The ratio k = Cs/CL of the solute concentrations in the solid and liquid at a solidifying interface. It is the ratio of the solidus and liquidus compositions, and a solute with k < 1 is rejected into the liquid and causes segregation.
Japanese name: 平衡分配係数
Equilibrium partition coefficient appears in 3 practice questions across 1 topic, including Solidification and cast structures.
The equilibrium partition coefficient k is the ratio Cs/CL of the compositions given by the solidus and liquidus at a given temperature. In systems where both the liquidus and the solidus fall toward the solute-rich side, the solidus lies on the solute-poor side, so k < 1; it is about 0.17 for Al-Cu and about 0.2 for δ iron in Fe-C. Conversely, in systems where adding solute raises the melting point, k > 1. If the lines of the phase diagram are approximated as straight, k can be treated as nearly independent of temperature, and the freezing range can be written using k as ΔT₀ = mC₀(1 − k)/k. The farther k is from 1, the larger the difference in composition between solid and liquid and the wider the freezing range.
In an alloy with k < 1, the first solid to form has a concentration of kC₀, lower than the alloy average, and the excess solute is rejected from the interface into the liquid. Because diffusion in the solid is slow, the dendrite cores stay lean, and the outer parts that solidify from the enriched liquid have higher concentrations. This creates uneven concentration between the dendrite cores and the interdendritic regions (microsegregation), and low-melting eutectics and compounds collect in the parts that solidify last. Solute piled up ahead of the interface also causes constitutional supercooling, which changes the shape of the interface.
Differences in k are also used to purify materials. If only part of a rod is melted and the molten zone is moved slowly in one direction, impurities with k < 1 are pushed into the liquid, carried along with the zone and collected at the end of the rod. Repeating this, known as zone refining (zone melting), has been used to purify germanium and silicon for semiconductors. Because k differs from impurity to impurity, impurities with k close to 1 are harder to remove. The change in dopant concentration along the length of a crystal pulled from the melt is a result of the same partitioning.
Answer: B. The solid can take up only about 0.17 times the solute in the liquid, so solute is rejected into the liquid ahead of the interface and the regions that solidify last are richest in solute
k is the ratio of the solidus and liquidus compositions at the same temperature on the phase diagram, and it is less than 1 in systems where both the liquidus and the solidus fall toward the solute-rich side. The first solid to form in an Al-4%Cu alloy contains only about 0.17 × 4 = 0.68% Cu, and the excess solute is rejected into the liquid, which gradually becomes richer, so the dendrite cores are lean and the interdendritic regions and grain boundaries that solidify last are rich. Regions that solidify first being richest in solute is the case k > 1. k is not a ratio of diffusion coefficients, and because diffusion in the solid is slow, the concentration differences do not vanish by themselves after solidification; heat treatment is needed to homogenize them. The farther k is from 1, the larger the segregation, and zone refining makes use of this property.
Answer: C. About 7.8%
Writing the remaining liquid fraction as fL = 1 − fs, 33.2 = 4.0 × fL^(−0.83), so fL = (33.2/4.0)^(−1/0.83) = 8.3^(−1.20) ≈ 0.078, about 7.8%. In equilibrium solidification, 4.0 wt%Cu is below the α solubility limit of 5.65 wt%, so the alloy is all α at the end of solidification and no eutectic forms. 0% is the mistake of applying this equilibrium reasoning to real solidification, where diffusion in the solid is slow. About 12% is 4.0/33.2, which assumes that all the copper in the alloy collects in liquid of eutectic composition and ignores the copper left in the solid. Real castings likewise show eutectic containing Al₂Cu between the dendrites even at compositions within the solubility limit, so the solution treatment temperature is chosen below the melting point of this eutectic.
The equilibrium partition coefficient is the ratio Cs/CL when equilibrium holds at the interface. In real solidification a solute-rich layer forms ahead of the interface, so the ratio of the solid concentration to the concentration in the bulk liquid (the effective partition coefficient) approaches 1 as the growth rate increases.
The solid takes up more solute than the liquid, so the parts that solidify first are richest in solute. The liquid concentration falls as solidification proceeds, giving the reverse segregation with rich dendrite cores and lean interdendritic regions.
In systems where the solidus and liquidus are not straight it changes with temperature. However, when the freezing range is narrow or for estimates it is often treated as a constant, and the Scheil equation is also derived with constant k.
The Scheil equation is derived by assuming that solute does not diffuse at all in the solid once it has formed, that the liquid is completely mixed and uniform through convection and the like, and that at the solid–liquid interface solute always partitions in the ratio of the equilibrium partition coefficient k.
Which is a correct assumption behind the Scheil equation Cs…Scheil equation Dendrite Lever rule Chvorinov's rule Constitutional supercooling Dendrite arm spacing Eutectic modification Macrosegregation