Evolution of force networks in dense particulate media

Stochastic methods for slip prediction in a sheared granular system

We consider a sheared granular system experiencing intermittent dynamics of stick-slip type via discrete element simulations. The considered setup consists of a two-dimensional system of soft frictional particles sandwiched between solid walls, one of which is exposed to a shearing force. The slip events are detected using stochastic state space models applied to various measures describing the system. The amplitudes of the events spread over more than four decades and present two distinctive peaks, one for the microslips and the other for the slips. We show that the measures describing the forces between the particles provide earlier detection of an upcoming slip event than the measures based solely on the wall movement. By comparing the detection times obtained from the considered measures, we observe that a typical slip event starts with a local change in the force network. However, some local changes do not spread globally over the force network. For the changes that become global, we find that their size strongly influences the further behavior of the system. If the size of a global change is large enough, then it triggers a slip event; if it is not, then a much weaker microslip follows. Quantification of the changes in the force network is made possible by formulating clear and precise measures describing their static and dynamic properties.

On intermittency in sheared granular systems

We consider a system of granular particles, modeled by two dimensional frictional soft elastic disks, that is exposed to externally applied time-dependent shear stress in a planar Couette geometry. We concentrate on the external forcing that produces intermittent dynamics of stick-slip type. In this regime, the top wall remains almost at rest until the applied stress becomes sufficiently large, and then it slips. We focus on the evolution of the system as it approaches a slip event. Our main finding is that there are two distinct groups of measures describing system behavior before a slip event. The first group consists of global measures defined as system-wide averages at a fixed time. Typical examples of measures in this group are averages of the normal or tangent forces acting between the particles, system size and number of contacts between the particles. These measures do not seem to be sensitive to an approaching slip event. On average, they tend to increase linearly with the force pulling the spring. The second group consists of the time-dependent measures that quantify the evolution of the system on a micro (particle) or mesoscale. Measures in this group first quantify the temporal differences between two states and only then aggregate them to a single number. For example, Wasserstein distance quantitatively measures the changes of the force network as it evolves in time while the number of broken contacts quantifies the evolution of the contact network. The behavior of the measures in the second group changes dramatically before a slip event starts. They increase rapidly as a slip event approaches, indicating a significant increase in fluctuations of the system before a slip event is triggered.

Controlling disorder in self-assembled colloidal monolayers via evaporative processes

Monolayers of assembled nano-objects with a controlled degree of disorder hold interest in many optical applications, including photovoltaics, light emission, sensing, and structural coloration. Controlled disorder can be achieved through either top-down or bottom-up approaches, but the latter is more suited to large-scale, low-cost fabrication. Disordered colloidal monolayers can be assembled through evaporatively driven convective assembly, a bottom-up process with a wide range of parameters impacting particle placement. Motivated by the photonic applications of such monolayers, in this review we discuss the quantification of monolayer disorder, and the assembly methods that have been used to produce them. We review the impact of particle and solvent properties, as well as the use of substrate patterning, to create the desired spatial distributions of particles.

Understanding slow compression and decompression of frictionless soft granular matter by network analysis

We consider dense granular systems in three spatial dimensions exposed to slow compression and decompression, below, during, above and well above jamming. The evolution of granular systems under slow deformation is non-trivial and involves smooth, continuous, reversible (de)compression periods, interrupted by fast, discontinuous, irreversible transition events. These events are often, but not always, associated with rearrangements of particles and of the contact network. How many particles are involved in these transitions between two states can range from few to almost all in the system. An analysis of the force network that is built on top of the contact network is carried out using the tools of persistent homology. Results involve the observation that kinetic energy is correlated with the intensity of rearrangements, while the evolution of global mechanical measures, such as pressure, is strongly correlated with the evolution of the topological measures quantifying loops in the force network. Surprisingly, some transitions are clearly detected by persistent homology even though motion/rearrangement of particles is much weaker, i.e., much harder to detect or, in some cases, not observed at all.

Topological persistence machine of phase transitions

The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass-liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed "topological persistence machine," to construct the shape of data from correlations in states, so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis. We demonstrate the efficacy of the approach in detecting the Berezinskii-Kosterlitz-Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose-Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide practical insights for exploring the phases of experimental physical systems.

Quantitative measure of memory loss in complex spatiotemporal systems

History dependence of the evolution of complex systems plays an important role in forecasting. The precision of the predictions declines as the memory of the systems is lost. We propose a simple method for assessing the rate of memory loss that can be applied to experimental data observed in any metric space X. This rate indicates how fast the future states become independent of the initial condition. Under certain regularity conditions on the invariant measure of the dynamical system, we prove that our method provides an upper bound on the mixing rate of the system. This rate can be used to infer the longest time scale on which the predictions are still meaningful. We employ our method to analyze the memory loss of a slowly sheared granular system with a small inertial number I. We show that, even if I is kept fixed, the rate of memory loss depends erratically on the shear rate. Our study suggests the presence of bifurcations at which the rate of memory loss increases with the shear rate, while it decreases away from these points. We also find that the rate of memory loss is closely related to the frequency of the sudden transitions of the force network. Moreover, the rate of memory loss is also well correlated with the loss of correlation of shear stress measured at the system scale. Thus, we have established a direct link between the evolution of force networks and the macroscopic properties of the considered system.

Evolution of local motifs and topological proximity in self-assembled quasi-crystalline phases

Using methods from the field of topological data analysis, we investigate the self-assembly and emergence of three-dimensional quasi-crystalline structures in a single-component colloidal system. Combining molecular dynamics and persistent homology, we analyse the time evolution of persistence diagrams and particular local structural motifs. Our analysis reveals the formation and dissipation of specific particle constellations in these trajectories, and shows that the persistence diagrams are sensitive to nucleation and convergence to a final structure. Identification of local motifs allows quantification of the similarities between the final structures in a topological sense. This analysis reveals a continuous variation with density between crystalline clathrate, quasi-crystalline, and disordered phases quantified by 'topological proximity', a visualization of the Wasserstein distances between persistence diagrams. From a topological perspective, there is a subtle, but direct connection between quasi-crystalline, crystalline and disordered states. Our results demonstrate that topological data analysis provides detailed insights into molecular self-assembly.

Failure of confined granular media due to pullout of an intruder: from force networks to a system wide response

We investigate computationally the pullout of a spherical intruder initially buried at the bottom of a granular column. The intruder starts to move out of the granular bed once the pulling force reaches a critical value, leading to material failure. The failure point is found to depend on the diameter of the granular column, pointing out the importance of particle-wall interactions in determining the material response. Discrete element simulations show that prior to failure, the contact network is essentially static, with only minor rearrangements of the particles. However, the force network, which includes not only the contact information, but also the information about the interaction strength, undergoes nontrivial evolution. An initial insight is obtained by considering the relative magnitudes of normal and tangential forces between the particles, and in particular the proportion of contacts that reach Coulomb threshold. More detailed understanding of the processes leading to failure is reached by the analysis of both spatial and temporal properties of the force network using the tools of persistent homology. We find that the forces between the particles undergo intermittent temporal variations ahead of the failure. In addition to this temporal intermittency, the response of the force network is found to be spatially dependent and influenced by proximity to the intruder. Furthermore, the response is modified significantly by the interaction strength, with the relevant measures describing the response showing differing behaviors for the contacts characterized by large interaction forces.

Topological data analysis of continuum percolation with disks

We study continuum percolation with disks, a variant of continuum percolation in two-dimensional Euclidean space, by applying tools from topological data analysis. We interpret each realization of continuum percolation with disks as a topological subspace of [0,1]^{2} and investigate its topological features across many realizations. Specifically, we apply persistent homology to investigate topological changes as we vary the number and radius of disks, and we observe evidence that the longest persisting invariant is born at or near the percolation transition.

Granular response to impact: Topology of the force networks

The impact of an intruder on granular matter leads to the formation of mesoscopic force networks, which were seen particularly clearly in the recent experiments carried out with photoelastic particles [Clark et al., Phys. Rev. Lett. 114, 144502 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.144502]. These force networks are characterized by complex structure and evolve on fast time scales. While it is known that total photoelastic activity in the granular system is correlated with the acceleration of the intruder, it is not known how the structure of the force network evolves during impact, and if there are dominant features in the networks that can be used to describe the intruder's dynamics. Here, we use topological tools, in particular persistent homology, to describe these features. Persistent homology allows quantification of both structure and time evolution of the resulting force networks. We find that there is a clear correlation of the intruder's dynamics and some of the topological measures implemented. This finding allows us to discuss which properties of the force networks are most important when attempting to describe the intruder's dynamics. In particular, we find that the presence of loops in the force network, quantified by persistent homology, is strongly correlated to the deceleration of the intruder. In some cases, particularly for the impact on soft particles, the measures derived from the persistence analysis describe the deceleration of the intruder even better than the total photoelastic activity. We are also able to define an upper bound on the relevant time scale over which the force networks evolve.

Force percolation transition of jammed granular systems

The mechanical and transport properties of jammed materials originate from an underlying percolating network of contact forces between the grains. Using extensive simulations we investigate the force-percolation transition of this network, where two particles are considered as linked if their interparticle force overcomes a threshold. We show that this transition belongs to the random percolation universality class, thus ruling out the existence of long-range correlations between the forces. Through a combined size and pressure scaling for the percolative quantities, we show that the continuous force percolation transition evolves into the discontinuous jamming transition in the zero pressure limit, as the size of the critical region scales with the pressure.

Quantifying disorder in colloidal films spin-coated onto patterned substrates

Polycrystals of thin colloidal deposits, with thickness controlled by spin-coating speed, exhibit axial symmetry with local 4-fold and 6-fold symmetric structures, termed orientationally correlated polycrystals (OCPs). While spin-coating is a very facile technique for producing large-area colloidal deposits, the axial symmetry prevents us from achieving true long-range order. To obtain true long-range order, we break this axial symmetry by introducing a patterned surface topography and thus eliminate the OCP character. We then examine symmetry-independent methods to quantify order in these disordered colloidal deposits. We find that all the information in the bond-orientational order parameters is well captured by persistent homology analysis methods that only use the centers of the particles as input data. It is expected that these methods will prove useful in characterizing other disordered structures.

Penetration in bimodal, polydisperse granular material

We investigate the impact penetration of spheres into granular media which are compositions of two discrete size ranges, thus creating a polydisperse bimodal material. We examine the penetration depth as a function of the composition (volume fractions of the respective sizes) and impact speed. Penetration depths were found to vary between δ=0.5D_{0} and δ=7D_{0}, which, for mono-modal media only, could be correlated in terms of the total drop height, H=h+δ, as in previous studies, by incorporating correction factors for the packing fraction. Bimodal data can only be collapsed by deriving a critical packing fraction for each mass fraction. The data for the mixed grains exhibit a surprising lubricating effect, which was most significant when the finest grains [d_{s}∼O(30) μm] were added to the larger particles [d_{l}∼O(200-500) μm], with a size ratio, ε=d_{l}/d_{s}, larger than 3 and mass fractions over 25%, despite the increased packing fraction. We postulate that the small grains get between the large grains and reduce their intergrain friction, only when their mass fraction is sufficiently large to prevent them from simply rattling in the voids between the large particles. This is supported by our experimental observations of the largest lubrication effect produced by adding small glass beads to a bed of large sand particles with rough surfaces.

Stretch-induced network reconfiguration of collagen fibres in the human facet capsular ligament

Biomaterials can display complex spatial patterns of cellular responses to external forces. Revealing and predicting the role of these patterns in material failure require an understanding of the statistical dependencies between spatially distributed changes in a cell's local biomechanical environment, including altered collagen fibre kinematics in the extracellular matrix. Here, we develop and apply a novel extension of network science methods to investigate how excessive tensile stretch of the human cervical facet capsular ligament (FCL), a common source of chronic neck pain, affects the local reorganization of collagen fibres. We define collagen alignment networks based on similarity in fibre alignment angles measured by quantitative polarized light imaging. We quantify the reorganization of these networks following macroscopic loading by describing the dynamic reconfiguration of network communities, regions of the material that display similar fibre alignment angles. Alterations in community structure occur smoothly over time, indicating coordinated adaptation of fibres to loading. Moreover, flexibility, a measure of network reconfiguration, tracks the loss of FCL's mechanical integrity at the onset of anomalous realignment (AR) and regions of AR display altered community structure. These findings use novel network-based techniques to explain abnormal collagen fibre reorganization, a dynamic and coordinated multivariate process underlying tissue failure.

Evolution of network architecture in a granular material under compression

As a granular material is compressed, the particles and forces within the system arrange to form complex and heterogeneous collective structures. Force chains are a prime example of such structures, and are thought to constrain bulk properties such as mechanical stability and acoustic transmission. However, capturing and characterizing the evolving nature of the intrinsic inhomogeneity and mesoscale architecture of granular systems can be challenging. A growing body of work has shown that graph theoretic approaches may provide a useful foundation for tackling these problems. Here, we extend the current approaches by utilizing multilayer networks as a framework for directly quantifying the progression of mesoscale architecture in a compressed granular system. We examine a quasi-two-dimensional aggregate of photoelastic disks, subject to biaxial compressions through a series of small, quasistatic steps. Treating particles as network nodes and interparticle forces as network edges, we construct a multilayer network for the system by linking together the series of static force networks that exist at each strain step. We then extract the inherent mesoscale structure from the system by using a generalization of community detection methods to multilayer networks, and we define quantitative measures to characterize the changes in this structure throughout the compression process. We separately consider the network of normal and tangential forces, and find that they display a different progression throughout compression. To test the sensitivity of the network model to particle properties, we examine whether the method can distinguish a subsystem of low-friction particles within a bath of higher-friction particles. We find that this can be achieved by considering the network of tangential forces, and that the community structure is better able to separate the subsystem than a purely local measure of interparticle forces alone. The results discussed throughout this study suggest that these network science techniques may provide a direct way to compare and classify data from systems under different external conditions or with different physical makeup.

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.

Structure of force networks in tapped particulate systems of disks and pentagons. II. Persistence analysis

In the companion paper [Pugnaloni et al., Phys. Rev. E 93, 062902 (2016)10.1103/PhysRevE.93.062902], we use classical measures based on force probability density functions (PDFs), as well as Betti numbers (quantifying the number of components, related to force chains, and loops), to describe the force networks in tapped systems of disks and pentagons. In the present work, we focus on the use of persistence analysis, which allows us to describe these networks in much more detail. This approach allows us not only to describe but also to quantify the differences between the force networks in different realizations of a system, in different parts of the considered domain, or in different systems. We show that persistence analysis clearly distinguishes the systems that are very difficult or impossible to differentiate using other means. One important finding is that the differences in force networks between disks and pentagons are most apparent when loops are considered: the quantities describing properties of the loops may differ significantly even if other measures (properties of components, Betti numbers, force PDFs, or the stress tensor) do not distinguish clearly or at all the investigated systems.

Structure of force networks in tapped particulate systems of disks and pentagons. I. Clusters and loops

The force network of a granular assembly, defined by the contact network and the corresponding contact forces, carries valuable information about the state of the packing. Simple analysis of these networks based on the distribution of force strengths is rather insensitive to the changes in preparation protocols or to the types of particles. In this and the companion paper [Kondic et al., Phys. Rev. E 93, 062903 (2016)10.1103/PhysRevE.93.062903], we consider two-dimensional simulations of tapped systems built from frictional disks and pentagons, and study the structure of the force networks of granular packings by considering network's topology as force thresholds are varied. We show that the number of clusters and loops observed in the force networks as a function of the force threshold are markedly different for disks and pentagons if the tangential contact forces are considered, whereas they are surprisingly similar for the network defined by the normal forces. In particular, the results indicate that, overall, the force network is more heterogeneous for disks than for pentagons. Such differences in network properties are expected to lead to different macroscale response of the considered systems, despite the fact that averaged measures (such as force probability density function) do not show any obvious differences. Additionally, we show that the states obtained by tapping with different intensities that display similar packing fraction are difficult to distinguish based on simple topological invariants.

Scaling properties of force networks for compressed particulate systems

We consider, computationally and experimentally, the scaling properties of force networks in the systems of circular particles exposed to compression in two spatial dimensions. The simulations consider polydisperse and monodisperse particles, both frictional and frictionless, and in experiments we use monodisperse and bidisperse frictional particles. While for some of the considered systems we observe consistent scaling exponents describing the behavior of the force networks, we find that this behavior is not universal. In particular, we find that frictionless systems, independently of whether they partially crystallize under compression or not, show scaling properties that are significantly different compared to the frictional disordered ones. The findings of nonuniversality are confirmed by explicitly computing fractal dimension for the considered systems. The results of the physical experiments are consistent with the results obtained in simulations of frictional disordered systems.

Network flow model of force transmission in unbonded and bonded granular media

An established aspect of force transmission in quasistatic deformation of granular media is the existence of a dual network of strongly versus weakly loaded particles. Despite significant interest, the regulation of strong and weak forces through the contact network remains poorly understood. We examine this aspect of force transmission using data on microstructural fabric from: (I) three-dimensional discrete element models of grain agglomerates of bonded subspheres constructed from in situ synchrotron microtomography images of silica sand grains under unconfined compression and (II) two-dimensional assemblies of unbonded photoelastic circular disks submitted to biaxial compression under constant volume. We model force transmission as a network flow and solve the maximum flow-minimum cost (MFMC) problem, the solution to which yields a percolating subnetwork of contacts that transmits the "maximum flow" (i.e., the highest units of force) at "least cost" (i.e., the dissipated energy from such transmission). We find the MFMC describes a two-tier hierarchical architecture. At the local level, it encapsulates intraconnections between particles in individual force chains and in their conjoined 3-cycles, with the most common configuration having at least one force chain contact experiencing frustrated rotation. At the global level, the MFMC encapsulates interconnections between force chains. The MFMC can be used to predict most of the force chain particles without need for any information on contact forces, thereby suggesting the network flow framework may have potential broad utility in the modeling of force transmission in unbonded and bonded granular media.

Hidden geometry of traffic jamming

We introduce an approach based on algebraic topological methods that allow an accurate characterization of jamming in dynamical systems with queues. As a prototype system, we analyze the traffic of information packets with navigation and queuing at nodes on a network substrate in distinct dynamical regimes. A temporal sequence of traffic density fluctuations is mapped onto a mathematical graph in which each vertex denotes one dynamical state of the system. The coupling complexity between these states is revealed by classifying agglomerates of high-dimensional cliques that are intermingled at different topological levels and quantified by a set of geometrical and entropy measures. The free-flow, jamming, and congested traffic regimes result in graphs of different structure, while the largest geometrical complexity and minimum entropy mark the edge of the jamming region.

Extraction of force-chain network architecture in granular materials using community detection

Force chains form heterogeneous physical structures that can constrain the mechanical stability and acoustic transmission of granular media. However, despite their relevance for predicting bulk properties of materials, there is no agreement on a quantitative description of force chains. Consequently, it is difficult to compare the force-chain structures in different materials or experimental conditions. To address this challenge, we treat granular materials as spatially-embedded networks in which the nodes (particles) are connected by weighted edges that represent contact forces. We use techniques from community detection, which is a type of clustering, to find sets of closely connected particles. By using a geographical null model that is constrained by the particles' contact network, we extract chain-like structures that are reminiscent of force chains. We propose three diagnostics to measure these chain-like structures, and we demonstrate the utility of these diagnostics for identifying and characterizing classes of force-chain network architectures in various materials. To illustrate our methods, we describe how force-chain architecture depends on pressure for two very different types of packings: (1) ones derived from laboratory experiments and (2) ones derived from idealized, numerically-generated frictionless packings. By resolving individual force chains, we quantify statistical properties of force-chain shape and strength, which are potentially crucial diagnostics of bulk properties (including material stability). These methods facilitate quantitative comparisons between different particulate systems, regardless of whether they are measured experimentally or numerically.