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Crystal polymorphism is the existence of different crystal structures for the same chemical substance. Each crystal polymorph can have its own mass density, melting behavior and range of stability on a phase diagram. A different external shape of the same crystal structure alone does not establish a different crystal polymorph.^[ref]

At fixed temperature and pressure, compare the chemical potentials of the possible crystal polymorphs. The lower value determines the stable bulk thermodynamic phase; equality permits phase coexistence when neither form is beaten by another thermodynamic phase. A higher-chemical potential form can persist through metastability, so finding it in a sample does not by itself locate an equilibrium boundary.

The eutectic composition is the liquid concentration at the eutectic temperature, where liquid and two solid phases coexist. It is a composition coordinate on a phase diagram, not the overall concentration of every sample in an eutectic system.

The lever rule finds phase fractions from a horizontal tie line on a phase diagram. For solid and liquid concentrations $C_s,C_l$ and bulk concentration $\overline{C}$, conservation gives $\overline{C}=(1-\varphi )C_s+\varphi C_l$. Thus $\varphi =(\overline{C}-C_s)/(C_l-C_s)$ is the liquid fraction, provided $C_s\le \overline{C}\le C_l$. For pure solid, $\varphi =\overline{C}/C_l$.

The liquidus is the locus in a phase diagram above which the equilibrium state is entirely liquid. In a dilute binary solution it is often approximated locally by a linear relation such as $T_L=T_0-mC$.

For a local illustrative ice–water melting calculation, stipulate $T_m=273.15\mathrm{K}$, latent heat of fusion $L=334\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$, solid mass density ${\rho }_s=917\mathrm{kg}{\mathrm{m}}^{-3}$, and liquid mass density ${\rho }_l=1000\mathrm{kg}{\mathrm{m}}^{-3}$. These are explicitly stipulated rounded inputs, not measurements made here. The specific volume change on melting isThe common-pressure melting-point slope therefore givesA small $1\mathrm{MPa}$ increase in the common pressure thus lowers the melting temperature by about $0.074\mathrm{K}$ in this local model. Positive latent heat does not force a positive phase coexistence curve slope: ice contracts on conversion to liquid water. Increasing pressure favors the smaller specific volume. This calculation compares equilibrium states of the same two thermodynamic phases; extending its tangent to large pressures would require updated mass densities, latent heat of fusion, and the relevant ice phase diagram.

Write $H=L/c_p$, $k=\rho c_p\kappa$ and $ϵ=\sqrt{D/\kappa }$. The requested model neglects density differences and uses the same thermal transport coefficients in the interfacial balances. Let $C_b=C(b(t),t)$ and $T_b=T_m-m{C}_b$. With no imposed heat gradient in the fresh liquid, its temperature remains $T_m$. The prescribed linear temperature in the ice isThe salt and temperature fields in the lower liquid satisfy diffusion equations, $C_t=D{C}_{zz}$ and $T_t=\kappa T_{zz}$. For constant interfacial values and the proposed similarity solution, their decaying solutions areThese follow by substituting a variable $z/(2\sqrt{\chi t})$ into the diffusion equation, which gives $U^{\prime \prime }+2\eta U^{\prime }=0$; integrating gives the complementary error function. Their values at $b$ and at infinity give the required boundary data.

For the upward-moving fresh-water/ice boundary, the Stefan condition is $\rho L\dot{a}=k{T}_{i,z}$: negative $\dot{a}$ means fresh water freezes. At the lower boundary the signed heat balance isSalt cannot enter the pure ice. With no bulk liquid motion its interfacial conservation law isThe derivatives and interface velocities areDefine $F(x)=\sqrt{\pi }x{e}^{x^2}\mathrm{erfc}x$. Substitution into all three balances givesTogether with $T_b=T_m-m{C}_b$, these determine the interfacial constants. In particular $C_b=C_0/[1-F({\lambda }_b)]$.

The physical branch has ${\lambda }_a<{\lambda }_b<0$. Since $F$ is negative at negative arguments, the salt relation gives $0<C_b<C_0$: lower-surface melting dilutes the brine. Combining the two heat balances eliminates the ice gradient and givesThus $T_m>T_b>T_{\infty }>T_L(C_0)$. The interface is at local liquidus equilibrium with diluted brine, while the undiluted far-field brine remains above its own liquidus. The similarity model describes a layer starting with zero thickness, or the long-time limit when an initial finite thickness no longer sets the dominant scale.

For the small-diffusivity-ratio limit, put $A=ϵ{\lambda }_a$ and $\Delta =T_m-T_{\infty }$, keeping $S=H/\Delta$ and the imposed concentrations fixed. Seek $A=O(1)$ and ${\lambda }_b=O(1)$. Then $F(ϵ{\lambda }_b)\sim \sqrt{\pi }ϵ{\lambda }_b$ and ${\lambda }_b-{\lambda }_a\sim -{\lambda }_a$. The upper heat balance givesIn the lower balance the two terms of order ${ϵ}^{-1}$ must cancel, givingConsequentlyand $T_b-T_{\infty }\sim -\sqrt{\pi }HA$. Adding the two temperature differences yields $2A^2-\sqrt{\pi }A=1/S$. The negative root isThe resulting $C_b=(T_m-T_b)/m$ lies strictly between zero and $C_0$. The salt relation therefore determines a finite negative ${\lambda }_b$, confirming its $O(1)$ scaling. The top interface travels on the thermal scale $\sqrt{\kappa t}$, whereas the lower melting interface travels on the shorter solutal scale $\sqrt{Dt}$. This is the brine-ice sandwich with disparate diffusion scales; $S$ is the latent-to-sensible heat ratio, the reciprocal of the usual sensible-to-latent Stefan number convention.

The sketches show constant fresh-water temperature, a declining linear ice temperature, and a much broader thermal than solutal boundary layer in the brine. Concentration is zero in freshwater and pure ice, jumps to $C_b$ across the lower interface, and rises to $C_0$. Accordingly the local liquidus is $T_m$ on the salt-free side, jumps to $T_b$ and falls rapidly to $T_L(C_0)$. In the phase diagram, the brine starts on the liquidus at $(C_b,T_b)$ and ends above it at $(C_0,T_{\infty })$; the pure ice has concentration zero, and the interface tie-line joins it to the diluted brine.

Physically, the cold saline liquid permits freshwater to freeze above, but salt lowers the liquidus and allows the existing ice to melt into brine below even though that brine is colder than $T_m$. Dilution raises the lower interfacial freezing temperature. Heat released by upper freezing supplies the lower melting and the conductive loss to the colder deep brine. With $|{\lambda }_a|\gg |{\lambda }_b|$, upper freezing exceeds lower melting and the ice layer thickens as $b-a\propto \sqrt{t}$.

A phase diagram divides a space of control parameters into equilibrium thermodynamic phases, marking phase coexistence boundaries and thermodynamic critical points. Its dimension counts independently variable controls, not spatial dimensions. An explicit three-dimensional example uses the controls $(r,u,h)$ of a sextic even Landau potential:Here $M$ is a scalar order parameter, $h$ its conjugate field, and $r$ a thermal control. The equilibrium magnetization minimizes $V$ globally. At $h=0$, $u>0$, the boundary $r=0$ is a continuous phase transition. At $h=0$, $u<0$, the boundary $r=3u^2/(16v)$ is a first-order phase transition between $M=0$ and $M=\pm \sqrt{-3u/(4v)}$. The two boundaries meet at the tricritical point $r=u=h=0$. A fixed strictly positive quartic coupling alone cannot produce this tricritical point; the stabilizing sextic term is essential when the quartic coefficient is tuned through zero.

The full phase diagram also contains the $h=0$ phase coexistence sheet between positive and negative ordered phases. Crossing it changes the sign of $M$ discontinuously. For $u<0$, the three-phase line $r=3u^2/(16v)$ is the junction of this sheet and two tricritical wings, on which two unequal same-sign equilibrium magnetizations coexist at nonzero $h$. The tricritical wing critical edges are lines of ordinary thermodynamic critical points; they terminate the first-order sheets and meet at the tricritical point. Away from these boundaries, varying $h$ produces a smooth crossover.

For an explicit construction of the tricritical wings, write their two coexisting order parameters as $M_1=s-d$ and $M_2=s+d$, with $s>0$ and $0<d\le s$. Equality of their Landau free energies and first derivatives givesIndeed, for these controls the tricritical wing coexistence factorization isThe last factor is $(M+2s{)}^2+s^2-d^2\ge 0$, proving that both specified minima are global. The reflected wing has negative $h$ and negative $M$. Letting $d\to 0$ gives the tricritical wing critical edge $u=-10v{s}^2/3$, $r=5v{s}^4$, $h=8v{s}^5/3$; there $V^{\prime \prime }(s)=V^{\prime \prime \prime }(s)=0$ and $V^{\prime \prime \prime \prime }(s)=40v{s}^2>0$. Letting $d=s$ recovers the three-phase line. This supplies the first-order sheets and their continuous boundary curves, rather than only a zero-field section.

A tricritical point occurs when a first-order boundary meets a continuous-transition boundary. In the spin-inversion-symmetric sextic Landau free energy, two independent parameters must tune $r=u=0$, while $v>0$ keeps the potential stable. In the $h=0$ plane, the continuous boundary is $r=0,u>0$, and the first-order boundary is $r=3u^2/(16v),u<0$. They meet at the tricritical point; the order-parameter jump on the first-order side tends to zero as $u\uparrow 0$. The first panel shows this phase diagram, including the metastability limits derived above.

Along the tuned line $u=0,h=0$, the equation of state is $M(r+v{M}^4)=0$. Thus $M=(-r/v{)}^{1/4}$ on a selected ordered branch, rather than the ordinary square-root behavior. Differentiation gives inverse magnetic susceptibility $r$ above and $4|r|$ below, while at criticality $h=v{M}^5$. Hence the mean-field tricritical exponent calculation yields $\beta =1/4$, $\gamma =1$, $\delta =5$; the minimized potential $-|r{|}^{3/2}/(3\sqrt{v})$ also gives $\alpha =1/2$. These tricritical powers require tuning the quartic coefficient, not merely varying a generic temperature path. The ordinary scalar exponents requested separately are calculated next.

In a first-order phase transition, two distinct global minima of the Landau free energy have equal values and exchange stability. The equilibrium order parameter jumps, and a first derivative of the equilibrium free energy is discontinuous. Depending on the direction through the phase diagram, that derivative can be magnetization, entropy or another conjugate quantity; a first-order transition need not have latent heat along every chosen path.

In a continuous phase transition, the equilibrium order parameter changes continuously while the curvature at the critical minimum vanishes. For $u>0$ and $h=0$, $M=0$ is stable for $r>0$, but for $r<0$ the stable minima have $M\simeq \pm \sqrt{-r/u}$. The order parameter therefore goes continuously to zero at $r=0$; the inverse magnetic susceptibility vanishes there.

For $u<0$, the sextic term stabilizes the Landau free energy. The nonzero minima can become favorable while $M=0$ still has positive curvature. At $r=3u^2/(16v)>0$ they coexist with $M=0$, so the order parameter jumps to magnitude $\sqrt{-3u/(4v)}$. Finding a stationary point alone does not locate the transition: equilibrium requires comparing the values of all competing minima. Spinodal boundaries describe loss of local stability, not coexistence.

A tricritical point requires two independent even coefficients in the Landau free energy to vanish: $r=u=0$, with $v>0$ providing stability. At $h=0$, the ordinary critical line $r=0,u>0$ meets the tricritical three-phase line $r=3u^2/(16v),u<0$. The order parameter discontinuity along the first-order line is $\sqrt{-3u/(4v)}$, which goes to zero at their meeting.

The positive sextic coefficient makes this a different leading local potential from an ordinary quartic critical point. Consequently its mean-field critical exponents, crossover behavior and upper critical dimension differ. There are two relevant even tuning directions near the tricritical point, in addition to the conjugate odd field. The root solution and labelled phase diagram display the three controls and the tricritical wings; varying $h$ can give a first-order reversal within the ordered coexistence sheet, a first-order weak/strong transition on a wing, or a continuous transition at a wing's ordinary critical edge.

A phase diagram partitions a space of thermodynamic controls into regions with distinct equilibrium phases. Its boundaries indicate phase coexistence or continuous phase transitions; a critical point is a termination of coexistence where the distinction between phases disappears. A concrete three-dimensional control space is $(T,\Delta ,h)$ for the Blume–Capel model: a spin can be $0$ or $\pm 1$, the exchange favors aligned nonzero spins, and $\Delta$ controls the energetic cost of nonzero spins. Varying $\Delta$ changes the balance between a continuous ordering transition and a first-order phase transition. Their meeting is a tricritical point.

Locally, the same structure is displayed by the Landau free energywith three controls $(r,u,h)$. For $u>0$, the line $r=h=0$ consists of ordinary critical points. For $u<0$, equal free energies of $M=0$ and a nonzero stationary point requireEliminating $r$ gives the tricritical three-phase lineHere $M=0$ and both signs of $M$ coexist, with a discontinuous order parameter. Below this line, or below $r=0$ when $u>0$, the plane $h=0$ is an ordered phase coexistence sheet: crossing it reverses the sign of $M$. Two tricritical wings also extend to nonzero $h$, separating weakly and strongly magnetized states of the same sign. Their ordinary critical points satisfy $f^{\prime }=f^{\prime \prime }=f^{\prime \prime \prime }=0$. Since $f^{\prime \prime \prime }=6uM+20v{M}^3$, their tricritical wing critical edges areAt these edges $f^{\prime \prime \prime \prime }=-12u>0$, giving an ordinary quartic critical expansion around $M_c$. All these structures end at the tricritical point $r=u=h=0$. The diagrams show both the zero-field section and the genuinely three-dimensional coexistence structure.

For the final critical exponent calculation, take $r=at$ with $a>0$, $t=(T-T_c)/T_c$, and hold the other controls fixed. At an ordinary critical point, the quartic term dominates the sextic term. For $h=0$ and $r<0$,whereas on the disordered side the singular contribution is zero. Two temperature derivatives give a finite jump in heat capacity, so $\alpha =0$. At $r=0$, the equation of state is $h=u{M}^3+O(M^5)$, so $\delta =3$.

Along the tricritical trajectory $u=0$, the nonzero minimum obeys $M^4=-r/v$ andThus two temperature derivatives produce $|t{|}^{-1/2}$. At $r=u=0$, $h=v{M}^5$. Therefore the mean-field critical exponents areWith $A$ the thermodynamic free energy, the physical sign is $C=-T{\partial }_T^{2}A$, rather than the positive sign printed in the paper. This correction affects the sign of the heat capacity, not its critical exponent. The tricritical result assumes that the independent quartic control is tuned to zero; a generic trajectory through nearby ordinary critical points need not have these exponents.

A tricritical point occurs when the continuous and first-order transition lines meet. In the symmetric sextic Landau free energy this requires tuning two coefficients, $r=u=0$, while $v>0$ stabilizes the potential. For $u>0$, the zero-field transition lies on $r=0$ and is continuous. For $u<0$, the coexistence curve is $r=3u^2/(16v)$. The order-parameter jump satisfies $M^2=-3u/(4v)$ and shrinks to zero as the tricritical point is approached. The labelled phase diagram below uses $(u,r)$ as local coordinates; two independent physical controls, such as temperature and composition, can provide these coordinates.

Along the tricritical tuning $u=0$, the equation of state is $h=rM+v{M}^5$. At zero field the ordered solution is $M=(-r/v{)}^{1/4}$, and the critical isotherm has $M=\mathrm{sgn}(h)(|h|/v{)}^{1/5}$. The inverse magnetic susceptibility is $r$ above and $4|r|$ below. The minimized singular potential is $-|r{|}^{3/2}/(3\sqrt{v})$ below, so differentiating twice thermally gives a $|t{|}^{-1/2}$ singularity. These calculations giveThe correlation-length and correlation-function exponents follow from the same quadratic gradient term. A generic path with a nonzero positive quartic coefficient instead has ordinary critical behaviour; observing tricritical powers requires the extra tuning. The sextic coupling is marginal in three dimensions, giving tricritical upper critical dimension three and logarithmic corrections there.

A phase diagram locates stable pure thermodynamic phases and their phase coexistence boundaries in a space of control parameters. For a concrete three-dimensional example, take the temperature, crystal-field parameter, and magnetic field of the Blume–Capel model. Its spins take values $0,\pm 1$; changing the cost of the zero-spin state can turn a continuous ordering transition into a discontinuous one. A local scalar description near the resulting tricritical point is the Landau-Ginzburg theoryThe three local control coordinates are $(r,u,h)$, with $r$ varying linearly with the thermal scaling field. For $u>0$, $h=0$, the transition at $r=0$ is continuous. For $u<0$, zero-field coexistence is at $r=3u^2/(16v)$, and the magnetization jumps between zero and $\pm \sqrt{-3u/(4v)}$. The two loci meet at $r=u=h=0$, the tricritical point.

The full three-dimensional phase diagram has an ordered coexistence sheet at $h=0$, where the two opposite magnetizations exchange stability when $h$ changes sign. On its three-phase line it meets two nonzero-field tricritical wings. These wings are first-order phase transition surfaces between minima of the same sign but different magnitude, terminating at ordinary thermodynamic critical points. Their tricritical wing critical edges are found by $V^{\prime }=V^{\prime \prime }=V^{\prime \prime \prime }=0$:Indeed $V^{\prime \prime \prime }=6uM+20v{M}^3$ fixes $M_c^2$, and substituting into $V^{\prime \prime }=0$ and $V^{\prime }=0$ gives $r_c,h_c$; $V^{\prime \prime \prime \prime }=-12u>0$ ensures a stable critical quartic minimum. The two signs give the two wings. All transition surfaces and critical edges end together at the tricritical point. The following plot uses $v=1$; its wing surfaces come from the exact tricritical wing coexistence factorization, rather than schematic unconstrained surfaces.

The unheaded requests for critical exponents follow from the uniform equation of stateFor an ordinary thermodynamic critical point, $u>0$ is fixed and $r=At$, $A>0$. At $h=0$, $t<0$, the small ordered minimum obeys $M^2\sim -r/u$, so the order-parameter critical exponent is $\beta =1/2$. Above the transition $\chi =(\partial M/\partial h{)}_{h=0}=1/r$, giving the magnetic-susceptibility critical exponent $\gamma =1$. At $r=0$, $h\sim u{M}^3$, so the critical-isotherm exponent is $\delta =3$. At the tuned tricritical point, $u=0$, instead $M^4=-r/v$ below the transition and $h=v{M}^5$ on its critical isotherm, while the disordered susceptibility remains $1/r$. ThereforeBelow the transition the inverse susceptibility is $r+3u{M}^2+5v{M}^4$, giving $2|r|$ in the ordinary limit and $4|r|$ in the tricritical limit. Thus both sides give the same susceptibility power, with different amplitudes.

To derive the general Ginzburg criterion, tune away all lower even nonlinearities and let the leading stabilizer be $v_n{M}^{2n}/(2n)$, $n\ge 2$. The equation of state gives $M_0^2\asymp |r{|}^{1/(n-1)}$, and its curvature at the ordered saddle is $2(n-1)|r|$, so $\xi \asymp |r{|}^{-1/2}$. The fluctuations of the order parameter averaged over a correlation volume are estimated by its long-wavelength quadratic covariance:The integral has a fixed dimensionless upper limit after $q=p/\xi$; this is a coarse-volume fluctuation estimate, not the ultraviolet-dominated point variance. Consequently the multicritical Ginzburg ratio isThe Landau approximation is asymptotically controlled by this test only when the ratio tends to zero. This gives the upper critical dimensionThe same threshold follows by giving $M$ its Gaussian engineering dimension $(D-2)/2$: the coupling to $M^{2n}$ has scaling exponent $D-n(D-2)=2n-(n-1)D$, which becomes relevant below $D_c$.

For $D<D_c$, the fluctuation ratio diverges and the saddle calculation cannot predict the generic interacting critical exponents. At $D=D_c$ the test is marginal, not a proof of changed power exponents: the ratio has zero thermal power and marginal interactions can produce logarithmic corrections. Thus the PDF's inclusion of equality in a blanket claim about wrong power exponents needs this qualification. The Landau-Ginzburg theory as a fluctuating field theory remains useful; what loses validity is its fluctuation-free Landau approximation.

A phase diagram partitions a space of control parameters into regions with distinct equilibrium phases. Its boundaries mark nonanalyticities of the thermodynamic-limit free energy; a first-order phase transition has a jump in a first derivative, whereas a continuous phase transition has a continuously vanishing order parameter and singular fluctuations.

An explicit three-parameter example is the scalar Landau free energywith parameter axes $(r,u,h)$. These can be realized locally by the temperature, single-ion anisotropy and magnetic field of the Blume–Capel model. The three dimensions here refer to control-parameter space, not necessarily to three spatial dimensions. At $h=0$, minimizing this potential gives a continuous phase transition at $r=0$ for $u>0$. For $u<0$, coexistence occurs at $r=3u^2/(16v)$, whereThere are three global minima, $M=0$ and $M=\pm \sqrt{-3u/(4v)}$. Their order parameter jump decreases to zero as $u\uparrow 0$. Thus $(r,u,h)=(0,0,0)$ is a tricritical point. The two transition curves in the $h=0$ plane meet there.

In the three-dimensional phase diagram, crossing $h=0$ inside the ordered region switches between opposite signs of $M$, so that region is itself a first-order phase transition sheet. For $u<0$, two tricritical wings at opposite nonzero fields separate weakly and strongly magnetized phases. Their wing critical edges can be obtained rather than merely sketched: require $V^{\prime }=V^{\prime \prime }=V^{\prime \prime \prime }=0$. Since $V^{\prime \prime \prime }=6uM+20v{M}^3$, the nonzero solutions satisfyThe fourth derivative there is $-12u>0$, giving ordinary continuous critical endpoints. The wings meet the zero-field coexistence sheet along the tricritical three-phase line and terminate at these edges. Their exact tricritical wing coexistence factorization supplies the surfaces in the figure: for two nonnegative minima $a\le b$, set $s=a+b$, $p=ab$ andDirect expansion gives $V(M)-V(a)=(v/6)(M-a{)}^2(M-b{)}^2[(M+s{)}^2+p]\ge 0$, proving that these surfaces describe coexistence of global, not merely local, minima.

For the unheaded Ising model calculation, assume ferromagnetic exchange $J>0$ and let $M$ denote the mean Ising spin. On the hypercubic lattice the coordination number of a lattice is $z=2D$, even though the original positive-direction bond sum counts each bond only once. The mean-field approximation neglects products of deviations from $M$:There are $ND$ bonds, so the approximate energy is $H_{\mathrm{MF}}=-2DJM{\sum }_i{\sigma }_i+NDJ{M}^2$. Each independent Ising spin therefore sees the effective field $2DJM$. With ${\beta }_{\mathrm{th}}=1/(k_{B}T)$, averaging its two Boltzmann weights gives the self-consistency equationThe slope of its right side at zero is $T_c/T$, whereAbove $T_c$, $\tanh x<x$ for $x>0$ excludes a nonzero solution. Below $T_c$ the initial slope exceeds one, and concavity of the hyperbolic tangent on the positive axis gives one positive solution and its negative partner. The Ising auxiliary mean-field free energy $DJ{M}^2-k_{B}T\log [2\cosh (2{\beta }_{\mathrm{th}}DJM)]$ has positive quartic coefficient near $T_c$, while its quadratic coefficient changes sign there. Its stable minima therefore move continuously away from zero; this is a continuous phase transition in the mean-field theory of the Ising model.

Put $\tau =(T_c-T)/T_c>0$ and $c=T_c/T=(1-\tau {)}^{-1}$. The Taylor expansion $\tanh x=x-x^3/3+O(x^5)$ gives, on the nonzero branch,Consequently $M\sim \sqrt{3}{\tau }^{1/2}$ and the mean-field order-parameter exponent is $\beta =1/2$. The inverse-temperature symbol ${\beta }_{\mathrm{th}}$ used above is distinct from this order-parameter critical exponent.

Take $G,V,D,m,C_0>0$ and $T_E<0$, with the liquidus referenced to the pure-water melting point. The solid-liquid segregation coefficient $k_D=0$ means that newly formed ice contains no salt. A steady sharp front producing only pure ice cannot remove the incoming solute flux. More explicitly, in the liquid the steady solute flux is $J_C=-VC-D{C}_z=-V{C}_0$, whereas in pure solid it is zero. Interfacial salt conservation would require $V{C}_0=0$, a contradiction. An equilibrium mushy layer avoids this by carrying concentrated pore liquid down to the eutectic region, where a salt-bearing solid assemblage can form.

Use equal densities and common downward translation velocity $-V$. Write $q=1-\varphi$ for the liquid fraction; $\varphi$ here is the solid fraction. Local liquidus equilibrium in the mush requiresAbove a stationary roof $h_0$, the liquid advection-diffusion equation is $-V{C}_z=D{C}_{zz}$, with solutionThe marginal equilibrium at a mush-liquid boundary condition is $-m{C}_l^{\prime }(h_0)=G$, hence $C_i=C_0+GD/(mV)$. Matching the temperature and liquidus at the roof givesA positive equilibrium mush requires $(-m{C}_0-T_E)/G>D/V$; otherwise this positive-thickness construction does not apply.

Including solute diffusion in the liquid pore space, the steady solute flux in the mush isThus $q[C+(D/V)C_z]=C_0$. Inserting $C_z=-G/m$ and the value of $h_0$ yieldsThis is the diffusive steadily pulled equilibrium mush. Omitting the diffusive contribution would give a different solid-fraction distribution.

The imposed temperature and local liquidus coincide throughout the mush. In the liquid put $y=V(z-h_0)/D\ge 0$. Their difference iswith equality and matching first derivatives at the roof. The inequality follows because the derivative of $y-1+e^{-y}$ is $1-e^{-y}\ge 0$. Thus the liquid is not constitutionally supercooled. In the phase diagram, the pore-liquid trajectory follows the liquidus from the eutectic composition $C_E=-T_E/m$ to $(C_i,T_i)$, then curves into the liquid region and approaches $C=C_0$ as height increases. Pure-ice crystals have composition zero; the trajectory denotes the liquid composition, not the bulk mush composition.

After pulling stops, the imposed temperature keeps the pore-liquid composition fixed wherever the material remains mush. Solute conservation becomesBecause $C_{zz}=0$, this reduces toThe method of characteristics uses $dz/dt=-D/(z+T_E/G)$ and $d\varphi /dt=0$. HenceThese are the solute-diffusion characteristics in a fixed-temperature mush; their validity is local to the coexisting solid–liquid region.

For an explicit initial-data solution, define $H=-T_E/G$, $c=m{C}_0/G$ and $d=D/V$, so $h_0=H-c-d$. A characteristic through $(z,t)$ has initial footWhile that foot lies in the original mush and no phase boundary has removed the point,This recovers the initial distribution and shows that interior ice fraction increases as salt diffuses upward from colder, more concentrated pore liquid.

If one simply transports the initial zero-solid-fraction level, $f(\eta (h_0,0))=0$ implies the printed formal lawOn the physical concentration branch $h<H$, this is $h(t)=H-\sqrt{(H-h_0{)}^2-2Dt}$, so $\dot{h}=D/(H-h)>0$. It starts moving above $h_0$ immediately.

This formal continuation cannot be the physical mush–liquid roof of a column sealed to salt by the membrane. The no-flux constraint in a fixed-temperature mush is $J_C=-Dq{C}_z=0$ at the membrane. At a smooth zero-fraction equilibrium roof, however, $q=1$ and $C_z=-G/m$, giving $J_C=DG/m\ne 0$. Moreover the formal roof has left the membrane-bounded domain. The printed interface law is incompatible with a salt-impermeable membrane when interpreted as the physical roof. Its derivation above is a characteristic extrapolation; it does not enforce the stated boundary condition. If “impermeable” were intended to block bulk flow but allow salt diffusion, a different boundary condition would need to be specified.

The closed-column physical sketch is consequently different. Salt transported upward accumulates below the membrane and produces a liquid brine cap; the cap composition need not lie on the liquidus at every height. Its lower boundary can have a finite jump in solid fraction, so it need not follow the initial zero-fraction characteristic. Near the cold solid base, zero salt supply from below requires the liquid fraction to vanish, and a pure-ice region develops. In the surviving mush, the characteristic formula increases $\varphi$ relative to its initial value. An intermediate profile therefore has pure ice at the bottom, a remaining mush with elevated solid fraction, and liquid at the top. The plotted intermediate roof is schematic, not claimed to obey the inconsistent formal law.

One can characterize the final state quantitatively if the solid base also admits no salt diffusion, as in the usual pure-solid/no-solid-diffusion closure. In a stationary column, solute flux is constant and is zero at the sealed roof. A finite-width equilibrium mush would have $J_C=DGq/m$, forcing $q=0$ everywhere in that region; it therefore cannot remain mush. The final state is pure ice for $0<z<h_f$ and a uniform-composition liquid cap for $h_f<z<h_0$. At their boundary,The cap is liquid because $T(z)>-m{C}_f=T(h_f)$ above its base. The fixed external thermal field can supply or remove the heat involved; this is a solute-budget determination rather than an isolated enthalpy budget.

The final salt budget of a sealed mush column determines $h_f$. Initiallysince $(1-{\varphi }_0)C_m=C_0[1+d/(c+h_0-z)]$. Conservation requires $C_f(h_0-h_f)=N$. Writing the final cap depth ${\ell }_f=h_0-h_f$, the equation and its unique positive root areThe initial salt integral is less than $C_E{h}_0$, so $0<{\ell }_f<h_0$ and $0<h_f<h_0$; the resulting $C_f<C_E$ remains on the ice–brine branch above the eutectic. If the basal eutectic region is instead allowed to exchange solute with the column, this salt budget must include that flux and the numerical final height is not fixed by the printed data alone. No universal physical $h(t)$ follows from the interior characteristic equation without the cap and basal boundary conditions.

Use a planar saline Stefan problem with upward coordinate $z$, the initial contact at $z=0$, and phase boundaries $z=a(t)$ and $z=b(t)$ enclosing pure ice, where $a<b$. Fresh liquid occupies $z>b$ and brine occupies $z<a$. We assume negligible bulk flow, equal constant mass density $\rho$, specific heat capacity $c_p$ and thermal diffusivity $\kappa$ in all regions, and zero salt content and salt transport in the ice. Equal densities remove phase-change volume flow; suppressing convection isolates the molecular-transport mechanism. Different material properties would change the numerical coefficients. Set $K=\rho c_p\kappa$ for the common thermal conductivity and $\Delta T=T_m-T_{\infty }>0$. Denote the brine salt diffusion coefficient by $D$.

Use a linear ice liquidus, $T_L(C)=T_m-mC$, with $m>0$ and $C$ the salt mass fraction. Assume the initial brine is a stable liquid, $T_{\infty }>T_L(C_0)$, and all concentrations considered are below the eutectic composition. Thus $R=m{C}_0/\Delta T>1$. These assumptions exclude independent nucleation in a supercooled brine or an eutectic system event, neither of which is specified by the initial data alone. Start with a negligible seed of ice and use local phase equilibrium without interfacial kinetics or curvature. Temperature differences may be in Celsius, since only differences enter this calculation.

WriteThe paper calls $S$ a Stefan number; it is the latent-to-sensible heat ratio, reciprocal to another common Stefan number convention. At the lower phase boundary, define $T_i=T_L(C_i)$ and letThe liquidus relation is $\vartheta =Rc$.

A similarity solution gives an explicit calculation of the two positions. Putand use $u=z/(2\sqrt{\kappa t})$, $v=z/(2\sqrt{Dt})$. The temperatures and salinities areThese error function and complementary error function profiles satisfy the heat equation and the salt diffusion equation, their far-field conditions, and the interfacial temperature conditions. The fresh-water temperature is constant because the interface and the far field are both at $T_m$.