The Lagrangian theory of polymer solutions at intermediate concentrations

De Gennes has shown that the properties of an isolated polymer in a solution (a chain with excluded volume) can be deduced within the framework of a Lagrangian theory for a zero component field in the absence of an external field. This result in generalized to the case of polymer solutions at intermediate concentrations. It is shown that a grand ensemble of polymers can be described by using a Lagrangian theory for a zero component field coupled to an external field. The concentrations Cp of polymers (chains) and C m of monomers (links) are fixed by two chemical potentials. It is shown that the osmotic pressure obeys a scaling law of the form (P/KTCp) = F(Cp N3ν) where N is the mean number of monomers per polymer (N = Cp/C m) and ν the critical index defining the size of a long isolated polymer. The function F(λ) can be expanded in powers of λ and it is given implicitly by the generating functional of the zero-momentum vertex functions derived from the Lagrangian theory. The results seem to be in good agreement with experiments.