Phase-field study of three-dimensional steady-state growth shapes in directional solidification

Abstract

We use a quantitative phase-field approach to study directional solidification in various three-dimensional geometries for realistic parameters of a transparent binary alloy. The geometries are designed to study the steady-state growth of spatially extended hexagonal arrays, linear arrays in thin samples, and axisymmetric shapes constrained in a tube. As a basis to address issues of dynamical pattern selection, the phase-field simulations are specifically geared to identify ranges of primary spacings for the formation of the classically observed "fingers" (deep cells) with blunt tips and "needles" with parabolic tips. Three distinct growth regimes are identified that include a low-velocity regime with only fingers forming, a second intermediate-velocity regime characterized by coexistence of fingers and needles that exist on separate branches of steady-state growth solutions for small and large spacings, respectively, and a third high-velocity regime where those two branches merge into a single one. Along the latter, the growth shape changes continuously from fingerlike to needlelike with increasing spacing. These regimes are strongly influenced by crystalline anisotropy with the third regime extending to lower velocity for larger anisotropy. Remarkably, however, steady-state shapes and tip undercoolings are only weakly dependent on the growth geometry. Those results are used to test existing theories of directional finger growth as well as to interpret the hysteretic nature of the cell-to-dendrite transition.

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

“…This correction was successfully realized by introducing an additional diffusion flux called the antitrapping current into the diffusion equation [10]. The quantitative model with the antitrapping current has been extended to deal with more general cases [12][13][14] and such models are increasingly utilized for investigations of solidification microstructures [15][16][17][18][19][20][21][22][23][24].…”

Analysis of thermodiffusive cellular instabilities in continuum combustion fronts

“…Tip splitting as a microstructure selection mechanism is also well known in the context of crystal growth [45][46][47][48][49][50][51][52][53]. Figure 9 shows the evolution of a flame front with the same model parameters and system size as in Fig.…”

Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy

“…The simulation results of quantitative models rapidly converge to unique values (the solutions of the free-boundary problem) with decreasing W , indicating good numerical performance [11,14]. Quantitative phase-field models are increasingly utilized in studies on the formation processes of solidification microstructures [18][19][20][21][22][23].…”