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Thermodynamics of Confined Knotted lattice Polygons

EJ Janse van Rensburg, E Orlandini, MC Tesi

Abstract

A ring polymer in a confining space may exhibit at least two phases, namely an expanded (or solvent-rich phase) if its concentration is small, or a collapsed (or polymer-rich phase) when it is concentrated and compressed. These phases are discussed in reference \cite{deG79}, and have been modelled, traditionally, in the mean field using Flory-Huggins theory \cite{Flory42,Huggins42}. In three dimensions the ring polymer may also be knotted, or linked, and have its conformational degrees of freedom constrained by its topology. In a lattice model of confined knotted ring polymers there are indications that the thermodynamic properties of the ring polymer (for example, the osmotic pressure \cite{GJvR18,JvR19}) is a function of its topology. In this paper we explore a lattice knot model of a confined ring polymer as a function of its chemical potential. We show that a well-defined phase transition occurs between solvent-rich and polymer-rich phases when the lattice knot exhibits either the unknot topology or any other fixed knot type. Furthermore, we observe small yet significant variations in the free energy near the critical point when comparing trefoil knots with other non-trivial knot types. These findings indicate that the thermodynamic properties of confined ring polymers depend on their topological entanglement characteristics (namely, their knot type).

Thermodynamics of Confined Knotted lattice Polygons

Abstract

A ring polymer in a confining space may exhibit at least two phases, namely an expanded (or solvent-rich phase) if its concentration is small, or a collapsed (or polymer-rich phase) when it is concentrated and compressed. These phases are discussed in reference \cite{deG79}, and have been modelled, traditionally, in the mean field using Flory-Huggins theory \cite{Flory42,Huggins42}. In three dimensions the ring polymer may also be knotted, or linked, and have its conformational degrees of freedom constrained by its topology. In a lattice model of confined knotted ring polymers there are indications that the thermodynamic properties of the ring polymer (for example, the osmotic pressure \cite{GJvR18,JvR19}) is a function of its topology. In this paper we explore a lattice knot model of a confined ring polymer as a function of its chemical potential. We show that a well-defined phase transition occurs between solvent-rich and polymer-rich phases when the lattice knot exhibits either the unknot topology or any other fixed knot type. Furthermore, we observe small yet significant variations in the free energy near the critical point when comparing trefoil knots with other non-trivial knot types. These findings indicate that the thermodynamic properties of confined ring polymers depend on their topological entanglement characteristics (namely, their knot type).
Paper Structure (15 sections, 1 theorem, 43 equations, 23 figures, 6 tables)

This paper contains 15 sections, 1 theorem, 43 equations, 23 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The model
  3. The free energy
  4. Lattice polygon
  5. Lattice knots
  6. Confined lattice knots
  7. The limiting free energy
  8. Scaling of the free energy
  9. Numerical sampling of confined lattice knots
  10. Finite size scaling
  11. The lattice unknot $0_1$
  12. The right-handed lattice trefoil $3_1^{+}$
  13. The granny and square knots
  14. Other prime knots
  15. Discussion

Key Result

Theorem 1

The limit $\chi_{0_1} (x) = \lim_{L\to\infty} \frac{1}{L^3} \log Z_{{0_1},L}(x)$ exists and defines the limiting free energy of confined lattice unknots. Moreover, since $Z_{{0_1},L}(x) \geq 1$, it follows that $\chi_{0_1} (x) \geq 0$.

Figures (23)

  • Figure 1: A lattice knot $\omega$ of knot type $5_1$ inside a cube in the cubic lattice of side-length $L$ sites and volume $L^3$ lattice sites. If $\omega$ has length $n$ (steps or lattice sites), then the concentration of the vertices (monomers) in the lattice knot is $\phi=n/L^3$.
  • Figure 2: Schematic diagrams of a knotted ring polymer in a cubical cavity at low concentration. The conformation on the left shows a knot confined in a small volume inside the cavity, while the location of the knot on the right fills the volume of the cavity.
  • Figure 3: Schematic diagrams of a knotted ring polymer in a cubical cavity at intermediate and high concentrations. The knot is still visible in the intermediate concentration on the left as a tight tangle towards the centre, but if it does not remain tight with increasing concentration, then it relaxes to fill the entire cavity in the high concentration regime on the right.
  • Figure 4: Finite size free energies $f_{0_1,L}(x)$ (see equation (\ref{['e14']})) of the unknot for $x\in[0,1]$. The values of $L$ increases from $L=7$ in steps of $2$ to $L=15$. The curves accumulate on zero when $x$ is small, but diverges once $x$ is larger than a a critical point $x_{0_1}$, consistent with equation (\ref{['e18']}).
  • Figure 5: Finite size energy densities (see equation (\ref{['e19']})) as a function of $x$ for the unknot. The curves accumulate on zero for $x$ small, but diverge sharply when $x$ increases beyond the critical point. With increasing $x$${\mathcal{E}}_{0_1}$ approaches $1$, for example, if $x=2$, then ${\mathcal{E}}_{0_1} \approx 0.93$ if $L=15$. As noted above equation (\ref{['e19']}), the density of polymer in the confining box is given by ${\mathcal{E}}_{K,L}(x)$.
  • ...and 18 more figures

Theorems & Definitions (1)

  • Theorem 1