Phase boundary

A phase boundary is an interface separating distinct phases of matter. Its temperature, velocity, and flux jumps are constrained by phase equilibrium and conservation laws.

A planar freezing front can push a spherical particle steadily when repulsive sphere-plane dispersion force balances lubrication resistance. The thin parabolic gap gives hydraulic drag $6\pi \mu V{a}^2/d$, and balancing it against $Aa/(6d^2)$ gives the displayed speed. This balance assumes a nonretarded interaction, small gap and negligible inertial and other particle forces.

For a sphere partially below a planar melting isotherm, write $c=\cos {\theta }_0$ and depth $a(\cos \theta -c)$. Local premelting gives film thickness $d^3=K/(\cos \theta -c)$. Conservation in the thin spherical film gives flux $Va\sin \theta /2$ and pressure $p=3\mu V{a}^2(\cos \theta -c{)}^2/K$. Integrating vertical repulsive and hydraulic stresses over the wetted cap yields the displayed speed. The unknown thermodynamic constant $K$ cancels, but a narrow rim where the film ceases to be thin is excluded from the leading approximation.

The solid-liquid segregation coefficient is the ratio of the solute concentration in newly formed solid to that in the adjacent equilibrium liquid. A coefficient of zero means complete solute rejection: newly formed crystals are pure solvent. It does not mean the bulk solute in a mushy layer vanishes, because its pore liquid can remain concentrated.

A crystal's interfacial free energy and attachment kinetics depend on the interface normal $\mathbf{n}$. The orientation-dependent energy $\gamma$ and mobility $M$ select directions even when bulk diffusion is isotropic. This supplements the transport mechanism of crystal dendrites; isotropic transport alone does not specify the number of preferred arms.

Ordinary atmospheric snow crystals are predominantly hexagonal ice Ih. Rotating by sixty degrees around its crystallographic axis leaves the equilibrium orientation dependence unchanged, giving six equivalent basal-plane directions. Temperature and supersaturation determine whether plates, columns or crystal dendrites develop. Dendritic growth of a snowflake needs this crystal symmetry as well as the diffusion instability.

Premelting is the persistence of a liquid layer at a surface or interface below the bulk melting temperature. Molecular disjoining pressure and phase equilibrium can stabilize the film. Its hydraulic connection to a reservoir permits frost heave when water freezes onto the base of loaded ice.

Thermomolecular pressure is the temperature-dependent force per area transmitted through a thin liquid film between a solid and a substrate. For a repulsive ice-film interaction $p_T=A/(6\pi a^3)$, it can drive water through a porous material and support a load. Its thermodynamic relation to undercooling is the Clapeyron pressure relation for equal-density phases.

For equal solid and liquid mass densities, $d\mu =-sdT+dp/\rho$ and $s_l-s_s=L/T_m$ imply the displayed relation from equality of chemical potentials. $T_m$ must be absolute temperature. The phase pressures differ; this should not be confused with differentiating a coexistence curve along a common pressure.

Frost heave is displacement caused by segregated ice growth supplied by liquid flow, rather than solely by the expansion of water on freezing. A premelting film allows liquid access beneath the ice; thermomolecular pressure draws supply against viscous resistance and load.

In an equal-density, no-slip lubrication approximation, repulsive disjoining pressure $A/(6\pi d^3)$ sustains a premelting film of thickness $d$ beneath a rigid disk of radius $a$. With imposed radial undercooling $\Delta T(1-r^2/a^2)$, local thermomolecular pressure is $P_T=\rho L\Delta T(1-r^2/a^2)/T_m$, and $d^3=A/(6\pi P_T)$. A constant vertical lifting rate $U$ requires inward volume flux $q_r=-Ur/2$. Integrating the resulting lubrication pressure and disk force balance gives the displayed rate, where $W$ is the apparent load, $\mu$ is liquid dynamic viscosity, $L$ is specific latent heat, and $T_m$ is the absolute bulk melting temperature. Positive lifting requires $W<\pi a^2\rho L\Delta T/(2T_m)$. For fixed positive load, the rate rises from zero at that threshold towards $A/(6\pi \mu a^2)$ as undercooling increases. That plateau is an asymptotic limit within this constitutive model, not a claim that continuum-film physics persists at arbitrarily low temperatures.

A porous sheet of thickness $d$ and permeability of a porous medium $\Pi$ supplies a thin premelting film. The Darcy flow law equates water supply to ice growth for equal densities. With equal film and ice conductivities, negligible latent heat relative to the through-going conductive flux, and $a\ll h$, phase equilibrium gives $p_T=(\rho L\Delta T/T_m{)}^{3/4}(A/(6\pi h^3){)}^{1/4}$. The increasing gravitational load limits growth. A quasi-steady temperature field alone does not imply the negligible-latent-heat approximation.

For ${\eta }^{\prime }={\eta }^{-3/4}-\eta$, set $u={\eta }^{7/4}$ to get $u^{\prime }=(7/4)(1-u)$. The displayed solution approaches the stable equilibrium $\eta =1$. From the formal zero start it behaves as $(7\tau /4{)}^{4/7}$ at early times and $1-(4/7)e^{-7\tau /4}$ at late times. The zero-start asymptote is an intermediate reduced-model regime, since the physical thin-film approximation fails at vanishing ice thickness.

Combine the full quasi-steady Stefan condition above with $\delta T=T_m{p}_T/(\rho L)$, $a=(A/(6\pi p_T){)}^{1/3}$ and the Darcy flow law to obtain an implicit pressure-growth system. Even when latent heat is negligible, a conductivity ratio $K_s/K_l=r$ changes the leading molecular driving pressure by $r^{3/4}$. Thus a reduced equal-conductivity formula requires an explicit material and rate approximation.

The excess free energy per area of a solid-liquid phase boundary. Curvature changes the equilibrium temperature through the Gibbs--Thomson relation. Crystal anisotropy can make this energy and its associated surface stiffness depend on orientation.