must reconcile the smooth variation of the diffuse interface along the direction normal to the interface with the corner or edge singularities that occur in the transverse directions along the interface.
In the early 1970's Cahn and Hoffman introduced a useful construction, the
so-called 
-vector, to characterize equilibrium shapes and missing
orientations of sharp interfaces. We have recently shown how the 
-vector
can be generalized to describe in a similar fashion the treatment of
anisotropic diffuse interface models as well. The generalized 
-vector is
useful in simplifying the construction of numerical algorithms for anisotropic
diffuse interface models, and is also quite useful in asymptotic analyses of
diffuse interface theories that relate them to sharp interface theories. In
particular, a generalized 
-vector has also been found for
multiple-order-parameter models of atomic ordering in solid-solid phase
transitions, where the anisotropy of interphase and antiphase boundaries has
recently been shown to lead to missing orientations in kinetic growth shapes.
In future work, the study of anisotropy in multiple-order parameter models will
be broadened to include crystals with symmetries other than face-centered-cubic
structure, and will incorporate more c omprehensive thermodynamic descriptions
in order to better model realistic phase diagrams.