Entropic tightening of vibrated chains

Abstract

We investigate experimentally the distribution of configurations of a ring with an elementary topological constraint, a "figure-8" twist. Using vibrated granular chains, which permit controlled preparation and direct observation of such a constraint, we show that configurations where one of the loops is tight and the second is large are strongly preferred. This agrees with recent predictions for equilibrium properties of topologically-constrained polymers. However, the dynamics of the tightening process weakly violate detailed balance, a signature of the nonequilibrium nature of this system. 81.05.Rm, 83.10Nn Topological constraints such as knots and entanglements constantly form and relax in polymer systems and in biomolecules such as DNA [1][2][3][4][5][6]. These constraints increase relaxation time scales and additionally restrict the phase space accessible by the macromolecule. Whereas the role entanglements play in chain dynamics is well appreciated [7,8], even more basic questions concerning effects of topological constraints on static properties such as the chain structure remain largely unanswered. Recent theoretical studies predict that in equilibrium, a knotted polymer will generally favor configurations where the knot is "tight", i.e., localized to a small region of the chain [9,10]. Numerical simulations and scaling analysis support this prediction [11], but direct experimental tests are difficult [12]. In this study, we examine this interesting prediction experimentally using vertically vibrated granular chains.Granular chains consist of spherical beads connected by rods, and their backbone enforces the same geometrical constraints as in a polymer system. A vibrating plate supplies the system with energy, balancing the energy dissipation due to inelastic collisions experienced by beads [13][14][15][16]. This polymer system is well suited for studying topological constraints as it allows control of the chain size and the constraint type, as well as direct observations of the chain conformation in contrast with traditional polymer systems. Recent studies have successfully used vibrated chains to study diffusive relaxation [17] and spontaneous formation [18] of knots.We considered the simplest possible topology, a "figure-8": a once-twisted ring consisting of two loops, separated by a single crossing point, which functions as the topological constraint (see Fig. 1). Under appropriate vibration amplitude, the system is effectively twodimensional and the crossing point hops along the chain without flipping open the figure-8. Surprisingly, despite the highly nonequilibrium drive applied to the system, we observed strong entropic tightening. The microscopic degrees of freedom, the beads, experience periodic drive and dissipative collisions with the plate, rods, and other beads, as well as frictional forces. Remarkably the macroscopic observable, the loop size, obeys a statistical mechanics. Detailed balance is only weakly violated and the empirical loop-size distribution is close to that conjectured on...

Tightness of slip-linked polymer chains

“…From the plot, we realise that the scaling be- haviour is surprisingly well fulfilled, and that the predicted value is reproduced in good agreement. This result was corroborated experimentally for a figure-eight necklace chain on a vibrating plate [74].…”

Effect of Knotting on the Shape of Polymers

“…This hypothesis also is consistent with observations that, on average, as the polymer length increases, the knotted portion of the chain gets smaller in comparison to the overall length of the chain, and therefore, the large unknotted portion of the polymer dominates the average configuration. [40][41][42] There is numerical evidence that knotting is, on average, weakly local. 43 Thus, the influence of simple knots, such as a single trefoil, on the average shape is quite small asymptotically.…”

Effective interactions between inclusions in complex fluids driven out of equilibrium

“…Extensions of the concepts of "effective potential" and "entropic forces" to out of equilibrium situations have been scarce [12,13,14]. Indeed effective potentials can not be simply derived from a free energy in an out-of-equilibrium context, and only the instantaneous force acting on the external objects for a given configuration of the medium can be defined.…”