Flory–Huggins solution theory

Remarks

The first two terms are the entropy of placing the chains, and the last term is a mean-field estimate of the energy of polymer-solvent contacts. The mixture is locally unstable where $\Delta F'' < 0$, and the critical point solves $\Delta F'' = \Delta F''' = 0$. This gives $\varphi_c = 1/(1 + \sqrt{N})$ and $\chi_c = \frac12 (1 + 1/\sqrt{N})^2$. Check: $\Delta F''/(k_B T) = \frac{1}{N \varphi} + \frac{1}{1 - \varphi} - 2\chi$ and $\Delta F'''/(k_B T) = -\frac{1}{N \varphi^2} + \frac{1}{(1 - \varphi)^2}$. The second vanishes when $1 - \varphi = \sqrt{N} \varphi$, and then the first vanishes when $2\chi = (1 + \sqrt{N})^2 / N$. Example: for $N = 1$ the model is the regular solution model, with $\varphi_c = 1/2$ and $\chi_c = 2$. As $N \to \infty$, $\chi_c \to 1/2$ and $\varphi_c \to 0$. Non-example: real molecules do not sit on a lattice of equal-sized sites, so for them the lattice is only a heuristic. Then $\chi$ is usually fitted to data and may depend on $\varphi$ and $T$.