= Dilute-polymer free-energy expansion
{title2=$f/(k_BT)=(\phi/N)\ln\phi+(\chi-1)\phi+(1/2-\chi)\phi^2+\phi^3/6+O(\phi^4)$}
For a single-site solvent and polymer length $N$, expand $(1-\phi)\ln(1-\phi)=-\phi+\phi^2/2+\phi^3/6+O(\phi^4)$ 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 $\phi f'-f$ yields the <osmotic pressure in Flory-Huggins solution theory>, whose cubic coefficient is $1/3$ rather than the free energy's $1/6$.
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