# Adsorbing staircase polygons subject to a force

## TL;DR

This paper analyzes simplified models of ring polymers represented by staircase polygons on a rotated lattice, revealing four distinct phases influenced by surface interaction and applied force, with exact free energy calculations.

## Contribution

It provides the first exact determination of the free energy for staircase polygons under combined surface interaction and force, identifying four unique phases.

## Key findings

- Four different phases including an adsorbed, ballistic, and mixed phase.
- Exact free energy expression valid across all surface interaction and force parameters.
- Demonstration of phase transitions driven by surface affinity and applied force.

## Abstract

We study several models of staircase polygons on the $45^\circ$ rotated square lattice, which interact with an impenetrable surface while also being pushed towards or pulled away from the surface by a force. The surface interaction is governed by a fugacity $a$ and the force by a fugacity $y$. Staircase polygons are simplifications of more general self-avoiding polygons, a well-studied model of interacting ring polymers. For this simplified case we are able to exactly determine the limiting free energy in the full $a$-$y$ plane, and demonstrate that staircase polygons exhibit four different phases, including a "mixed" adsorbed-ballistic phase.