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

This paper describes and provides Z1+, the successor of the Z- and Z1-codes for topological analyses of mono- and polydisperse entangled linear polymeric systems, in the presence or absence of confining surfaces or nano-inclusions. In contrast to its predecessors, Z1+ makes use of adaptive neighbor lists, and keeps the number of temporary nodes relatively large, yielding improved performance for large system sizes. Z1+ also includes several features its predecessors lacked, including several that are advantageous for analyses of semi-crystalline systems, brushes, nano-composites, and flowing liquids. It offers a graphical user interface that can be used to run Z1+ and visualize the results, and a PPA+ option that allows Z1+ to perform a primitive path analysis more efficiently than the standard procedure (PPA option). In addition to describing Z1+'s and PPA+'s implementation and computational performance in detail, we use it to show that it yields entanglement lengths that agree quantitatively with both a recently proposed unified analytic theory for flexible and semiflexible polymer-melt entanglement and with the available experimental data for these systems. Finally we show that the associated theoretical expressions, which express reduced entanglement-related quantities in terms of the scaled Kuhn segment density Λ, need not describe results for model polymer solutions of different "chemistries", i.e. different angular and dihedral interactions but the same Λ. Program title: Z1+ CPC Library link to program files: https://doi.org/10.17632/m425t6xtwr.1 Licensing provisions: Apache-2.0 Programming language: fortran 90, perl (standalone batch version) or java (interactive online version) External requirements: perl 5+ (standalone batch version), jre 1.8.0+ (interactive online version) Nature of problem: Starting from a multiple-chain configuration that consists of the coordinates of particles forming linear chains, as well as the dimensions of the rectangular or monoclinic (periodic or closed) cell containing the particles, the problem is to find the shortest multiple disconnected path obtained from the starting configuration upon (i) fixing chain ends at their original positions, (ii) disallowing bond crossings, and (iii) monotonically decreasing the path length. Solution method: We use local geometric operations that fulfill the constraints (i)-(iii) and apply them as long as the path is still shrinking. This ultimately leads to disconnected polygons, i.e., paths characterized by nodes interconnected by straight lines. Each node has a corresponding 'entangled' node responsible for its existence. These pairs as well as the full configuration of the shortest path is reported.