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Relative knot probabilities in confined lattice polygons.

E J Janse van Rensburg1, E Orlandini2, M C Tesi3

  • 1York University, Department of Mathematics and Statistics, Toronto, Ontario M3J 1P3, Canada.

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This study investigates ring polymer knotting in confined spaces. Results show that unknotted polymers dominate even at higher concentrations, with knotting probabilities remaining low.

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Area of Science:

  • Polymer Physics
  • Computational Chemistry
  • Statistical Mechanics

Background:

  • Ring polymers exhibit complex topological states.
  • Confining polymers influences their conformational properties and knotting behavior.

Purpose of the Study:

  • To investigate the relative knotting probabilities of ring polymers in a cubic lattice confinement.
  • To determine how polymer concentration affects knot formation in confined systems.

Main Methods:

  • Utilized a lattice model of ring polymers within a cubic cavity.
  • Employed Monte Carlo simulations to estimate conformation counts.
  • Calculated relative knotting probabilities (ρ_{n,L}(K/0₁)) for various knot types.

Main Results:

  • Relative knotting probabilities remain low for a wide range of monomer concentrations (φ=n/V) within the studied confinement sizes (L≤12).
  • The model is predominantly characterized by unknotted lattice polygons.
  • Knotting probability increases with concentration, eventually plateauing as the Hamiltonian state is approached.

Conclusions:

  • Confinement significantly suppresses knot formation in ring polymers.
  • The concentration-dependent knotting behavior follows a predictable trend, saturating at high densities.