Solidification Science and Engineering Practice
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A Simple and Useful Model Dendritic growth occurs either in "columnar" or "equiaxed" form (Figure 1), in which a single dendrite with many dendrite arms ultimately grows to become a cast "grain." The simplest model ! describing this solidification is often j termed the "Scheil equation," which is a I simple solute mass balance as described below. The model is based on a number of simplifying assumptions, including local equilibrium, except with no solid diffusion, macroscopic mass transport, and no effect of interface curvature on melting point.1 Figure 2 shows a solidifying front (a) and the temperature distribution across that front (b). As a result of the assumption of local equilibrium, Figure 2c, the liquid composition within the liquidsolid zone is directly related to the curve in Figure 2b by the liquidus line of the appropriate phase diagram. With this solidification model, we can carry out the mass balance, on a "volume element" that is on the order of a few dendrite arm spacings. This mass balance constitutes the basic Scheil relation. The relation provides fraction solid versus position in the liquid-zone, as in the curve in Figure 2d. It also gives the solid composition forming at any time during solidification (Figure 3c). Because no diffusion occurs in the solid, the solid composition at each interface position becomes the final solute distribution across a dendrite arm (i.e., the microsegregation; see Figure 3a). This basic model has proven over the years to be a most useful engineering tool, both for calculating microsegregation and for understanding the nature of the liquid-solid zone in a solidifying
LIQUID
SOLID a cold chill wall: (a) columnar solidification,
MRS BULLETIN/MAY 1998
Solidification Science and Engineering Practice
Equiaxed Dendritic
Solid SM
Liquid Liquid
Distance
Liquid g
8
Liquid
ca
XE
XE Distance
XL
Distance
Figure 2. Model of equiaxed solidification: (a) dendrite structure; (b) temperature versus distance; (c) liquid composition versus distance; and (d) fraction solid versus distance.1 JL andJE are the liquidus and eutectic temperatures respectively; CE and Co are eutectic and overall compositions; XE and XLare locations of the eutectic isotherm and the liquidus isotherm, respectively.
J
CE
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