Dilute-polymer free-energy expansion ($f/(k_{B}T)=(\varphi /N)\ln \varphi +(\chi -1)\varphi +(1/2-\chi ){\varphi }^2+{\varphi }^3/6+O({\varphi }^4)$)

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 $1/6$.