The Z1+ package: Shortest multiple disconnected path for the analysis of entanglements in macromolecular systems

Microstructural evolution and reverse flow in shear-banding of entangled polymer melts

“…Theoretically, one can calculate the equilibrium chain entanglement number by Z = hLi/a t = N K b 2 /a t 2 = hLi 2 /hR 2 i, where hLi and a t are the average primitive chain contour length and the tube diameter at equilibrium, respectively. The entanglement network topological analysis using the Z1-code of Kro ¨ger [65][66][67][68] provides the L and a t for calculation of Z. Moreover, the Z1-code introduces an alternative topological quantifier, the number of kinks per chain, Z k .…”

Topological Analysis and Recovery of Entanglements in Polymer Melts

“…Conversely, the more the rings are entangled, the less they will shrink. Then we apply (see section for details) the Z1 algorithm ,− to the shrunken structures and estimate N e thereby. Figure (main panel, solid lines) shows the values of N e as a function of N and for different bending stiffness values, κ bend .…”

“…By following the approach by Bobbili and Milner for molecular dynamics simulations of a melt of seemingly shrunk and randomly linked ring polymers, we estimate N e by applying a recent version (Z1+) of the Z1 algorithm. ,− The Z1 algorithm consists of the implementation of a series of geometrical operations that transform the entangled polymer chains in a collection of straight segments that are sharply bent at the entanglement points, and then one may estimate N e as the average length of these straight segments. In particular, the Z1+ version takes explicitly into account the role of chain self-entanglements (knots) during the determination of N e .…”

“…The authors are indebted to M. Kröger for sharing with us the Z1+ algorithm before its official release 40 and P. Dabrowski-Tumanski, who gave us invaluable technical advice regarding the Topoly package. 38 The authors also acknowledge networking support by COST Action CA17139 (EUTOPIA).…”

Molecular dynamics simulation insight into topological structure dependence of self-healing polymer nanocomposites

“…Snapshots reveal that this situation occurs because Crosslinkers accumulate locally in the PNCs due to their steric hindrance effect during the dispersion process, while overall dispersion in the PNCs is relatively uniform. The entanglement of matrix chains in PNCs with NPs of different topological structures was explored using the Z1+ code 49,56 The entanglement of matrix chains obtained through the Z1+ code operation is presented in Table 2.…”