  The dependence on volume fraction $\varphi$ of the Flory-Huggins
$\chi_{\mathrm{wp}}\left(\varphi\right)$ describing the free energy of mixing
of polymers in water is obtained by exploiting the connection of
$\chi_{\mathrm{wp}}\left(\varphi\right)$ to the chemical potential of the
water, for which quasi-chemical theory is satisfactory. We test this
theoretical approach with simulation data for aqueous solutions of capped PEO
oligomers. 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 and
water-poor solutions at $T$ = 300K and $p$ = 1atm. Direct observation of the
coexistence of these two solutions on simulation time scales supports that
prediction for the system studied. This approach directly provides the osmotic
pressures. The osmotic second virial coefficient for these chains is positive,
reflecting repulsive interactions between the chains in the water, a good
solvent for these chains.
