Flory–Huggins solution theory ($f/(k_{B}T)=(\varphi /N)\ln \varphi +(1-\varphi )\ln (1-\varphi )+\chi \varphi (1-\varphi )$)

Flory–Huggins solution theory is an incompressible lattice description of mixing a polymer with a solvent. A chain occupies $N$ connected sites, while a solvent molecule occupies one; the displayed mixing free energy is per site. The first two terms are translational entropy of mixing contributions and the last is a mean-field excess contact interaction. A positive interaction parameter $\chi$ penalizes polymer–solvent contacts. If the site volume is $v_0$, the physical free-energy density is $f/v_0$ and the polymer volume fraction gives monomer number density $\varphi /v_0$. Composition-dependent interaction parameters require extra derivative terms in stability calculations.

Let $f(\varphi )$ be the mixing free energy per lattice site in Flory–Huggins solution theory, $v_0$ the site volume, and $\varphi$ the polymer volume fraction. The solvent mixing chemical potential is $f-\varphi f^{\prime }$, so its balance against the pure-solvent pressure contribution gives $\Pi =(\varphi f^{\prime }-f)/v_0$. For composition-independent $\chi$,The first term is the ideal chain translational contribution; the second encodes the effective two-segment excluded volume. At the theta temperature condition it vanishes but the positive third-order term remains. Treating $f$ as a free-energy density without dividing by $v_0$ loses the physical dimensions of osmotic pressure.

The Flory interaction parameter measures the excess free-energy cost of unlike contacts in Flory–Huggins solution theory. For a simple lattice of coordination number $z_c$ and contact energies ${ϵ}_{ps}$, ${ϵ}_{pp}$ and ${ϵ}_{ss}$,Real effective parameters may also contain entropic contributions and composition dependence. With site volume $v_0$ and composition-independent $\chi$, the effective two-segment excluded volume is $v_{\mathrm{eff}}=v_0(1-2\chi )$. Positive, zero and negative $v_{\mathrm{eff}}$ correspond respectively to a good solvent, the theta temperature condition and a poor solvent. In particular, $\chi =1/2$ means cancellation of leading repulsive and attractive contributions; it does not mean that every molecular interaction vanishes.

For composition-independent $\chi$, the spinodal of Flory–Huggins solution theory is $2\chi =1/(N\varphi )+1/(1-\varphi )$. At its thermodynamic critical point, a second condition $-1/(N{\varphi }^2)+1/(1-\varphi {)}^2=0$ gives $1-{\varphi }_c=\sqrt{N}{\varphi }_c$, proving the displayed values. Hence ${\varphi }_c\sim N^{-1/2}$ and ${\chi }_c-1/2\sim N^{-1/2}$. Under $\chi =C/T$, the theta temperature is $\Theta =2C$ and $T_c=\Theta /(1+N^{-1/2}{)}^2$. Thus $T_c$ approaches a constant while $\Theta -T_c$ scales as $N^{-1/2}$. Converting ${\chi }_c$ to a temperature always requires a specified or inferred $\chi (T)$ law.