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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. [ABSTRACT FROM AUTHOR]

We attempt to provide direct evidence for the suggested behavior of dendronized polymers as molecular objects (i.e., single shape persistent macromolecules). For this purpose, the microscopic structure of dendronized polymers adsorbed onto mica has been investigated using atomistic molecular dynamics simulations. We find that the shape of the second to fourth generation dendronized polymers is basically kept upon adsorption due to substantial backfoldingwithin their interior. The fluctuation strength of the polymer backbones, which is seen to decrease with increasing generation, also indicates that these individual macromolecules exhibit molecular object behavior in the nanosize range.

Inspired by molecular mechanisms that cells exploit to sense mechanical forces and convert them into biochemical signals, chemists dream of designing mechanochemical switches integrated into materials. Using the adhesion protein fibronectin, whose multiple repeats essentially display distinct molecular recognition motifs, we derived a computational model to explain how minimalistic designs of repeats translate into the mechanical characteristics of their fibrillar assemblies. The hierarchy of repeat-unfolding within fibrils is controlled not only by their relative mechanical stabilities, as found for single molecules, but also by the strength of cryptic interactions between adjacent molecules that become activated by stretching. The force-induced exposure of cryptic sites furthermore regulates the nonlinearity of stress-strain curves, the strain at which such fibers break, and the refolding kinetics and fraction of misfolded repeats. Gaining such computational insights at the mesoscale is important because translating protein-based concepts into novel polymer designs has proven difficult.

The properties of polymer layers end-grafted to the inner surface of nanopores connected to solvent reservoirs are studied theoretically as a function of solvent quality and pore geometry. Our systematic study reveals that nanoconfinement is affected by both pore radius and length and that the conformations of the polymer chains strongly depend on their grafting position along the nanopore and on the quality of the solvent. In poor solvent, polymer chains can collapse to the walls, form a compact plug in the pore, or self-assemble into domains of different shape due to microphase separation. The morphology of these domains (aggregates on pore walls or stacked micelles along the pore axis) is mainly determined by the relationship between chain length and pore radius. In other cases the number of aggregates depends on pore length. The presence of reservoirs decreases confinement at pore edges due to the changes in available volume and introduces new organization strategies not available for infinite nanochannels. In good solvent conditions, chains grafted at the pore entrances stretch out of the pore, relieving the internal osmotic pressure and increasing the entropy of the polymers. Our study also addresses the experimentally relevant case of end-grafted chains on the outer walls of the membrane surrounding the nanopore. The effect of these polymer chains on the organization within the nanopore depends on solvent quality. For good solvents the outer chains increase the confinement of the chains at the entrance of the pore; however, the effect does not result in new structures. For poor solvents the presence of the outer polymer layer may lead to changes in the morphology of the microphase-separated domains. Our results show the complex interplay between the different interactions in a confined environment and the need to develop theoretical and experimental tools for their study.

Abstract: We cast the Jacobi identity for continuous fields into a local form which eliminates the need to perform any partial integration to the expense of performing variational derivatives. This allows us to test the Jacobi identity definitely and efficiently and to provide equations between different components defining a potential Poisson bracket. We provide a simple MathematicaTM notebook which allows to perform this task conveniently, and which offers some additional functionalities of use within the framework of nonequilibrium thermodynamics: reversible equations of change for fields, and the conservation of entropy during the reversible dynamics. Program summary: Program title: Poissonbracket.nb Catalogue identifier: AEGW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGW_v1_0.html Program obtainable from: CPC Program Library, Queen''s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 227 952 No. of bytes in distributed program, including test data, etc.: 268 918 Distribution format: tar.gz Programming language: MathematicaTM 7.0 Computer: Any computer running MathematicaTM 6.0 and later versions Operating system: Linux, MacOS, Windows RAM: 100 Mb Classification: 4.2, 5, 23 Nature of problem: Testing the Jacobi identity can be a very complex task depending on the structure of the Poisson bracket. The MathematicaTM notebook provided here solves this problem using a novel symbolic approach based on inherent properties of the variational derivative, highly suitable for the present tasks. As a by product, calculations performed with the Poisson bracket assume a compact form. Solution method: The problem is first cast into a form which eliminates the need to perform partial integration for arbitrary functionals at the expense of performing variational derivatives. The corresponding equations are conveniently obtained using the symbolic programming environment MathematicaTM. Running time: For the test cases and most typical cases in the literature, the running time is of the order of seconds or minutes, respectively. [Copyright &y& Elsevier]

Abstract: A package has been developed for calculating analytic expressions for forces and torques onto an arbitrarily shaped convex tracer (aerosol) particle small compared to the mean free path of the surrounding nonequilibrium gas. The package Phoretic allows to compute analytical (and also numerical) expressions for forces and torques stemming from elastic and diffusive scattering processes parameterized by an accommodation coefficient. The method is based on calculating half-sphere integral tensors of arbitrary rank and on integrating forces and torques acting on surface elements. The surrounding gas is completely specified by an arbitrarily shaped velocity distribution function. Accordingly, Phoretic requires two inputs: A particle (surface) geometry and a velocity distribution function. For example, the particle may be a cylinder with flat end caps, and the distribution function the one of Maxwell (isotropic) or Grad (13th moment approximation). The package reproduces analytic results for spheres which were available in the literature, and the ones for other geometries (cylinders, cuboids, ellipsoids) which were, however, only partially available (some works considered only elastic collisions, others temperature, or pressure, or only velocity gradients, etc.). In addition, Phoretic takes into account angular velocities which have been usually neglected and become relevant for non-spherical particles. The package is geared towards the implementation of dynamical equations for aerosol particles suspended in dilute or semidilute gases and as such helps to obtain concentration profiles and mobilities of aerosol particles depending on their shape (distribution) and environmental conditions. Program summary: Title of program: Phoretic Catalogue identifier:ADYI_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADYI_v1_0 Program obtainable from: CPC Program Library, Queen''s University of Belfast, N. Ireland Licensing provisions: Persons requesting the program must sign the standard CPC-non-profit use license (see license agreement printed in every issue) Computer for which the program is designed and others on which it has been tested: All platforms with a monitor Operating systems or monitors under which the program has been tested: Linux, Windows XP, Unix, Mac-OS Program language used: Mathematica®, version 5.2 or later. Phoretic makes use of the DiscreteMath‘Combinatorica’ Mathematica® package Memory required to execute with typical data: 10 MByte No. of lines in distributed program, including test data, etc.: 22 410 No. of bytes in distributed program, including test data, etc.: 114 657 Distribution format:tar.gz Nature of physical problem: Starting from a non-uniform velocity distribution function of a gas in terms of its moments, i.e. field variables, and field gradients such as temperature, pressure, or velocity field, the problem is to analytically calculate forces and torques acting onto arbitrarily shaped convex tracer (aerosol) particles small in size compared to the mean free path of the gas. The collision process is modeled as a superposition of elastic and diffusive scattering processes (parameterized by ). Method of solution: We implemented the solution to this problem in the symbolic programming language Mathematica®. The program allows to specify an arbitrary shape of the tracer particle and an arbitrary distribution function of the gas and returns symbolic or numerical expressions for forces and torques. The solution requires the calculation of half-sphere and base surface integrals and subsequent symbolic algebraic and tensorial manipulations. Restrictions on the complexity of the problem: Not known. In case the software cannot calculate surface integrals analytically it offers the possibility to proceed with a numerical evaluation of the corresponding terms. Typical running time: Typical running times mostly depend on the shape of the tracer particle. For all examples coming together with the software distribution run times are below 5 minutes on a modern single-processor platform. [Copyright &y& Elsevier]

A large number of well-equilibrated atomistic configurations of linear, strictly monodisperse polyethylene (PE) melts of molecular length ranging from C24up to C1000, obtained through extensive Monte Carlo simulations based on chain connectivity altering algorithms, have been subjected to a detailed topological analysis. Primitive paths are geometrically constructed connecting the two ends of a polymer chain (which in all cases are considered as fixed in space) under the constraint of no chain crossability, such that the multiple disconnected (coarse-grained) path has minimum contour length. When applied to a given, dense polymer configuration in 3-D space, the algorithm returns the primitive path (PP) and the related number and positions of entanglements (kinks) for all chains in the simulation box, thus providing extremely useful information for the topological structure (the primitive path network) hidden in bulk PE. In particular, our analysis demonstrates that once a characteristic chain length value (around C200) is exceeded, the entanglement molecular length for PE at T 450 K reaches a plateau value, characteristic of the entangled polymeric behavior. We further validate recent analytical predictions Schieber, J. D. J. Chem. Phys.2003, 118, 5162 about the shape of the distribution for the number of strands in a chain at equilibrium. At the same time, we show that the number of entanglements obtained by assuming random walk statistics Everaers, R. et al. Science2004, 303, 823 deviates significantly from these predictions, which we regard as a clear evidence that by directly counting the entanglements and their distribution functions, as proposed here, offers advantages for a quantitative analysis of the statistical nature of entanglements in polymeric systems.

Abstract: We present an algorithm which returns a shortest path and related number of entanglements for a given configuration of a polymeric system in 2 or 3 dimensions. Rubinstein and Helfand, and later Everaers et al. introduced a concept to extract primitive paths for dense polymeric melts made of linear chains (a multiple disconnected multibead ‘path’), where each primitive path is defined as a path connecting the (space-fixed) ends of a polymer under the constraint of non-interpenetration (excluded volume) between primitive paths of different chains, such that the multiple disconnected path fulfills a minimization criterion. The present algorithm uses geometrical operations and provides a—model independent—efficient approximate solution to this challenging problem. Primitive paths are treated as ‘infinitely’ thin (we further allow for finite thickness to model excluded volume), and tensionless lines rather than multibead chains, excluded volume is taken into account without a force law. The present implementation allows to construct a shortest multiple disconnected path (SP) for 2D systems (polymeric chain within spherical obstacles) and an optimal SP for 3D systems (collection of polymeric chains). The number of entanglements is then simply obtained from the SP as either the number of interior kinks, or from the average length of a line segment. Further, information about structure and potentially also the dynamics of entanglements is immediately available from the SP. We apply the method to study the ‘concentration’ dependence of the degree of entanglement in phantom chain systems. Program summary: Title of program:Z Catalogue number:ADVG Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVG Program obtainable from: CPC Program Library, Queen''s University of Belfast, N. Ireland Computer for which the program is designed and others on which it has been tested: Silicon Graphics (Irix), Sun (Solaris), PC (Linux) Operating systems or monitors under which the program has been tested: UNIX, Linux Program language used: USANSI Fortran 77 and Fortran 90 Memory required to execute with typical data: 1 MByte No. of lines in distributed program, including test data, etc.: 10 660 No. of bytes in distributed program, including test data, etc.: 119 551 Distribution formet:tar.gz Nature of physical problem: The problem is to obtain primitive paths substantiating a shortest multiple disconnected path (SP) for a given polymer configuration (chains of particles, with or without additional single particles as obstacles for the 2D case). Primitive paths are here defined as in [M. Rubinstein, E. Helfand, J. Chem. Phys. 82 (1985) 2477; R. Everaers, S.K. Sukumaran, G.S. Grest, C. Svaneborg, A. Sivasubramanian, K. Kremer, Science 303 (2004) 823] as the shortest line (path) respecting ‘topological’ constraints (from neighboring polymers or point obstacles) between ends of polymers. There is a unique solution for the 2D case. For the 3D case it is unique if we construct a primitive path of a single chain embedded within fixed line obstacles [J.S.B. Mitchell, Geometric shortest paths and network optimization, in: J.-R. Sack, J. Urrutia (Eds.), Handbook of Computational Geometry, Elsevier, Amsterdam, 2000, pp. 633–701]. For a large 3D configuration made of several chains, short is meant to be the Euclidean shortest multiple disconnected path (SP) where primitive paths are constructed for all chains simultaneously. While the latter problem, in general, does not possess a unique solution, the algorithm must return a locally optimal solution, robust against minor displacements of the disconnected path and chain re-labeling. The problem is solved if the number of kinks (or entanglements Z), explicitly deduced from the SP, is quite insensitive to the exact conformation of the SP which allows to estimate Z with a small error. Efficient method of solution: Primitive paths are constructed from the given polymer configuration (a non-shortest multiple disconnected path, including obstacles, if present) by first replacing each polymer contour by a line with a number of ‘kinks’ (beads, nodes) and ‘segments’ (edges). To obtain primitive paths, defined to be uncrossable by any other objects (neighboring primitive paths, line or point obstacles), the algorithm minimizes the length of all primitive paths consecutively, until a final minimum Euclidean length of the SP is reached. Fast geometric operations rather than dynamical methods are used to minimize the contour lengths of the primitive paths. Neighbor lists are used to keep track of potentially intersecting segments of other chains. Periodic boundary conditions are employed. A finite small line thickness is used in order to make sure that entanglements are not ‘lost’ due to finite precision of representation of numbers. Restrictions on the complexity of the problem: For a single chain embedded within fixed line or point obstacles, the algorithm returns the exact SP. For more complex problems, the algorithm returns a locally optimal SP. Except for exotic, probably rare, configurations it turns out that different locally optimal SPs possess quite an identical number of nodes. In general, the problem constructing the SP is known to be NP-hard [J.S.B. Mitchell, Geometric shortest paths and network optimization, in: J.-R. Sack, J. Urrutia (Eds.), Handbook of Computational Geometry, Elsevier, Amsterdam, 2000, pp. 633–701], and we offer a solution which should suffice to analyze physical problems, and gives an estimate about the precision and uniqueness of the result (from a standard deviation by varying the parameter: cyclicswitch). The program is NOT restricted to handle systems for which segment lengths of the SP exceed half the box size. Typical running time: Typical running times are approximately two orders of magnitude shorter compared with the ones needed for a corresponding molecular dynamics approach, and scale mostly linearly with system size. We provide a benchmark table. [Copyright &y& Elsevier]

A method is described for obtaining information about the local order existing in monoatomic model solids or real materials based on their atomistic configurations. An efficient algorithmic implementation is provided. The shape of the polyhedra formed by ‘relevant’ neighbors of each atom enter a pattern recognition method to resolve the type of the (usually non-ideal) crystal structure to which atoms surrounded by their relevant neighbors belong: hexagonal close-packed, face-centered cubic or body-centered cubic. Further, this approach allows for the analysis of icosahedral structure which preferably occurs in amorphous solids. Results of a molecular dynamics computer simulation illustrate how this method can be applied to contribute to an understanding of the mechanical and structural properties of solids (i) undergoing a steady shear stress and (ii) upon increasing temperature. [Copyright &y& Elsevier]

In this note, we propose a mapping from the spatial coordinates of an atomistic polymer chain to its ‘primitive path’ (PP), a concept being frequently used in the framework of reptation models. For the model to be presented, the projection preserves as much structure of the atomistic chain as appropriate to replace an atomistic chain on a prescribed (parameterized) coarse-grained level. We present an efficient numerical method to extract a PP as well as an analytic approach to study the conformational properties of the coarse-grained chain in an approximate fashion. The knowledge of the PP is a prerequisite to facilitate tests of mesoscopic descriptions of polymeric fluids, in particular in the framework of nonequilibrium thermodynamics, and allows for a thorough analysis of atomistic chain configurations on a ‘relevant’ coarse-grained level.