Quantitativeness of phase-field simulations for directional solidification of faceted silicon monograins in thin samples.

We report the results of a two-dimensional reference model for the formation of facets on the left and right side of a silicon monograin that is solidified by pulling a thin sample in a constant temperature gradient. Anisotropy functions of both the surface energy and the kinetic attachment coefficient are adapted from a recent model for free growth of silicon micrometer-sized grains [Boukellal et al., J. Cryst. Growth 522, 37 (2019)0022-024810.1016/j.jcrysgro.2019.06.005.]. More precise estimates of the physical parameters entering these functions are obtained by reanalyzing available experimental results. We show that the reference model leads to a differential equation for the shape of the solid-liquid interface. The numerical solutions of this equation give a reference law Λ(V_{f}) relating the facet length Λ to the facet normal velocity V_{f}. In parallel, phase-field simulations of the reference model are performed for two growth orientations, [001] and [011]. Facet lengths Λ obtained from simulations at different facet velocities are first extrapolated to the limit of vanishing interface width. This extrapolation is made possible by constructing a master curve common to the whole range of V_{f} values considered. The extrapolated Λ values are then compared with the ones predicted by the Λ(V_{f}) reference law. Both sets give comparable values, with an accuracy of a few percent, which confirms that the phase-field model can give quantitative results for faceted solidification of silicon.

Oscillatory-nonoscillatory transitions for inclined cellular patterns in three-dimensional directional solidification.

The oscillatory behavior of cellular patterns produced by directional solidification of a transparent alloy under microgravity conditions was recently observed to depend on the misorientation of the main crystal axis with respect to the direction of the imposed thermal gradient [Pereda et al., Phys. Rev. E 95, 012803 (2017)2470-004510.1103/PhysRevE.95.012803]. To characterize the oscillatory-nonoscillatory transition resulting from the variations of the crystal misorientation, new experiments performed in DECLIC-DSI onboard the International Space Station and phase-field simulations are analyzed and combined in the present study. Experimental results are extracted from movies showing regions that extend on both sides of a boundary between two grains with respective misorientations of roughly 3 and 7 degrees. A set of tools are developed to analyze the experimental data and the same analysis is reproduced for the numerical data. A number of points are addressed in the simulations, like the effects of the system dimensions. The oscillatory state is found to be favored by the increase of the geometrical degrees of freedom. In bulk samples, a good agreement is found between the experimental and the numerical oscillatory-nonoscillatory threshold given by the ratio of the drift time to the oscillation period at the transition. The existence and the origin of bursts of localized groups of oscillating cells within a globally nonoscillatory pattern are characterized. A qualitative description of the physical mechanism that governs the oscillatory-nonoscillatory transition is provided.

Phase-field study of crystal growth in three-dimensional capillaries: Effects of crystalline anisotropy.

Phase-field simulations are performed to explore the thermal solidification of a pure melt in three-dimensional capillaries. Motivated by our previous work for isotropic or slightly anisotropic materials, we focus here on the more general case of anisotropic materials. Different channel cross sections are compared (square, hexagonal, circular) to reveal the influence of geometry and the effects of a competition between the crystal and the channel symmetries. In particular, a compass effect toward growth directions favored by the surface energy is identified. At given undercooling and anisotropy, the simulations generally show the coexistence of several growth modes. The relative stability of these growth modes is tested by submitting them to a strong spatiotemporal noise for a short time, which reveals a subtle hierarchy between them. Similarities and differences with experimental growth modes in confined geometry are discussed qualitatively.

Crystal growth in a channel: pulsating fingers, merry-go-round patterns, and seesaw dynamics.

We perform phase-field simulations of unsteady crystal growth in a three-dimensional capillary. Motivated by the appearance of chirality-symmetry breaking periodic states in our preceding study, we here focus on more general dynamic states. Most of these are obtained in the limit of isotropic surface tension, but we test genericity by looking at a few cases with weak anisotropy. Whereas steady states are similar for all channel shapes studied so far, including channels with circular, hexagonal, quadratic, and triangular cross sections, there is a stronger dependence on the cross section for time-dependent states. Various oscillatory modes are identified and discussed, including rotating and swinging patterns as well as pulsating modes containing one, two, and four fingers, respectively.

Phase-field study of solidification in three-dimensional channels.

Three-dimensional solidification of a pure material with isotropic properties of the solid phase is studied in cylindrical capillaries of various cross sections (circular, hexagonal, and square). As the undercooling is increased, we find, depending on the width of the capillary, a number of different growth modes and dynamical behaviors, including stationary symmetric single fingers, stationary asymmetric fingers, and oscillating double and quadruple fingers. Chaotic states are also observed, some of them in unexpected parameter regions. Our simulations suggest that the bifurcation from symmetric to asymmetric fingers is supercritical. We discuss the nature of the oscillatory states, one of which is chirality breaking, and the origin of the unexpected chaotic finger. Bifurcation diagrams are given comparing three different ratios of capillary length to channel width in the hexagonal channel as well as the three different geometries.

Modelling the two-dimensional polymerization of 1,4-benzene diboronic acid on a Ag surface.

Modelling of the two-dimensional polymerization of 1,4-benzene diboronic acid molecules on the Ag(111) surface, which leads to the formation of a covalent organic framework, is reported. An estimation of free enthalpy is given that takes into account the constraints induced by the molecular adsorption on the surface. The various thermodynamic functions, enthalpies, entropies, and free enthalpies, are obtained from DFT calculations. The entropic effect of the surface plays an important role in the polymerization free energy. A germination threshold is obtained.

Growth patterns in a channel for singular surface energy: phase-field model.

We study solidification in a two-dimensional channel for faceted materials whose facets correspond to cusps in the gamma plot. The main result is the existence of three growth modes, according to the anisotropy strength: a single faceted finger at high anisotropies, two faceted fingers in the intermediate range, and an oscillating mode at low anisotropies. Simple geometrical and dynamical models are proposed to explain the nature of the observed modes. In particular, the one-finger patterns are shown to be similar to free dendrites while the two-finger patterns correspond to confined solidification fingers.

Phase-field approach for faceted solidification.

We extend the phase-field approach to model the solidification of faceted materials. Our approach consists of using an approximate gamma plot with rounded cusps that can approach arbitrarily closely the true gamma plot with sharp cusps that correspond to faceted orientations. The phase-field equations are solved in the thin-interface limit with local equilibrium at the solid-liquid interface [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The convergence of our approach is first demonstrated for equilibrium shapes. The growth of faceted needle crystals in an undercooled melt is then studied as a function of undercooling and the cusp amplitude delta for a gamma plot of the form gamma=gamma0[1+delta(/sin theta/+/cos theta/)]. The phase-field results are consistent with the scaling law Lambda approximately V(-1/2) observed experimentally, where Lambda is the facet length and V is the growth rate. In addition, the variation of V and Lambda with delta is found to be reasonably well predicted by an approximate sharp-interface analytical theory that includes capillary effects and assumes circular and parabolic forms for the front and trailing rough parts of the needle crystal, respectively.

Measuring kinetic coefficients by molecular dynamics simulation of zone melting.

Molecular dynamics simulations are performed to measure the kinetic coefficient at the solid-liquid interface in pure gold. Results are obtained for the (111), (100), and (110) orientations. Both Au(100) and Au(110) are in reasonable agreement with the law proposed for collision-limited growth. For Au(111), stacking fault domains form, as first reported by Burke, Broughton, and Gilmer [J. Chem. Phys. 89, 1030 (1988)]. The consequence on the kinetics of this interface is dramatic: the measured kinetic coefficient is three times smaller than that predicted by collision-limited growth. Finally, crystallization and melting are found to be always asymmetrical and here again the effect is much more pronounced for the (111) orientation.