Scaling properties of the entropy of polymer solutions

Direct renormalisation group equations are derived and solved for the entropy S(c p , L, (P(x)), o v ) of a polymer solution with specified polydispersity c p (x) identical to c p (1/L)P(x/L). A two-parameter description of the universal scaling function S is developed which characterises the scaling behaviour of the entropy for all temperatures T>or= theta , 'the Flory temperature', and all concentrations c p . Explicit representations are constructed parametrically for all values of the 'overlap' o v approximately uL 2 d p to O( epsilon 2 ) in the epsilon expansion. Dilute solution expressions are reproduced, the limit o v >>1 of semi-dilute physics is discussed and a simple phenomenological model is proposed.