Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A2

For equal-size molecules, let $f$ be the mixing Helmholtz free energy per lattice site. Flory–Huggins solution theory reduces to the regular solution model:The first two terms are minus the combinatorial entropy of mixing; the last is the excess cost of unlike contacts. A positive Flory interaction parameter penalizes such contacts. Additive terms linear in composition change the chosen pure-component reference but do not change stability.

A small composition fluctuation at fixed mean composition costs energy proportional to $f^{\prime \prime }$. Thus local stability requires $f^{\prime \prime }>0$, and the spinodal isInside this curve, $f^{\prime \prime }<0$ and arbitrarily small composition fluctuations grow. The minimum of ${\chi }_s$ is at $\varphi =1/2$, giving the critical value ${\chi }_c=2$. At the thermodynamic critical point, $f^{(2)}=f^{(3)}=0$ while $f^{(4)}>0$.

For $\chi =0$, $f$ is strictly convex with its minimum at $1/2$. At $\chi =2$, the central quadratic term vanishes but a positive quartic term remains. At $\chi =2.5$, there are two minima and a central maximum; the spinodal compositions are $(1\pm \sqrt{1-2/\chi })/2\simeq 0.2764,0.7236$. The common-tangent construction gives coexisting compositions ${\varphi }_b\simeq 0.1448$ and $1-{\varphi }_b$, satisfying $\ln [{\varphi }_b/(1-{\varphi }_b)]+\chi (1-2{\varphi }_b)=0$. Between a binodal endpoint and the spinodal, the homogeneous state is locally stable but metastable; global stability requires the convex envelope of $f$.

For two species of chains with the same polymerization degree $N$, the number of independently translating objects per occupied site is reduced by $N$. The symmetric Flory-Huggins polymer blend hasThe symmetry still places the thermodynamic critical point at ${\varphi }_c=1/2$, but nowThe entropy of mixing per monomer is much smaller than for simple liquids, so even a weak unfavorable interaction can cause phase separation for long chains. If the energetic interaction parameter is $\chi =A/T$ with $A>0$, this model gives $T_c=AN/2$; the conclusion relies on that specified temperature law.

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.

Write $z_c$ for the lattice coordination number and ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ for contact energies. Counting the excess of unlike contacts relative to the unmixed reference gives the Flory interaction parameterUnlike contacts that are less favorable than the mean like contact give $\chi >0$. For temperature-independent contact energies, $\chi$ is proportional to $1/T$, so heating weakens this energetic preference relative to the entropy of mixing. Effective parameters in real solutions can also include entropic terms, often represented schematically by $A/T+B$; the pure contact-energy model does not explain every possible temperature dependence.