arXiv subjects

M. F. Thorpe

Publications and source records attributed to M. F. Thorpe.

At least 19 recordsLinked to original sources

arXiv2021

Realizations of Isostatic Material Frameworks

This paper studies the set of equivalent realizations of isostatic frameworks in two dimensions, and algorithms for finding all such realizations. We show that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. We enumerate the complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions and study the impact of boundary length on the emergence of these realizations. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints; alternative methods - based on constraint reduction and distance-based covering map or Cayley parameterization of the search space - are presented. The application of these methods is studied on atomic clusters, a model two-dimensional glasses, and jamming.

cond-mat.dis-nn↗

arXiv2018

Disordered auxetic networks with no re-entrant polygons

It is widely assumed that disordered auxetic structures (i.e. structures with a negative Poisson's ratio) must contain re-entrant polygons in $2$D and re-entrant polyhedra in $3$D. Here we show how to design disordered networks in $2$D with only convex polygons. The design principles used allow for any Poisson ratio $-1 < ν< 1/3$ to be obtained with a prescriptive algorithm. By starting from a Delaunay triangulation with a mean coordination $ \simeq 6$ and $ν\simeq 0.33$ and removing those edges that decrease the shear modulus by the least without creating any re-entrant polygons, the system evolves monotonically towards the isostatic point with $ \simeq 4$ and $ν\simeq -1$.

cond-mat.dis-nn↗

arXiv2018

Jamming in Perspective

Jamming occurs when objects like grains are packed tightly together (e.g. grain silos). It is highly cooperative and can lead to phenomena like earthquakes, traffic jams, etc. In this Letter we point out the paramount importance of the underlying contact network for jammed systems; the network must have one contact in excess of isostaticity and a finite bulk modulus. Isostatic means that the number of degrees of freedom are exactly balanced by the number of constraints. This defines a large class of networks that can be constructed without the necessity of packing particles together compressively (either in the lab or computationally). One such construction, which we explore here, involves setting up the Delaunay triangulation of a Poisson disk sampling and then removing edges to maximize the bulk modulus until the isostatic plus one point is reached. This construction works in any dimensions and here we give results in 2D where we also show how such networks can be transformed into a disk pack.

cond-mat.dis-nn↗

arXiv2017

Ring correlations in random networks

We examine the correlations between rings in random network glasses in two dimensions as a function of their separation. Initially, we use the topological separation (measured by the number of intervening rings), but this leads to pseudo-long-range correlations due to a lack of topological charge neutrality in the shells surrounding a central ring. This effect is associated with the non-circular nature of the shells. It is, therefore, necessary to use the geometrical distance between ring centers. Hence we find a generalization of the Aboav-Weaire law out to larger distances, with the correlations between rings decaying away when two rings are more than about 3 rings apart.

cond-mat.soft↗

arXiv2017

Anchored boundary conditions for locally isostatic networks

Finite pieces of locally isostatic networks have a large number of floppy modes because of missing constraints at the surface. Here we show that by imposing suitable boundary conditions at the surface, the network can be rendered effectively isostatic. We refer to these as anchored boundary conditions. An important example is formed by a two-dimensional network of corner sharing triangles, which is the focus of this paper. Another way of rendering such networks isostatic, is by adding an external wire along which all unpinned vertices can slide (sliding boundary conditions). This approach also allows for the incorporation of boundaries associated with internal holes and complex sample geometries, which are illustrated with examples. The recent synthesis of bilayers of vitreous silica has provided impetus for this work. Experimental results from the imaging of finite pieces at the atomic level needs such boundary conditions, if the observed structure is to be computer-refined so that the interior atoms have the perception of being in an infinite isostatic environment.

cond-mat.dis-nn↗

arXiv2017

Refining glass structure in two dimensions

Recently determined atomistic scale structures of near-two dimensional bilayers of vitreous silica (using scanning probe and electron microscopy) allow us to refine the experimentally determined coordinates to incorporate the known local chemistry more precisely. Further refinement is achieved by using classical potentials of varying complexity; one using harmonic potentials and the second employing an electrostatic description incorporating polarization effects. These are benchmarked against density functional calculations. Our main findings are that (a) there is a symmetry plane between the two disordered layers; a nice example of an emergent phenomenon, (b) the layers are slightly tilted so that the Si-O-Si angle between the two layers is not $180^{\circ}$ as originally thought but rather $175 \pm 2 ^{\circ}$ and (c) while interior areas that are not completely imagined can be reliably reconstructed, surface areas are more problematical. It is shown that small crystallites that appear are just as expected statistically in a continuous random network. This provides a good example of the value that can be added to disordered structures imaged at the atomic level by implementing computer refinement.

cond-mat.dis-nn↗

arXiv2015

Rigidity Loss in Disordered Systems: Three Scenarios

We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a protocol that avoids the appearance of floppy regions. The marginal state of jammed and stress-relieved networks are globally isostatic, while marginal randomly diluted networks show both overconstrained and underconstrained regions. When a single bond is added to or removed from these isostatic systems, jammed networks become globally overconstrained or floppy, whereas the effect on stress-relieved networks is more local and limited. These differences are also reflected in the linear elastic properties and point to the highly effective and unusual role of global self-organization in jammed sphere packings.

cond-mat.soft↗

arXiv2014

Ring statistics of silica bilayers

The recent synthesis and characterization of bilayers of vitreous silica has produced valuable new information on ring sizes and distributions. In this paper, we compare the ring statistics of experimental samples with computer generated samples. The average ring size is fixed at six by topology, but the width, skewness and other moments of the distribution of ring edges are characteristics of particular samples. We examine the Aboav-Weaire law that quantifies the propensity of smaller rings to be adjacent to larger rings, and find similar results for available experimental samples which however differ somewhat from computer-generated bilayers currently. We introduce a new law for the areas of rings of various sizes.

cond-mat.mtrl-sci↗

arXiv2013

Universal Behavior for Bond Percolation

We collect together results for bond percolation on various lattices from two to fourteen dimensions which, in the limit of large dimension $d$ or number of neighbors $z$, smoothly approach a randomly diluted Erdős-Rényi graph. We include new results on bond diluted hyper-sphere packs in up to nine dimensions, which show the mean coordination, excess kurtosis and skewness evolving smoothly with dimension towards the Erdős-Rényi limit.

cond-mat.stat-mech↗

arXiv2013

Modeling vitreous silica bilayers

We computer model a free-standing vitreous silica bilayer which has recently been synthesized and characterized experimentally in landmark work. Here we model the bilayer using a computer assembly procedure that starts from a single layer of amorphous graphene, generated using a bond switching algorithm from an initially crystalline graphene structure. Next each bond is decorated with an oxygen atom and the carbon atoms are relabeled as silicon. This monolayer can be now thought of as a two dimensional network of corner sharing triangles. Next each triangle is made into a tetrahedron, by raising the silicon atom above each triangle and adding an additional singly coordinated oxygen atom at the apex. The final step is to mirror reflect this layer to form a second layer and then attach the two layers together to form the bilayer. We show that this vitreous silica bilayer has the additional macroscopic degrees of freedom to easily form a network of identical corner sharing tetrahedra if there is a symmetry plane through the center of the bilayer going through the layer of oxygen ions that join the upper and lower layers. This has the consequence that the upper rings lie exactly above the lower rings, which are tilted in general. The assumption of a network of perfect corner sharing tetrahedra leads to a range of possible densities that we have previously characterized in three dimensional zeolites as a flexibility window. Finally, using a realistic potential, we have relaxed the bilayer to determine the density, and other structural characteristics such as the Si-Si pair distribution functions and the Si-O-Si bond angle distribution, which are compared to the experimental results obtained by direct imaging.

cond-mat.dis-nn↗

arXiv2012

Insulating Behavior of an Amorphous Graphene Membrane

We investigate the charge transport properties of planar amorphous graphene that is fully topologically disordered, in the form of sp2 three-fold coordinated networks consisting of hexagonal rings, but also including many pentagons and heptagons distributed in a random fashion. Using the Kubo transport methodology and the Lanczos method, the density of states, mean free paths and semiclassical conductivities of such amorphous graphene membranes are computed. Despite a large increase in the density of states close to the charge neutrality point, all electronic properties are dramatically degraded, evidencing an Anderson insulating state caused by topological disorder alone. These results are supported by Landauer-Buttiker conductance calculations, which show a localization length as short as 5 nanometers

cond-mat.mes-hall↗

arXiv2011

Pentagonal puckering in a sheet of amorphous graphene

Ordered graphene has been extensively studied. In this paper we undertake a first density functional study of it topologically disordered analogues of graphene, in the form of a random network, consisting predominantly of hexagonal rings, but also including pentagons and heptagons. After some preliminaries with crystalline material, we relax various random network models and find that the presence of carbon pentagons induce local curvature, thus breaking the initial planar symmetry, in some analogy with the case of fullerenes. Using density functional theory to calculate the total energy, we find that while the planar state is locally stable, there is a puckered state that has lower energy. The scale of the puckering is consistent with that expected with local maxima and minima associated with pentagons surrounded by larger rings; forming local "buckyball domes".

cond-mat.dis-nn↗

arXiv2009

Electronic structure of a realistic model of amorphous graphene

In this note, we calculate the electronic properties of a realistic atomistic model of amorphous graphene. The model contains odd membered rings, particularly five and seven membered rings and no coordination defects. We show that odd-membered rings increase the electronic density of states at the Fermi level relative to crystalline graphene; a honeycomb lattice with semi-metallic character. Some graphene samples contain amorphous regions, which even at small concentrations, may strongly affect many of the exotic properties of crystalline graphene, which arise because of the linear dispersion and semi-metallic character of perfectly crystalline graphene. Estimates are given for the density of states at the Fermi level using a tight-binding model for the $π$ states.

cond-mat.str-el↗

arXiv2009

The static structure factor of amorphous silicon and vitreous silica

Liquids are in thermal equilibrium and have a non-zero static structure factor S(Q->0) = [ - ^2]/ = rho*k_B*T*Chi_T where rho is the number density, T is the temperature, Q is the scattering vector and Chi_T is the isothermal compressibility. The first part of this result involving the number N (or density) fluctuations is a purely geometrical result and does not involve any assumptions about thermal equilibrium or ergodicity and so is obeyed by all materials. From a large computer model of amorphous silicon, local number fluctuations extrapolate to give S(0) = 0.035+/-0.001. The same computation on a large model of vitreous silica using only the silicon atoms and rescaling the distances gives S(0) = 0.039+/-0.001, which suggests that this numerical result is robust and similar for all amorphous tetrahedral networks. For vitreous silica, we find that S(0) = 0.116+/-0.003, close to the experimental value of S(0) = 0.0900+/-0.0048 obtained recently by small angle neutron scattering. More detailed experimental and modelling studies are needed to determine the relationship between the fictive temperature and structure.

cond-mat.soft↗

arXiv2007

Hierarchical Rigidity from Pair Distance Fluctuations

Often, experiments, observations or simulations generate large numbers of snapshots of the configurations of complex many-particle systems. It is important to find methods of extracting useful information from these ensembles of snapshots in order to document the motion as the system evolves. The most interesting information is contained in the correlated motions of individual constituents rather than in their absolute motion. We present a statistical method to identify hierarchies of correlated motions from a series of two or more snapshot configurations. This method is demonstrated in a number of systems, including freely-jointed polymer chains, hard plastic spheres, water, and proteins. These concepts are implemented as TIMME, the Tool for Identifying Mobility in Macromolecular Ensembles.

cond-mat.soft↗

arXiv2007

Algorithms for 3D rigidity analysis and a first order percolation transition

A fast computer algorithm, the pebble game, has been used successfully to study rigidity percolation on 2D elastic networks, as well as on a special class of 3D networks, the bond-bending networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on BCC and FCC lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first order rigidity percolation transition, which is in contrast to the second order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed for Bethe lattices and networks with ``chemical order'', this is the first time it has been seen for a regular randomly diluted network. In the case of site dilution, the transition is also first order for BCC, but results for FCC suggest a second order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent non-universality, make this phase transition highly unusual.

cond-mat.dis-nn↗

arXiv2005

Self-Organization and the Physics of Glassy Networks

Network glasses are the physical prototype for many self-organized systems, ranging from proteins to computer science. Conventional theories of gases, liquids, and crystals do not account for the strongly material-selective character of the glass-forming tendency, the phase diagrams of glasses, or their optimizable properties. A new topological theory, only 25 years old, has succeeded where conventional theories have failed. It shows that (probably all slowly quenched) glasses, including network glasses, are the result of the combined effects of a few simple mechanisms. These glass-forming mechanisms are topological in nature, and have already been identified for several important glasses, including chalcogenide alloys, silicates (window glass, computer chips), and proteins.

cond-mat.dis-nn↗

arXiv1999

High real-space resolution measurement of the local structure of Ga_1-xIn_xAs using x-ray diffraction

High real-space resolution atomic pair distribution functions (PDF)s from the alloy series Ga_1-xIn_xAs have been obtained using high-energy x-ray diffraction. The first peak in the PDF is resolved as a doublet due to the presence of two nearest neighbor bond lengths, Ga-As and In-As, as previously observed using XAFS. The widths of nearest, and higher, neighbor pairs are analyzed by separating the strain broadening from the thermal motion. The strain broadening is five times larger for distant atomic neighbors as compared to nearest neighbors. The results are in agreement with model calculations.

cond-mat.mtrl-sci↗