285 research outputs found
Electronic States of Graphene Grain Boundaries
We introduce a model for amorphous grain boundaries in graphene, and find
that stable structures can exist along the boundary that are responsible for
local density of states enhancements both at zero and finite (~0.5 eV)
energies. Such zero energy peaks in particular were identified in STS
measurements [J. \v{C}ervenka, M. I. Katsnelson, and C. F. J. Flipse, Nature
Physics 5, 840 (2009)], but are not present in the simplest pentagon-heptagon
dislocation array model [O. V. Yazyev and S. G. Louie, Physical Review B 81,
195420 (2010)]. We consider the low energy continuum theory of arrays of
dislocations in graphene and show that it predicts localized zero energy
states. Since the continuum theory is based on an idealized lattice scale
physics it is a priori not literally applicable. However, we identify stable
dislocation cores, different from the pentagon-heptagon pairs, that do carry
zero energy states. These might be responsible for the enhanced magnetism seen
experimentally at graphite grain boundaries.Comment: 10 pages, 4 figures, submitted to Physical Review
Recursive generation of IPR fullerenes
We describe a new construction algorithm for the recursive generation of all
non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm
stays entirely within the class of IPR fullerenes, that is: every IPR fullerene
is constructed by expanding a smaller IPR fullerene unless it belongs to
limited class of irreducible IPR fullerenes that can easily be made separately.
The class of irreducible IPR fullerenes consists of 36 fullerenes with up to
112 vertices and 4 infinite families of nanotube fullerenes. Our implementation
of this algorithm is faster than other generators for IPR fullerenes and we
used it to compute all IPR fullerenes up to 400 vertices.Comment: 19 pages; to appear in Journal of Mathematical Chemistr
Shape and symmetry determine two-dimensional melting transitions of hard regular polygons
The melting transition of two-dimensional (2D) systems is a fundamental
problem in condensed matter and statistical physics that has advanced
significantly through the application of computational resources and
algorithms. 2D systems present the opportunity for novel phases and phase
transition scenarios not observed in 3D systems, but these phases depend
sensitively on the system and thus predicting how any given 2D system will
behave remains a challenge. Here we report a comprehensive simulation study of
the phase behavior near the melting transition of all hard regular polygons
with vertices using massively parallel Monte Carlo simulations
of up to one million particles. By investigating this family of shapes, we show
that the melting transition depends upon both particle shape and symmetry
considerations, which together can predict which of three different melting
scenarios will occur for a given . We show that systems of polygons with as
few as seven edges behave like hard disks; they melt continuously from a solid
to a hexatic fluid and then undergo a first-order transition from the hexatic
phase to the fluid phase. We show that this behavior, which holds for all
, arises from weak entropic forces among the particles. Strong
directional entropic forces align polygons with fewer than seven edges and
impose local order in the fluid. These forces can enhance or suppress the
discontinuous character of the transition depending on whether the local order
in the fluid is compatible with the local order in the solid. As a result,
systems of triangles, squares, and hexagons exhibit a KTHNY-type continuous
transition between fluid and hexatic, tetratic, and hexatic phases,
respectively, and a continuous transition from the appropriate "x"-atic to the
solid. [abstract truncated due to arxiv length limitations]
Whirling Hexagons and Defect Chaos in Hexagonal Non-Boussinesq Convection
We study hexagon patterns in non-Boussinesq convection of a thin rotating
layer of water. For realistic parameters and boundary conditions we identify
various linear instabilities of the pattern. We focus on the dynamics arising
from an oscillatory side-band instability that leads to a spatially disordered
chaotic state characterized by oscillating (whirling) hexagons. Using
triangulation we obtain the distribution functions for the number of pentagonal
and heptagonal convection cells. In contrast to the results found for defect
chaos in the complex Ginzburg-Landau equation and in inclined-layer convection,
the distribution functions can show deviations from a squared Poisson
distribution that suggest non-trivial correlations between the defects.Comment: 4 mpg-movies are available at
http://www.esam.northwestern.edu/~riecke/lit/lit.html submitted to New J.
Physic
Sizes of pentagonal clusters in fullerenes
Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes
Connectivity Control for Quad-Dominant Meshes
abstract: Quad-dominant (QD) meshes, i.e., three-dimensional, 2-manifold polygonal meshes comprising mostly four-sided faces (i.e., quads), are a popular choice for many applications such as polygonal shape modeling, computer animation, base meshes for spline and subdivision surface, simulation, and architectural design. This thesis investigates the topic of connectivity control, i.e., exploring different choices of mesh connectivity to represent the same 3D shape or surface. One key concept of QD mesh connectivity is the distinction between regular and irregular elements: a vertex with valence 4 is regular; otherwise, it is irregular. In a similar sense, a face with four sides is regular; otherwise, it is irregular. For QD meshes, the placement of irregular elements is especially important since it largely determines the achievable geometric quality of the final mesh.
Traditionally, the research on QD meshes focuses on the automatic generation of pure quadrilateral or QD meshes from a given surface. Explicit control of the placement of irregular elements can only be achieved indirectly. To fill this gap, in this thesis, we make the following contributions. First, we formulate the theoretical background about the fundamental combinatorial properties of irregular elements in QD meshes. Second, we develop algorithms for the explicit control of irregular elements and the exhaustive enumeration of QD mesh connectivities. Finally, we demonstrate the importance of connectivity control for QD meshes in a wide range of applications.Dissertation/ThesisDoctoral Dissertation Computer Science 201
The Clar Structure of Fullerenes
A fullerene is a 3-regular plane graph consisting only of pentagonal and hexagonal faces. Fullerenes are designed to model carbon molecules. The Clar number and Fries number are two parameters that are related to the stability of carbon molecules. We introduce chain decompositions, a new method to find lower bounds for the Clar and Fries numbers. In Chapter 3, we define the Clar structure for a fullerene, a less general decomposition designed to compute the Clar number for classes of fullerenes. We use these new decompositions to understand the structure of fullerenes and achieve several results. In Chapter 4, we classify and give a construction for all fullerenes on |V| vertices that attain the maximum Clar number |V|/6 - 2. In Chapter 5, we settle an open question with a counterexample: we construct an infinite family of fullerenes for which a set of faces attaining the Clar number cannot be a subset of a set of faces that attains the Fries number. We develop a method to calculate the Clar number directly for many infinite families of fullerene
The topology of fullerenes
Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website
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