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

TL;DR

This study investigates how the number of contacts between a floating polymer chain in fractal containers and a surface-adsorbed chain varies with the fractal structure, using Monte Carlo methods to compute critical exponents.

Contribution

It introduces a Monte Carlo Renormalization Group approach to calculate contact exponents for polymers on fractals and proposes a codimension additivity formula for these exponents.

Findings

The contact exponent decreases as the fractal parameter increases.

The proposed CA formula aligns well with the Monte Carlo data.

Contact behavior trends towards Euclidean limits as fractal dimension increases.

Abstract

We study the interaction problem 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 $\phi$, 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 $2\le b\le 40$. Besides, we propose the codimension additivity (CA) argument formula for $\phi$, and compare its predictions with our reliable set of the MCRG data. We find that $\phi$ monotonically decreases with increasing $b$, that is, with increase of the container fractal dimension. Finally, we discuss the relations between different contact exponents, and analyze their possible behaviour in the fractal-to-Euclidean crossover region $b\to\infty$.