Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section A / A3 / a / Solution

Use polymer volume fraction $\varphi$ and interaction parameter $\chi$, increasing toward poor-solvent conditions. The Flory–Huggins solution theory free energy per lattice site isThe common-tangent construction gives the binodal; $f^{\prime \prime }=0$ gives the spinodal, $2{\chi }_s=1/(N\varphi )+1/(1-\varphi )$. Its critical point is ${\varphi }_c=1/(1+\sqrt{N})$, ${\chi }_c=\frac{1}{2}(1+N^{-1/2}{)}^2$. Above the binodal in this $\chi$ plot, the solution separates into polymer-poor and polymer-rich phases; between binodal and spinodal it is metastable, while inside the spinodal a homogeneous state is unstable.

The line $\chi =1/2$ is the theta solvent condition for a long dilute chain. Below it is the good solvent side; above it is the poor solvent side. Poor solvent does not imply that every composition is already phase separated. Independently, a homogeneous solution is dilute for $\varphi \ll {\varphi }^{\ast }$, semidilute for $\varphi \gg {\varphi }^{\ast }$ below close packing, and concentrated as $\varphi$ approaches one.