On the number of contacts of a floating polymer chain cross-linked with a surface adsorbed chain on fractal structures

We study the problem of the interaction of a linear polymer chain, floating in fractal containers that belong to the three-dimensional Sierpinski gasket (3D SG) family of fractals, with a surface adsorbed linear polymer chain. Each member of the 3D SG fractal family has a fractal impenetrable 2D adsorbing surface, which appears to be 2D SG fractal. The two-polymer system is modelled by two mutually crossing self-avoiding walks. By applying the Monte Carlo renormalization group (MCRG) method, we calculate the critical exponents, rho, associated with the number of contacts of the 3D SG floating polymer chain, and the 2D SG adsorbed polymer chain, for a sequence of SG fractals with 2 = b = 40. Also, we propose a codimension additivity argument formula for rho, and compare its predictions with our reliable set of MCRG data. We find that rho decreases monotonically with increasing b, that is, with increase of the container fractal dimension. Finally, we discuss the relations between different contact exponents, and analyse their possible behaviour in the fractal-to-Euclidean crossover region b -> infinity.