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For composition-independent $\chi$, the spinodal of Flory–Huggins solution theory is $2\chi =1/(N\varphi )+1/(1-\varphi )$. At its thermodynamic critical point, a second condition $-1/(N{\varphi }^2)+1/(1-\varphi {)}^2=0$ gives $1-{\varphi }_c=\sqrt{N}{\varphi }_c$, proving the displayed values. Hence ${\varphi }_c\sim N^{-1/2}$ and ${\chi }_c-1/2\sim N^{-1/2}$. Under $\chi =C/T$, the theta temperature is $\Theta =2C$ and $T_c=\Theta /(1+N^{-1/2}{)}^2$. Thus $T_c$ approaches a constant while $\Theta -T_c$ scales as $N^{-1/2}$. Converting ${\chi }_c$ to a temperature always requires a specified or inferred $\chi (T)$ law.

For a homogeneous polymer solution, examine the curvature of the mixing Helmholtz free energy with respect to composition. A composition fluctuation into equal volumes at $\varphi \pm \delta \varphi$ changes the average free energy byThus $f^{\prime \prime }>0$ locally resists fluctuations and $f^{\prime \prime }<0$ amplifies them, leading to spinodal decomposition. The spinodal curve is the locus $f^{\prime \prime }=0$, separating locally stable or metastable homogeneous compositions from locally unstable ones. The coexistence binodal instead follows a common tangent to the free energy. Between the binodal and spinodal, separation needs nucleation.

In Flory–Huggins solution theory, $\varphi$ is the volume fraction, $N$ is the number of equal-sized lattice segments per chain, $T$ is temperature, $k_{\mathrm{B}}$ is the Boltzmann constant, and $\chi$ is the dimensionless Flory interaction parameter. The stated $f$ is the mixing Helmholtz free energy per lattice site. The first two terms describe mixing entropy; the last describes the contact-energy contribution.

At fixed $T$ and composition-independent $\chi$,Hence the spinodal isThe homogeneous state is locally unstable when $\chi >{\chi }_s(\varphi )$. The minimum gives the critical point of a Flory–Huggins polymer solution:If $\chi$ depends on temperature, the equation $\chi (T)={\chi }_s(\varphi )$ converts this result into a temperature–composition boundary.

Differentiating the Landau theory free energy givesIts stationary values arewhen the discriminant is nonnegative. At $P_0$, the curvature is $a(T-T_c)$: a local minimum above $T_c$ and a local maximum below it. Exactly at $T_c$, $F^{\prime \prime \prime }(0)=-2b\ne 0$, so the origin is a stationary inflection point, not a stable minimum.

At a nonzero stationary point, eliminating $T-T_c$ yields $F^{\prime \prime }=P(2cP-b)$. Between $T_c$ and the upper spinodal, $P_{+}$ is a minimum and $P_{-}$ a maximum. Below $T_c$, $P_{-}<0$ becomes another local minimum, while $P_{+}>0$ is the lower minimum. Thus loss of local stability at $T_c$ is distinct from the equilibrium transition temperature.

The two nonzero stationary points merge when their discriminant vanishes, giving the upper spinodalAt a nonzero stationary point, $a(T-T_c)=bP-c{P}^2$, so its free energy reduces to $F=P^3(b/6-cP/4)$. Equality with $F(0)=0$ therefore requires $P=2b/(3c)$. Substitution givesBoth have positive curvature $2b^2/(9c)$ at coexistence. The intervening maximum is at $P=b/(3c)$. For $T_2<T<T_1$ the positive-polarization minimum is metastable; for $T_c<T<T_2$ the zero-polarization minimum is metastable.

A first-order phase transition has a discontinuity in a first derivative of the equilibrium free energy, such as entropy or the order parameter; a thermal first-order transition normally has latent heat. At a continuous phase transition the order parameter goes continuously to zero and there is no latent heat, although response functions and higher derivatives of the free energy can be singular.

For the symmetric scalar Landau free energy at zero conjugate field, minimizeIf $u>0$, the sixth-order term is unnecessary near $r=0$. For $r>0$ the minimum is $M=0$, whereas for $r<0$ there are two symmetry-related minima with $M^2=-r/u$ to leading order. The order parameter vanishes continuously as $r\uparrow 0$. The equilibrium free-energy density changes from zero to $-r^2/(4u)$, so its first temperature derivative is continuous and its second derivative has a jump. This is the ordinary mean-field continuous phase transition.

If $u<0$, keep $v>0$ for stability. A nonzero stationary order parameter, with $s=M^2$, obeys $r+us+v{s}^2=0$. For phase coexistence with $M=0$ its free energy must also vanish. Substituting $r=-us-v{s}^2$ into $V$ givesThe nonzero solution of $V=0$ is thereforeThe competing minima exchange stability at positive $r_t$, with a finite jump of the order parameter. This is a first-order phase transition. The metastability region is also visible: nonzero stationary points first appear at the spinodal $r=u^2/(4v)$, and the disordered local minimum loses stability at $r=0$.

For example, holding $u,v$ fixed and taking $r=a(T-T_0)$, the ordered phase has entropy lower than the disordered phase by $a{M}_t^2/2$ at coexistence. Heating through the transition absorbs latent heat per volume $T_{t}a{M}_t^2/2$. More generally the temperature dependence of all coefficients contributes to this entropy difference. With no inversion symmetry, a cubic term can also produce a first-order phase transition, as comparison of the competing minima of the Landau free energy shows.

Pressure and its physical terms. Solving the molar Van der Waals equation for pressure givesThe excluded-volume parameter $b$ reduces the volume available for translational motion, raising the kinetic pressure relative to $RT/V$. The attraction parameter $a$ lowers the pressure because attractive interactions reduce the momentum transferred to the boundary. This mean-field model treats $a,b$ as temperature-independent material constants. For one mole, $V$ is the sample volume with the molar amount already fixed; $b$ has units of that volume and $a$ of pressure times volume squared.

volume dependence of entropy. The Helmholtz free energy has differential $dF=-SdT-PdV$. Equality of mixed derivatives yields the Maxwell relationFor $a=0$ this derivative is $R/(V-b)$, givingEquivalently, $S=R\ln (V-b)+f(T)$ with a common reference volume understood inside the logarithm. The equation of state fixes this volume dependence, not the otherwise unspecified function of temperature.

Excess entropy at fixed volume and temperature. The ideal gas has $(\partial S_{\mathrm{id}}/\partial V{)}_T=R/V$. ThereforeIntegrate from infinite volume, where the entropy difference is specified to vanish:It is negative because excluding configurations reduces the accessible state count. It tends to zero as $V\to \infty$ and has leading dilute limit $-Rb/V$. This is the entropy excess of a van der Waals gas.

Effect of attraction. Restoring a temperature-independent $a$ changes the pressure by $-a/V^2$, but not its derivative with respect to temperature. With the same infinite-volume entropy reference, it therefore does not change $\Delta S^{\mathrm{ex}}$ at fixed $V,T$. This can also be seen fromThe attraction affects internal energy and the equilibrium volume, so it can affect entropy indirectly when different state points are compared. If $a$ were temperature dependent, its derivative would contribute directly to the entropy; that is outside the stipulated constant-parameter model.

Coexistence isotherm. Below the Van der Waals critical point, the formal homogeneous isotherm has a local minimum and maximum. Its middle branch has $\partial P/\partial V>0$, hence negative isothermal compressibility, and is mechanically unstable. Portions of the negative-slope branches can be metastable rather than globally stable. The equilibrium isotherm replaces the loop between liquid volume $V_l$ and vapour volume $V_g$ by a constant saturation pressure $P_s$. The two volume are fixed byThe Maxwell equal-area construction expresses equality of the two phases' chemical potentials. Intermediate total volume are mixtures: changing the vapour fraction changes total volume at the same pressure, rather than following the unstable homogeneous curve. The spinodals are the turning points of the formal isotherm; they are distinct from the coexistence endpoints.

Dense-phase volume at low temperature. At a general subcritical temperature, the two coexistence equations above must be solved together. For this equation of state the area condition is explicitlyThis supplies an exact implicit determination of the liquid volume even when a simple closed form is unavailable.

For the asymptotically low-temperature limit, put $\tau =RTb/a\ll 1$. The liquid is dense, $V_l\simeq b$, while the vapour is dilute, $P_s{V}_g\simeq RT$ and $V_g\gg V_l$. The area condition then givesThus the vapour volume grows exponentially in $1/\tau$, and the saturation pressure is exponentially small. It is consequently consistent to neglect $P_s$ in the liquid equation to algebraic order in $\tau$:Choose the root near $b$, not the large-volume zero-pressure root:Expanding gives the leading requested dense-phase behavior,This is the low-temperature liquid volume of a van der Waals gas. Setting $P_s=0$ is an asymptotic approximation, not an exact coexistence condition at every temperature below $T_c$. The figure illustrates generic subcritical coexistence; its finite-temperature liquid volume is determined by the full equal-area equations. The van der Waals liquid also ceases to be a realistic description at sufficiently low physical temperature, where other phases or quantum effects can matter.

For equal-size molecules, let $f$ be the mixing Helmholtz free energy per lattice site. Flory–Huggins solution theory reduces to the regular solution model:The first two terms are minus the combinatorial entropy of mixing; the last is the excess cost of unlike contacts. A positive Flory interaction parameter penalizes such contacts. Additive terms linear in composition change the chosen pure-component reference but do not change stability.

A small composition fluctuation at fixed mean composition costs energy proportional to $f^{\prime \prime }$. Thus local stability requires $f^{\prime \prime }>0$, and the spinodal isInside this curve, $f^{\prime \prime }<0$ and arbitrarily small composition fluctuations grow. The minimum of ${\chi }_s$ is at $\varphi =1/2$, giving the critical value ${\chi }_c=2$. At the thermodynamic critical point, $f^{(2)}=f^{(3)}=0$ while $f^{(4)}>0$.

For $\chi =0$, $f$ is strictly convex with its minimum at $1/2$. At $\chi =2$, the central quadratic term vanishes but a positive quartic term remains. At $\chi =2.5$, there are two minima and a central maximum; the spinodal compositions are $(1\pm \sqrt{1-2/\chi })/2\simeq 0.2764,0.7236$. The common-tangent construction gives coexisting compositions ${\varphi }_b\simeq 0.1448$ and $1-{\varphi }_b$, satisfying $\ln [{\varphi }_b/(1-{\varphi }_b)]+\chi (1-2{\varphi }_b)=0$. Between a binodal endpoint and the spinodal, the homogeneous state is locally stable but metastable; global stability requires the convex envelope of $f$.

Use polymer volume fraction $\varphi$ and interaction parameter $\chi$, increasing toward poor-solvent conditions. The Flory–Huggins solution theory free energy per lattice site isThe common-tangent construction gives the binodal; $f^{\prime \prime }=0$ gives the spinodal, $2{\chi }_s=1/(N\varphi )+1/(1-\varphi )$. Its critical point is ${\varphi }_c=1/(1+\sqrt{N})$, ${\chi }_c=\frac{1}{2}(1+N^{-1/2}{)}^2$. Above the binodal in this $\chi$ plot, the solution separates into polymer-poor and polymer-rich phases; between binodal and spinodal it is metastable, while inside the spinodal a homogeneous state is unstable.

The line $\chi =1/2$ is the theta solvent condition for a long dilute chain. Below it is the good solvent side; above it is the poor solvent side. Poor solvent does not imply that every composition is already phase separated. Independently, a homogeneous solution is dilute for $\varphi \ll {\varphi }^{\ast }$, semidilute for $\varphi \gg {\varphi }^{\ast }$ below close packing, and concentrated as $\varphi$ approaches one.

The Flory interaction parameter measures the energetic preference for unlike rather than like contacts in Flory–Huggins solution theory. For polymer volume fraction $\varphi$ and $N$ lattice sites per chain, the dimensionless mixing Helmholtz free energy per site isThe logarithmic terms represent combinatorial entropy; polymer connectivity reduces the chain's translational mixing entropy by the factor $N$. In a simple contact model $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$, where $z$ is the coordination number. Positive $\chi$ penalizes polymer-solvent contacts. Increasing it can overcome mixing entropy and cause phase separation; a negative second derivative of the mixing free energy marks the spinodal region. For a long dilute chain, $\chi <1/2$ corresponds to a good solvent, $\chi =1/2$ to the idealized theta condition, and $\chi >1/2$ to a poor solvent. Real effective parameters can depend on temperature and concentration, rather than being universal chemical constants.

Use polymer volume fraction $\varphi$ horizontally and the Flory interaction parameter $\chi$ vertically. Lower $\chi$ gives better solvent quality. The simplest Flory–Huggins solution theory assigns the dimensionless mixing free energy per lattice siteThus $\chi <1/2$ gives positive excluded volume and a good-solvent coil, $\chi =1/2$ is the long-chain theta condition, and $\chi >1/2$ is the poor-solvent side. It is the net two-segment interaction that vanishes at theta, not every molecular force.

At fixed good-solvent quality, the polymer overlap volume fraction ${\varphi }^{\ast }\sim N{a}^3/R_g^3$ separates isolated coils from a semidilute polymer solution. Above overlap, many chains interpenetrate, but there is still solvent on the segment scale. As $\varphi$ approaches order unity the solution is concentrated; these concentration-regime boundaries are crossovers, not first-order phase transitions. The overlap fraction increases from order $N^{-4/5}$ in a strong good solvent toward order $N^{-1/2}$ at theta. The crossover can be expressed as ${\varphi }^{\ast }\sim N^{-4/5}(1-2\chi {)}^{-3/5}$ while $1-2\chi \gg N^{-1/2}$; it saturates at the theta scaling when that condition fails.

On the poor-solvent side there can instead be coexistence of polymer-poor and polymer-rich phases. The binodal follows a common tangent of $f$; the spinodal followsThe critical point of a Flory–Huggins polymer solution isFor finite $N$, ${\chi }_c$ lies above $1/2$. Therefore poor solvent does not automatically mean demixing at every concentration, and an overlap line must not be mistaken for a coexistence boundary. The original figure uses the stated ideal model for an illustrative chain length; it does not assert a measured phase boundary. A vertical temperature axis would additionally require a material-specific relation $\chi (T)$.

In a regular solution, ideal combinatorial entropy competes with a nonzero unlike-contact energy. For equal molecular volumes and mole fraction $\varphi$ of species A, write mixing free energy per site aswhereHere $z$ is the coordination number and ${ϵ}_{ij}$ a pair-contact energy counted consistently in the lattice model. A positive $\chi$ penalizes unlike contacts; a negative value favors them. The logarithmic terms lower free energy on mixing. The sign of the enthalpy of mixing alone therefore does not determine miscibility.

At fixed temperature and composition, a homogeneous phase is globally stable only if no division into two compositions lowers its free energy. The common-tangent construction selects coexisting compositions ${\varphi }_a,{\varphi }_b$, equivalent to equality of both species' chemical potentials. Local stability to infinitesimal fluctuations instead requiresZero curvature gives the spinodal, while the coexistence boundary is the binodal. Between them a homogeneous state can be metastable and require nucleation; inside the spinodal its small composition fluctuations grow. The symmetric regular solution hasBelow this contact penalty, the free energy is convex at every composition. Above it a miscibility gap is possible around the central composition. If $\chi =A/T$ with $A>0$, heating promotes mixing, but specific interactions or temperature-dependent noncombinatorial terms can produce other trends.

For two polymers, let $N_A,N_B$ count equal-volume segments per chain, and now let $\varphi$ be the segment volume fraction of A. Connecting segments into chains reduces their independent translational entropy. The binary Flory-Huggins polymer blend hasThe energetic contact term remains extensive in the number of segments, while the favorable entropy per segment is reduced by $1/N_A$ and $1/N_B$. This is the key long-chain limit of polymer-blend miscibility: a small positive incompatibility can overcome the mixing entropy. The lattice treatment is a controlled model for this argument, rather than a claim that every real blend has composition-independent $\chi$.^[ref]

The local stability condition and critical interaction parameter becomeFor equal chain lengths, ${\chi }_c=2/N$. Similar segment chemistry and matched cohesive interactions can reduce the unlike-contact penalty; favorable specific interactions, such as complementary hydrogen bonding, can make mixing favorable even for long chains. Lower molar mass increases mixing entropy. temperature can help when it lowers the effective $\chi$, but heating is not universally sufficient. A suitable common solvent can facilitate processing or ternary mixing, and a block-copolymer compatibilizer can stabilize fine domains; neither guarantees that the final solvent-free binary material is one equilibrium molecular phase.

An oil–water emulsion illustrates emulsion stability versus molecular miscibility. A surfactant adsorbs with its water-affine head toward water and its oil-affine tail toward oil, lowering interfacial tension and the work $\gamma dA$ needed to make additional interface. Breaking a fixed oil volume $V_o$ into spherical droplets of radius $R$ gives total area $3V_o/R$, so creating small droplets still requires substantial area and usually mechanical work.

Adsorbed surfactants can oppose droplet coalescence through charged double-layer repulsion, hydrated-head interactions or steric repulsion from sufficiently extended groups. Adequate coverage, adsorption time, ionic conditions and film drainage affect the lifetime. A lower interfacial tension does not itself remove the bulk miscibility gap, and a protected emulsion is commonly kinetically stabilized. A thermodynamically stable microemulsion is a distinct situation requiring an appropriate composition and free-energy balance. The two liquids must not be called molecularly miscible simply because droplets disperse visibly.

For the capillary, balance upward contact-line force against hydrostatic column weight, neglecting meniscus volume and gas density. For inner radius $r$,The contact angle is measured through the liquid. complete wetting means ${\theta }_2=0$, so the contact-angle change of capillary rise givesThis follows from the stipulated wetting change, holding radius, bulk liquid surface tension and density fixed. One should not replace that premise with a general claim that grease always improves glass wetting; if contamination also changes the liquid surface tension, the original height and new angle alone would not determine the new height.

For two wide vertical plates separated by gap $d$, use unit width. There are two upward contact-line contributions and liquid cross-sectional area $d$, giving the capillary rise between parallel plates:Only one principal meniscus curvature contributes between wide plates, whereas a circular tube has two. For the same fluid and angle,If the plate gap equals the tube diameter, $d=2r$, the rise is half as large: the initial and completely wetting heights would be $2.50$ cm and $3.05$ cm. If instead $d=r$, the heights are equal. In either geometry, changing only the angle from $35^{\circ }$ to zero multiplies the height by $1/\cos 35^{\circ }$. Since the question specifies no plate spacing, the general gap-dependent relation is the unambiguous result; the half-height answer requires the equal-gap-to-diameter convention.

The Flory interaction parameter $\chi$ measures the excess cost of unlike segment contacts. In a simple lattice with coordination number $z$, contact energies ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ and temperature $T$,A positive Flory interaction parameter penalizes unlike contacts; a negative value favors them. Mixing still depends on the balance with entropy of mixing, rather than the sign of $\chi$ alone.

For two polymers of degrees of polymerization $N_A,N_B$, let $\varphi$ be the volume fraction of species $A$. The binary Flory-Huggins polymer blend has mixing free energy per lattice siteThe logarithmic terms favor mixing. Their factors $1/N_A$ and $1/N_B$ arise because an entire chain translates as one object; connecting many segments greatly reduces the mixing entropy per segment. Hence long polymers can demix for a much smaller positive Flory interaction parameter than small molecules.

For composition-independent $\chi$, the local stability criterion isPositive curvature means stability to small composition fluctuations, zero curvature locates the spinodal, and negative curvature means spontaneous growth of fluctuations. Actual two-phase coexistence is determined by the common-tangent construction, or equality of both species' chemical potentials; its boundary is the binodal. A locally stable but globally unfavourable homogeneous mixture can lie between the binodal and spinodal.

Setting the second and third derivatives of $f$ to zero givesFor the symmetric Flory-Huggins polymer blend, ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. For a polymer solution with $N_B=1$ and long chains of species $A$, ${\chi }_c\to 1/2$, consistent with the theta solvent limit. An effective Flory interaction parameter can depend on temperature and composition; in the simple energetic case $\chi \propto 1/T$, heating encourages mixing, but that temperature trend is not universal.

The original illustration uses stipulated $N_A=N_B=100$. At $\chi =0.01$ the mixing free energy is convex, whereas at $\chi =0.03$ a central unstable region appears. The horizontal common-tangent construction in the latter case joins the symmetric coexisting compositions; its contacts differ from the spinodal boundaries. These are model curves, not supplied experimental data.

Local compositional stability requires positive curvature of the mixing free energy. Differentiating twice at fixed temperature, with $\chi$ independent of composition as in the given model, givesThe spinodal occurs where this vanishes:Between the two spinodal compositions at sufficiently large $\chi$, $f^{\prime \prime }<0$ and a homogeneous mixture is unstable to infinitesimal composition fluctuations. The binodal is instead fixed by the common-tangent construction, and encloses a larger coexistence region. Between binodal and spinodal the mixture is locally stable but metastable.

A phase diagram with a conventional upper critical temperature additionally assumes that $\chi$ decreases with $T$, for example $\chi =C/T$ with $C>0$. Then high temperature gives a uniform mixture; cooling produces an asymmetric coexistence dome whose critical composition is on the dilute-polymer side. The spinodal lies inside the binodal dome. An arbitrary $\chi (T)$ need not give this orientation, so the free-energy formula alone does not specify a unique temperature axis.

Interpret the stated transition as an upper critical miscibility temperature: cooling increases the effective $\chi$ through ${\chi }_c$. Above $T_c$, the homogeneous mixing free energy is convex throughout $0<\varphi <1$, with no two-phase common tangent. Below $T_c$, it acquires a region with $f^{\prime \prime }<0$, bordered by the spinodal. A lower straight line touches the curve at two binodal compositions ${\varphi }_{-}$ and ${\varphi }_{+}$; this common-tangent construction gives the equilibrium polymer-poor and polymer-rich phases.

The precise raw curve need not have two equal minima: subtracting an affine function of $\varphi$ changes reference chemical potentials without changing curvature or coexistence compositions. The free-energy panels above subtract the tangent at ${\varphi }_c$ from the upper-temperature curve and the common tangent from the lower-temperature curve. This makes the common tangent and the convex-versus-nonconvex distinction visible.

If the mean initial composition ${\varphi }_0$ lies inside the low-temperature spinodal region, an arbitrarily small separation into equal-volume regions with compositions ${\varphi }_0\pm \delta \varphi$ changes the mean free energy bySmall fluctuations therefore grow spontaneously: this is spinodal decomposition. Interfacial costs limit very short-wavelength fluctuations, and the polymer-poor and polymer-rich domains subsequently coarsen towards their binodal compositions.

If ${\varphi }_0$ lies just outside a spinodal branch but still inside the binodal, then $f^{\prime \prime }({\varphi }_0)>0$. Small fluctuations increase free energy, so the uniform mixture is metastable. Separation requires a sufficiently large nucleus to overcome the interfacial barrier, followed by nucleation and growth. There is no guarantee that this happens rapidly. If the initial composition is outside the binodal as well, it remains a stable uniform phase after cooling.

For a mean composition in the coexistence interval, conservation of polymer fixes the volume fraction $\alpha$ of the rich phase by the lever rule:The temperatures alone do not determine which mechanism occurs without the initial composition. Numerical free-energy curves additionally require $\chi (T)$; the plotted example uses $\chi (T)={\chi }_c{T}_c/T$ with absolute temperatures $T_c=298.15\mathrm{K}$, $T_1=303.15\mathrm{K}$ and $T_2=293.15\mathrm{K}$.

Fix $c>0$. Stationary points satisfy $m(a+b{m}^2+c{m}^4)=0$. With $x=m^2$, the nonzero candidates areKeep only real positive roots. At such a point,so $x_{+}$ gives local minima and $x_{-}$ gives local maxima when both exist. The solution $m=0$ is a local minimum for $a>0$. To choose the phase, compare free energies: using the stationary point relation $a=-bx-c{x}^2$ givesFor $b>0$, this expression is negative whenever a nonzero stationary minimum exists. The paramagnetism phase is stable for $a>0$, and the ferromagnetism phase with $m=\pm \sqrt{x_{+}}$ is stable for $a<0$. The phase transition at $a=0$, $b>0$, is continuous and second order.

For $b<0$, equality with $F(0)$ requires $x=-3b/(4c)$. Substitution into the stationary equation gives the first-order coexistence curveThe disordered phase is the global minimum above this curve, and the ordered phase is the global minimum below it. At coexistence the factorizationexhibits three equally low minima, $m=0$ and $m=\pm \sqrt{-3b/(4c)}$, proving the discontinuous jump.

The ordered spinodal is $a=b^2/(4c)$, $b<0$, where its local minimum meets the intervening local maximum. The disordered spinodal is $a=0$, $b<0$. Thus the ordered phase is metastable for $a_{\mathrm{t}}<a<b^2/(4c)$; the disordered phase is metastable for $0<a<a_{\mathrm{t}}$. These spinodals are limits of metastability, not equilibrium transition curves.

The first-order curve meets the continuous transition line at the tricritical point $(a,b)=(0,0)$. On the specially tuned path $b=0$, the equilibrium ordered solution is $m=\pm (-a/c{)}^{1/4}$ for $a<0$ and tends continuously to zero at the origin.

A regular solution keeps the ideal combinatorial entropy of mixing while allowing a nonzero energetic preference for unlike contacts. For a symmetric binary mixture with equal molecular sizes, the displayed free energy per site combines mixing entropy with a contact penalty $\chi \varphi (1-\varphi )$. Its spinodal satisfies $1/\varphi +1/(1-\varphi )-2\chi =0$. At the symmetric thermodynamic critical point, the third composition derivative also vanishes, giving ${\varphi }_c=1/2$ and ${\chi }_c=2$. If $\chi =A/T$, the critical temperature is $A/2$. This is the equal-size limit of Flory–Huggins solution theory.

The spinodal is the boundary where a metastable phase loses local stability. In a mean-field free-energy description the relevant curvature vanishes there, separating local minima from unstable configurations.

Spinodal decomposition is spontaneous growth of composition fluctuations in a homogeneous mixture whose local mixing free energy has negative composition curvature. For a mean $\varphi$, equal regions at $\varphi \pm \delta \varphi$ change the mean free energy by $f^{\prime \prime }(\varphi )(\delta \varphi {)}^2/2+O((\delta \varphi {)}^4)$, which is negative inside the spinodal. Interfacial costs suppress sufficiently short wavelengths, while conservation of composition constrains the dynamics. Outside the spinodal but inside the binodal, the uniform state is instead metastable and requires nucleation to begin separation.