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.

An incompressible, equal-site-volume mean-field approximation for two polymers of degrees of polymerization $N_A,N_B$. Here $f$ is mixing free energy per site and $\varphi$ is the volume fraction of species $A$. For a composition-independent Flory interaction parameter, local stability requires $1/(N_A\varphi )+1/[N_B(1-\varphi )]-2\chi >0$. The thermodynamic critical point is ${\varphi }_c=\sqrt{N_B}/(\sqrt{N_A}+\sqrt{N_B})$ and ${\chi }_c=(N_A^{-1/2}+N_B^{-1/2}{)}^2/2$.

Connecting segments into chains reduces their independent translational mixing entropy per segment by factors $1/N_A$ and $1/N_B$. Consequently the tolerable positive contact penalty tends to zero as both chains become long. Equal chain lengths have ${\chi }_c=2/N$; favorable unlike interactions can still produce a miscible polymer pair.

For a single-site solvent and polymer length $N$, expand $(1-\varphi )\ln (1-\varphi )=-\varphi +{\varphi }^2/2+{\varphi }^3/6+O({\varphi }^4)$ in Flory–Huggins solution theory. The displayed result follows. Its linear term changes reference chemical potentials; its quadratic term sets the effective two-segment excluded volume. The latter vanishes at the theta temperature, while the positive cubic term remains. Taking $\varphi f^{\prime }-f$ yields the osmotic pressure in Flory-Huggins solution theory, whose cubic coefficient is $1/3$ rather than the free energy's $1/6$.

For two equal-site-volume polymers of chain length $N$, the mixing free energy per lattice site is $f/(k_{B}T)=[\varphi \ln \varphi +(1-\varphi )\ln (1-\varphi )]/N+\chi \varphi (1-\varphi )$. Each chain is one translating object, reducing the entropy of mixing per monomer by $N$ relative to the regular solution model. Its curvature is $f^{\prime \prime }/(k_{B}T)=1/[N\varphi (1-\varphi )]-2\chi$. The symmetric thermodynamic critical point has ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. Long chains can therefore demix under a very small unfavorable Flory interaction parameter.

For a strongly segregated, symmetric long-chain polymer blend, take the square-gradient functional $F/(k_{B}T{\rho }_0)=\int [\chi \varphi (1-\varphi )+a^2({\varphi }^{\prime }{)}^2/(24\varphi (1-\varphi ))]dz$. Its first integral equates the two terms, giving ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$. Thus $\varphi =[1+\tanh (z/\ell )]/2$ with $\ell =a/\sqrt{6\chi }$. The inverse-central-slope width $w=1/{\varphi }^{\prime }(0)=2\ell$ differs by a factor two from the tanh-argument scale. The continuum approximation is marginal when these lengths approach the statistical segment size. An observed width can additionally include roughness and instrument broadening.

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.