Consider the solidification of a bar of a binary alloy as discussed in class. The alloy possesses an ideal phase diagram, in which the liquidus and solidus are all straight lines (see Lect 9 slides). Let X denote the molar fraction of B atoms in the alloy. We now focus on the A-rich alloy (i.e., when X is small). The melting temperature of pure A is Tm. Consider the composition X = X0, which is sufficiently small such that in the case of equilibrium solidification, the system does not have a Eutectic transition. For this composition, under equilibrium conditions, the solidification of the alloy starts at T1, and ends at T3. Let k = Xs/XL over the temperature range [T1, T3], as discussed in class, for this ideal phase diagram, k is a constant over the specified temperature range.

(1) Determine the equation of the solidus and liquids lines in the phase diagram, i.e., T(Xs) and T(XL). (hint: the intersection of the lines with T axis is at X = 0 and T = Tm; the slopes can be computed using the equilibrium compositions associated with temperatures T1 and T3).

(2) For equilibrium solidification, derive the relation of solid phase fraction fs and temperature T, i.e., fs(T) with T in [T1, T3]. (hint: first use lever rule to find fs(Xs, XL, X0), then use the results obtained in (1)).

(3) For the non-equilibrium solidification discussed in class (i.e., no diffusion in solid, perfect mixing in liquid), determine the average composition of the solid Xs_ave as a function of fs (hint: Xs_ave = total amount of solutes in solid/amount of solid fs, the former can be obtained by an integration of the Xs(fs))

(4) Assuming the same relation fs(T) obtained in equilibrium solidification (1) also holds in the non-equilibrium solidification case (3). Determine the equation of the non-equilibrium branch T(Xs_ave) in the phase diagram.course name

Phase transformation, kinetics, and diffusion in solids topic name : alloy soldification