Linear and branched polymers on fractals

TL;DR

This paper reviews the critical properties of linear and branched polymers on deterministic fractals, using real-space renormalization to analyze phenomena like self-avoiding walks, collapse transitions, and adsorption.

Contribution

It provides exact calculations of critical exponents and phase transitions for polymers on fractals, extending understanding of polymer behavior in complex geometries.

Findings

Critical exponents for self-avoiding walks on fractals are determined.

Collapse transition of linear polymers on certain fractals is characterized.

Analytical study of adsorption and zipping transitions of polymers on fractals.

Abstract

This is a pedagogical review of the subject of linear polymers on deterministic finitely ramified fractals. For these, one can determine the critical properties exactly by real-space renormalization group technique. We show how this is used to determine the critical exponents of self-avoiding walks on different fractals. The behavior of critical exponents for the $n$-simplex lattice in the limit of large $n$ is determined. We study self-avoiding walks when the fractal dimension of the underlying lattice is just below 2. We then consider the case of linear polymers with attractive interactions, which on some fractals leads to a collapse transition. The fractals also provide a setting where the adsorption of a linear chain near on attractive substrate surface and the zipping-unzipping transition of two mutually interacting chains can be studied analytically. We also discuss briefly the critical properties of branched polymers on fractals.