= Solution
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, gives
$$
\frac1{k_BT}\frac{\partial^2f}{\partial\phi^2}
=\frac1{N\phi}+\frac1{1-\phi}-2\chi.
$$
The <spinodal> occurs where this vanishes:
$$
\boxed{\chi_s(\phi)=\frac12\left[\frac1{N\phi}+\frac1{1-\phi}\right].}
$$
Between the two spinodal compositions at sufficiently large $\chi$, $f''<0$ 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.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2021-q6-demixing.png]
{title=<Flory–Huggins solution theory>: coexistence, instability and free-energy curvature}
{description=Original calculations for $N=300$ with the illustrative law $\chi=C/T$. Left: binodal, spinodal and critical point near the critical <temperature>; horizontal dotted lines mark the two temperatures used in part (e). The other panels show free energies above and below the critical <temperature> after subtraction of affine references: the tangent at $\phi_c$ for the upper <temperature> and the common tangent for the lower <temperature>. These subtractions leave phase stability and coexistence unchanged. The low-temperature panel displays the common tangent and spinodal boundaries.}
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