Pulling-force-induced elongation and alignment effects on entanglement and knotting characteristics of linear polymers in a melt

Abstract

We employ a primitive path (PP) algorithm and the Gauss linking integral to study the degree of entanglement and knotting characteristics of linear polymer model chains in a melt under the action of a constant pulling force applied to selected chain ends. Our results for the amount of entanglement, the linking number, the average crossing number, the writhe of the chains and their PPs and the writhe of the entanglement strands all suggest a different response at the length scale of entanglement strands than that of the chains themselves and of the corresponding PPs. Our findings indicate that the chains first stretch at the level of entanglement strands and next the PP (tube) gets oriented with the "flow." These two phases of the extension and alignment of the chains coincide with two phases related to the disentanglement of the chains. Soon after the onset of external force the PPs attain a more entangled conformation, and the number of nontrivially linked end-to-end closed chains increases. Next, the chains disentangle continuously to attain an almost unentangled conformation. Using the linking matrix of the chains in the melt, we furthermore show that these phases are accompanied by a different scaling of the homogeneity of the global entanglement in the system. The homogeneity of the end-to-end closed chains first increases to a maximum and then decreases slowly to a value characterizing a completely unlinked system.

A topological study of protein folding kinetics

“…Specifically, a measure of entanglement that can be directly applied to open chains and is a continuous function of the chain coordinates is the Gauss linking integral. This measure has been successfully applied to polymers in order to study polymer entanglement and it has been shown that it correlates with the physical properties of polymeric material [19][20][21][22]. The Gauss linking integral has likewise been used to classify protein conformations successfully [3,25] and it was shown that the maximum linking between any two parts of a protein correlates with its folding rate [4].…”

Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity

“…The number of kinks has been shown to increase with viscosity [9,10]. A similar inverse behavior between the Writhe of the chains and the number of kinks was observed in the initial time of an elongation of the chains [37].…”

“…In non-equilibrium conditions, the relation becomes more complex [37]. The viscosity can be obtained as the limit η=limω→0G2(ω)/ω.…”

“…These measures have been applied to study polymer entanglement in both equilibrium and non-equilibrium conditions. More precisely, it was shown that the writhe in combination with the Z1 algorithm can provide a new estimator of the entanglement length with several advantages over other estimators and our proposed measures have also been used to understand the dis-entanglement of polymer chains in a melt under an elongational force [36,37,38]. Recently, the Gauss linking integral has been also used to study protein folding kinetics [39].…”

The linking number in systems with Periodic Boundary Conditions

“…Exactly the same considerations apply for the linking number and the self-linking number. Indeed, computer experiments indicate that the linking number and the writhe are effective indirect measures of whatever one might call "entanglement", especially in systems of "random" filaments [41,27,43,44,45,46,47,48,42,49,50,51,52,53,54] and it has been shown that they can provide information relevant to the tube model [52,53].…”