Differential Geometry of Polymer Models: Worm-like Chains, Ribbons and Fourier Knots

TL;DR

This paper compares different continuum polymer models, revealing that worm-like chains and ribbons have diverging torsion, while Fourier knots are smooth enough to be described by the Frenet-Serret equations, highlighting their geometric properties.

Contribution

It demonstrates the geometric differences among polymer models, showing that Fourier knots are sufficiently smooth for FS description, unlike worm-like chains and ribbons.

Findings

Worm-like chains have diverging torsion.

Ribbons have finite twist rate but diverging torsion.

Fourier knots have finite curvature and torsion.

Abstract

We analyze several continuum models of polymers: worm-like chains, ribbons and Fourier knots. We show that the torsion of worm-like chains diverges and conclude that such chains can not be described by the Frenet-Serret (FS) equation of space curves. While the same holds for ribbons as well, their rate of twist is finite and, therefore, they can be described by the generalized FS equation of stripes. Finally, Fourier knots have finite curvature and torsion and, therefore, are sufficiently smooth to be described by the FS equation of space curves.