= Solution
The <Flory interaction parameter> $\chi$ measures the excess cost of unlike segment contacts. In a simple lattice with coordination number $z$, contact <energies> $\epsilon_{AA},\epsilon_{BB},\epsilon_{AB}$ and <temperature> $T$,
$$
\chi=\frac{z}{k_BT}\left[\epsilon_{AB}-\frac{\epsilon_{AA}+\epsilon_{BB}}2\right].
$$
A positive <Flory interaction parameter> penalizes unlike contacts; a negative value favors them. Mixing still depends on the balance with <entropy of mixing>, rather than the sign of $\chi$ alone.
For two <polymers> of degrees of polymerization $N_A,N_B$, let $\phi$ be the <volume fraction> of species $A$. The <binary Flory-Huggins polymer blend> has mixing <free energy> per lattice site
$$
\frac{f(\phi)}{k_BT}=\frac\phi{N_A}\ln\phi+
\frac{1-\phi}{N_B}\ln(1-\phi)+\chi\phi(1-\phi).
$$
The logarithmic terms favor mixing. Their factors $1/N_A$ and $1/N_B$ arise because an entire chain translates as one object; connecting many segments greatly reduces the mixing <entropy> per segment. Hence long <polymers> can demix for a much smaller positive <Flory interaction parameter> than small <molecules>.
For composition-independent $\chi$, the local stability criterion is
$$
\frac{f''(\phi)}{k_BT}=
\frac1{N_A\phi}+\frac1{N_B(1-\phi)}-2\chi.
$$
Positive curvature means stability to small composition fluctuations, zero curvature locates the <spinodal>, and negative curvature means spontaneous growth of fluctuations. Actual two-phase coexistence is determined by the <common-tangent construction>, or equality of both species' <chemical potentials>; its boundary is the <binodal>. A locally stable but globally unfavourable homogeneous <mixture> can lie between the <binodal> and <spinodal>.
Setting the second and third <derivatives> of $f$ to zero gives
$$
\phi_c=\frac{\sqrt{N_B}}{\sqrt{N_A}+\sqrt{N_B}},\qquad
\chi_c=\frac12\left(N_A^{-1/2}+N_B^{-1/2}\right)^2.
$$
For the <symmetric Flory-Huggins polymer blend>, \b[$\phi_c=1/2$ and $\chi_c=2/N$]. For a <polymer solution> with $N_B=1$ and long chains of species $A$, $\chi_c\to1/2$, consistent with the <theta solvent> limit. An effective <Flory interaction parameter> can depend on <temperature> and composition; in the simple energetic case $\chi\propto1/T$, heating encourages mixing, but that temperature trend is not universal.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2016/paper-1-polymer-mixing.png]
{title=Increasing the <Flory interaction parameter> can destroy stability of a symmetric <polymer> blend}
The original illustration uses stipulated $N_A=N_B=100$. At $\chi=0.01$ the mixing <free energy> is convex, whereas at $\chi=0.03$ a central unstable region appears. The horizontal <common-tangent construction> in the latter case joins the symmetric coexisting compositions; its contacts differ from the <spinodal> boundaries. These are model curves, not supplied experimental data.
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