= Solution
Use <polymer> <volume fraction> $\phi$ horizontally and the <Flory interaction parameter> $\chi$ vertically. Lower $\chi$ gives better solvent quality. The simplest <Flory–Huggins solution theory> assigns the dimensionless mixing <free energy> per lattice site
$$
f(\phi)=\frac\phi N\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi),
\qquad v_{\rm eff}\simeq a^3(1-2\chi).
$$
Thus $\chi<1/2$ gives positive <excluded volume> and a good-solvent coil, $\chi=1/2$ is the long-chain theta condition, and $\chi>1/2$ is the poor-solvent side. It is the net two-segment interaction that vanishes at theta, not every molecular <force>.
At fixed good-solvent quality, the <polymer overlap volume fraction> $\phi^*\sim Na^3/R_g^3$ separates isolated coils from a <semidilute polymer solution>. Above overlap, many chains interpenetrate, but there is still solvent on the segment scale. As $\phi$ approaches order unity the solution is concentrated; these concentration-regime boundaries are crossovers, not first-order phase transitions. The overlap fraction increases from order $N^{-4/5}$ in a strong <good solvent> toward order $N^{-1/2}$ at theta. The crossover can be expressed as $\phi^*\sim N^{-4/5}(1-2\chi)^{-3/5}$ while $1-2\chi\gg N^{-1/2}$; it saturates at the theta scaling when that condition fails.
On the poor-solvent side there can instead be coexistence of polymer-poor and polymer-rich phases. The <binodal> follows a common tangent of $f$; the <spinodal> follows
$$
f''(\phi)=\frac1{N\phi}+\frac1{1-\phi}-2\chi=0.
$$
The <critical point of a Flory–Huggins polymer solution> is
$$
\boxed{\phi_c=\frac1{1+\sqrt N},\qquad
\chi_c=\frac12(1+N^{-1/2})^2.}
$$
For finite $N$, $\chi_c$ lies above $1/2$. Therefore <poor solvent> does not automatically mean demixing at every concentration, and an overlap line must not be mistaken for a coexistence boundary. The original figure uses the stated ideal model for an illustrative chain <length>; it does not assert a measured phase boundary. A vertical <temperature> axis would additionally require a material-specific relation $\chi(T)$.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2012/paper-1-polymer-regimes.png]
{title=Polymer solution regimes}
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