This talk will discuss the analytical and computational challenges involved in using the phase-field method as a "quantitative" tool to simulate solidification patterns in parameter ranges that match experiments. Recent results for dendritic growth in pure materials will be surveyed. Moreover, a new phase-field formulation of alloy solidification will be presented that makes it possible to accurately predict both the interface evolution and the history-dependent microsegregation pattern of solute in the solid that is the crucial determinant of materials properties.
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