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The dependence on volume fraction $\varphi$ of the Flory-Huggins$\chi_{\mathrm{wp}}\left(\varphi\right)$ describing the free energy of mixingof polymers in water is obtained by exploiting the connection of$\chi_{\mathrm{wp}}\left(\varphi\right)$ to the chemical potential of thewater, for which quasi-chemical theory is satisfactory. We test thistheoretical approach with simulation data for aqueous solutions of capped PEOoligomers. For CH$_3$(CH$_2$-O-CH$_2$)$_m$CH$_3$ ($m$=11),$\chi_{\mathrm{wp}}\left(\varphi\right)$ depends strongly on $\varphi$,consistent with experiment. These results identify coexisting water-rich andwater-poor solutions at $T$ = 300K and $p$ = 1atm. Direct observation of thecoexistence of these two solutions on simulation time scales supports thatprediction for the system studied. This approach directly provides the osmoticpressures. The osmotic second virial coefficient for these chains is positive,reflecting repulsive interactions between the chains in the water, a goodsolvent for these chains.

Concentration dependence of the Flory-Huggins interaction parameter in aqueous solutions of capped PEO chains