Abstract
Understanding the dynamical and rheological properties of solutions of long ring polymers is of primary importance in several areas of soft matter, material science and biophysics (Cremer and Cremer 2001; Kapnistos et al. 2008; Halverson et al. 2011b, 2013). As mentioned in Chap. 2, ring polymers do not follow the standard reptation theory and in order to make progress it seems that the scientific community will require innovative and unconventional approaches to analyse their properties.
It looks as if it was a bride, walking down the isle, while her dress is being pulled back by flower girls whose dresses are also being pulled by flower girls whose dresses are pulled by other flower girls ...
A.Y. Grosberg
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Notes
- 1.
It is interesting to notice that in the ensemble I study in this chapter, the picture of “rings in a gel”, or “in a background of obstacles”, is more appropriate than in the case of a pure melt.
- 2.
These could be closed at infinity, but anywhere away from the cell would suffice.
- 3.
It is fair to say that the similarity between these two methods became apparent, at least to me, only after we finalised the algorithm to detect threadings.
- 4.
\(\langle T_{0.1} \rangle \) can be conceptually compared to the time taken by a linear polymer to renew its confining tube, begin threadings a sort of “tube” (or “cage”) for rings.
- 5.
It is interesting to notice that the same question is also frequently asked when studying more conventional glass-forming materials (Berthier and Biroli 2011).
- 6.
Since the mobile chain leaves a hole that produces an uniformity in the system density and could, in principle, interfere with a perfectly free diffusion.
- 7.
It is perhaps interesting to think about what would happen if one were to relax the uncrossability condition imposed on the chains and substitute it with a potential barrier of finite height A. In this scenario, one expects the scattering function to re-establish its decay although only after a time \(t \sim \langle Th\rangle \exp {A}\) that corresponds to an activated process where at least some of the threadings must be by-passed. In this picture, the system would probably display a long-time \(\alpha \) relaxation which decouples from the local rattling proportionally to the number of threadings in the system.
- 8.
and is broadly analogous, although in a much less coarse-grained sense, to what happens in glass-forming systems belonging to the class of “Kinetically Constrained Models” (Palmer et al. 1984).
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Michieletto, D. (2016). Threading Rings. In: Topological Interactions in Ring Polymers. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41042-5_4
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