Stochastic Modeling of Grain Structure Formation in Solidification Processes
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in the modeling of solidification processes and thus is receiving increased attention.1 The microstructures which develop during the solidification of metallic alloys can be divided into four main classes depending on whether they are dendritic or eutectic, columnar or equiaxed.2 (The special cases of peritectic alloys and of nodular cast iron will not be discussed here.) In casting, columnar microstructures are usually initiated at the surface of the mold by heterogeneous nucleation, but in welding or laser remelting, they can grow epitaxially from the unmelted grains. Equiaxed microstructures are the result of the continuous nucleation of grains in the bulk of the liquid and their growth. In most solidification processes, fine equiaxed grain structures are preferred because of their improved isotropic mechanical properties. On the other hand, columnar structures are desired in directionally solidified or single-crystal turbine blades because such parts are mechanically stressed along a given direction. In other solidification processes, such as the continuous casting of steel for which there is no effective inoculant, both columnar and equiaxed microstructures might be present. The occurrence of columnar and equiaxed structures and their associated transition (the so-called "columnar-to-equiaxed transition" or CET) is strongly affected by the composition of the alloy, the inoculation conditions (purity of the melt, intentional addition of nucleating agents to the alloy), the convection in the melt, and the thermal field. Three scales can be distinguished in
a solidification process: the dimension of the process (1-100 cm), the typical spacing characterizing the dendritic/eutectic microstructure (1-100 fxm), and in between these two scales the size of the grain (0.1-10 mm). A numerical simulation of the whole process with a spatial resolution at the scale of the microstructure would require an unrealistic number of nodal points (at least 1012 in three dimensions)! Therefore, reasonable approximations must be found. One of the latest techniques for predicting microstructures in solidification processes is to use rather coarse meshes for solving, with finite-element (FE) or finite-difference (FD) methods, the continuity equations at the macroscopic scale of the process, a finer grid for the stochastic modeling of nucleation and growth of grains (mesoscale), and analytical models of solidification for the prediction of the microstructural spacings (microscale). This article will treat only the macro-aspects and meso-aspects of dendritic solidification even though the analytical models of dendrite growth, whose details can be found in References 2 and 3, are used in some of the meso-models. As will be shown, the coupling of these different scales can produce realistic computed micrographs of the dendritic grain structure in casting. Grain Structures in Castings Figure 1 presents the grain structure seen in a longitudinal section of an aluminum 4.5 wt% copper alloy that has been solidified in a mold of slightly conica
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