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In Flory–Huggins solution theory, $\varphi$ is the volume fraction, $N$ is the number of equal-sized lattice segments per chain, $T$ is temperature, $k_{\mathrm{B}}$ 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 $\chi$,Hence the spinodal isThe homogeneous state is locally unstable when $\chi >{\chi }_s(\varphi )$. The minimum gives the critical point of a Flory–Huggins polymer solution:If $\chi$ depends on temperature, the equation $\chi (T)={\chi }_s(\varphi )$ converts this result into a temperature–composition boundary.

With polymer degree $N_A$ and single-site solvent molecules, Flory–Huggins solution theory gives $f/(k_{B}T)=(\varphi /N_A)\ln \varphi +(1-\varphi )\ln (1-\varphi )+\chi \varphi (1-\varphi )$. Expanding the solvent entropy at low concentration gives the dilute-polymer free-energy expansionThe linear term sets a reference chemical potential. The quadratic term controls the leading effective two-segment excluded volume, $v_{\mathrm{eff}}=v_0(1-2\chi )$ for site volume $v_0$.

In a good solvent, $\chi <1/2$ and positive excluded volume swells a dilute chain: the Flory approximation gives $R\sim b{N}_A^{3/5}$ in three dimensions. In a poor solvent, $\chi >1/2$ favors segment contacts and a sufficiently long isolated chain forms a polymer globule, with compact scaling $R\sim b{N}_A^{1/3}$ up to density-dependent factors.

At the theta temperature, the leading two-segment interaction cancels, $\chi =1/2$ in this model, and the long-chain conformation is approximately an ideal chain, $R\sim b\sqrt{N_A}$. The positive cubic term remains. The critical point of a Flory–Huggins polymer solution at finite $N_A$ has ${\chi }_c=\frac{1}{2}(1+N_A^{-1/2}{)}^2$; its limiting value, rather than every finite-chain critical point, is the theta condition.

Use polymer volume fraction $\varphi$ 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 siteThus $\chi <1/2$ 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 ${\varphi }^{\ast }\sim N{a}^3/R_g^3$ separates isolated coils from a semidilute polymer solution. Above overlap, many chains interpenetrate, but there is still solvent on the segment scale. As $\varphi$ 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 ${\varphi }^{\ast }\sim N^{-4/5}(1-2\chi {)}^{-3/5}$ 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 followsThe critical point of a Flory–Huggins polymer solution isFor 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 $\chi (T)$.

At the thermodynamic critical point, the two spinodal branches merge: $f^{\prime \prime }=0$ and $f^{(3)}=0$. The third derivative isFor $0<\varphi <1$, its zero gives $1-{\varphi }_c=\sqrt{N}{\varphi }_c$. Substitution in the spinodal relation yieldsAn absolute critical temperature requires an interaction–temperature relation. Under the usual energetic approximation $\chi =C/T$, define the theta temperature $\Theta =2C$, at which an infinite chain has $\chi =1/2$. ThenThus $T_c$ tends to a nonzero constant, $T_c\sim N^0$, while $\Theta -T_c\sim N^{-1/2}$. It is the temperature deficit, not $T_c$ itself, that scales as $N^{-1/2}$.

More generally, if a smooth $\chi (T)$ obeys $\chi (\Theta )=1/2$ and ${\chi }^{\prime }(\Theta )\ne 0$, thenThis is the critical point of a Flory–Huggins polymer solution. The same leading chain-length exponent follows for a smooth temperature law, while its coefficient depends on the omitted solvent thermodynamics. The lattice interpretation connects the chain-length dependence of mixing entropy to the theta condition $\chi =1/2$.^[ref].