Regularized anisotropic motion-by-curvature in phase-field theory: Interface phase separation of crystal surfaces

The kinetic equation for anisotropic motion-by-curvature is ill posed when the surface energy is strongly anisotropic. In this case, corners or edges are present on the Wulff shape, which span a range of missing orientations. In the sharp-interface problem the surface energy is augmented with a curvature-dependent term that rounds the corners and regularizes the dynamic equations. This introduces a new length scale in the problem, the corner size. In phase-field theory, a diffuse description of the interface is adopted. In this context, an approximation of the Willmore energy can be added to the phase-field energy so as to regularize the model. In this paper, we discuss the convergence of the Allen-Cahn version of the regularized phase-field model toward the sharp-interface theory for strongly anisotropic motion-by-curvature in three dimensions. Corners at equilibrium are also compared to theory for different corner sizes. Then we investigate the dynamics of the faceting instability, when initially unstable surfaces decompose into stable facets. For crystal surfaces with trigonal symmetry, we find the following scaling law L∼t^{1/3}, for the growth in time t of a characteristic morphological length scale L, and coarsening is found to proceed by either edge contraction or cube removal, as in the sharp-interface problem. Finally, we study nucleation of crystal surfaces in a two-phase system, as for a terrace-and-step surface. We find that, as compared with saddle-point nucleation, ridge crossing is dynamically favored. However, the induced nucleation mechanism, when a facet induces at its wake formation of additional facets, is not evidenced with a type-A dynamics.

Corners in phase-field theory

Phase-field models for strongly anisotropic surface energy need to be regularized to remove the ill posedness of the dynamic equations. Regularization introduces a new length scale, the corner size, also called the bending length. For large corner size, with respect to interface thickness, the phase-field method is known to converge asymptotically toward the sharp-interface theory when the appropriate approximation of the Willmore energy is used. In this work we study the opposite limit, i.e., corner size smaller than the interface width, and show that the shape of corners, at equilibrium, differs from the sharp-interface picture. However, we find that the phase transition at the interface is preserved and presents the same properties as the classical problem.

A regularized phase-field model for faceting in a kinetically controlled crystal growth

At equilibrium, the shape of a strongly anisotropic crystal exhibits corners when for some orientations the surface stiffness is negative. In the sharp-interface problem, the surface free energy is traditionally augmented with a curvature-dependent term in order to round the corners and regularize the dynamic equations that describe the motion of such interfaces. In this paper, we adopt a diffuse interface description and present a phase-field model for strongly anisotropic crystals that is regularized using an approximation of the Willmore energy. The Allen-Cahn equation is employed to model kinetically controlled crystal growth. Using the method of matched asymptotic expansions, it is shown that the model converges to the sharp-interface theory proposed by Herring. Then, the stress tensor is used to derive the force acting on the diffuse interface and to examine the properties of a corner at equilibrium. Finally, the coarsening dynamics of the faceting instability during growth is investigated. Phase-field simulations reveal the existence of a parabolic regime, with the mean facet length evolving in t , with t the time, as predicted by the sharp-interface theory. A specific coarsening mechanism is observed: a hill disappears as the two neighbouring valleys merge.

Nanowire Y-junction formation during self-faceting on high-index GaAs substrates

In this contribution, we report on the observation of high-order and bi-dimensional surface mechanisms that allows the self-assembling of an alternating array of straight and bifurcated nanowires.