= Solution
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 $\phi\pm\delta\phi$ changes the average free energy by
$$
\frac{f(\phi+\delta\phi)+f(\phi-\delta\phi)}2-f(\phi)
=\frac12 f''(\phi)(\delta\phi)^2+O((\delta\phi)^4).
$$
Thus $f''>0$ locally resists fluctuations and $f''<0$ amplifies them, leading to <spinodal decomposition>. \b[The <spinodal> curve is the locus $f''=0$], 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>.
\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1-polymer-spinodal.png]
{title=Local instability boundary of a Flory–Huggins polymer solution}
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