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

TL;DR

This study investigates how self-interacting polymers adsorb onto fractal surfaces in three-dimensional Sierpinski gaskets, using exact and Monte Carlo methods to analyze critical exponents and phase diagrams across various fractal dimensions.

Contribution

It provides the first detailed calculation of critical exponents for polymer adsorption on 3D fractals, revealing how these exponents vary with fractal dimension and identifying multiple phases and multicritical points.

Findings

Critical exponents decrease with increasing fractal dimension.

Exponents cross Euclidean values at certain dimensions.

Six phases identified in phase diagrams for specific fractals.

Abstract

We study the problem of adsorption of self-interacting linear polymers situated in fractal containers that belong to the three-dimensional (3d) 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 $b$ ($2\le b\le\infty$). 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 $2\le b\le4$ (exactly) and $2\le b\le40$ (Monte Carlo). We find that both $\nu$ and $\phi$ monotonically decrease with increasing $b$ (that is, with increasing 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 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.