Interacting Linear Polymers on Three–dimensional Sierpinski Fractals

Using self–avoiding walk model on three–dimensional Sierpinski fractals (3d SF) we have studied critical properties of self–interacting linear polymers in porous environment, via exact real–space renormalization group (RG) method. We have found that RG equations for 3d SF with base b = 4 are much more complicated than for the previously studied b = 2 and b = 3 3d SFs. Numerical analysis of these equations shows that for all considered cases there are three fixed points, corresponding to the high–temperature extended polymer state, collapse transition, and the low–temperature state, which is compact or semi–compact, depending on the value of the fractal base b. We discuss the reasons for such different low–temperature behavior, as well as the possibility of establishing the RG equations beyond b = 4. Introduction At low temperatures and in poor solvents linear polymers are in compact globule state, while at high temperatures and in good solvents they have extended coil configuration [1]. Self–interacting self–avoiding walk (SISAW) on a lattice is a good model for linear polymer in solvent. If −ε < 0 is the attractive energy per pair of nearest–neighbors monomers (not directly connected) and w =exp(ε/kBT ) the corresponding Boltzman factor, then for N–step SISAW and N ≫ 1 the following behavior of the mean-squared end-to-end distance 〈R2 N 〉 is expected: 〈R N 〉 ∼ N 2ν . (1) There is a critical value of interaction w = wc, such that for any w < wc critical exponent ν = νSAW , which corresponds to extended coil phase, whereas for any w > wc exponent ν has smaller value, equal to 1/d for d–dimensional homogeneous lattices. The value w = wc corresponds to the