Mathematical modeling of solidification paths in ternary alloys: Limiting cases of solute redistribution
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ON processing is used in a variety of fabrication methods such as casting, welding, laser surface treatment, and crystal growth. In many cases, products prepared from these processes are used in the as-solidified condition. In such applications, the mechanical properties and corrosion performance of the component are strongly influenced by the distribution of alloying elements and relative fraction of secondary constituents that form during the solidification process. Thus, it is often useful to make quantitative estimates of the resultant solidification microstructure as a means for controlling the ultimate component performance. Many models have been developed in the solidification literature on solute redistribution and microstructural development in binary alloys that are capable of accounting for factors such as dendrite tip undercooling, coarsening, and back diffusion, and several review articles have been published on the subject.[1–4] While these models are quite useful, it is worth noting that most all conditions of solute redistribution in binary alloys must fall within two very simple cases: the lever law and so-called nonequilibrium Scheil equation.[5] The first condition assumes that solute diffusion is infinitely fast in both the liquid and solid phases, equilibrium is maintained at the solid/liquid interface, and there is no undercooling during nucleation or growth. The latter condition invokes the same assumptions, except that solute diffusion is negligible in the solid. The relation between liquid composition and fraction liquid for equilibrium solidification is given simply by CL ¼
Co ð1 # kÞf L 1 k
[1]
J.N. DuPONT, Associate Professor, is with the Department of Materials Science & Engineering, Lehigh University, Bethlehem, PA 18015-3195. Contact e-mail: [email protected] Manuscript submitted October 7, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS A
where CL is the liquid composition at any value of fraction liquid, fL; Co is the nominal alloy composition; and k is the equilibrium distribution coefficient, defined as the ratio of solid to liquid compositions and assumed constant in Eq. [1]. The corresponding relation between CL and fL for the Scheil condition is given by ðk#1Þ
CL ¼ Co f L
[2]
These two limiting cases of solute redistribution in binary alloys are quite useful as they allow the possible ranges of microstructural development to be easily bound. As an example, consider a binary eutectic A-B system shown in Figure 1, which exhibits linear solidus and liquidus lines, a k value (for solute element B) of 0.2, and a eutectic composition of 20 wt pct B. Figure 2 shows the variation in liquid composition during solidification under equilibrium (Eq. [1]) and nonequilibrium (Eq. [2]) conditions for two alloys: one below the maximum solid solubility with Co 5 2 wt pct B (Figure 2(a)) and one above the maximum solid solubility with Co 5 5 wt pct B (Figure 2(b)). For the 2 wt pct B alloy, the solidification conditions under each extreme are quite different. Under equilibrium conditions, the liquid compo
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