Constitutional supercooling in directional solidification calculator

H. Yan and H. K. D. H. Bhadeshia, Queen Mary University of London

Steady-state solute redistribution, liquidus depression, and the Tiller–Jackson–Rutter–Chalmers criterion [Acta metallurgica, 1 (1953)428-437]. This original work estimated the redistribution of solute atoms during the solidification of a binary solution under transient and steady-state conditions, and derived mathematical expressions for the extent of the constitutionally supercooled zone in the liquid adjacent to an advancing solid-liquid interface. Constitutional supercooling is related to the distribution of solute in the liquid ahead of an advancing interface.

Thermodynamic and kinetic foundations

During the directional solidification of a binary alloy at steady-state pulling velocity $v$ , solute partitioning at the advancing solid/liquid interface builds up a solute-enriched boundary layer in the melt. Because the equilibrium liquidus temperature depends directly on composition, this concentration gradient creates a corresponding spatial gradient in the local freezing point. When the imposed thermal gradient in the liquid is shallower than the liquidus temperature gradient, a liquid zone ahead of the interface finds itself below its equilibrium freezing point, a state termed constitutional supercooling.

1. Steady-state solute redistribution ahead of a planar interface

Let coordinate $z$ denote the spatial distance ahead of the planar solid/liquid interface ( $z = 0$ at the moving interface, and $z > 0$ extending into the liquid). Solute conservation across the interface is governed by the equilibrium partition coefficient:

k p = C SL C LS

At steady state, the solid forming at the interface must have a uniform composition equal to the far-field nominal alloy composition, $C SL \approx C 0$ . Consequently, the liquid composition directly at the interface ( $z = 0$ ) builds up to:

C LS = C 0 k p

In the moving reference frame attached to the interface advancing at velocity $v$ , the one-dimensional steady-state diffusion equation in the liquid is:

D L d 2 C L d z 2 + v d C L d z = 0

Applying the boundary conditions $C L {0} = C 0 / k p$ and $C L {z \to \infty} = C 0$ yields the classical exponential solute profile:

C L {z} = C 0 [1 + 1 - k p k p exp (- v D L z)] with characteristic boundary layer thickness δ c = D L v

2. Liquidus profile and constitutional supercooling criterion

Assuming a linear liquidus line of slope $m > 0$ (where liquidus temperature drops with solute addition, $T liq (C) = T m - m C$ ), the equilibrium freezing point profile ahead of the interface is:

T liq {z} = T m - m C L {z}

The actual liquid temperature is dictated by the furnace temperature gradient $G = d T / d z$ . Assuming equilibrium at the interface, the actual temperature at $z = 0$ is fixed at the interface liquidus point $T i = T m - m (C 0 / k p)$ :

T L {z} = T i + G z = [T m - m C 0 k p] + G z

Constitutional supercooling exists whenever the actual liquid temperature is lower than the local equilibrium liquidus temperature ( $T L {z} < T liq {z}$ ). The threshold for the onset of instability at the interface is the Tiller–Jackson–Rutter–Chalmers criterion:

G < (d T liq d z) z = 0 = m C 0 (1 - k p) v k p D L or G v < m C 0 (1 - k p) k p D L

Interactive constitutional supercooling dashboard

Modify the alloy parameters, growth velocity, and liquid thermal gradient below. The synchronous plots dynamically trace the solute enrichment boundary layer $C L {z}$ and compare the actual thermal profile $T L {z}$ against the equilibrium liquidus $T liq {z}$ . The shaded red region highlights the extent of constitutional supercooling ahead of the interface.

Solidification parameters

Alloy preset system

Nominal composition (

C 0

) 2.00 wt%

Partition coefficient (

k p

) 0.14

Liquidus slope magnitude (

m

) 2.60 K/wt%

Solvent melting point (

T m

) 660.0 °C

Solidification velocity (

v

) 20.0 µm/s

Liquid diffusivity (

D L

) 3.0 × 10⁻⁹ m²/s

Liquid thermal gradient (

G

) 3.50 K/mm

Stability status (G vs G_crit) UNSTABLE (Supercooled)

Critical gradient (G_crit = (dT_liq/dz)|z=0) 5.32 K/mm

Interface liquid concentration (C_LS = C_0 / k) 14.29 wt%

Solute boundary layer thickness (δ_c = D_L / v) 150.0 µm

Max constitutional supercooling (ΔT_max) 0.78 °C

Constitutional supercooled zone length (z_cs) 78.4 µm

Concentration in liquid ahead of interface, C_L{z} C_L{z} profile

Actual temperature T_L vs equilibrium liquidus T_liq

T_L Actual T_liq Liquidus Supercooled

Physical significance of constitutional supercooling

In directional solidification, constitutional supercooling is the primary origin of morphological interface instability, dictating whether a growing crystal maintains a smooth planar front, develops cellular ripples, or transitions into complex dendritic arrays:

Planar front stability ( $G \geq G crit$ ): When a sufficiently steep thermal gradient $G$ is maintained or the growth velocity $v$ is low, the actual liquid temperature $T L {z}$ lies everywhere above the local liquidus temperature $T liq {z}$ . If a random perturbation forms on the planar interface and extends into the melt, it encounters a positive thermodynamic superheating, which melts the protrusion back down and stabilises the planar interface.
Instability and cellular/dendritic growth ( $G < G crit$ ): When $G$ is shallow or growth velocity $v$ is high, the actual temperature line intersects and falls below the equilibrium liquidus profile, establishing a constitutionally supercooled layer ( $Δ T cs = T liq - T L > 0$ ). Any bump or protrusion projecting ahead into this supercooled liquid finds itself thermodynamically undercooled, accelerating its forward growth. The planar front breaks down sequentially into hexagonal cells, then primary dendrites, and finally branched side arms with substantial microsegregation.