Scheil–Gulliver Segregation

Freeze an alloy and the first solid to form is not the alloy's composition. It is the solidus composition at that temperature, which for a solute that lowers the melting point is much leaner. The solute the solid refuses to accept is rejected into the liquid, which becomes richer; the liquid being richer means the next solid to form is richer too; and the process runs away, so that the last liquid to freeze is far more concentrated than anything the phase diagram says the alloy contains. Gulliver set the argument out in 1913 and Scheil put it in its modern form in 1942.

The derivation is a mass balance on the solute rejected at the moving interface, and the whole content is in two assumptions. The liquid is perfectly mixed — convection and diffusion keep it uniform right up to the interface — and there is NO diffusion in the solid at all: every layer is frozen in composition the instant it forms. Integrating gives C_S = kC₀(1 − f_S)^(k−1), with k the equilibrium partition coefficient C_S/C_L read off the tie line.

Set this against the lever rule and the contrast is the whole teaching point of the pair. Both start from the same tie line and the same mass balance. They differ in exactly one assumption: the lever rule allows diffusion in the solid to keep all the already-frozen material at equilibrium, and Scheil allows none. Everything else is identical. Work a real case — aluminium with 4.5 % copper, on the tie line running from a solidus of 1.5 % Cu to a liquidus of 12 % Cu, so k = 0.125 — and the lever rule says the casting is 71.4 % solid at that temperature while Scheil says 67.4 %. Four percentage points of extra liquid are still unfrozen, and they are carrying all the copper the solid refused to take.

That residue is what appears as interdendritic eutectic in the finished casting, in an alloy the phase diagram says should be single-phase. It is also why a cast alloy can begin to melt below its nominal solidus — incipient melting — which is what ruins a solution treatment when the furnace is set from the equilibrium diagram rather than the real one. And the composition gradient across each dendrite arm is coring, visible on an etched as-cast section as banding inside each grain.

The equation's most famous feature is a failure. As f_S approaches 1 it predicts C_S going to infinity: an infinitely concentrated last drop. That cannot happen, and what actually happens is that the enriched liquid reaches the eutectic composition and freezes as eutectic at the eutectic temperature. So a Scheil calculation is always TRUNCATED at that point, and the solid fraction at which it truncates is the fraction of primary phase — everything remaining freezes as eutectic. Running the equation at the eutectic composition to find that truncation point is one of its most useful applications, because it predicts how much eutectic a casting will contain.

Reality sits between Scheil and the lever rule. Substitutional solutes in a fast-cooled casting are close to Scheil, because they barely diffuse in the solid on the timescale of freezing. Interstitials — carbon and nitrogen in steel — are close to equilibrium, because they diffuse fast enough to homogenise as the front advances. The Brody–Flemings and Clyne–Kurz modifications add a back-diffusion parameter that interpolates between the two extremes, and they are the honest next model when a measured microprobe profile comes out flatter than Scheil predicts, which it usually does a little.

The cure for all of it is a homogenisation soak, and Chvorinov's rule connects to that directly. Homogenisation time scales as the square of the dendrite arm spacing, and the arm spacing scales with the freezing time — so a section that froze quickly is very much cheaper to homogenise than one that froze slowly. That is one more reason chills and thin sections are worth designing for, quite apart from the shrinkage porosity they prevent.