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Statistical Mechanics of Confined Polymer Networks

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Abstract

We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical \(\Theta \)-point. In the \(\Theta \)-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, \(\gamma _b^{\Theta }\), to that of terminally-attached arches, \(\gamma _{11}^{\Theta },\) and to the correlation length exponent \(\nu ^{\Theta }.\) We find \(\gamma _b^{\Theta } =\gamma _{11}^{\Theta }+\nu ^{\Theta }\). In the case of the special transition, we find \(\gamma _b^{\Theta }(\mathrm{sp}) =\frac{1}{2}[\gamma _{11}^{\Theta }(\mathrm{sp})+\gamma _{11}^{\Theta }]+\nu ^{\Theta }\). For general networks, the explicit expression of configurational exponents then naturally involves bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm–Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the cases of ordinary, mixed and special surface transitions, and of the \(\Theta \)-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.

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Acknowledgements

We wish to acknowledge the hospitality of the Erwin Schrödinger International Institute for Mathematical Physics where this work was initiated, during the programme on Combinatorics, Geometry and Physics in June, 2014. AJG wishes to thank the Australian Research Council for supporting this work through Grant DP120100931, and more recently ACEMS, the ARC Centre of Excellence for Mathematical and Statistical Frontiers. We also wish to warmly thank Hans Werner Diehl for pointing out a number of references relevant to surface transitions, and Emmanuel Guitter for his kind help with the figures.

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  1. Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 91191, Gif-sur-Yvette, France

    Bertrand Duplantier

  2. School of Mathematics and Statistics, The University of Melbourne, Victoria, 3010, Australia

    Anthony J. Guttmann

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  1. Bertrand Duplantier
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Duplantier, B., Guttmann, A.J. Statistical Mechanics of Confined Polymer Networks. J Stat Phys 180, 1061–1094 (2020). https://doi.org/10.1007/s10955-020-02584-2

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