2D Packings Research Articles

Graph topological transformations in space-filling cell aggregates.

Cell rearrangements are fundamental mechanisms driving large-scale deformations of living tissues. In three-dimensional (3D) space-filling cell aggregates, cells rearrange through local topological transitions of the network of cell-cell interfaces, which is most conveniently described by the vertex model. Since these transitions are not yet mathematically properly formulated, the 3D vertex model is generally difficult to implement. The few existing implementations rely on highly customized and complex software-engineering solutions, which cannot be transparently delineated and are thus mostly non-reproducible. To solve this outstanding problem, we propose a reformulation of the vertex model. Our approach, called Graph Vertex Model (GVM), is based on storing the topology of the cell network into a knowledge graph with a particular data structure that allows performing cell-rearrangement events by simple graph transformations. Importantly, when these same transformations are applied to a two-dimensional (2D) polygonal cell aggregate, they reduce to a well-known T1 transition, thereby generalizing cell-rearrangements in 2D and 3D space-filling packings. This result suggests that the GVM's graph data structure may be the most natural representation of cell aggregates and tissues. We also develop a Python package that implements GVM, relying on a graph-database-management framework Neo4j. We use this package to characterize an order-disorder transition in 3D cell aggregates, driven by active noise and we find aggregates undergoing efficient ordering close to the transition point. In all, our work showcases knowledge graphs as particularly suitable data models for structured storage, analysis, and manipulation of tissue data.

Competitive random sequential adsorption of binary mixtures of disks and discorectangles

The two-dimensional (2D) packings of binary mixtures of disks with diameter d and discorectangles with aspect ratio ɛ (length-to-width ratio ε=l/d ) were studied numerically. The competitive random sequential adsorption (RSA) with simultaneous deposition of particles was considered. The aspect ratio was changed within the range ε=1−10 . In the competitive model, the particle was selected with probability pd (disks) and pε=1−pd (discorectangles), and then they were placed sequentially on a solid surface without overlapping with previously placed particles. Behavior of the total coverage in jamming state ϕ T at different values of pd and ɛ was analyzed. For core–shell structure of the particles the percolation connectivity of films was also discussed.

Temporal evolution of electrical resistance through the granular packing of Ni beads

In this paper we investigate the temporal evolution of the electrical resistance through different two-dimensional (2D) packings of Ni beads when 1 mA current is injected in them. In the first stages of measurements, resistance decreases towards a saturation value and it can be fitted with Mittag-Leffler function. Fitting parameters show that the relaxation dynamics does not depend on the type of the packing. Different packings show differences in the initial values of the resistance which is attributed to the formation of new microcontacts during the formation of the new packing. Pauses in the flow of the current cause the resistance either to decrease, increase or remain the same, depending on the packing. Longer measurements show, that after the initial drop, the resistance starts to rise which can probably be attributed to the deterioration of microcontacts between the beads. Small variations in temperature do not affect the temporal evolution of the resistance.

Effect of lateral confinement on the apparent mass of granular dampers

We study, via discrete element method simulations, the apparent mass (m, the ratio of a driving force to a resulting acceleration) and loss factor ( $$\eta$$ , the ratio of dissipated to stored energy) of granular dampers attached to a vertically driven, single degree of freedom mechanical system. Granular dampers (or particle dampers) consist in receptacles that contain macroscopic particles which dissipate energy, when they are subjected to vibration, thanks to the inelastic collisions and friction between them. Although many studies focus on $$\eta$$ , less work has been devoted to m. The apparent mass of granular dampers is an important characteristic since the grains, which are free to move or collide inside their receptacle, act as a non-constant and time-dependent mass which alters the mass of the main vibrating system in a non-trivial way. In particular, it has been recently demonstrated (Masmoudi et al. in Granul Matter 18:71, 2016) that m non-linearly depends on the driving acceleration $$\varGamma$$ according to a power law, $$m\propto \varGamma ^k$$ . Experiments using three-dimensional (3D) packings of particles suggest $$k=-2$$ . However, simulations with one-dimensional (1D) columns of particles on a vibrating plate and theoretical predictions based on the inelastic bouncing ball model (IBBM) suggest that $$k=-1$$ . These findings opened questions as to whether the apparent mass, which relies on how linear momentum is transferred from the damper to the primary system, depends on the dimensionality of the packing or on lateral interactions between walls and grains. Interestingly, $$\eta$$ was shown to follow a universal curve, $$\eta \propto \varGamma ^{-1}$$ , whatever the dimensionality and the constraints in the motion of the grains. In this work, we consider granular dampers without a lid under different confinement conditions in the motion of the particles (1D, quasi-1D, quasi-2D and full 3D). We find that the mechanical response of the granular damper (m and $$\eta$$ ) is not sensitive to the lateral confinement or dimensionality. However, we have observed two distinct regimes, depending on whether the driving frequency is above or below the resonant frequency of the primary system. In the inertial regime, $$\eta$$ decays according to the IBBM for all dimensions, $$\eta \propto \varGamma ^{-1}$$ , while m falls as $$\varGamma ^{-2}$$ for all dimensions, in agreement with Masmoudi’s experiments. However, the power law for m is valid only for moderate acceleration, before becoming negative at high accelerations. In the quasi-static regime, both m and $$\eta$$ display a complex behavior as functions of the excitation amplitude, but the mean trend is consistent with the IBBM predictions, i.e., $$m\propto \varGamma ^{-1}$$ and $$\eta \propto \varGamma ^{-1}$$ .

Optimal packing in 2D and 3D granular systems: Density, connectivity, and force distributions

In this work, we explore the influence of the grain size distribution (GSD) on density, connectivity and internal forces distributions, for both 2D and 3D granular packings built mechanically. For power law GSDs, we show that there is an exponent for which density and connectivity are optimized, and this exponent is close to those that characterize other well known GSDs such as the Fuller and Thompson distribution and the Appollonian packing. In addition, we studied the distributions of normal forces, finding that these can be well described by a power-law tail, specially for the GSDs with large size span. These results highlight the role of the GSD on internal structure and suggest important consequences on macroscopic properties.

Link between packing morphology and the distribution of contact forces and stresses in packings of highly nonconvex particles.

An external load on a particle packing is distributed internally through a heterogeneous network of particle contacts. This contact force distribution determines the stability of the particle packing and the resulting structure. Here, we investigate the homogeneity of the contact force distribution in packings of highly nonconvex particles both in two-dimensional (2D) and three-dimensional (3D) packings. A recently developed discrete element method is used to model packings of nonconvex particles of varying sphericity. Our results establish that in 3D packings the distribution of the contact forces in the normal direction becomes increasingly heterogeneous with decreasing particle sphericity. However, in 2D packings the contact force distribution is independent of particle sphericity, indicating that results obtained in 2D packings cannot be extrapolated readily to 3D packings. Radial distribution functions show that the crystallinity in 3D packings decreases with decreasing particle sphericity. We link the decreasing homogeneity of the contact force distributions to the decreasing crystallinity of 3D packings. These findings are complementary to the previously observed link between the heterogeneity of the contact force distribution and a decreasing packing crystallinity due to an increasing polydispersity of spherical particles.

Characterising 3D spherical packings by principal component analysis

Purpose The purpose of this paper is to extend the previous study [Computer Methods in Applied Mechanics and Engineering 340: 70-89, 2018] on the development of a novel packing characterising system based on principal component analysis (PCA) to quantitatively reveal some fundamental features of spherical particle packings in three-dimensional. Design/methodology/approach Gaussian quadrature is adopted to obtain the volume matrix representation of a particle packing. Then, the digitalised image of the packing is obtained by converting cross-sectional images along one direction to column vectors of the packing image. Both a principal variance (PV) function and a dissimilarity coefficient (DC) are proposed to characterise differences between different packings (or images). Findings Differences between two packings with different packing features can be revealed by the PVs and DC. Furthermore, the values of PV and DC can indicate different levels of effects on packing caused by configuration randomness, particle distribution, packing density and particle size distribution. The uniformity and isotropy of a packing can also be investigated by this PCA based approach. Originality/value Develop an alternative novel approach to quantitatively characterise sphere packings, particularly their differences.

Can a Large Packing be Assembled from Smaller Ones?

We consider zero temperature packings of soft spheres that undergo a jamming to unjamming transition as a function of packing fraction. We compare differences in the structure, as measured from the contact statistics, of a finite subsystem of a large packing to a whole packing with periodic boundaries of an equivalent size and pressure. We find that the fluctuations of the ensemble of whole packings are smaller than those of the ensemble of subsystems. Convergence of these two quantities appears to occur at very large systems, which are usually not attainable in numerical simulations. Finding differences between packings in two dimensions and three dimensions, we also consider four dimensions and mean-field models, and find that they show similar system size dependence. Mean-field critical exponents appear to be consistent with the 3D and 4D packings, suggesting they are above the upper critical dimension. We also find that the convergence as a function of system size to the thermodynamic limit is characterized by two different length scales. We argue that this is the result of the system being above the upper critical dimension.

Particle rotations and energy dissipation during mechanical compression of granular materials

We present new in-situ measurements of particle rotations and energy dissipation during compression of 3D packings of stiff, frictional particles. Two confined, uniaxial compression tests with different degrees of lateral confinement are discussed. X-ray computed tomography and 3D X-ray diffraction were combined to provide inter-particle forces, slip and roll distances, twist angles, and energy dissipation at all inter-particle contacts. Each of these measured quantities followed exponential distributions above their respective mean values and power-law distributions below their mean values in both experiments. Changes in these distributions during experiments suggest that the quantities generally became more homogeneous with increasing overall sample stress. Contact roll and slip distances, twist angles, and energy dissipation were all more heterogeneous than inter-particle normal force magnitudes in both experiments. Energy dissipation due to inter-particle slipping accounted for 95% of the total energy dissipated in both experiments. Dissipation mechanisms at inter-particle contacts bearing more than the mean normal force were responsible for approximately 70% of each sample’s dissipated energy at each load step, even though these contacts constituted approximately 40% of the total number of contacts.

Parallel implicit contact algorithm for soft particle systems

This paper presents a numerical technique to model soft particle materials in which the particles can undergo large deformations. It combines an implicit finite strain formalism of the Material Point Method and the Contact Dynamics method. In this framework, the large deformations of individual particles as well as their collective interactions are treated consistently. In order to reduce the computational cost, this method is parallelised using the Message Passing Interface (MPI) strategy. Using this approach, we investigate the uniaxial compaction of 2D packings composed of particles governed by a Neo-Hookean material behaviour. We consider compressibility rates ranging from fully compressible to incompressible particles. The packing deformation mechanism is a combination of both particle rearrangements and large deformations, and leads to high packing fractions beyond the jamming state. We show that the packing strength declines when the particle compressibility decreases, and the packing can deform considerably. We also discuss the evolution of the connectivity of the particles and particle deformation distributions in the packing.

Modeling of the self-organization processes in crystal-forming systems. Tetrahedral metal clusters and the self-assembly of crystal structures of intermetallic compounds

A combinatorial and topological modeling of 1D, 2D, and 3D packings of symmetrically linked A4 tetrahedra has been performed. Three types of 1D chains with tetrahedra connectivities of 4, 6, and 8 were used to model 2D layers L-1, L-2, and L-3 and 3D frameworks FR-1, FR-2, FR-3, and FR-4. A family of tetrahedral structures with FR-1, FR-2, and FR-3 frameworks has been selected among the intermetallic compounds with chemical compositions of A 3 B, A 2 B 2, AB 3, A 2 BC, AB 2 C, and ABCD; this family includes more than 1900 compounds (TOPOS program package). It is found that the topological models of tetrahedral 3D frameworks are in correspondence with all types of the crystal structures formed in Au–Cu binary systems (FR-1 for Cu3Au (auricupride), Cu2Au2 (tetraauricupride), and CuAu3 (bogdanovite)), in the Mg–Cd system (FR-3 for Mg3Cd, Mg2Cd2, and MgCd3), in the Li–Hg system (FR-2 for Li3Hg and Li2Hg2 and FR-3 for LiHg3), in the Li–Ag–Al ternary system (FR-2 for LiAg2Al and Li2AgAl), and in the Li–Mg–Pd–Sn quaternary system (FR-2 for LiMgPdSn). Framework FR-4 has been established in ternary intermetallic compounds A(Li2Sn2); A = Cu, Ag, Au.

Modeling root growth in granular soils: effects of root stiffness and packing fraction

We use molecular dynamics simulations to investigate the effects of root bending stiffness and packing fraction on the path followed by a growing root in 2D packings of grains representing a soil. The root is modeled as a chain of elements that can grow in length and change their direction depending on the forces exerted by soil grains. We show that the root shape is mainly controlled by the bending stiffness of its apex. At low stiffness, the root randomly explores the pore space whereas at sufficiently high stiffness, of the order of soil hardness multiplied by mean grain size, the root follows a straight path across the soil. Between these two limits, the root shape can be characterized by the standard deviation of its re-directions at the scale of soil grains. We find that this shape parameter varies as a power-law function of the normalized bending stiffness.

Linear and nonlinear elastic properties of dense granular packings: a DEM exploration

Discrete Element Method modeling is used to study the frequency spectrum of particle motion in dense 3D packings of glass beads with Hertz-Mindlin contacts. Frequency sweeps show a dependency of the resonant frequencies on the drive amplitude and confining stress on the system, showing material changes in the system. The amplitude dependency of the second thickness mode 3λ /4 as identified by the internal strain field scales as f ∝ σ1/6 while the confining stress dependency scales as f ∝ σ 2/3 , as predicted by Hertzian theory.

Structure of hexapod 3D packings: understanding the global stability from the local organization

Aggregates of non-convex particles have shown to be particularly stable which makes them good candidates to design new lightweight and reversible structures. However, few is known about the fundamental reason of their stability. In this paper we presents a novel experimental method to investigate the local structure of piles made of hexapod particles. This method is based on X-ray scanning and on an accurate homemade particle detection code. It permits to get the position and orientation of each particle as well as to detect their contact points. Measurement of the coordination numbers, statistics of the contact positions and local density evaluation for different packing configurations show a good agreement with the previous studies carried out at the global scale and permits to explain the main local mechanisms leading to stable structures.

Effect of particle size distribution on 3D packings of spherical particles

We use molecular dynamics simulations of frictionless spherical particles to investigate a class of polydisperse granular materials in which the particle size distribution is uniform in particle volumes. The particles are assembled in a box by uniaxial compaction under the action of a constant stress. Due to the absence of friction and the nature of size distribution, the generated packings have the highest packing fraction at a given size span, defined as the ratio α of the largest size to the smallest size. We find that, up to α = 5, the packing fraction is a nearly linear function of α . While the coordination number is nearly constant due to the isostatic nature of the packings, we show that the connectivity of the particles evolves with α . In particular, the proportion of particles with 4 contacts represents the largest proportion of particles mostly of small size. We argue that this particular class of particles occurs as a result of the high stability of local configurations in which a small particle is stuck by four larger particles.

Topological and geometric measurements of force-chain structure.

Developing quantitative methods for characterizing structural properties of force chains in densely packed granular media is an important step toward understanding or predicting large-scale physical properties of a packing. A promising framework in which to develop such methods is network science, which can be used to translate particle locations and force contacts into a graph in which particles are represented by nodes and forces between particles are represented by weighted edges. Recent work applying network-based community-detection techniques to extract force chains opens the door to developing statistics of force-chain structure, with the goal of identifying geometric and topological differences across packings, and providing a foundation on which to build predictions of bulk material properties from mesoscale network features. Here we discuss a trio of related but fundamentally distinct measurements of the mesoscale structure of force chains in two-dimensional (2D) packings, including a statistic derived using tools from algebraic topology, which together provide a tool set for the analysis of force chain architecture. We demonstrate the utility of this tool set by detecting variations in force-chain architecture with pressure. Collectively, these techniques can be generalized to 3D packings, and to the assessment of continuous deformations of packings under stress or strain.

Crystal chemical analysis of structures of the unsaturated tetraazamacrocyclic gold(III) complexes

The detailed comparative study is carried out for crystal structure packings of the following gold(III) complexes with unsaturated ligands: [Au(C14H22N4)]Br (I), [Au(C14H23N4)](ClO4)2 (II), [Au(C14H24N4)](H3O)(ClO4)4 (III). The determining role in the topological pattern of packings I–III belongs to the compositions and structures of the cations along with the ability of the ions of the complexes to act as donors and acceptors of hydrogen bonds. The 3D packings of complexes I and II containing iminate six-membered rings are determined and stabilized by a wide network of weak hydrogen bonds (C–H…π, C–H…Au, and C–H…Br(O)) and short contacts Au(N)…O (in structure II). The structure of imine complex III is characterized by one-dimensional piles formed due to hydrogen bonds O(w)–H…O and contacts Au…O of the О(2) atom of anion Сl(1)О4- with cations (H3O)+ and [Au(C14H24N4)]3+ (CIF files CCDC 251258 (I), 276132 (II), and 287784 (III)).

Modeling of self-organization processes in crystal-forming systems: Symmetry and topology code for cluster self-assembly of crystal structures for molecular and framework compounds

This is a review over the state of art in self-organization in crystal-forming systems, where a longrange order appears spontaneously in the arrangement of nanoscale building units of any nature (atomic clusters and molecules), which initially existed in a dynamic state as a chaotic mixture. The focus is on bimolecular compounds where building units are M-octahedra M(O, OH, H2O) and T-tetrahedra T(O, OH)4 bonded by OH–O hydrogen bonds and/or bridging oxygen atoms O. The combinatorial–topological modeling of 1D and 2D packings of suprapolyhedral clusters (MT)n has been carried out. These packings are symmetrically and topologically conceivable types of molecular MT clusters S03, from which chains S13 and 2D microlayers S23 are formed. The theoretically derived model MT-clusters (MT)n are compared with known types of precursor clusters in the molecular and framework MT-structures of polyvalent metal sulfates and their analogues. Combinatorial–topological analysis algorithms are provided to restore, from structure data, the convergent matrix crystal structure self-assembly code in the form of the sequence of significant elementary events. Crystal structure modeling involves combinatorial–topological analysis methods, whose underlying idea is the recogniition of packing-forming precursor nanoclusters and the construction of the relevant basal 2D and 3D networks in the form of graphs where nodes correspond to the positions of their centroids. The model is universal and has first been used to model the cluster self-assembly of molecular MT structures for V(2+)(H2O)4(SO4) (V-Rozenite), V(4+)O(H2O)5(SO4)(H2O) (SYN1), V(4+)O(H2O)5(SO4) (Orthominasragrite, Minasragrite, Anorthominasragrite), V(4+)O(H2O)3(SO4) (Bobjonesite); framework MT structures V(4+)O(SO4)(H2(SO4) (SYN2), V(4+)O(SO4) (Pauflerite and SYN3), V(4+)2O3(SO4)2 (SYN4), V(4+)2O3(SeO4)2 (SYN5), Fe(3+)2(SO4)3 (LIPHOS), and Fe(3+)2(SO4)3 (NASICON); and for templated framework MT structures (NH4)2[Ga2(PO4)2F2] (KTP), (NH4)2[Ga2(PO4)2(HF)F2] (p-KTP), TlMo2(PO4)3 (SYN06), and K2Mg2(SO4)3 (Langbeinite). Frequency analysis of topological and symmetry pathways in the formation and evolution of clusters (from primary chain S13 through microlayer S23 to microframework S33) elucidates new crystal-formation trends in diverse chemical systems at the microscopic level.

Patchy particle packing under electric fields.

Colloidal particles equipped with two, three, or four negatively charged patches, which endow the particles with 2-fold, 3-fold, or tetrahedral symmetries, form 1D chains, 2D layers, and 3D packings when polarized by an AC electric field. Two-patch particles, with two patches on opposite sides of the particle (2-fold symmetry) pack into the cmm plane group and 3D packings with I4mm space group symmetry, in contrast to uncharged spherical or ellipsoidal colloids that typically crystallize into a face-centered ABC layer packing. Three-patch particles (3-fold symmetry) form chains having a 21 screw axis symmetry, but these chains pair in a manner such that each individual chain has one-fold symmetry but the pair has 21 screw axis symmetry, in an arrangement that aligns the patches that would favor Coulombic interactions along the chain. Surprisingly, some chain pairs form unanticipated double-helix regions that result from mutual twisting of the chains about each other, illustrating a kind of polymorphism that may be associated with nucleation from short chain pairs. Larger 2D domains of the three-patch particles crystallize in the p6m plane group with alignment (with respect to the field) and packing densities that suggest random disorder in the domains, whereas four-patch particles form 2D domains in which close-packed rows are aligned with the field.

Simulation of self-organization processes in crystal-forming systems: Supramolecular cyclic R6 cluster precursors and self-assembly of TeO2-TEL (Tellurite) and TeO2-PAR (Paratellurite) structures

The supramolecular chemistry of oxides of sp elements (SO2, SeO2, and TeO2) is considered. The self-assembly of TeO2-TEL (Tellurite) and TeO2-PAR (Paratellurite) crystal structures is simulated. Methods of combinatorial and topological analysis (TOPOS program package) are applied which are based on constructing a basis 3D network of the structure in the form of a graph, the sites of which correspond to the positions of centroids of TeO2 molecules and the edges characterize bonds between them. The topological type of the basis 2D network in the TeO2-TEL structure corresponds to graphite (C-GRA), while in the TeO2-PAR structure the basis network corresponds to the 3D diamond network (C-DIA). A nanocluster precursor of cyclic type (R6) composed of six covalently bound TeO2 molecules (chair conformation) is established for both structures. The desymmetrization of the cyclic structure of the R6 cluster in TeO2-PAR is related to the formation of Te-Te bonds with lengths of 3.824 and 4.062 A. The symmetry and topology code of the processes of self-assembly of 3D structures from nanocluster precursors is completely reconstructed into the form “primary chain → microlayer → microframework.” In both structures R6 clusters form 2D packings with a coordination number of 6. The cluster self-assembly model explains the specific features of the morphogenesis of TeO2-TEL and TeO2-PAR (phases with low and high crystallization temperatures, respectively): platelike shape, perfect cleavage in the (110) plane, and preferred growth in the primar-chain direction [100] in the former case and growth in the direction of the primary [001] axis with the preferred formation of tetragonal prism faces (110) in the latter case.