On scaling theories of polymer solutions

Abstract

We present a rigorous derivation of a scaling theory of the thermodynamics of polymer solutions at finite concentrations. The derivation proceeds directly in 3(2, 4, etc.) dimensions from the expression for the partition function for a solution of monodisperse continuous Gaussian chains with excluded volume. There is no need for the use of renormalization group methods or for the extrapolation of results from calculations near four-dimensions. Nethertheless, the importance of four dimensions emerges directly from the scaling theory as does a proof that only the binary excluded volume parameter appears in the equation of state, average chain dimensions, etc., in the limit of long enough chains. The irrelevance of the range of the binary segment interaction and the sufficiency of the two parameter theory is thus established by the scaling theory in a straightforward and elementary fashion. We demonstrate that the power law dependence, ν in 〈R2〉∝L2ν, of the mean square end-to-end distance 〈R2〉 on the chain length L in good solutions can be calculated directly in three (two, etc.) dimensions from the scaling relations. Applications are presented to good solutions at finite concentrations, to confined chains in good solvents, and to the properties of solutions of block copolymers.