Solute redistribution in cellular solidification
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Mathematical solutions and experimental results are presented which describe solute r e distribution during cellular or dendritic solidification at high G / R (thermal gradient divided by growth rate) of binary and t e r n a r y alloys. The solutions assume negligible constitutional supercooling in the vicinity of the growing cells or dendrites and negligible effects of c u r v a ture and interface kinetics. Data are in the form of measurements of tip temperature and composition, and, for the t e r n a r y system, temperature at which two-phase cells or dendrites start to form. In all cases agreement between experiment and theory is good.
IN a s e r i e s
of papers in recent years, Flemings and c o - w o r k e r s have used a simple model to describe solute redistribution in dendritic solidification of binary and t e r n a r y alloys, t-4 This model is used here to describe cellular or dendritic solidification at high G / R (thermal gradient divided by growth rate). The basic assumptions of the model as employed here are: 1) The solid-liquid interface is at equilibrium. 2) The effect of radius of curvature on the melting point is negligible. 3) Solid state diffusion is negligible. 4) No convection occurs either in the bulk liquid or in intercellular spaces. 5) Solidification is unidirectional with planar isotherms. 6) There is negligible constitutional supercooling in intercellular regions; that is, t r a n s v e r s e solute gradients a r e negligibly small so isoconcentrates are e s sentially planar. Another way of stating this last a s sumption is that it is assumed the cells are sufficiently thin and close together that their diffusion fields o v e r lap to great extent at the given growth rate.
aCL
ax
G -
(for x E < x < x t )
m
[1]
where m is the slope of the liquidus. Solute diffuses down this concentration gradient and out in front of the cell tips. The isoconcentrate at the cell tips is C t . At steady state, the requirement of no solute accumulation within the whole liquid-solid region now gives
c0cLh
[2]
R ( C t - Co) = - D \ ~ ]
where R is the growth velocity and D, the liquid diffusion coefficient. Combining Eqs. [1] and [2], we obtain an expression originally derived by Bower e t a l s (C t _ Co ) _ -mR DG
[3]
We can now define a p a r a m e t e r kt, an effective p a r t i tion ratio for the cell tips, kCt
kt = Co
[4]
where k is the equilibrium partition ratio. Then, BINARY ALLOYS 'When cells or dendrites grow into a binary alloy melt, the intercellular or interdendritic liquid becomes increasingly enriched in solute with distance back from the tips, as shown in Fig. 1. For a simple eutectic system, assuming solid state diffusion to be negligible, the maximum composition reached in the intercellular r e gion is always the eutectic composition CE . This is at the eutectlc isotherm, x = x E . T h e liquid composition CL at x along the growth axis in a fixed co-ordinate system decreases to Ct at the dendrite tips, x = x t , and finally to the bulk liquid composition Co at a distance far
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