= Solution
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>:
$$
\boxed{\frac f{k_BT}=\phi\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi).}
$$
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 $f''$. Thus local stability requires $f''>0$, and the <spinodal> is
$$
\boxed{\frac{f''}{k_BT}=\frac1{\phi(1-\phi)}-2\chi,\qquad \chi_s(\phi)=\frac1{2\phi(1-\phi)}.}
$$
Inside this curve, $f''<0$ and arbitrarily small composition fluctuations grow. The minimum of $\chi_s$ is at $\phi=1/2$, giving \b[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 $(1\pm\sqrt{1-2/\chi})/2\simeq0.2764,0.7236$. The <common-tangent construction> gives coexisting compositions $\phi_b\simeq0.1448$ and $1-\phi_b$, satisfying $\ln[\phi_b/(1-\phi_b)]+\chi(1-2\phi_b)=0$. Between a binodal endpoint and the spinodal, the homogeneous state is locally stable but metastable; global stability requires the convex envelope of $f$.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2006-mixture-free-energy.png]
{title=Mixing free energies and the onset of phase separation}
{description=Original lattice-model curves with independent vertical scales. The two-phase free energy for $\chi=2.5$ is the horizontal common tangent between the two marked binodal compositions; shaded regions identify negative curvature.}
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