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In this paper, we show how the method of field theoretical renormalizationgroup may be used to analyze universal shape properties of long polymer chainsin porous environment. So far such analytical calculations were primarilyfocussed on the scaling exponents that govern conformational properties ofpolymer macromolecules. However, there are other observables that along withthe scaling exponents are universal (i.e. independent of the chemical structureof macromolecules and of the solvent) and may be analyzed within therenormalization group approach. Here, we address the question of shape which isacquired by the long flexible polymer macromolecule when it is immersed in asolvent in the presence of a porous environment. This question is of relevancefor understanding of the behavior of macromolecules in colloidal solutions,near microporous membranes, and in cellular environment. To this end, weconsider a previously suggested model of polymers in d-dimensions [V.Blavats'ka, C. von Ferber, Yu. Holovatch, Phys. Rev. E, 2001, 64, 041102] in anenvironment with structural obstacles, characterized by a pair correlationfunction h(r), that decays with distance r according to a power law: h(r) \simr-a. We apply the field-theoretical renormalization group approach and estimatethe size ratio <R_e^2>/<R_G^2 > and the asphericity ratio \hat{A}_d up to thefirst order of a double \epsilon=4-d, \delta=4-a expansion.

Shapes of macromolecules in good solvents: field theoretical renormalization group approach