An Adaptation of the Hoshen-Kopelman Cluster Counting Algorithm for Honeycomb Networks

We develop a simplified implementation of the Hoshen-Kopelman cluster counting algorithm adapted for honeycomb networks. In our implementation of the algorithm we assume that all nodes in the network are occupied and links between nodes can be intact or broken. The algorithm counts how many clusters there are in the network and determines which nodes belong to each cluster. The network information is stored into two sets of data. The first one is related to the connectivity of the nodes and the second one to the state of links. The algorithm finds all clusters in only one scan across the network and thereafter cluster relabeling operates on a vector whose size is much smaller than the size of the network. Counting the number of clusters of each size, the algorithm determines the cluster size probability distribution from which the mean cluster size parameter can be estimated. Although our implementation of the Hoshen-Kopelman algorithm works only for networks with a honeycomb (hexagonal) structure, it can be easily changed to be applied for networks with arbitrary connectivity between the nodes (triangular, square, etc.). The proposed adaptation of the Hoshen-Kopelman cluster counting algorithm is applied to studying the thermal degradation of a graphene-like honeycomb membrane by means of Molecular Dynamics simulation with a Langevin thermostat. ACM Computing Classification System (1998): F.2.2, I.5.3

Rigidity and Elasticity beyond Isostaticity

Isotatic mechanical structures, where numbers of constraints arising from physical interactions balance the number of internal degrees of freedom, are on the verge of mechanical instability. Isostatic structures exhibit fascinating phenomena, showing criticality in mechanical responses and other properties. Such critical mechanical structures, including jamming, rigidity percolation (RP) and Maxwell lattices, have been widely explored. In those structures, the emergence of rigidity is controlled by the isostatic point and the density is moderately high. This dissertation focuses on critical mechanical phenomena when the connectivity of the structure is away from isostaticity. The study of these systems unravel the complexity of rigidity in a broad range of materials. The first act of this dissertation discusses the emergence of rigidity in ultra-low-density systems with the introduction of positional correlations, including two projects related to correlated rigidity percolation. In addition to low-density solids, high-density solids like glasses exhibit interesting phenomena and different mechanical behaviors compared with isostatic systems. The second act of this dissertation studies high-density glasses and includes a project discussing stressed elasticity in over-isostatic region. The first project presented in this dissertation concerns RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which is introduced by attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecular dynamics simulations, we show that increasing attraction in colloidal gelation lowers the RP transition, and this phenomena agrees with experiments. Hence, the emergence of rigidity at colloidal gelation can be understood as a RP transition, but occurs at volume fractions far below values predicted by the classical RP, due to attractive interactions which induce structural correlation. Motivated by the experimentally observed fractal nature of materials like colloidal gels and disordered fiber networks, the second project discussed in this dissertation studies RP in a fractal network where intrinsic correlation in particle positions is controlled by fractal iteration. Specifically, we calculate the critical packing fractions of site-diluted lattices of Sierpinski Gasket's (SGs) with varying degrees of fractal iteration. Our results suggest that although the correlation length exponent and fractal dimension of the RP of these lattices are identical to those of the regular triangular lattice, the critical volume fraction is dramatically lower due to the fractal nature of the network. Our results characterize rigidity in ultra-low-density fractal networks. The third project presents a systematic method based on States of Self-Stress (SSSs) to investigate how prestress affects elastic response of amorphous solids. Using a triangular lattice model with varying prestress, and also in amorphous configuration of compressed repulsive particles, as a model for a colloidal soft solid, we show how prestress determines the response of glasses to both macroscopic shear strain and local dipole forces, where they display behaviors qualitatively different from un-stressed random networks with the same geometry. We also use this method to study the dependence of the stress-bearing ability of the system on the preparation protocol, which changes the microscopic prestress distribution, as well as signatures of the spatial evolution of stress under strain. The heterogeneity of stress change and mechanical responses are accurately depicted from the SSS calculation when prestress is included. Our results characterize the elasticity for prestressed amorphous solids.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169649/1/zhshang_1.pd

CHAMPION: Chalmers Hierarchical Atomic, Molecular, Polymeric & Ionic Analysis Toolkit

We present CHAMPION: a software developed to automatically detect time-dependent bonds between atoms based on their dynamics, classify the local graph topology around them, and analyze the physicochemical properties of these topologies by statistical physics. In stark contrast to methodologies where bonds are detected based on static conditions such as cut-off distances, CHAMPION considers pairs of atoms to be bound only if they move together and act as a bound pair over time. Furthermore, the time-dependent global bond graph is possible to split into dynamically shifting connected components or subgraphs around a certain chemical motif and thereby allow the physicochemical properties of each such topology to be analyzed by statistical physics. Applicable to condensed matter and liquids in general, and electrolytes in particular, this allows both quantitative and qualitative descriptions of local structure, as well as dynamical processes such as speciation and diffusion. We present here a detailed overview of CHAMPION, including its underlying methodology, implementation and capabilities.Comment: 11 pages, 8 figure

Piezoresistive Hybrid Nanocomposites for Strain and Damage Sensing: Experimental and Numerical Analysis

Carbon nanomaterials such as carbon nanotubes (CNTs) and graphite nanoplatelets (GNPs) demonstrate remarkable electrical and mechanical properties, which suggest promising structural and functional applications as fillers for polymer nanocomposites. The piezoresistive behavior of these nanocomposites makes them ideal for sensing applications. Besides, hybrid nanocomposites with multiple fillers like carbon nanotubes (CNTs) and graphite nanoplatelets (GNPs) are known to exhibit improved electrical and mechanical performance when compared to mono-filler composites. To comprehensively understand the mechanisms of electrical percolation, conductivity, and piezoresistivity in hybrid nanocomposites, the author develops a two-dimensional (2D) and a three-dimensional (3D) computational Monte Carlo percolation network models for hybrid nanocomposites with CNT and GNP fillers. In the experimental studies correlated to the computational models, the author fabricates the hybrid nanocomposites made of both fillers using resin infiltration techniques and show an improvement of their electromechanical performance when compared to CNT nanocomposites. Due to the limitations of the resin infiltration techniques, the author develops an inkjet printing procedure with a new water-based CNT ink to fabricated printed nanocomposites on both polyimide film (Kapton) and paper with high device-todevice reproducibility. The ink formulation, as well as the substrate surface treatment, have been optimized to obtain conductive and piezoresistive devices. The author shows the effectiveness of the printed devices as strain sensors and impact damage sensors respectively under mechanical strains and hypervelocity impact damages. Devices printed with the minimum number of ink deposited layers lead to the best sensing performance

Understanding Disordered Systems Through Numerical Simulation and Algorithm Development

Disordered systems arise in many physical contexts. Not all matter is uni- form, and impurities or heterogeneities can be modeled by fixed random disor- der. Numerous complex networks also possess fixed disorder, leading to appli- cations in transportation systems [1], telecommunications [2], social networks [3, 4], and epidemic modeling [5], to name a few. Due to their random nature and power law critical behavior, disordered systems are difficult to study analytically. Numerical simulation can help overcome this hurdle by allowing for the rapid computation of system states. In order to get precise statistics and extrapolate to the thermodynamic limit, large systems must be studied over many realizations. Thus, innovative al- gorithm development is essential in order reduce memory or running time requirements of simulations. This thesis presents a review of disordered systems, as well as a thorough study of two particular systems through numerical simulation, algorithm de- velopment and optimization, and careful statistical analysis of scaling proper- ties. Chapter 1 provides a thorough overview of disordered systems, the his- tory of their study in the physics community, and the development of tech- niques used to study them. Topics of quenched disorder, phase transitions, the renormalization group, criticality, and scale invariance are discussed. Several prominent models of disordered systems are also explained. Lastly, analysis techniques used in studying disordered systems are covered. In Chapter 2, minimal spanning trees on critical percolation clusters are studied, motivated in part by an analytic perturbation expansion by Jackson and Read [6] that I check against numerical calculations. This system has a direct mapping to the ground state of the strongly disordered spin glass [7]. We compute the path length fractal dimension of these trees in dimensions d = {2, 3, 4, 5} and find our results to be compatible with the analytic results suggested by Jackson and Read. In Chapter 3, the random bond Ising ferromagnet is studied, which is es- pecially useful since it serves as a prototype for more complicated disordered systems such as the random field Ising model and spin glasses. We investigate the effect that changing boundary spins has on the locations of domain walls in the interior of the random ferromagnet system. We provide an analytic proof that ground state domain walls in the two dimensional system are de- composable, and we map these domain walls to a shortest paths problem. By implementing a multiple-source shortest paths algorithm developed by Philip Klein [8], we are able to efficiently probe domain wall locations for all possible configurations of boundary spins. We consider lattices with uncorrelated dis- order, as well as disorder that is spatially correlated according to a power law. We present numerical results for the scaling exponent governing the probabil- ity that a domain wall can be induced that passes through a particular location in the system’s interior, and we compare these results to previous results on the directed polymer problem