Past exam of the quantum condensed matter course of the University of Cambridge / 2009 / Section A / A2 / a

The Flory interaction parameter measures the energetic preference for unlike rather than like contacts in Flory–Huggins solution theory. For polymer volume fraction $\varphi$ and $N$ lattice sites per chain, the dimensionless mixing Helmholtz free energy per site isThe logarithmic terms represent combinatorial entropy; polymer connectivity reduces the chain's translational mixing entropy by the factor $N$. In a simple contact model $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$, where $z$ is the coordination number. Positive $\chi$ penalizes polymer-solvent contacts. Increasing it can overcome mixing entropy and cause phase separation; a negative second derivative of the mixing free energy marks the spinodal region. For a long dilute chain, $\chi <1/2$ corresponds to a good solvent, $\chi =1/2$ to the idealized theta condition, and $\chi >1/2$ to a poor solvent. Real effective parameters can depend on temperature and concentration, rather than being universal chemical constants.