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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 equationwith $T(r,0)=T_{\infty }$ 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 writewhere $\gamma$ 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 isThe 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, ${\partial }_r{T}^{(s)}(0,t)=0$, and $T^{(s)}(a^{-},t)=T(a^{+},t)$. The two-phase condition is $\rho \mathcal{L}\dot{a}=k_s{T}_r^{(s)}(a^{-},t)-k_{\ell }{T}_r(a^{+},t)$. 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 ratioIt 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 $S{a}_0^2/\kappa$. More explicitly, with $R=r/a_0$, $\theta =(T-T_{\infty })/\Delta T$ and slow time $\tau =\kappa t/(S{a}_0^2)$, the liquid equation isFor $S\gg 1$, 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:The small initial diffusion adjustment is omitted at this order. Inserting $T_r(a)=-\Delta T/a+\Gamma /a^2$ into the Stefan condition givesThis 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 $\kappa /(S{a}_c^2)>0$. 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 $a_0$. HenceIt cannot remain positive beyond $S{a}_0^3/[3\kappa (a_c-a_0)]$, proving finite-time extinction of a subcritical spherical seed. The exact extinction time in the reduced model is obtained by separating variables:With ${\eta }_0=a_0/a_c\in (0,1)$, this isNear disappearance, the capillary term dominates and $\dot{a}\sim -\kappa a_c/(S{a}^2)$, so $\boxed{a^3\sim 3\kappa a_c(t_e-t)/S}$. For $a_0\ll a_c$, the lifetime is $t_e\sim S{a}_0^3/(3\kappa a_c)$; 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.