Thermal and Topological Crossover of Ring Polymers
Abstract
We use Monte Carlo simulations of off-lattice hard-sphere-monomer rings with N = 100-16 000 segments, classifying every sampled conformation by knot type, to show that topology reshapes the finite-size route to the good-solvent regime without changing the leading asymptotic behavior. For the topology-unrestricted ensemble, every nonzero hard-sphere diameter flows to the same good-solvent fixed point, but the leading correction to ⟨R 2 g ⟩/N 2ν vanishes at d * /b = 0.431(10), in agreement with the linear-chain magic value 0.433(1). The thermal swelling factor is governed by the same Zimm-Stockmayer-Fixman variable z as for linear chains. Fixed-z extrapolations and fixed-d/b data determine a common Monte Carlo crossover curve, which is described without adjustable parameters by the independently derived one-loop field theory for the ring topology. Its good-solvent amplitude gives the universal mean-square ring-to-linear size ratio H = 0.539(1)(4). Conditioning on a knot type changes this picture at finite length. The leading good-solvent exponent remains the same, but the correction amplitude and hence the magic diameter become knot dependent, shifting systematically to smaller values as the knot becomes more complex: d * /b = 0.409(7) for the unknot (01), 0.314(7) for the trefoil (31), and 0.237( 11) for the figureeight (41). Because the knotting length varies strongly with excluded volume, the available chains probe different stages of the topological crossover; the observed ordering of the sector magic diameters is therefore a finite-length effect of topological conditioning. To isolate the thermal effect, we normalize each sector by the size of phantom rings with the same knot type. Even after this normalization, fixed-knot rings swell more slowly at small z than the unrestricted ensemble. The measured small-probe factors SK = 0.43(7)-0.81(3) shift the data left by 19-57% and bring the sector-resolved curves much closer to the unrestricted crossover, showing that the displacement arises largely from the altered near-ideal response.
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BibTeX
@misc{dadamo2026thermal,
title = {{Thermal and Topological Crossover of Ring Polymers}},
author = {Giuseppe d'Adamo},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05790674v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Thermal and Topological Crossover of Ring Polymers
AU - Giuseppe d'Adamo
PY - 2026
DA - 2026/09/22
PB - HAL
UR - https://hal.science/hal-05790674v1
AB - We use Monte Carlo simulations of off-lattice hard-sphere-monomer rings with N = 100-16 000 segments, classifying every sampled conformation by knot type, to show that topology reshapes the finite-size route to the good-solvent regime without changing the leading asymptotic behavior. For the topology-unrestricted ensemble, every nonzero hard-sphere diameter flows to the same good-solvent fixed point, but the leading correction to ⟨R 2 g ⟩/N 2ν vanishes at d * /b = 0.431(10), in agreement with the linear-chain magic value 0.433(1). The thermal swelling factor is governed by the same Zimm-Stockmayer-Fixman variable z as for linear chains. Fixed-z extrapolations and fixed-d/b data determine a common Monte Carlo crossover curve, which is described without adjustable parameters by the independently derived one-loop field theory for the ring topology. Its good-solvent amplitude gives the universal mean-square ring-to-linear size ratio H = 0.539(1)(4). Conditioning on a knot type changes this picture at finite length. The leading good-solvent exponent remains the same, but the correction amplitude and hence the magic diameter become knot dependent, shifting systematically to smaller values as the knot becomes more complex: d * /b = 0.409(7) for the unknot (01), 0.314(7) for the trefoil (31), and 0.237( 11) for the figureeight (41). Because the knotting length varies strongly with excluded volume, the available chains probe different stages of the topological crossover; the observed ordering of the sector magic diameters is therefore a finite-length effect of topological conditioning. To isolate the thermal effect, we normalize each sector by the size of phantom rings with the same knot type. Even after this normalization, fixed-knot rings swell more slowly at small z than the unrestricted ensemble. The measured small-probe factors SK = 0.43(7)-0.81(3) shift the data left by 19-57% and bring the sector-resolved curves much closer to the unrestricted crossover, showing that the displacement arises largely from the altered near-ideal response.
ER -