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Beyond Flory theory: Distribution functions for interacting lattice trees

Angelo Rosa and Ralf Everaers
Phys. Rev. E 95, 012117 – Published 12 January 2017

Abstract

While Flory theories [J. Isaacson and T. C. Lubensky, J. Physique Lett. 41, 469 (1980); M. Daoud and J. F. Joanny, J. Physique 42, 1359 (1981); A. M. Gutin et al., Macromolecules 26, 1293 (1993)] provide an extremely useful framework for understanding the behavior of interacting, randomly branching polymers, the approach is inherently limited. Here we use a combination of scaling arguments and computer simulations to go beyond a Gaussian description. We analyze distribution functions for a wide variety of quantities characterizing the tree connectivities and conformations for the four different statistical ensembles, which we have studied numerically in [A. Rosa and R. Everaers, J. Phys. A: Math. Theor. 49, 345001 (2016) and J. Chem. Phys. 145, 164906 (2016)]: (a) ideal randomly branching polymers, (b) 2d and 3d melts of interacting randomly branching polymers, (c) 3d self-avoiding trees with annealed connectivity, and (d) 3d self-avoiding trees with quenched ideal connectivity. In particular, we investigate the distributions (i) pN(n) of the weight, n, of branches cut from trees of mass N by severing randomly chosen bonds; (ii) pN(l) of the contour distances, l, between monomers; (iii) pN(r) of spatial distances, r, between monomers, and (iv) pN(r|l) of the end-to-end distance of paths of length l. Data for different tree sizes superimpose, when expressed as functions of suitably rescaled observables x=r/r2(N) or x=l/l(N). In particular, we observe a generalized Kramers relation for the branch weight distributions (i) and find that all the other distributions (ii–iv) are of Redner-des Cloizeaux type, q(x)=C|x|θexp(K|x|)t. We propose a coherent framework, including generalized Fisher-Pincus relations, relating most of the RdC exponents to each other and to the contact and Flory exponents for interacting trees.

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  • Received 14 October 2016

DOI:https://doi.org/10.1103/PhysRevE.95.012117

©2017 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & ThermodynamicsPolymers & Soft Matter

Authors & Affiliations

Angelo Rosa*

  • SISSA-Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy

Ralf Everaers

  • Univ Lyon, Ens de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique and Centre Blaise Pascal, F-69342 Lyon, France

  • *anrosa@sissa.it
  • ralf.everaers@ens-lyon.fr

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Issue

Vol. 95, Iss. 1 — January 2017

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