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Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots

We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connec...

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Autores principales: Deguchi, Tetsuo, Uehara, Erica
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2017
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6432503/
https://www.ncbi.nlm.nih.gov/pubmed/30970929
http://dx.doi.org/10.3390/polym9070252
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author Deguchi, Tetsuo
Uehara, Erica
author_facet Deguchi, Tetsuo
Uehara, Erica
author_sort Deguchi, Tetsuo
collection PubMed
description We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs “topological polymers”. Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology “topological polymers”, for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer–Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it “topological swelling”.
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spelling pubmed-64325032019-04-02 Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots Deguchi, Tetsuo Uehara, Erica Polymers (Basel) Review We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs “topological polymers”. Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology “topological polymers”, for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer–Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it “topological swelling”. MDPI 2017-06-28 /pmc/articles/PMC6432503/ /pubmed/30970929 http://dx.doi.org/10.3390/polym9070252 Text en © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Review
Deguchi, Tetsuo
Uehara, Erica
Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
title Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
title_full Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
title_fullStr Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
title_full_unstemmed Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
title_short Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots
title_sort statistical and dynamical properties of topological polymers with graphs and ring polymers with knots
topic Review
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6432503/
https://www.ncbi.nlm.nih.gov/pubmed/30970929
http://dx.doi.org/10.3390/polym9070252
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