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

We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (313) 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 less than or equal to b less than or equal to infinity). By applying the exact and Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents nu (associated with the mean-squared end-to-end distance of polymers) and phi (associated with the number of adsorbed monomers), for a sequence of fractals with 2 less than or equal to b less than or equal to 4 (exactly) and 2 less than or equal to b less than or equal to 40 (Monte Carlo). We find that both v and 0 monotonically decrease with increasing b (that is, with increase of the container fractal dimension d(f)), and the interesting fact that both functions, nu(b) and phi(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.