Concentration and excluded volume dependence of coherent scattering functions for polymers: Chain conformational space renormalization group

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Abstract
The concentration dependent chain conformational space renormalization group theory is extended to a consideration of the explicit dependence on the strength of the excluded volume interaction in the nonscaling, ‘‘cross-over,’’ region between the single chain Gaussian and self-avoiding limits. Generalized ‘‘scaling laws’’ in this cross-over domain are derived without the use of perturbation theory or ε expansions. In the dilute and semidilute region these generalized scaling laws have distances scaled by the infinite dilution radius of gyration RG(ζ) where ζ is a scaling variable which vanishes in the Gaussian chain limit and tends to infinity in the good solvent limit. Hence, concentrations c are scaled by the overlap concentration c*ζ evaluated from the cross-over RG(ζ). Quantities like the coherent scattering function are then functions of kRG(ζ), c/c*ζ, and ζ with k the momentum transfer vector. The explicit ζ dependence persists through the cross-over regime for all c and is eliminated in the simple scaling limits ζ→0 or ∞. We use the theory to evaluate the coherent scattering functions for a single tagged polymer chain, for a set of tagged chains at concentration cT<c, as well as the full polymer solution. The concentration and excluded volume dependences of a tagged chain radius of gyration and the full solution correlation length are evaluated, and the former is studied as a function of the ratio of the tagged chain molecular weight to that for an average solution chain.