Solidification cellular

K. Kassner, J.-M. Debierre, B. Billia, N. Noel, H. Jamgotchian. Cellular structures in three-dimensional directional solidification simulation and analysis. Phys Rev E 57 2849, 1998. 

L. Biihler, S. H. Davis. Flow induced changes of the morphological stability in directional solidification localized morphologies. J Crystal Growth 756 629, 1998 Y.-J. Chen, S. H. Davis, Directional solidification of a binary alloy into a cellular convective flow (unpublished). Applied Math Technical Report No. 9708, Northwestern University, Evanston, IL 60208. 

Figure 2. Photographs of cellular and dendritic structures in a thin-film solidification experiment of an organic alloy (succinonitrile-acetone) reported by Ref. 6.

The presentation in this paper concentrates on the use of large-scale numerical simulation in unraveling these questions for models of two-dimensional directional solidification in an imposed temperature gradient. The simplest models for transport and interfacial physics in these processes are presented in Section 2 along with a summary of the analytical results for the onset of the cellular instability. The finite-element analyses used in the numerical calculations are described in Section 3. Steady-state and time-dependent results for shallow cell near the onset of the instability are presented in Section 4. The issue of the presence of a fundamental mechanism for wavelength selection for deep cells is discussed in Section 5 in the context of calculations with varying spatial wavelength. 

CELLULAR AND DENDRITIC SOLIDIFICATION 22.2.1 Formation of Cells and Dendrites... 

Connection between Transport Processes and Solid Microstructure. The formation of cellular and dendritic patterns in the microstructure of binary crystals grown by directional solidification results from interactions of the temperature and concentration fields with the shape of the melt-crystal interface. Tiller et al. (21) first described the mechanism for constitutional supercooling or the microscale instability of a planar melt-crystal interface toward the formation of cells and dendrites. They described a simple system with a constant-temperature gradient G (in Kelvins per centimeter) and a melt that moves only to account for the solidification rate Vg. If the bulk composition of solute is c0 and the solidification is at steady state, then the exponential diffusion layer forms in front of the interface. The elevated concentration (assuming k < 1) in this layer corresponds to the melt that solidifies at a lower temperature, which is given by the phase diagram (Figure 5) as... 

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