Exact study of surface critical exponents of polymer chains grafted to adsorbing boundary of fractal lattices embedded in three-dimensional space

TL;DR

This paper calculates exact surface critical exponents for polymers grafted to fractal boundaries in 3D, using renormalization group methods on Sierpinski gasket fractals, revealing their relations and specific values.

Contribution

It provides the first exact determination of surface critical exponents for polymer adsorption on 3D fractal lattices using RG techniques.

Findings

Exact values of critical exponents for b=2,3,4 fractals.

Relations between surface exponents and bulk exponents.

Insights into polymer conformations near fractal boundaries.

Abstract

We study the adsorption problem of linear polymers, when the container of the polymer--solvent system is taken to be a member of the three dimensional Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer $b$ ($2\le b\le\infty$), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the critical exponents $\gamma_1, \gamma_{11}$, and $\gamma_s$ which, within the self--avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends grafted on the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method, for $2\le b\le 4$, we have obtained specific values for these exponents, for various type of polymer conformations. We discuss their mutual relations and their relations with other critical exponents pertinent to SAWs on the SG fractals.