Question 8 of 8: Effects of Increasing Undercooling; Why Metals Do Not Solidify Exactly at $T_m$; Equilibrium versus Effective Distribution Coefficients

Question 8: Effects of Increasing Undercooling; Why Metals Do Not Solidify Exactly at $T_m$; Equilibrium versus Effective Distribution Coefficients (20 marks: a–6, b–8, c–6)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

8.1 — (a) Three effects of increasing undercooling

1. Increased nucleation rate and finer as-cast grain size. From Question 4, both $r^*=2\gamma_{SL}/\Delta G_v$ and $\Delta G^*=16\pi\gamma_{SL}^3/3\Delta G_v^2$ fall rapidly as undercooling (and hence $\Delta G_v$) increases, and nucleation rate varies exponentially with $\Delta G^*$. A larger undercooling therefore produces many more nuclei per unit volume in a given time, which grow to impingement and yield a correspondingly FINER as-solidified grain size — the basis of grain refinement by rapid chilling.
2. A shift in interface growth morphology, planar → cellular → dendritic. Greater undercooling (particularly constitutional supercooling ahead of the interface, Question 4(d)) increasingly destabilizes an initially planar solid–liquid interface, first into cells and, at still larger undercooling/solute buildup or thermal supercooling, into fully developed dendrites with secondary and tertiary arms.
3. Increased interface growth velocity, up to solute trapping at extreme undercooling. A larger undercooling supplies a larger driving force for the interface to advance, so the growth velocity $R$ increases with undercooling. At very large (rapid-solidification-level) undercooling, growth can become so fast that solute diffusion in the liquid cannot keep pace at all: the effective distribution coefficient $k_{eff}\to1$ (Question 8(c)), solute is trapped in the solid without segregating, and, at extreme rates, the transformation can even bypass long-range diffusion entirely, producing metastable or amorphous (glassy) solidification products instead of the equilibrium crystalline phase.

8.2 — (b) Why solidification requires a finite undercooling, not $T=T_m$ exactly

At $T=T_m$, the liquid and solid are in exact thermodynamic EQUILIBRIUM by definition ($G_L=G_S$), so the volumetric driving force for the transformation, $\Delta G_v=G_L-G_S$, is EXACTLY ZERO. From Question 4(a), the nucleation barrier is $\Delta G^*=16\pi\gamma_{SL}^3/3\Delta G_v^2$: as $\Delta G_v\to0$, $\Delta G^*\to\infty$, i.e. the energy required to create a stable solid nucleus becomes infinite. Physically, forming ANY new solid–liquid interface costs a positive surface energy $\gamma_{SL}$ that must be repaid by the volumetric free-energy release of the material inside the nucleus — with zero volumetric driving force available at $T_m$, there is nothing to repay that surface-energy cost with, so no finite nucleus can ever become thermodynamically stable. A finite undercooling $\Delta T=T_m-T$ below $T_m$ is therefore not a kinetic inconvenience but a THERMODYNAMIC NECESSITY: it is what supplies the nonzero $\Delta G_v\approx L\Delta T/T_m$ needed to make $\Delta G^*$ finite and achievable within a practical timescale.

The temperature RANGE over which solidification is actually observed is set by several compounding factors, not a single sharp threshold: (i) the magnitude of undercooling needed to nucleate at an observable rate is very different for homogeneous nucleation (typically $\Delta T\sim0.2\,T_m$, i.e. tens to a couple hundred degrees, since no catalytic substrate is available to reduce $\Delta G^*$) versus heterogeneous nucleation (often only a few degrees of undercooling, since container walls, oxide films, or deliberately added inoculants provide a wetting-angle reduction $S(\theta)<1$ to the barrier, Question 4(c)); (ii) once nucleation begins, the latent heat of fusion released at the growing interface, and/or solute rejected ahead of it, locally raise the interface temperature back toward $T_m$ (recalescence) or change the local equilibrium temperature, so the transformation does not proceed at one fixed instantaneous temperature but spreads over a temperature (and time) interval as heat and solute redistribute; and (iii) for an ALLOY (rather than a pure metal), the phase diagram itself specifies a genuine liquidus-to-solidus temperature INTERVAL over which L and $\alpha$ coexist even at full equilibrium, which is a separate, additional source of an observed solidification range on top of the nucleation-undercooling effect.

8.3 — (c) Equilibrium ($k_0$) versus effective ($k_{eff}$) distribution coefficients

The equilibrium distribution coefficient is a purely thermodynamic ratio read directly off the phase diagram at a given temperature, $$k_0 = \frac{C_S^*}{C_L^*}$$ the ratio of solidus to liquidus compositions in LOCAL equilibrium exactly at the solid–liquid interface. It assumes the liquid immediately adjacent to the interface has had time to fully equilibrate — i.e. either the growth rate is slow enough, or diffusion/convective mixing in the liquid is fast enough, that no composition gradient builds up ahead of the interface, and $k_0$ is a single number for a given alloy and temperature, independent of how fast solidification actually proceeds.

The effective distribution coefficient is instead an OPERATIONAL ratio defined against the bulk (far-field) liquid composition rather than the local interfacial one, $$k_{eff} = \frac{C_S}{C_0}$$ and captures what actually happens under REAL (finite growth rate $R$) solidification conditions: solute rejected at the interface (for $k_0<1$) piles up in a boundary layer of thickness $\delta$ because liquid-phase diffusion ($D_L$) cannot fully re-homogenize it at the imposed growth rate, so the liquid actually touching the solid, $C_L^*$, is enriched above the bulk value $C_0$ far from the interface. The Burton–Prim–Slichter (BPS) relation connects the two:

In the limit of very slow growth or very efficient mixing ($R\to0$, or $\delta\to0$), the exponential $\to1$ and $k_{eff}\to k_0$: the effective coefficient recovers the true equilibrium value once the boundary layer is negligible. In the opposite limit of very fast growth or very sluggish liquid diffusion ($R\to\infty$, or $D_L\to0$), the exponential $\to0$ and $k_{eff}\to1$: the solid ends up with essentially the SAME composition as the bulk liquid (complete solute trapping, no segregation at all), because diffusion simply cannot redistribute the rejected solute fast enough to build up any boundary-layer enrichment. $k_{eff}$ therefore always lies between $k_0$ and 1, and is the coefficient that actually predicts the macrosegregation pattern (e.g. via the Scheil equation, which uses $k_{eff}$ in place of $k_0$) in a real, finite-rate casting or ingot, whereas $k_0$ alone would only be correct in the unreachable limit of infinitely slow solidification.