# Osmotic pressure of compressed lattice knots

## TL;DR

This study uses numerical simulations to explore how the osmotic pressure of compressed lattice knots varies with knot type and concentration, revealing unique equilibrium states and divergence at high concentrations.

## Contribution

It introduces a detailed numerical analysis of osmotic pressure in lattice knots, highlighting the dependence on knot type and identifying multiple equilibrium points.

## Key findings

- Osmotic pressure depends on knot type and entanglement.
- Unknot has two equilibrium states, one stable and one unstable.
- Non-trivial knots have a single stable equilibrium.

## Abstract

A numerical simulation shows that the osmotic pressure of compressed lattice knots is a function of knot type, and so of entanglements. The osmotic pressure for the unknot goes through a negative minimum at low concentrations, but in the case of non-trivial knot types $3_1$ and $4_1$ it is negative for low concentrations. At high concentrations the osmotic pressure is divergent, as predicted by Flory-Huggins theory. The numerical results show that each knot type has an equilibrium length where the osmotic pressure for monomers to migrate into or our of the lattice knot is zero. Moreover, the lattice unknot is found to have two equilibria, one unstable, and one stable, whereas the lattice knots of type $3_1$ and $4_1$ have one stable equilibrium each.

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Source: https://tomesphere.com/paper/1902.04490