Ring Polymers in Melts and Solutions: Scaling and Crossover

Abstract

We propose a simple mean-field theory for the structure of ring polymer melts. By combining the notion of topological volume fraction and a classical van der Waals theory of fluids, we take into account many body effects of topological origin in dense systems. We predict that although the compact statistics with the Flory exponent ν = 1/3 is realized for very long chains, most practical cases fall into the crossover regime with the apparent exponent ν = 2/5 during which the system evolves toward a topological dense-packed limit.PACS numbers: 61.25. 83.80.Sg Statistics of melts and concentrated solutions of ring polymers is a longstanding problem in polymer physics [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Unlike linear chain systems, their physical properties crucially depends on the preparation history during which the topology of the system is frozen. Then, the non-crossing requirement creates topological constraints which impose nontrivial restrictions on the phase space of the system, hence, have a strong influence even in statics.In the linear polymer counterpart, a well-known Flory theorem states that the chain conformation is Gaussian characterized by the Flory exponent ν = 1/2 due to the screening of excluded-volume interactions [15][16][17]. This simple (but surprising) result, however, no longer holds for the ring polymer melt. The most basic question arises in the melt of ring polymers free from any mutual-linking and self-knotting. There have been several experimental [6,7] and numerical studies [8][9][10][11][12][13][14] in this direction, but the clear-cut conclusion has not been attained yet.In their seminal paper, Cates and Deutsch (C-D) argued that the unconcatenated rings in the melt may have statistics intermediate between those of collapsed (ν = 1/3) and Gaussian (ν = 1/2) chains [1]. Specifically, they proposed a conjecture on the scaling exponent ν = 2/5 based on Flory-type mean-field theory. While this leading theoretical guide seems to be supported by following numerical simulations [8][9][10][11][12], some of more recent observations claim the collapsed statistics ν = 1/3 for sufficiently long rings [13,14]. The latter result was hypothesized by Khokhlov and Nechaev based on the analogy with lattice animals [2]. Closely related to this is the crumpled globule (CG) model [3,4] It is important to keep in mind that the topological effect manifests itself in the scale larger than some characteristic length ξ 1 . Individual rings in concentrated solutions of small molecular weight N < g 1 ≡ (ξ 1 /a) 2 φ 1/4 (N is the number of monomers in each ring, a is the monomer size and φ is the monomeric volume fraction) thus show Gaussian behaviors with the size R 0 ≃ ξ(N/g) 1/2 ≃ aN 1/2 φ −1/8 where ξ ≃ ag 3/5 ≃ aφ −3/4 is the correlation length of concentration fluctuation. In the present paper, we introduce the notion of the topological volume fraction and construct a mean-filed theory for the concentrated solution of noncatenated long ring polymers. Requiring the theory to be compatible ...

Effect of Topology on the Conformations of Ring Polymers

“…11,14,16 Thus, we expect the following power laws, R ∼ ϕ x , for ϕ ≫ ϕ* Here, the first case corresponds to interpenetrating rings and is equivalent with the behavior of linear chains in semidilute solutions. 6 The other two cases correspond to the two suggested regimes of the noninterpenetrating rings.…”

“…We briefly note that dropping the topological excluded volume term in eq V.5 leads to R ∼ N 1/4 , a result that formally corresponds to ideal hyperbranched polymers and was found previously for nonconcatenated rings in an array of obstacles. 44 Grosberg et al 5 and Sakaue 16,45 discussed that the unknotting constraint leads to a free energy contribution that turns into ≈N 3 /R 6 for a large compression of rings. This term results from the then dominating three-body interactions among the monomers of the ring and can be pictured as a simultaneous compression in length and width of an R-tube 5 that confines the ring.…”

On the tree-like structure of rings in dense solutions

“…It is nowadays well accepted that, in the limit of large polymerisation index M , rings in the melt assume configurations which display a typical size R g scaling as [8][9][10][11] R g ∼ M ν , with ν = 1/3. This value of the metric exponent ν is usually associated with collapsed polymers in poor solvents, which tightly fold onto themselves expelling other chains and solvent from their interior volume.…”

Communication: When does a branched polymer become a particle?

“…Ring polymers tend to have relative contracted molecular conformations meaning that their exponent is smaller than a random walk chain (ν = 1/2), 1,50 but larger than a polymer in collapsed globular state (ν = 1/3). 51,52 Given the discussion above regarding the average ring shape in the melt, the natural comparison we make is to randomly branched polymers with screened excluded volume interactions where ν ≈ 0.4, i.e., branched polymers at their θ-point. 47 This value accords with our simulations and measured estimates for ring melts, 47 but recent simulations have suggested that ν might approach smaller values, ν = 0.36 or even ν = 1/3 in the limit of extremely long melt rings.…”