= Osmotic pressure in Flory-Huggins solution theory
{title2=$\Pi v_0/(k_BT)=\phi/N-\ln(1-\phi)-\phi-\chi\phi^2$}
Let $f(\phi)$ be the mixing <free energy> per lattice site in <Flory–Huggins solution theory>, $v_0$ the site volume, and $\phi$ the polymer <volume fraction>. The solvent mixing <chemical potential> is $f-\phi f'$, so its balance against the pure-solvent pressure contribution gives $\Pi=(\phi f'-f)/v_0$. For composition-independent $\chi$,
$$
\frac{\Pi v_0}{k_BT}=\frac\phi N-\ln(1-\phi)-\phi-\chi\phi^2
=\frac\phi N+\left(\frac12-\chi\right)\phi^2+\frac{\phi^3}{3}+O(\phi^4).
$$
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>.
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