Critical exponents for numbers of differently anchored polymer chains on fractal lattices with adsorbing boundaries

Abstract

We study the problem of polymer adsorption in a good solvent when the container of the polymer-solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integerb (2≤b≤∞), and it is assumed that one side of each SG fractal is an impenetrable adsorbing boundary. We calculate the critical exponents γ₁, γ₁₁, and γ _s , which, within the self-avoiding walk model (SAW) of the polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends anchored to the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method for 2≤b≤8 and the Monte Carlo renormalization group (MCRG) method for a sequence of fractals with 2≤b≤80, we obtain specific values for these exponents. The obtained results show that all three critical exponents γ₁, γ₁₁, and γ _s , in both the bulk phase and crossover region are monotonically increasing functions withb. We discuss their mutual relations, their relations with other critical exponents pertinent to SAWs on the SG fractals, and their possible asymptotic behavior in the limitb→∞, when the fractal dimension of the SG fractals approaches the Euclidean value 2.