The size of a polymer in a symmetric solvent

Using a simple thermodynamic model, we derive an expression for the excluded volume parameter v of a polymer chain in a symmetric solvent (solvated by its own monomers). For a chain with a given segment length and number of monomers, this parameter determines whether the chain is collapsed or expanded. For the latter it determines the degree of expansion. Using a simple off-lattice version of Flory's model [P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953)] and relaxing the assumption of incompressibility, we obtain the result v=(1-kappa)rho(0), where kappa is the dimensionless compressibility and rho(0) the number density of solvent. In the incompressible limit (in the sense that kappa-->0) the chain is expanded and the inverse of the solvent number density determines the degree of expansion of the chain. Using the van der Waals equation of state to estimate kappa (allowing for nonzero compressibility in a system that can undergo a gas-liquid phase transition), the model predicts that upon raising the temperature at constant pressure there is both a lower (coil to globule) and upper (globule to coil) Flory temperature. This is in quantitative agreement with experiment and computer simulations.