= Solution
In <Flory–Huggins solution theory>, $\phi$ is the <volume fraction>, $N$ is the number of equal-sized lattice segments per chain, $T$ is <temperature>, $k_{\rm B}$ is the <Boltzmann constant>, and $\chi$ is the dimensionless <Flory interaction parameter>. The stated $f$ is the mixing <Helmholtz free energy> per lattice site. The first two terms describe mixing <entropy>; the last describes the contact-energy contribution.
At fixed $T$ and composition-independent $\chi$,
$$
\frac{f'}{k_{\rm B}T}=\frac{\ln\phi+1}{N}-\ln(1-\phi)-1+\chi(1-2\phi),
\qquad
\frac{f''}{k_{\rm B}T}=\frac1{N\phi}+\frac1{1-\phi}-2\chi.
$$
Hence the <spinodal> is
$$
\boxed{\chi_s(\phi)=\frac12\left[\frac1{N\phi}+\frac1{1-\phi}\right],\quad0<\phi<1.}
$$
The homogeneous state is locally unstable when $\chi>\chi_s(\phi)$. The minimum gives the <critical point of a Flory–Huggins polymer solution>:
$$
\boxed{\phi_c=\frac1{1+\sqrt N},\qquad\chi_c=\frac12(1+N^{-1/2})^2.}
$$
If $\chi$ depends on <temperature>, the equation $\chi(T)=\chi_s(\phi)$ converts this result into a <temperature>–composition boundary.
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