Engineering & Science
Scheil-Gulliver Solidification Equation: Dendrite Coring & Non-Equilibrium Eutectic Fraction
Calculate local solid composition C_S = k * C0 * (1 - f_S)^(k - 1) during casting solidification and predict non-equilibrium interdendritic eutectic fraction f_E = (C_E / C0)^(-1 / (1 - k)).
Explain why aerospace Al-4.5%Cu castings form brittle eutectic even below the 5.65% solubility limit!
Equilibrium Lever Rule solidification assumes infinitely fast diffusion in BOTH liquid and solid—predicting that Al-4.5 wt% Cu (2024/2014 aerospace alloy family, well below the 5.65 wt% maximum equilibrium solid solubility) freezes into 100% single-phase alpha-aluminum! In real foundry castings and laser welds, solid-state solute diffusion is nearly ZERO while liquid mixing remains rapid—causing solute rejection ("coring") that enriches the remaining interdendritic liquid until it hits the eutectic composition C_E = 33.2 wt% Cu!
With partition coefficient k = C_S / C_L (= 0.17 for Cu in Al), the Scheil-Gulliver equation gives instantaneous solid composition freezing at solid fraction f_S as C_S = k * C0 * (1 - f_S)^(k - 1), and the exact non-equilibrium eutectic liquid fraction remaining when C_L reaches C_E is f_E = (C_E / C0) ^ (-1 / (1 - k))! For Al-4.5 wt% Cu (k = 0.17, C_E = 33.2%), the first dendrite core freezes at k*C0 = 0.765 wt% Cu, rises to 1.360 wt% Cu at f_S = 0.50 and 2.909 wt% Cu at f_S = 0.80, and traps f_E = 0.0900 (9.0% brittle Al-Al2Cu eutectic requiring solution heat treatment)!
| Solidification Stage (Al-4.5%Cu, k = 0.17) | Solid Fraction f_S | Freezing Solid Composition C_S | Metallurgical Consequence |
|---|---|---|---|
| Dendrite Core Nucleation | f_S = 0.00 | 0.17 * 4.5 = 0.765 wt% Cu | Solute-depleted dendrite spine |
| Half-Solidified Mush | f_S = 0.50 | 1.360 wt% Cu | Remaining liquid enriched to 8.0 wt% |
| Late Solidification | f_S = 0.80 | 2.909 wt% Cu | Liquid reaches 17.1 wt% Cu |
| Terminal Eutectic Arrest (C_E = 33.2%) | f_L = f_E = 0.0900 | 9.0% Non-Equilibrium Eutectic! | Requires homogenization anneal |
Scheil-Gulliver Instantaneous Solid Composition C_S(f_S) (wt%)
Evaluates k * C0 * (1 - f_S)^(k - 1) for partition coefficient k > 0 and solid fraction 0 <= f_S < 1.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B4 >= 1), "Invalid Scheil Parameters", ROUND(B3 * B2 * ((1 - B4) ^ (B3 - 1)), 3))Non-Equilibrium Terminal Eutectic Fraction f_E = (C_E / C0) ^ (-1 / (1 - k))
Predicts the fraction of casting volume that freezes as divorced/lamellar eutectic when C_L hits C_E (B5).
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B5 <= B2), "Invalid Scheil Parameters", ROUND((B5 / B2) ^ (-1 / (1 - B3)), 4))Try the example
Use the original values or change them to test your understanding.