Exact and Monte Carlo study of adsorption of a self-interacting polymer chain for a family of three-dimensional fractals

Abstract

We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3D) Sierpinski gasket (SG) family of fractals. Each member of the 3D SG fractal family has a fractal impenetrable 2D adsorbing surface (which is, in fact, 2D SG fractal) and can be labelled by an integer b (2 ≤ b ≤ ∞). By applying the exact and Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents ν (associated with the mean-squared end-to-end distance of polymers) and φ (associated with the number of adsorbed monomers), for a sequence of fractals with 2 ≤ b ≤ 4 (exactly) and 2 ≤ b ≤ 40 (Monte Carlo). We find that both ν and φ monotonically decrease with increasing b (that is, with increase of the container fractal dimension d f ), and the interesting fact that both functions, ν(b) and φ(b), cross the estimated Euclidean values. Besides, we establish the phase diagrams, for fractals with b = 3 and b = 4, which reveal the existence of six different phases that merge together at a multi-critical point, whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface.