Continuum theory of critical phenomena in polymer solutions: Formalism and mean field approximation

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Abstract
A theoretical description of the critical point of a polymer solution is formulated directly from the Edwards continuum model of polymers with two- and three-body excluded-volume interactions. A Hubbard–Stratonovich transformation analogous to that used in recent work on the liquid–vapor critical point of simple fluids is used to recast the grand partition function of the polymer solution as a functional integral over continuous fields. The resulting Landau–Ginzburg–Wilson (LGW) Hamiltonian is of the form of a generalized nonsymmetric n=1 component vector model, with operators directly related to certain connected correlation functions of a reference system. The latter is taken to be an ensemble of Gaussian chains with three-body excluded-volume repulsions, and the operators are computed in three dimensions by means of a perturbation theory that is rapidly convergent for long chains. A mean field theory of the functional integral yields a description of the critical point in which the power-law variations of the critical polymer volume fraction φc, critical temperature Tc, and critical amplitudes on polymerization index N are essentially identical to those found in the Flory–Huggins theory. In particular, we find φc ∼N−1/2, Tθ−Tc∼N−1/2 with (Tθ the theta temperature), and that the composition difference between coexisting phases varies with reduced temperature t as N−1/4t1/2. The mean field theory of the interfacial tension σ between coexisting phases near the critical point, developed by considering the LGW Hamiltonian for a weakly inhomogeneous solution, yields σ∼N−1/4t3/2, with the correlation length diverging as ξ∼N1/4t−1/2 within the same approximation, consistent with the mean field limit of de Gennes’ scaling form. Generalizations to polydisperse systems are discussed.