Polymers in solutions : principles and applications of a direct renormalization method

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Abstract
The configurational properties of long polymers in solution can be conveniently studied by using a direct renormalization method, recently introduced by the author. This method is described here in detail and applied to the two parameter model. The indices γ, ν and ω are calculated directly to second order in ε = 4 - d, where d is the space dimension. As expected, these expansions coincide with the expansions of the indices γ, ν and ω of the zero-component Landau-Ginzburg-Wilson field theory. The formalism may describe monodisperse or polydisperse ensembles and leads directly to scaling equations. The second virial coefficient, expressed in terms of a scaling length which is the end to end distance of an isolated polymer, has been exactly calculated, for a monodisperse system to second order in ε. The ratio of the radius of gyration to the end to end distance is calculated to first order in ε ; the result obtained is in agreement with the value calculated by T. Witten and L. Schäfer using field theory. The dependence of the expansion factor and of the virial coefficient, with respect to the interaction in the cross over domain, are studied to order ε2. An expression is also given for the entropy of an isolated chain