Topological effects in ring polymers. II. Influence of persistence length

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 61, Issue 4, 2000, Pages 4078-4089

  Muller M ^a,b   Wittmer, J P ^b,c   Cates, M E ^b  

^a JOHANNES GUTENBERG UNIVERSITY   (Germany)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

^c UNIVERSITÉ LYON 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; FRACTAL DIMENSION; INTELLIGENT SYSTEMS; MONTE CARLO METHODS; POLYMERS;

CHARACTERISTIC LENGTH; CONFORMATIONAL PROPERTIES; LOCAL FRACTAL DIMENSION; MEAN-SQUARE DISPLACEMENT; STATIC STRUCTURE FACTORS; THREE-DIMENSIONAL LATTICES; TOPOLOGICAL CONSTRAINTS; TOPOLOGICAL INTERACTIONS;

TOPOLOGY;

EID: 0000752441     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.61.4078     Document Type: Article

References (33)

1. Unknotted and unconcatenated ring polymers have actually been made and investigated experimentally. In Ref. 4 the reader will find a detailed account of results and technical difficulties related to ring synthesis and contamination with linear rings. We do not wish to repeat this here. (Also see Ref. 2.) All these studies focus on dynamical properties. No experimental study of the radius of gyration of rings in melt appears to have been made until now—despite the fact that any reasonable description of dynamics requires a good understanding of conformational properties.
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11. Note that ring closure is a necessary condition for the topological constraints we are focusing on. It is a relatively simple matter to conceive an algorithm for closed chains which violate topology. Hence closure and topology constraints are related, but distinct, properties.
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24. These estimates depend strongly on the further evolution of the effective Flory exponent (Formula presented) discussed below. However, the balance remains strongly favorable even if the computational overhead (to calculate the additional potential), the reduced acceptance rate A (see Table II), and (more importantly) the slower diffusion (see Fig. 1212) are taken into account.
25. Recently the effects of non-self-knotting for dilute rings with extremely weak excluded volume interactions were investigated by J. M. Deutsch, Phys. Rev. E 59, R2539 (1999). His Monte Carlo simulations indicate surprisingly that the extension of these chains still scale with the same exponent (Formula presented) due to the unknottedness constraint.
26. We have also tried values based on the effective bond lengths for Gaussian chains from Ref. 16 quoted in Table II. However, the results are virtually identical.
27. Let us assume that the ring consists of uncorrelated needlelike arms of a fixed length (Formula presented) Two segments on the same arm are either parallel or antiparallel; the directions of different arms are completely uncorrelated. For this crude model the bond-bond correlation function (Formula presented) decays linearly from 1 at (Formula presented) to (Formula presented) at (Formula presented) and then increases linearly to 0 at (Formula presented) For larger distances n along the chains there are no correlations by construction. This motivates the assumption that a minimum in the bond-bond correlation function is associated with a backfolding, and the position of the minimum is an estimate of the arm length. Certainly, there is a distribution of arm lengths, and the correlation along an arm is not perfect. This will broaden the minimum. More important, the above argument neglects the hierarchical structure of lattice animals and only captures local correlations.
28. Note that even for linear chains of length (Formula presented) we do not obtain a perfect Gaussian plateau, but rather a flattish hump [see Fig. 66(b)]. This provides an important warning. Even for the linear chains it is tricky to reach the asymptotic limit for the differential fractal dimension.
29. Admittingly a priori it is not excluded that b triggers a continuous spectrum of universality classes.
30. Accepting that all lines will merge eventually, we have to ask how they will do so. The simplest suggestion is that they all merge together at one point (Formula presented) Then no persistence length, e.g., the (Formula presented) line, is singled out. Otherwise one may characterize each persistence length by the point (Formula presented) where it joins the flexible ring line (Formula presented)
31. J. Klein, Macromolecules 19, 105 (1986).
32. Not to mention logarithmic corrections due to weak topological interactions in the (Formula presented) limit of the CD regime.
33. It was estimated in S. P. Obukhov, M. Rubinstein, and R. H. Colby, Macromolecules 27, 3191 (1994) that the (Formula presented) regime could last until (Formula presented)