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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.

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 $T_2=293.15\mathrm{K}$.

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.

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.