Geometric Property （T）

This paper discusses“geometric property （T）”. This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property （T） is a strong form of “expansion property”, in particular, for a sequence （Xn） of bounded degree finite graphs, it is strictly stronger than （Xn） being an expander in the sense that the Cheeger constants h（Xn） are bounded below. In this paper, the authors show that geometric property （T） is a coarse invariant, i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property （T） interacts with amenability, property （T） for groups, and coarse geometric notions of a-T-menability. In particular, it is shown that property （T） for a residually finite group is characterised by geometric property （T） for its finite quotients.     

Coarse‐grained A‐graft‐B model of poly(lactic acid) for molecular dynamics simulations

A mesoscopic model of poly(lactic acid) is developed where the polymer is represented as an A‐graft‐B chain with monomer units consisting of two covalently connected beads. A coarse‐graining algorithm is proposed to convert an atomistic model of PLA into a coarse‐grained one. The developed model is based on atomistic simulations of oligolactides to take into account terminal groups correctly. It was used for coarse‐grained simulations of polylactide. Gyration radii and end to end distances of polymer chains as well as the density of the polymer melt are shown to be in a good agreement with those obtained from atomistic simulations. The thermal expansion coefficients of the OLA melts calculated using the coarse‐grained model are in reasonable agreement with those obtained from all‐atom molecular dynamics. The model provides a 17‐fold speedup compared with atomistic calculations. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 604–612     

Calculating particle pair potentials from fluid‐state pair correlations: Iterative ornstein–zernike inversion

An iterative Monte Carlo inversion method for the calculation of particle pair potentials from given particle pair correlations is proposed in this article. The new method, which is best referred to as Iterative Ornstein–Zernike Inversion, represents a generalization and an improvement of the established Iterative Boltzmann Inversion technique (Reith, Pütz and Müller‐Plathe, J. Comput. Chem. 2003, 24, 1624). Our modification of Iterative Boltzmann Inversion consists of replacing the potential of mean force as an approximant for the pair potential with another, generally more accurate approximant that is based on a trial bridge function in the Ornstein–Zernike integral equation formalism. As an input, the new method requires the particle pair correlations both in real space and in the Fourier conjugate wavenumber space. An accelerated iteration method is included in the discussion, by which the required number of iterations can be greatly reduced below that of the simple Picard iteration that underlies most common implementations of Iterative Boltzmann Inversion. Comprehensive tests with various pair potentials show that the new method generally surpasses the Iterative Boltzmann Inversion method in terms of reliability of the numerical solution for the particle pair potential. © 2018 Wiley Periodicals, Inc.     

A descent principle for the Dirac-dual-Dirac method

Let G be a torsion-free discrete group with a finite-dimensional classifying space BG. We show that G has a dual-Dirac morphism if and only if a certain coarse (co-)assembly map is an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric, that is, coarse, invariant. We get results for groups with torsion as well.     

Polarizable coarse‐grained models for molecular dynamics simulation of liquid cyclohexane

Force field parameters for polarizable coarse‐grained (CG) supra‐atomic models of liquid cyclohexane are proposed. Two different bead sizes were investigated, one representing two fine‐grained (FG) CH₂r united atoms of the cyclohexane ring, and one representing three FG CH₂r united atoms. Electronic polarizability is represented by a massless charge‐on‐spring particle connected to each CG bead. The model parameters were calibrated against the experimental density and heat of vaporization of liquid cyclohexane, and the free energy of cyclohexane hydration. Both models show good agreement with thermodynamic properties of cyclohexane, yet overestimate the self‐diffusion. The dielectric properties of the polarizable models agree very well with experiment. © 2015 Wiley Periodicals, Inc.     

Phase Morphologies of Binary Polymer Blends Predicted by Systematically Coarse‐Grained Models

In this work, the combined iterative Boltzmann inversion/conditional reversible work scheme is extended with a little modifications to derive the systematically coarse‐grained (CG) potentials for simulating two typical atactic polymer blends composed of poly(methyl methacrylate) (PMMA) and poly(vinyl chloride) (PVC) or polystyrene (PS). Molecular dynamics simulations are extensively performed on the two blends with a wide formulation range. It is revealed by these simulations that, throughout the entire composition range, the PMMA/PVC blend is homogeneous whereas the PMMA/PS blend undergoes phase separation, which agrees well with the experimental observation that the former exhibits strong interactions that are absent in the latter. Depending upon the formulation, the immiscible PMMA/PS blend presents one single‐ or double‐continuous phase. It is further confirmed that intermolecular interactions play the key roles in forming the phase morphologies, which in turn can be inferred from only the three nonbonded CG potentials of one unlike pair and two like pairs.

     

Monte Carlo simulations of a coarse grained model for an athermal all-polystyrene nanocomposite system

The structure of a polystyrene matrix filled with tightly cross-linked polystyrene nanoparticles, forming an athermal nanocomposite system, is investigated by means of a Monte Carlo sampling formalism. The polymer chains are represented as random walks and the system is described through a coarse grained Hamiltonian. This approach is related to self-consistent-field theory but does not invoke a saddle point approximation and is suitable for treating large three-dimensional systems. The local structure of the polymer matrix in the vicinity of the nanoparticles is found to be different in many ways from that of the corresponding bulk, both at the segment and the chain level. The local polymer density profile near to the particle displays a maximum and the bonds develop considerable orientation parallel to the nanoparticle surface. The depletion layer thickness is also analyzed. The chains orient with their longest dimension parallel to the surface of the particles. Their intrinsic shape, as characterized by spans and principal moments of inertia, is found to be a strong function of position relative to the interface. The dispersion of many nanoparticles in the polymeric matrix leads to extension of the chains when their size is similar to the radius of the dispersed particles.     

The coarse Baum–Connes conjecture via coarse geometry

The C₀ coarse structure on a metric space is a refinement of the bounded structure and is closely related to the topology of the space. In this paper we will prove the C₀ version of the coarse Baum–Connes conjecture and show that K_*(C^*X₀) is a topological invariant for a broad class of metric spaces. Using this result we construct a ‘geometric’ obstruction group to the coarse Baum–Connes conjecture for the bounded coarse structure. We then show under the assumption of finite asymptotic dimension that the obstructions vanish, and hence we obtain a new proof of the coarse Baum–Connes conjecture in this context.     

Coarse‐Grained Simulations of Model Polymer Nanofibres

We describe the development of a coarse‐grained (CG) force field for nylon‐6 (polycaprolactam) and its application to the simulation of the structure and macromolecular dynamics within cylindrical fibres formed by this polymer, having diameters in the 14–28 nm range. Our CG model is based on the MARTINI force field for the non‐bonded interactions and on Boltzmann‐inverted gas‐phase atomistic simulations for intramolecular stretching and bending energies. The simulations are carried out on infinite, isolated nanofibres at temperatures of 300, 400 and 500 K, with different starting configurations. Starting from ordered chain‐extended configurations, we simulate the melting of the polymer in the nanofibres and, after cooling back to room temperature, its re‐crystallization in a chain‐folded lamellar configuration. This agrees with experimental observations on electrospun nylon‐6 nanofibres and demonstrated the suitability of the approach for the simulation of these systems. The effect of nanoscale confinement on the structure and dynamics of the polymer chains is extensively discussed.

     

() does not coarsely embed into a Hilbert space

We show that a Banach space with a normalized symmetric basis behaving like that of () cannot coarsely embed into a Hilbert space.