TEST OF THE BOUNDS ON THE CROSSOVER EXPONENT FOR POLYMER ADSORPTION ON FRACTALS

We study the problem of adsorption of linear chain polymers situated on fractal substrates that belong to the Sierpinski-gasket (SG) family. Each member of the SG family is labeled by an integer b (2 less than or equal to b less than or equal to infinity), and it is assumed that one side of each SG fractal is an impenetrable adsorbing wall. By applying the Monte Carlo renormalization-group (MCRG) method, we calculate the critical exponent phi, associated with the number of adsorbed monomers, for a sequence of SG fractals with 2 less than or equal to b less than or equal to 100. We find that our MCRG results deviate at mast 0.12% from the available (2 less than or equal to b less than or equal to 9) exact renormalization-group results. In addition, we test the bounds for phi, proposed recently on heuristic grounds by Bouchaud and Vannimenus [J. Phys. (Paris) 50, 2931 (1989)]. We demonstrate that their lower bound is violated for b greater than or equal to 12. Finally, we discuss a possible behavior of phi for large b, including the limit b --> infinity.