Predicting the Behaviour of Rings in Solution
Polymeric systems offer an incredible richness of behaviour. Depending on the solution concentration, its temperature or its quality and the polymers length, or topology, every system made of polymers can be categorised into a “universality class”, within
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Predicting the Behaviour of Rings in Solution
No man is obliged to learn and know every thing; [...]; yet all persons are under some obligation to improve their understanding; otherwise it will be a barren desert, or a forest with overgrown weed and brambles. I. Watts
Polymeric systems offer an incredible richness of behaviour. Depending on the solution concentration, its temperature or its quality and the polymers length, or topology, every system made of polymers can be categorised into a “universality class”, within which it finds a physical characterisation (scaling) of its macroscopic properties. The physical properties of polymers have been pioneered by Flory in the ‘50s, by Edwards in the ‘60s and ‘70s and by de Gennes in the ‘70s and ‘80s. The theories that they developed were mainly concerned with linear polymers and helped the understanding and realisation of many polymer compounds used nowadays. Both Edwards and de Gennes became, at some stage, interested in studying polymers displaying more complicated topologies. They mainly focused on branched and star polymers, although both of them turned their attention to ring polymers, sooner or later, during their lifetime. Edwards (1967, 1968) chose to tackle the matter from a field-theoretic point-of-view while de Gennes (1979), Raphael et al. (1997) chose a more practical “gedankenexperiment” in which he studied a gel made of linked polymer rings, broadly known as “Olympic gel”. Both these series of attempts were far from being the most successful and important contributions brought forward by these two giants which indicates the difficulty of the topic (and partially excuses the diffident approach that I will assume in tackling the matter). Even though understanding ring polymers is a difficult task, important advances in the field have been achieved in the past decades (Cates and Deutsch 1986; Rubinstein 1986; Grosberg et al. 1993; Obukhov and Rubinstein 1994): Ring polymers are nowadays well known for behaving very differently from their linear counterparts, although a full theoretical description of their static and dynamic properties is far from achieved.
© Springer International Publishing Switzerland 2016 D. Michieletto, Topological Interactions in Ring Polymers, Springer Theses, DOI 10.1007/978-3-319-41042-5_2
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2 Predicting the Behaviour of Rings in Solution
In this chapter I will briefly review the main theoretical findings regarding systems of ring polymers in dense solutions, melts and embedded in gels. In particular, I will separately treat static and dynamic properties, and I will introduce the key observables that I will use to characterise and investigate the systems in the following chapters.
2.1 Statics 2.1.1 The Size of a Crumpled Coil The size of a polymer coil has been investigated in various solvents and different concentrations in the past decades. The general assumption is that the size R of a polymer coil depends on the degree of polymerisation M as R ∼ Mν
(2.1)
where ν is also known as the entropic exponent and is rela
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