Planar Line Processes for Void and Density Statistics in Thin Stochastic Fibre Networks

Using results for the distribution of perimeters of random polygons arising from random lines in a plane, we obtain new analytic approximations to the distributions of areas and local line densities for random polygons and compute various limiting properties of random polygons. Using simulation, we show that the lengths of adjacent sides of polygons generated by random line processes in the plane are correlated with ρ=0.616±0.001.     

A new interface tracking method: The polygonal area mapping method

We present a new method, the polygonal area mapping (PAM) method, for tracking a non-diffusive, immiscible material interface between two materials in two-dimensional incompressible flows. This method represents material areas explicitly as piecewise polygons, traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon-clippings in a discrete manner. The new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately. In addition to high accuracy, the PAM method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications. The mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons. The results from a set of widely used benchmark tests show that the PAM method is superior to existing volume-of-fluid (VOF) methods.     

A planar carbon allotrope with linear bipentagon-octagon and hexagon arrangement

A two-dimensional (2D) metallic carbon allotrope is proposed, which consists of linearly aligned bipentagon-octagon and hexagon rings in a planar sheet. The relatively high percentage of hexagon and the regular arrangement of the polygons make it energetically more favorable than most of other predicted 2D carbon allotropes. Phonon dispersions without negative frequencies also indicate its stability. Electronic structure calculations show that its metallic nature is mainly due to the atoms shared by the pentagon, hexagon and octagon. Its lattice thermal conductivity is only about one fifth of that of graphene. Armchair- and zigzag-edged nanoribbons of this structure are also studied. The former is metallic while the latter has a small band gap due to the spin-polarized edge states. The appropriate band gap and the significantly reduced thermal conductivity suggest potential applications in thermoelectricity.     

Surface diffusion: the low activation energy path for nanotube growth

We present the temperature dependence of the growth rate of carbon nanofibers by plasma-enhanced chemical vapor deposition with Ni, Co, and Fe catalysts. We extrapolate a common low activation energy of 0.23-0.4 eV, much lower than for thermal deposition. The carbon diffusion on the catalyst surface and the stability of the precursor molecules, C2H2 or CH4, are investigated by ab initio plane wave density functional calculations. We find a low activation energy of 0.4 eV for carbon surface diffusion on Ni and Co (111) planes, much lower than for bulk diffusion. The energy barrier for C2H2 and CH4 dissociation is at least 1.3 eV and 0.9 eV, respectively, on Ni(111) planes or step edges. Hence, the rate-limiting step for plasma-enhanced growth is carbon diffusion on the catalyst surface, while an extra barrier is present for thermal growth due to gas decomposition.     

On the area of square lattice polygons

We consider the generating function of self-avoiding square lattice polygons grouped by both area and perimeter. The generating function for polygons of arean is found to diverge atx _c =0.251834, with an exponent of zero. The mean perimeter of polygons with arean is found to be proportional ton, while the mean area of polygons with perimetern is found to be proportional ton ^1.5.     

The mechanism investigation of the imidazolidinone catalyzed five‐membered ring synthesis reaction

The intramolecular asymmetric Michael addition reaction catalyzed by imidazolidinone is investigated using the density functional theory calculations. The details of the reaction mechanism, potential energy surfaces, and the influence of the acid additive are investigated. The reaction process includes two stages. The first stage is Michael addition, in which the enamine complex is created and then the Michael addition is carried out. The second stage is a product separation stage which includes an enol‐keto tautomerization and a two‐step hydrolysis. The enantioselectivity is controlled by the Michael addition step which involves a new carbon–carbon bond formation. The calculation results provide a general model which may explain the mechanism and enantioselectivity of the title reaction. Copyright © 2012 John Wiley & Sons, Ltd.     

Characterisation of molybdenum intermediate layers in Cu-C system with SIMS method

In the design of new high-speed chip generations a huge problem is bleeding off process heat during their operation. The installation of heat sinks onto such chips is necessary. Possible materials are copper-coated carbon composites. They combine high thermal conductivity with low density and a tailorable coefficient of thermal expansion (CTE). The low wettability of copper onto carbon necessitates a surface pretreatment.Flat slices of nitrogen-plasma etched vitreous carbon (Sigradur G) made up as a model system for carbon fiber material. The later serial fabrication of these fibers includes a hot pressing step after the deposition joining them to solid composites. It is simulated by a heat treatment step of the compound. The first sample series consisted of samples with 100 nm molybdenum and 500 nm copper layers (sputter deposited), as deposited and heat treated. The second run concludes samples without molybdenum layer but an additional 50 nm cap layer deposited after heat treatment.All samples were investigated with secondary ion mass spectrometry (SIMS), showing a diffusion of carbon into the molybdenum layer. Measuring MCs⁺ secondary ions, both matrix elements and trace elements were detectable sufficiently.     

Microstructure and anti-oxidation property of CrSi2-SiC coating for carbon/carbon composites

To protect carbon/carbon (C/C) composites from oxidation, a new type of oxidation protective coating has been produced by a two-step pack cementation technique. XRD and SEM analysis show, the coating obtained by the first step pack cementation was a porous β-SiC structure, and a new phase of CrSi₂ was generated in the porous SiC coating after heat-treatment according to the second step pack cementation process. Oxidation test shows that, the weight loss of the SiC coated C/C is up to 11.26% after 5 h oxidation in air at 1773 K, and the weight loss of the CrSi₂-SiC coated C/C composites is only 4.15% after oxidation in air at 1773 K for 34 h. The oxidation of C/C composites was primarily due to the reaction of C/C matrix and oxygen diffusing through the penetrable cracks in the coating.     

A motivic approach to phase transitions in Potts models

We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion–contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.     

碳纳米管封装Si20团簇的热稳定性研究

用分子动力学方法模拟一种特殊结构Si20 (表面异构的硅十二面体结构)填充到不同管径的扶手椅型碳纳米管中组成复合结构的热稳定性。通过能量分析和定量统计缺陷多边形数量等方法来研究这种结构在碳纳米管中的稳定性和结构演变情况。研究发现Si20的热稳定性和碳纳米管的直径关系密切;其在CNT(15, 15)中的热稳定性最好,当管径逐渐增加时,其热稳定性呈下降趋势;直至管径增加到CNT(21, 21),碳纳米管对Si20的空间限制作用变得很小,以至于不足以维持Si20的稳定。此外,Si原子因热振动替换碳纳米管中C原子而形成十二边形缺陷,这对碳纳米管的热稳定性有着明显的降低作用。     

Vector potential gauge for superconducting regular polygons

An approach to the Ginzburg-Landau problem of superconducting polygons is developed, based on the exact fulfillment of superconducting boundary conditions along the boundary of the sample. To this end an analytical gauge transformation for the vector potential A is found which gives A _n = 0 for the normal component along the boundary line of an arbitrary regular polygon. The use of the new gauge reduces the Ginzburg-Landau problem of superconducting polygons in external magnetic fields to an eigenvalue problem in a basis set of functions obeying Neumann boundary conditions. The advantages of this approach, especially for low magnetic fields, are illustrated and novel vortex patterns are obtained which can be probed experimentally. Received 28 February 2002 and Received in final form 12 April 2002 Published online 6 June 2002     

A green one-pot three-component cascade reaction: the synthesis of 2-amino-5,8-dihydro-3H-pyrido[2,3-D]pyrimidin-4-ones in aqueous medium

A one-pot three-component cascade reaction for the green synthesis of a new class of 2-amino-5,8-dihydro-3H-pyrido[2,3-d]pyrimidin-4-ones was developed from the condensation of aromatic aldehydes with 2,6-diaminopyrimidin-4(3H)-one and acetophenone derivatives or various cyclic ketones in the presence of a catalytic amount of sodium carbonate in a mixture of water and ethanol at 60  \({^{\circ }}\hbox {C}\). This reaction led to the construction of two carbon–carbon bonds and one carbon–nitrogen bond in a single synthetic step.     

Nonchiral BN Haeckelite nanotubes

A new class of boron-nitrogen (BN) nanotubes composed of tetragons, pentagons, hexagons, heptagons, and octagons is considered. By analogy with carbon nanotubes of the same topological structure, these nanotubes were called Haeckelites. The geometry, energetics, and electronic properties were studied in detail for two regular mutual arrangements of the polygons. It was found that Haeckelite nanotubes are dielectrics with the energy gap E_g = 3.24–4.09 eV. As the nanotube diameter increases, the energy gap E_g decreases, approaching the value for the corresponding planar Haeckelite layer. The ground-state energy of the Haeckelite BN nanotubes is 0.3 eV/atom higher than that of well-known hexagonal BN nanotubes.     

Carbon clusters near the step of Rh surface: implication for the initial stage of graphene nucleation

To understand the initial nucleation of graphene by chemical vapor deposition along metal step, carbon clusters (N = 1 ～ 24) near Rh (4 3 3) stepwise surface were systemically explored by first-principles calculations. Carbon chains are always more stable than carbon rings on stepped metal surface. Starting from C₆, carbon chains prefer two-end passivation on the metal step. A structural transition of carbon clusters from one-dimensional sp chains to two-dimensional s p ² networks are found at C₁₀, which corresponds to the nucleation size at a wide range of chemical potentials. According to these theoretical results, we proposed that appropriately controlling the chemical potential may be helpful for observing the four stable carbon clusters along metal step and improving the quality of graphene.     

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

On contour crossings in contour-advective simulations – part 1 – algorithm for detection and quantification

This is the first of two papers devoted to the analysis of contour crossing errors that occur in contour-advective simulations of fluid motion. Here an algorithm is presented for quantifying the error due to contour crossings. The first step is to determine the relative proximity of all possible pairs of contours. A digital representation of each contour is produced to aid in the corresponding calculation. Simple analysis of functions is then used to find any crossings between contours deemed close to each other by the digital representation method. Next, the area in error of a pair of crossing contours is calculated by identifying the polygon or polygons that approximately bound the erroneous region. Finally, some preliminary results of analysis of contour crossings that occur in contour-advective semi-lagrangian (CASL) simulations of single layer quasigeostrophic turbulence are presented. It is shown that the error due to contour crossings is small in the simulations considered here.     

The Pentagram Map: A Discrete Integrable System

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \mathbb Z{\mathbb Z} into \mathbbRP²{{\mathbb{RP}}^2} that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].     

Bond-counting rule for carbon and its application to the roughness of diamond (001)

Despite that carbon is tetravalent identical to silicon, first-principles calculations reveal that stable step structures on diamond (001) are entirely different from those on silicon. Moreover, pristine Si(001) is flat; pristine diamond (001) could be rough due to negative step formation energies. A generic bond-counting rule is established, which should apply to most carbon structures where sp2 and sp3 hybrids coexist: e.g., it provides a qualitative account of the step energy order without detailed calculation. Our findings agree with experimental observations.     

Texture rules for concentrated filled nematics

Defect textures in concentrated fiber-filled polygonal networks in nematic liquid crystals are analyzed using differential geometry and computational modeling based on Landau--de Gennes theory. Micron fibers exhibit singular cores of strength -1/2 for odd polygons and escaped cores of strength -(N-2)/2 for even polygons (N: number of sides), in agreement with experiments while simulations predict singular cores of strength -1/2 in submicron fibers. The computed textures satisfy physical and topological stability rules, and the total charge inside each polygon obeys the Poincaré-Brouwer theorem.     

Brane tilings and reflexive polygons

Reflexive polygons have attracted great interest both in mathematics and in physics. This paper discusses a new aspect of the existing study in the context of quiver gauge theories. These theories are 4d supersymmetric worldvolume theories of D3 branes with toric Calabi‐Yau moduli spaces that are conveniently described with brane tilings. We find all 30 theories corresponding to the 16 reflexive polygons, some of the theories being toric (Seiberg) dual to each other. The mesonic generators of the moduli spaces are identified through the Hilbert series. It is shown that the lattice of generators is the dual reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver gauge theories with the lattice of generators being the toric diagram of the dual and vice versa.