Semiempirical crossover analysis of phase separation in polymer solutions

The thermodynamic temperature variable appropriate for the description of the scaling behavior of chain dimensions R and density correlations ξ in polymer solutions in the critical through mean field limits is identified by setting up and solving the renormalization group (RG) equation for the system. The solution to the RG equation is obtained by expressing the set of renormalization constants that relate the bare parameters of the Hamiltonian to their renormalized counterparts as approximate resummed expansions in the coupling constants of the system. These expansions, which include unknown coefficients, are so defined that certain calculated asymptotic limits of R and ξ are reproduced. In satisfying these matching conditions, the unknown coefficients are determined self-consistently; they in turn fix the form of the relevant crossover variable. The predicted measure of distance to the critical temperature is found in this way to coincide with de Gennes's original proposal. This semiempirical approach to the calculation of asymptotic and crossover behavior in polymer solutions provides a more rigorous alternative to simple scaling prescriptions but eschews the elaborate mathematical machinery of field-theoretic methods.