Long-time-scale dynamics observed in directional solidification of a binary alloy

Abstract

Asymptotic analysis and finite-element simulations are used to explore the nonlinear dynamics present in cellular interface patterns that arise in thin-film directional solidification, as caused by the presence of a codimension-two bifurcation at a specific value of the spatial wavelength for the cells. For certain parameter sets, the quadratic nonlinearities that dominate the dynamics because of 2:1 spatial resonance give rise to secondary steady-state bifurcation and to time-periodic states with periods that scale as 5 ' ', where 5 measures the distance from the parameter values for the codimension-two state. The structure of this dynamics is explored in closed form for a modified Swift-Hohenberg equation and for the two-sided, solutal model for directional solidification. The asymptotic results for directional solidification are verified by finite-element numerical simulations for small amplitude cells.

Phase-field simulation during directional solidification of a binary alloy using adaptive finite element method

“…The average primary arm spacing for V ¼ 0:00015625 m=s is considerably small comparing to the value predicted by the scaling low. This result also agrees well with the previous study [25,26]. As the concentration profile schematically illustrated in Fig.…”

“…The wavelength of each cell varies during the simulation, and the interface never reaches steady-state growth. These phenomenon observed here agree well with those of the previous studies [25,26]. The detail studies of morphological evolution around this lower MS-loop boundary have been performed experimentally and numerically in Ref.…”

“…The onset of instability may, therefore, lie between V ¼ 0:00015625 and 0.000078125 m/s. Regarding V ¼ 0:00015625 m=s, the long time scale morphological dynamics is observed, because it is near the onset of instability from planner to cellular structures [16,25,26]. This problem needs very long time.…”

Phase-Field Study of the Cellular Bifurcation in Dilute Binary Alloys

“…It should be mentioned that several oscillatory instabilities that occur in directional solidification have been linked to the occurrence of a 1:2 spatial resonance, [30,31,32] that is, to a resonance between the fundamental mode of wavenumber q and its first harmonic 2q. However, in the simulation of the shallow cells shown in Figure 5, the 2q mode is still strongly stable (that is, it has a large negative growth rate).…”

An asymptotic analysis for directional solidification of a diffusion-dominated binary system

“…Since there is a close relationship between growth conditions and the microstructure and properties of solidified materials, there has been an extensive amount of investigation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] of sharp-interface, Stefan-type problems and directional solidification configurations. Exact solutions to these classes of problems are generally restricted to unbounded domains and are subject to limitations on the boundary conditions.…”