### Abstract

The relaxation of a single knotted ring polymer is studied by Brownian dynamics simulations. The relaxation rate λ_{q} for the wave number q is estimated by the least square fit of the equilibrium time-displaced correlation function Ĉ_{q}(t) = N^{-1} ∑_{i}∑_{j}(1/3)(R_{i}(t) · R _{j}(0)) exp[i2πq(j - i)/N] to a double exponential decay at long times. Here, N is the number of segments of a ring polymer and R_{i} denotes the position of the ith segment relative to the center of mass of the polymer. The relaxation rate distribution of a single ring polymer with the trivial knot appears to behave as λ_{q} ≃ A(1/N)^{x} for q = 1 and λ_{q} ≃ A′(q/N)^{x′} for q > 1, where x ≃ 2:10, x′ ≃ 2:17, and A < A′. These exponents are similar to that found for a linear polymer chain. The topological effect appears as the separation of the power law dependences for q = 1 and q > 1, which does not appear for a linear polymer chain. In the case of the trefoil knot, the relaxation rate distribution appears to behave as λ_{q} ≃ A(1/N)^{x} for q = 1 and λ_{q} ≃ A′(q/N)^{x}′ for q = 2 and 3, where x ≃ 2.61, x′ ≃ 2:02, and A > A′. The wave number q of the slowest relaxation rate λ_{q} for each N is given by q = 2 for small values of N, while it is given by q = 1 for large values of N. This crossover corresponds to the change of the structure of the ring polymer caused by the localization of the knotted part to a part of the ring polymer.

Original language | English |
---|---|

Article number | 034001 |

Journal | Journal of the Physical Society of Japan |

Volume | 77 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2008 Mar |

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### Keywords

- Brownian dynamics simulations
- Knot
- Relaxation modes
- Relaxation rates
- Ring polymer
- Single polymer
- Topological effects

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Journal of the Physical Society of Japan*,

*77*(3), [034001]. https://doi.org/10.1143/JPSJ.77.034001

**Relaxation of a single knotted ring polymer.** / Saka, Shinya; Takano, Hiroshi.

Research output: Contribution to journal › Article

*Journal of the Physical Society of Japan*, vol. 77, no. 3, 034001. https://doi.org/10.1143/JPSJ.77.034001

}

TY - JOUR

T1 - Relaxation of a single knotted ring polymer

AU - Saka, Shinya

AU - Takano, Hiroshi

PY - 2008/3

Y1 - 2008/3

N2 - The relaxation of a single knotted ring polymer is studied by Brownian dynamics simulations. The relaxation rate λq for the wave number q is estimated by the least square fit of the equilibrium time-displaced correlation function Ĉq(t) = N-1 ∑i∑j(1/3)(Ri(t) · R j(0)) exp[i2πq(j - i)/N] to a double exponential decay at long times. Here, N is the number of segments of a ring polymer and Ri denotes the position of the ith segment relative to the center of mass of the polymer. The relaxation rate distribution of a single ring polymer with the trivial knot appears to behave as λq ≃ A(1/N)x for q = 1 and λq ≃ A′(q/N)x′ for q > 1, where x ≃ 2:10, x′ ≃ 2:17, and A < A′. These exponents are similar to that found for a linear polymer chain. The topological effect appears as the separation of the power law dependences for q = 1 and q > 1, which does not appear for a linear polymer chain. In the case of the trefoil knot, the relaxation rate distribution appears to behave as λq ≃ A(1/N)x for q = 1 and λq ≃ A′(q/N)x′ for q = 2 and 3, where x ≃ 2.61, x′ ≃ 2:02, and A > A′. The wave number q of the slowest relaxation rate λq for each N is given by q = 2 for small values of N, while it is given by q = 1 for large values of N. This crossover corresponds to the change of the structure of the ring polymer caused by the localization of the knotted part to a part of the ring polymer.

AB - The relaxation of a single knotted ring polymer is studied by Brownian dynamics simulations. The relaxation rate λq for the wave number q is estimated by the least square fit of the equilibrium time-displaced correlation function Ĉq(t) = N-1 ∑i∑j(1/3)(Ri(t) · R j(0)) exp[i2πq(j - i)/N] to a double exponential decay at long times. Here, N is the number of segments of a ring polymer and Ri denotes the position of the ith segment relative to the center of mass of the polymer. The relaxation rate distribution of a single ring polymer with the trivial knot appears to behave as λq ≃ A(1/N)x for q = 1 and λq ≃ A′(q/N)x′ for q > 1, where x ≃ 2:10, x′ ≃ 2:17, and A < A′. These exponents are similar to that found for a linear polymer chain. The topological effect appears as the separation of the power law dependences for q = 1 and q > 1, which does not appear for a linear polymer chain. In the case of the trefoil knot, the relaxation rate distribution appears to behave as λq ≃ A(1/N)x for q = 1 and λq ≃ A′(q/N)x′ for q = 2 and 3, where x ≃ 2.61, x′ ≃ 2:02, and A > A′. The wave number q of the slowest relaxation rate λq for each N is given by q = 2 for small values of N, while it is given by q = 1 for large values of N. This crossover corresponds to the change of the structure of the ring polymer caused by the localization of the knotted part to a part of the ring polymer.

KW - Brownian dynamics simulations

KW - Knot

KW - Relaxation modes

KW - Relaxation rates

KW - Ring polymer

KW - Single polymer

KW - Topological effects

UR - http://www.scopus.com/inward/record.url?scp=54349100996&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=54349100996&partnerID=8YFLogxK

U2 - 10.1143/JPSJ.77.034001

DO - 10.1143/JPSJ.77.034001

M3 - Article

AN - SCOPUS:54349100996

VL - 77

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 3

M1 - 034001

ER -