Question 4 of 8: Classical Nucleation Theory — Critical Radius, Undercooling, Heterogeneous versus Homogeneous Nucleation, and Interface Growth Morphology

Question 4: Classical Nucleation Theory — Critical Radius, Undercooling, Heterogeneous versus Homogeneous Nucleation, and Interface Growth Morphology (20 marks: a–6, b–4, c–4, d–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.

4.1 — (a) Deducing $r^*$ and $\Delta G^*$ for a spherical nucleus

Forming a spherical solid nucleus of radius $r$ from the liquid changes the system's total free energy by a competition between a favourable VOLUME term (the solid is thermodynamically preferred below $T_m$, releasing energy $\Delta G_v$ per unit volume) and an unfavourable SURFACE term (creating new solid–liquid interface costs energy $\gamma_{SL}$ per unit area):

The volume term scales as $-r^3$ (favourable, grows without bound) while the surface term scales as $+r^2$ (unfavourable, grows more slowly) — for small $r$ the surface term dominates ($\Delta G$ rises with $r$), while for large $r$ the volume term dominates ($\Delta G$ falls with $r$), so $\Delta G(r)$ passes through a MAXIMUM at some critical radius $r^*$. Setting the derivative to zero:

Substituting $r^*$ back into $\Delta G(r)$ gives the activation-energy barrier for nucleation:

Embryos smaller than $r^*$ REDISSOLVE (shrinking lowers $\Delta G$, since they sit on the rising part of the curve), while embryos that fluctuate past $r^*$ grow spontaneously (growing lowers $\Delta G$ beyond the peak) — $r^*$ is therefore the size an embryo must randomly achieve, by thermal fluctuation, to become a stable, growing nucleus, and $\Delta G^*$ is the energy barrier that fluctuation must supply.

4.2 — (b) Why undercooling, and the $2\gamma/r^*$ term, matter

The volumetric driving force is proportional to the undercooling below the melting point, $\Delta G_v\approx \dfrac{L\,\Delta T}{T_m}$ (Turnbull's linear approximation, with $L$ the latent heat of fusion and $\Delta T=T_m-T$). At $T=T_m$ exactly, $\Delta T=0$ so $\Delta G_v=0$, and from part (a) $r^*=2\gamma_{SL}/\Delta G_v\to\infty$ and $\Delta G^*\to\infty$: with NO undercooling there is no finite critical radius and no achievable nucleation barrier — the liquid could in principle persist metastably forever. As undercooling increases, $\Delta G_v$ grows, and both $r^*=2\gamma_{SL}/\Delta G_v$ and $\Delta G^*=16\pi\gamma_{SL}^3/3\Delta G_v^2$ shrink rapidly (the barrier falls as $1/\Delta T^2$), which is why solidification is only ever OBSERVED at a finite undercooling below $T_m$, never exactly at it.

The term $2\gamma/r^*$ is the Gibbs–Thomson (capillarity) pressure — the extra pressure, and equivalently the depression in local equilibrium (melting) temperature, that a small curved solid–liquid interface of radius $r^*$ must sustain purely because of its own curvature. It is the quantitative reason SMALL embryos are less stable than the bulk solid: a tiny sphere has a much higher $2\gamma/r$ than a large one, so it "sees" an effectively lower local melting point and re-melts unless it is large enough ($r\ge r^*$) that this capillary penalty is outweighed by the bulk driving force. It is exactly this term that sets the size threshold in part (a) and, physically, is why nucleation — not just growth — requires undercooling: growth alone (advancing an already-existing flat interface) needs no capillary correction, but CREATING a new, small, curved interface from nothing does.

4.3 — (c) Why heterogeneous nucleation dominates despite an unchanged $r^*$

$r^*=2\gamma_{SL}/\Delta G_v$ depends only on the solid–liquid interfacial energy and the volumetric driving force — neither of which changes when nucleation occurs on a foreign substrate (a vessel wall) instead of in the bulk liquid, so $r^*$ is indeed identical in both cases, exactly as the question states. What DOES change is the SHAPE of the critical nucleus and hence the total new interfacial area it must create. A nucleus forming on a flat wall adopts a spherical-CAP shape (partially "borrowing" the existing solid–wall contact rather than being a complete sphere immersed in liquid), and the fraction of the homogeneous barrier this costs is given by a wetting-angle shape factor:

for wetting (contact) angle $\theta$ between the nucleus and the wall. $S(\theta)$ is always $\le 1$ (equal to 1 only for $\theta=180^\circ$, i.e. complete non-wetting, which recovers the homogeneous case exactly) and falls toward 0 as the wall wets the solid more effectively (small $\theta$) — because the wall then already supplies most of the "surface" the embryo would otherwise have to create for itself. Since $r^*$ is unchanged but $\Delta G^*$ is multiplied by $S(\theta)<1$, and the nucleation RATE varies exponentially with the barrier, $I\propto\exp(-\Delta G^*/kT)$, even a modest reduction in $\Delta G^*$ produces an enormous increase in nucleation rate. This is why heterogeneous nucleation on vessel walls, inclusions, or other substrates overwhelmingly dominates in practice, even though the critical EMBRYO SIZE required is exactly the same as for homogeneous nucleation — it is the ENERGY BARRIER to reach that size, not the size itself, that heterogeneous nucleation lowers.

4.4 — (d) Planar versus cellular interface growth and constitutional supercooling

(i) Planar growth. A flat solid–liquid interface remains STABLE (grows as a plane) only if any small protrusion that randomly advances into the liquid finds itself in liquid that is HOTTER than its own local equilibrium (liquidus) temperature — such a protrusion melts back, restoring flatness. This requires the actual temperature in the liquid to rise (or at least not fall below the liquidus) everywhere ahead of the interface, i.e. a temperature gradient $G_L$ in the liquid that is positive and steep enough that no zone of the liquid is undercooled relative to its own local liquidus temperature.

(ii) Cellular growth and constitutional supercooling. During alloy solidification, the advancing interface rejects (or absorbs) solute according to the partition coefficient $k<1$, building up a solute-enriched boundary layer in the liquid immediately ahead of the interface. Because the LIQUIDUS temperature of that boundary-layer liquid depends on its (locally elevated) composition, the LOCAL liquidus temperature is highest right at the interface and falls back to the bulk liquidus value away from it — the opposite sense to how the ACTUAL temperature profile behaves if $G_L$ is only modest. If the actual temperature profile falls BELOW this local-liquidus profile anywhere ahead of the interface, that zone of liquid is at a temperature below its own equilibrium freezing point even though it is still liquid — it is constitutionally supercooled, not thermally supercooled. The standard (Tiller) instability criterion is:

for growth rate $R$, liquidus slope $m_L$, bulk composition $C_0$ and liquid diffusivity $D_L$. When this holds, any small protrusion on the interface now advances INTO a region of constitutional supercooling, where its growth is thermodynamically favoured (more effective undercooling ahead of the tip than beside it), so the protrusion is amplified rather than melted back — the flat interface breaks down into CELLS (and, at still larger supercooling/solute buildup, into dendrites). The temperature gradient in the liquid is therefore important because it is the stabilizing term (steep $G_L$ suppresses constitutional supercooling for a given $R$), while constitutional supercooling is important because it is the destabilizing term that a low $G_L$, high $R$, or strongly partitioning solute ($k$ far from 1, large $m_LC_0$) can trigger — the balance between the two, not either alone, decides the interface morphology.