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A binary fluid mixture contains two components with a variable local composition. Temperature, pressure and interactions determine whether a uniform mixture is stable or separates into phases with different compositions. Composition is a conserved quantity in a closed sample even while individual molecules move.

A compositional order parameter can be the local concentration difference. A free energy with two competing minima favors phase separation, while interfacial tension penalizes the boundaries between domains. Conserved order-parameter dynamics describes diffusive redistribution; fluid motion additionally requires a hydrodynamic description such as Model H dynamics. Watching domain growth after a temperature change therefore probes both equilibrium driving forces and the transport processes that move material.

A compositionally enriched biomolecular assembly. Calling an assembly a condensate does not alone establish equilibrium liquid–liquid phase separation or a particular biological function.

Researcher in chromatin organization. Gibson and collaborators investigated intrinsic and regulated phase separation of chromatin, testing how interactions among its components can generate collective organization.^[ref]

The Gibbs free energy of mixing compares the final mixture's Gibbs free energy with that of its initially separate components at the same temperature and pressure. For an ideal solution, subtract the pure-component reference from $G={\sum }_i{n}_i{\mu }_i$:For one mole of the chosen A/B ideal solution at $300\mathrm{K}$ and $x_A=0.4$,The individual chemical potential changes are $RT\ln 0.4\approx -2.28554\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for A and $RT\ln 0.6\approx -1.27417\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ for B. Weighting these by $0.4\mathrm{mol}$ and $0.6\mathrm{mol}$ reproduces the total Gibbs free energy of mixing.

If ideality holds in a neighborhood of the specified temperature and pressure, differentiation at fixed component amounts givesThus the ideal mixing Gibbs free energy decrease comes entirely from the entropy increase; the mixing enthalpy and volume changes vanish. Ideality at a single isolated state alone does not determine temperature or pressure derivatives.^[ref]

The same formula rules out spontaneous separation into two liquid thermodynamic phases within this model. Per mole, the mixing Gibbs free energy $g_{\mathrm{mix}}=RT[x\ln x+(1-x)\ln (1-x)]$ hasAdding the pure-component reference contributions changes only a term linear in $x$. The total molar Gibbs free energy is therefore a strictly convex function: a separated pair of different compositions has a larger composition-weighted Gibbs free energy than the uniform sample with the same inventory. Such phase separation requires physics beyond this ideal solution model.

The Mixture measurements and the Gibbs-Duhem constraint article describes the inspected acetone–chloroform preparation, apparatus, paired readouts, controls and sampling limits.^[ref]

That existing apparatus account supplies the experimental controls and limitations. Its refractive index calibration yields mass fractions, which must be converted to mole fractions. For a measured component-A mass fraction $w_A$ and component molar masses $M_A,M_B$,Apply this conversion separately to the flask sample and to completely condensed vapor to obtain $x_A$ and $y_A$. Comparing $y_{i}P$ with $x_i{p}_i^{\ast }(T)$ tests Raoult's law at the measured temperature; using the changing temperature at each composition is essential. The chosen A/B values above are unrelated to the protocol's substances.

The more general diagnostic, retaining an ideal gas vapor and negligible pure-liquid pressure correction, isThis follows by replacing $\ln x_i$ with $\ln (x_i{\gamma }_i)$ in the chemical potential subtraction used for Raoult's law. A reproducible discrepancy from ${\gamma }_i=1$ can expose failure of the ideal solution approximation, but interpreting it requires checking measurement uncertainty, representative paired sampling and the ideal gas approximation. Apparent agreement within uncertainty supports an approximation over the tested conditions; it cannot establish exact ideality at every composition.

Positive or negative deviations of component partial pressures require activity coefficients different from one under these assumptions. An azeotrope is also absent from the chosen unequal-vapor pressure model: the explicit inequality in ideal binary vapor composition gives $y_A>x_A$ throughout $0<x_A<1$. An interior equality would require $p_A^{\ast }=p_B^{\ast }$; in that degenerate case every composition has $y_A=x_A$, rather than an isolated enrichment reversal. Neither an azeotrope nor liquid phase separation should be inferred from the ideal curves.

A partition function sums the Boltzmann weights of all states: $Z={\sum }_s{e}^{-{\beta }_{T}H(s)}$ for discrete states, or the corresponding integral for continuous fields. It normalizes equilibrium probabilities. Its physical Helmholtz free energy is $\mathcal{F}=-k_{B}T\log Z$, and its free-energy density is $\mathcal{F}/V$. With a momentum ultraviolet cutoff ${\Lambda }_0$, a scalar-field example isThe normalization of the functional measure and field-independent energy terms must be tracked when discussing absolute free energy.

Choose a smaller cutoff $\Lambda$ and split the field into retained modes ${\varphi }_{<}$ with $|q|<\Lambda$ and eliminated modes ${\varphi }_{>}$ with $\Lambda <|q|<{\Lambda }_0$. Define the Wilsonian coarse-grained statistical Hamiltonian bySubstitution into the partition function proves $Z=\int \mathcal{D}{\varphi }_{<}{e}^{-{\beta }_T{H}_{\Lambda }[{\varphi }_{<}]}$ for every intermediate cutoff. A source retained in this definition similarly preserves long-wavelength connected correlation functions and response. Thus the coefficients in a Hamiltonian density must run with $\Lambda$: the dependence induced by eliminated modes compensates the dependence of the remaining field integral. Holding the original bare coefficients fixed while changing the cutoff would generally change the system. Exact integration generates all interactions allowed by symmetry, including higher powers, derivative terms and an additive constant; a local derivative expansion or a truncation to a few couplings is an extra approximation.

Cutoff independence makes arbitrary intermediate coarse-graining scales unphysical. It allows calculations at the scale of the observable, avoids repeatedly resolving irrelevant short-distance details and exposes the renormalization-group fixed points responsible for universality. A microscopic cutoff, such as a Bravais lattice spacing, can still label a physical microscopic model; taking it to infinity as a continuum limit requires tuning appropriate bare couplings. This is distinct from lowering an intermediate Wilsonian cutoff in a fixed model.

To interpret the zero-cutoff limit, define $M=V^{-1}\int \varphi (x)d^{D}x$, the spatially uniform mode of the order parameter. Keep this mode fixed and integrate the others, rather than integrating over $M$ too. Write $\varphi =M+\eta$, with $\int \eta =0$, and define the constrained order-parameter free energy byIn a finite periodic box, once $\Lambda$ is below the smallest nonzero momentum, all nonconstant modes have been eliminated. The remaining effective energy is $H_{\Lambda }[M]=V{U}_{\Lambda }(M)$, so its defining integral is exactly $Z_M$. Therefore, with consistent normalization,When $\mathcal{H}(\Lambda ,M)$ denotes this uniform effective Hamiltonian density, this is the stated identification of $F(M)$ with its zero-cutoff limit. It requires a free-energy density constrained at fixed $M$, not the unconstrained extensive free energy; all integrated fluctuations and additive constants are included, the zero mode in field theory is retained, and a density for the uniform configuration is well defined. The finite-volume argument avoids an interchange of the thermodynamic and cutoff limits without justification. On a homogeneous pure-phase branch one may use a local potential description, while a thermodynamic limit allowing phase separation can convexify the constrained potential. A bare nonconvex Landau polynomial is accordingly a homogeneous-branch approximation, not a claim that the exact equilibrium Legendre potential stays nonconvex.

In a field $h$, the remaining integral is $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the consistently chosen zero-mode measure. The thermodynamic saddle then gives $f(h)={\inf }_M\{F_c(M)-hM\}$ and selects the equilibrium magnetization. Landau theory assumes that the relevant uniform potential can be approximated by a regular analytic expansion in $M$, with symmetry-constrained coefficients, and minimizes it. At tree level the integrated fluctuations are neglected, recovering the familiar bare polynomial. With fluctuations retained, the coefficients run and can become singular near criticality; the zero-cutoff identification of constrained free energy alone does not justify mean-field critical exponents.

For an ordinary short-range scalar thermodynamic critical point, the leading interaction is quartic. Absorb inverse temperature and the gradient coefficient into field normalization and writeScale $x=b{x}^{\prime }$ and $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$ to leave the gradient term invariant. The quartic term changes to $g{b}^{4-D}\int {\varphi }^{\prime 4}/4!$, so its engineering scaling exponent is $4-D$. It is irrelevant at the Gaussian fixed point for $D>4$, relevant for $D<4$, and marginal for $D=4$. HenceThe loop expansion makes the breakdown concrete. A quartic-vertex correction is proportional to $g^2{I}_2(r)$, whereFor $0<D<4$, rescaling $q=\sqrt{r}k$ gives $I_2(r)\sim C_D{r}^{(D-4)/2}$, diverging as $r\to 0$. At $D=4$, radial integration giveswhich is logarithmically divergent. Above four dimensions its infrared limit is finite; the leading cutoff-dependent constant can be absorbed into a coupling renormalization. The effective expansion parameter is thus of order $g{\xi }^{4-D}$ below four, or $g\log (\Lambda \xi )$ at four, since $\xi \sim r^{-1/2}$. It eventually becomes large for any fixed bare interaction in the lower-dimensional critical limit, defeating the uncorrected mean-field expansion. The Ginzburg criterion gives the same threshold by comparing correlation-volume fluctuations with the squared mean order parameter.

Below four dimensions an interacting renormalization-group fixed point can replace the Gaussian fixed point and change critical exponents, provided a continuous transition exists. At four dimensions the interaction is marginal and produces logarithmic corrections: the leading power exponents can remain their mean-field values, so “breakdown” there means that pure mean-field power laws omit those corrections, rather than necessarily that every power exponent changes. Above four the stabilizing quartic interaction is dangerously irrelevant for the ordered phase, explaining why mean-field thermodynamic exponents need not obey naive hyperscaling with the spatial dimension.

For two species of chains with the same polymerization degree $N$, the number of independently translating objects per occupied site is reduced by $N$. The symmetric Flory-Huggins polymer blend hasThe symmetry still places the thermodynamic critical point at ${\varphi }_c=1/2$, but nowThe entropy of mixing per monomer is much smaller than for simple liquids, so even a weak unfavorable interaction can cause phase separation for long chains. If the energetic interaction parameter is $\chi =A/T$ with $A>0$, this model gives $T_c=AN/2$; the conclusion relies on that specified temperature law.

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.

Pilong Li investigated phase separation produced by interactions among multivalent signaling proteins. The study connected the onset of droplet formation with a transition from small complexes to large assemblies, and related the threshold to the number of interaction sites.^[ref]

Researcher in membrane biophysics. With Sarah L. Veatch, Keller studied liquid-phase separation in vesicles containing phospholipids and cholesterol, connecting composition with coexistence of distinct membrane regions.^[ref]

Researcher in membrane biophysics. With Sarah L. Keller, Veatch investigated liquid-phase separation in giant vesicles, connecting observable membrane domains with physical interactions among lipids.^[ref]

W. Craig Carter is a materials scientist at MIT whose research applies physical analysis and computation to the evolution of microstructure. His departmental profile identifies work on material properties, phase transitions, fracture and battery-electrode materials.^[ref]

MIT's official course page identifies him as the instructor of its Fall2002 Thermodynamics of Materials course. The associated Unstable Solutions: MIT 3.00 Lecture 32 by W. C. Carter prints the byline W.C.Carter and develops the regular solution model, common-tangent construction and instability toward phase separation.^[ref]

An original dimensionless version of the regular solution model makes one teaching point explicit. Let $x$ be mole fraction andIts curvature isAt equal composition $x=1/2$, this becomes $4-2\chi$. For stipulated $\chi =3$, the value is $-2$. A small symmetric split of composition into $x\pm \delta$ changes the average free energy by $\frac{1}{2}{g}^{\prime \prime }(x){\delta }^2$ to leading order, which is negative here. The homogeneous state can therefore lower this model free energy through small composition fluctuations. This illustrates spinodal decomposition; it is a derivation in a chosen parameter convention, not a measurement of a named alloy.

The common-tangent construction separately determines equilibrium coexistence compositions. Carter's notes use these geometrical and thermodynamic conditions to distinguish stability questions from the eventual separated phases. His MIT faculty profile provides an institutional portrait.