= Solution
Let $a(t)$ be the particle radius and $\Delta T=T_s-T_\infty>0$ the supercooling. Use <spherical symmetry>, constant thermal properties, equal phase densities $\rho$, and negligible fluid advection. Let $\mathcal L$ be <latent heat> per unit mass, $c_p$ the liquid specific heat, $k_\ell$ its <thermal conductivity> and $\kappa=k_\ell/(\rho c_p)$ its <thermal diffusivity>. The exterior <temperature> satisfies the radial <heat equation>
$$
\boxed{T_t=\kappa\frac1{r^2}\frac\partial{\partial r}\left(r^2T_r\right),\qquad r>a(t),}
$$
with $T(r,0)=T_\infty$ for $r>a_0$, $a(0)=a_0$ and $T(r,t)\to T_\infty$ as $r\to\infty$.
The <Gibbs-Thomson effect> gives <curvature-induced melting-temperature depression>. For a spherical interface write
$$
\boxed{T(a(t),t)=T_s-\frac\Gamma{a(t)},\qquad
\Gamma=\frac{2\gamma T_s}{\rho\mathcal L}>0,}
$$
where $\gamma$ is solid-liquid interfacial energy per area and the factor two is the sphere's curvature $2/a$. Here $T_s$ in the thermodynamic formula is an absolute <temperature>. The notation $\Gamma$ also permits a given measured capillary coefficient without requiring its microscopic derivation.
In the one-phase <Stefan problem>, the solid has no internal <heat flux> and the <Stefan condition> is
$$
\boxed{\rho\mathcal L\dot a=-k_\ell T_r(a^+,t).}
$$
The sign follows from heat balance: positive outward liquid <heat flux> releases <latent heat> as the solid grows. If the solid's thermal storage is retained, its <temperature> $T^{(s)}$ instead satisfies its own radial <heat equation>, $\partial_rT^{(s)}(0,t)=0$, and $T^{(s)}(a^-,t)=T(a^+,t)$. The two-phase condition is $\rho\mathcal L\dot a=k_sT_r^{(s)}(a^-,t)-k_\ell T_r(a^+,t)$. It reduces to the one-phase condition in the quasistatic limit because a regular harmonic <temperature> inside a sphere is spatially constant. An initial solid <temperature> profile is additional data for a fully time-dependent two-phase problem.
Two reciprocal <Stefan number> conventions are common. The large parameter appropriate to the requested slow-interface limit is the <latent-to-sensible heat ratio>
$$
\boxed{S=\frac{\mathcal L}{c_p\Delta T}.}
$$
It measures the energy required for phase change relative to cooling or heating by the supercooling. The often-used sensible-to-latent definition is $\mathrm{Ste}=c_p\Delta T/\mathcal L=1/S$. Thus “large Stefan number” here means large $S$, not large $\mathrm{Ste}$ in that other convention.
The thermal diffusion time for scale $a_0$ is $a_0^2/\kappa$, whereas the interface time is $Sa_0^2/\kappa$. More explicitly, with $R=r/a_0$, $\theta=(T-T_\infty)/\Delta T$ and slow time $\tau=\kappa t/(Sa_0^2)$, the liquid equation is
$$
S^{-1}\theta_\tau=\frac1{R^2}(R^2\theta_R)_R.
$$
For $S\gg1$, <temperature> therefore equilibrates before the radius changes appreciably. The <quasistatic spherical Stefan problem with capillarity> has a harmonic exterior <temperature>, fixed by its interface and far-field values:
$$
\boxed{T(r,t)=T_\infty+\frac{\Delta T\,a(t)-\Gamma}{r}.}
$$
The small initial diffusion adjustment is omitted at this order. Inserting $T_r(a)=-\Delta T/a+\Gamma/a^2$ into the <Stefan condition> gives
$$
\boxed{\dot a=\frac\kappa S\frac{a-a_c}{a^2},\qquad
a_c=\frac\Gamma{\Delta T}=\frac{2\gamma T_s}{\rho\mathcal L(T_s-T_\infty)}.}
$$
This is the <critical radius for a supercooled spherical seed>. For $a_0>a_c$, the interface is warmer than the liquid and loses heat, driving growth. For $0<a_0<a_c$, its curvature-depressed interface <temperature> is colder than the liquid, so heat enters and the solid melts. At $a=a_c$, the uniform <temperature> $T_\infty$ supplies an equilibrium radius. It is unstable, since the radius-velocity derivative there is $\kappa/(Sa_c^2)>0$. For a growing particle with $a\gg a_c$, $a^2$ increases asymptotically as $2\kappa t/S$.
For a subcritical particle the radius decreases monotonically and remains below $a_0$. Hence
$$
\frac d{dt}a^3=-\frac{3\kappa}S(a_c-a)
\leq-\frac{3\kappa}S(a_c-a_0)<0.
$$
It cannot remain positive beyond $Sa_0^3/[3\kappa(a_c-a_0)]$, proving <finite-time extinction of a subcritical spherical seed>. The exact extinction time in the reduced model is obtained by separating variables:
$$
t_e=\frac S\kappa\int_0^{a_0}\frac{a^2}{a_c-a}\,da.
$$
With $\eta_0=a_0/a_c\in(0,1)$, this is
$$
\boxed{t_e=\frac{Sa_c^2}\kappa\left[-\log(1-\eta_0)-\eta_0-\frac12\eta_0^2\right]<\infty.}
$$
Near disappearance, the capillary term dominates and $\dot a\sim-\kappa a_c/(Sa^2)$, so $\boxed{a^3\sim3\kappa a_c(t_e-t)/S}$. For $a_0\ll a_c$, the lifetime is $t_e\sim Sa_0^3/(3\kappa a_c)$; it diverges logarithmically as $a_0\to a_c^-$, consistently with the unstable equilibrium. The threshold and lifetime above are conclusions of the specified large-$S$ quasistatic continuum model, rather than an assertion that its approximation is uniform through every finite-$S$ shrinking-interface transient.
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