Granular and Particulate Networks

Ternary liquid-liquid-solid systems exhibit a wide variety of different morphologies depending on the ratio of the three components. With small amounts of secondary fluid, the characteristic mechanical strength of such capillary suspensions arises due to the capillary force inducing a percolating network of particles bridged by small individual droplets of secondary fluid. These suspensions can exhibit a wide range of rheological behaviors that are closely linked to both the bulk particle structure as well as the microstructure including direct particle contacts. Spatial information on the structure of such particle networks can be obtained using 3D confocal microscopy on an index-matched model system and directly correlated to changes in the mechanical strength or mixing conditions. Graph theory offers methods and parameters that can be used to analyze complex structures. We analyze these networks using the coordination number and clustering coefficient. These parameters are compared to the measured storage and loss moduli.

In systems of multi-dimensional, interacting dynamical units, two types of inverse problems prevail. First, to find the interaction structure from observed time series of the units Second, to design the unit's interactions to achieve desired collective dynamics? Both constitute partially uncharted territory for theory and may offer a wide range of applications Solving an inverse problem means to identify the elements of a system (and its input) such that either (i) the system fits experimental observations and simultaneously satisfies known constraints or (ii) to enable the system to generate specific desired behavior. For instance (i) given observations about how a solid scatters incoming radiation reveals information about its structure and (ii) a robot's sensors, actuators, processors and mechanics or the structure of a complex synthetic material shall be designed such that they can fulfill certain tasks. Inverse problems have a long history in the natural sciences and mathematics and are implicit to many engineering problems. Yet, the vast majority of research on collective nonlinear dynamics of networks is still focusing on the “forward direction” of analysis or modeling and asks what types of collective dynamics emerge from a network of given units interacting via a given topology. We go beyond this forward perspective and address two new directions of research. One is network reconstruction, revealing the existence and possibly type of network interactions from accessing the time series of the collective dynamics of the system only [1]. A second is network design, setting up the interactions of a network such that it robustly exhibits a given collective dynamics and thus function. Interestingly, the theoretical framework we derived for network inference — one inverse problem — seems to also provide a natural viewpoint on how to design networks — a second inverse problem. In this talk, I will sketch an overview of how data-driven, model-free approaches may help addressing both of these challenges. [1] Phys. Rev. Lett. 2007; New J. Phys. 2011; Frontiers Comput. Neurosci. 2011; J. Phys. A 2014 (Topical Review); Science Adv. 2017; Nature Comm. 2017; Phys. Rev. Lett. 2018; Haehne et al., under review (2019) [2] Phys. Rev. Lett. 2006; Physica D 2006; New J. Phys. 2011; IEEE Trans. Autom. Control, 2017; Chapter in: Mathematical Technology of Networks (ed.: D. Mungolo), 2015 For complete references see http://networkdynamics.info

Frictional jamming is poorly understood compared to frictionless jamming, however it is arguably more relevant to real-world situations. In this talk, I will introduce an adaptation of rigidity percolation to frictional packings and show that it can predict the location of the transition. This (3,3) pebble game, valid in two dimensions where each particle has three degrees of freedom, takes into account that frictional forces contribute two constraints to each packing, while sliding contacts only contribute one. Beginning with slowly sheared simulations of two dimensional frictional packings, we show that a percolating rigid cluster emerges when the jamming transition is crossed, and that local forces correlate with the clusters. For real experimental packings where the jamming transition is crossed with a shear step protocol, we also find that jamming and the appearance of a percolating rigid cluster correspond with each other. Using the normal modes extracted from the dynamical matrix confirms our identification of coexisting rigid and floppy regions, and we are also able to correlate them to actual, finite, displacements. In parallel, using a simplified network model of frictional packings, we show that frictional rigidity percolation is a second order transition with exponents distinct from ordinary rigidity percolation.

A load applied to a jammed frictional granular system will be localized into a network of force chains making inter-particle connections throughout the system. Because such systems are typically under- constrained, the observed force network is not unique to a given particle configuration, but instead varies upon repeated formation. In this paper, we examine the ensemble of force chain configurations created under repeated assembly in order to develop tools to statistically forecast the observed force network. In experiments on a gently suspended 2D layer of photoelastic particles, we subject the assembly to hundreds of repeated cyclic compressions. As expected, we observe the non-unique nature of the force network, which differs for each compression cycle, by measuring all vector inter-particle contact forces using our open source PeGS software. We find that total pressure on each particle in the system correlates to its betweenness centrality value extracted from the geometric contact network. Thus, the mesoscale network structure is a key control on individual particle pressures.

Physicists have been thinking for a century about ordered solids. But ordered systems can’t compete with what biological systems such as proteins, which aren’t ordered, can do. We’ve discovered that relaxing the constraint of order increases design flexibility enormously, so that systems can be tuned to perform complex biologically-inspired tasks. Allostery in a protein is a phenomenon in which a molecule binding locally to one site affects the ability of another molecule to bind at a second distant site. Inspired by the long-range coupled conformational changes that constitute allosteric function in proteins, we tune in “allostery” into disordered mechanical networks derived from granular packings by modifying the network architecture to control the local strain at one point in such a network in response to a strain applied elsewhere in the system. In another biological example, the vascular network in the brain contracts and dilates blood vessels in order to direct enhanced blood flow to multiple specific regions of the brain as it performs multiple tasks at once. We have designed flow networks to accomplish similar complex tasks. In both examples, we address a key question: what, precisely, are we tuning into the structure in order to accomplish the function? Surprisingly, we find that the structure/function relationship is not geometrical, but topological in nature.

Rigidity is a central theme in soft matter as it controls how materials deform and flow under stress. Due to their complicated, disordered structures, characterizing rigidity of soft materials poses a grand challenge for theory. Classical theories of rigidity percolation and jamming are wonderful examples of understanding rigidity in disordered soft matter. However, new questions arise when we examine rigidity in more varieties of soft matter: Why colloidal gels have rigidity at volume fractions far below classical thresholds for rigidity? When low density soft materials fail under stress do they behave differently from crystals? We will discuss these questions in this talk, introduce mathematical frameworks for understanding rigidity, and explore scenarios where we can we use what we learn to control how soft matter solidify and flow.

Particle simulations are a standard tool to study the phase behavior of fluids and solids. Traditional methods evolve the system deterministically by solving Newton's equation of motion or statistically with Monte Carlo. But both simulation methods can be trapped in long-lived metastable states and often do not reach the time scales accessible in experiment. Examples are the aging of glasses and crystallization processes. Here we present advanced methods to speed up the simulation of complex particulate systems and discuss structure formation phenomena that can be investigated with them. Newtonian event-chain Monte Carlo is a collective move simulation method that updates particles along meandering chains of collision events. We apply Newtonian event chains to polyhedral frictionless particles that form hierarchical networks as well as to systems of granular particles with dissipative energy transfer. Another system of interest are disperse particle mixtures that are natural outcomes of syntheses (colloids, nanoparticles) or can develop dynamically through exchange of mass (micelles) or charge (atoms). We combine molecular dynamics with particle swap moves (static dispersity) or particle resize moves (dynamic dispersity). Our results reveal the bottlenecks of relaxation processes in particulate systems and the role of collective moves to efficiently update their geometric arrangement.

Over the last few years, the reach of topological band theory has been extended from topological insulators to photonics as well as to the focus of this talk, mechanical networks. After introducing the bulk-boundary correspondence principle which connects deformations and waves in materials to the mathematics of topology, I will discuss examples of how network architecture defines topological invariants. When time-reversal symmetry is preserved, the rigidity of an elastic network can be characterized by a topological invariant called the polarization. Materials with a uniform polarization display a dramatic range of edge softness depending on the orientation of the polarization relative to the terminating surface. In other examples, time-reversal symmetry is broken by using active fluids, which are composed of self-driven microbots that endow the liquid with a unique set of mechanical characteristics. I will discuss how confining an active fluid within a network of microfluidic channels can lead to topological states. These states are characterised by an invariant called the Chern number and allow for the propagation of sound waves without scattering due to topological protection. Such exotic active metamaterials blur the line between materials science and robotics.

Granular solids subjected to stress levels typical of reservoir environments display a range of microstructural alterations, such as pore collapse, grain rearrangement and crushing. Modern X-Ray digital imaging enables the direct visualization of these processes, but quantitative interpretation is necessary to unveil the relation between macroscopic deformation and grain-scale dynamics. In this research, a procedure to quantify concurrently grain kinematics and morphology is proposed. The goal of this research is to define a strategy applicable to compression tests. For this purpose, data from in situ experiments assisted by synchrotron X-ray micro tomography are used. The digital images are analyzed through a processing technique involving the identification and tracking of particles in consecutive time lapse images. In addition, the deformation response of the assembly is interpreted concurrently with the evolution of its microstructure on the basis of the datasets extracted from the experiments. It is shown that the proposed methodology enables the quantification of the evolving particle properties in sets of sequential images, as well as the characterization of its statistical variability. Furthermore, it is found that the simultaneous tracking of grain-scale kinematics facilitates the identification of families of crushed particles resulting from continuous fragmentation.

Amorphous, densely-packed systems are abundant in nature and utilized often in man-made materials. The way in which their disordered, multi-scale structure supports (or catastrophically fails to support) the application of stress is an area of active investigation with wide-reaching consequences in contexts ranging from earth science to cancer research. A particularly useful means by which to study jamming and yield is through observing a two-dimensional amorphous packing of colloids under the application of oscillatory shear. It has been found that these dynamics are quite rich, consisting of reversible, hysteretic particle rearrangements below the yield strain and irreversible rearrangements above the yield strain. We have collaborated with Professor Paulo Arratia’s group in the Department of Mechanical Engineering at University of Pennsylvania to characterize the structural evolution and rearrangement of these systems. We find that the crystalline quality of the local environment surrounding a particle serves as an informative indicator of individual particle rearrangement probability. Furthermore, we find that community detection in a mesoscale network, in which nodes represent system regions rather than particles, and edges reflect the structural similarity between regions as they evolve in time, is an illuminating means of distinguishing multiple separate and unique system responses to shear. Our work shows the utility of coarse-grained networks to represent dynamic, amorphous systems, and provides insight into local structural signatures that give rise to larger-scale rearrangement behavior.

Many networks including granular and particulate networks are spatial in that nodes are located in physical space and links more likely between mutually close nodes. There has been a huge amount of work on diverse kinds of spatial networks, but most mathematical models assume that the spatial distribution of nodes is uniform, or at least has a smooth density. Here we discuss networks with fractal node distributions, where the environment is highly inhomogeneous and has structure at many length scales. Interestingly, the inhomogeneity has greater qualitative and quantitative effects on node structure than the fractality.

Nature is rife with networks that are functionally optimized to propagate inputs in order to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. We investigate computationally the maximum complexity of tuned function that can be achieved in such networks as a function of network size. We find that both flow and mechanical networks display qualitatively similar phase transitions in the complexity of functions that can be tuned. Further, we identify the structural features responsible for function in these tuned networks. Using persistent homology, we show that networks tuned to perform such functions develop characteristic topological features that are similar for different networks that perform the same function, regardless of differences in the local link connectivity. These features correlate strongly with the tuned response, providing a clear topological relationship between structure and function.

I will describe how persistent homology can be used to quantify the dynamics of the interactions of forces between particles in granular systems

Predicting the response of frictional particulate networks to externally imposed forces, torques or structural deformations is of fundamental importance in the study of granular materials. In this talk I will describe network-based procedures to understand such a response for both the stresses as well as the geometry of these systems. Focusing on the shear jamming transition of quasi-statically sheared frictional disks, I will show that the shear jamming transition can be characterized by an abrupt jump in the number of force bearing contacts between particles. This mechanical or stress coordination differs crucially from the often-used geometric coordination, as it increases discontinuously at the transition. This is also accompanied by a diverging timescale that characterizes the time required by the system to attain force balance when subjected to a perturbation. Finally, I will describe procedures to extract such a mechanical coordination using purely geometric information from granular packings.

Many intriguing dynamical properties of complex systems, such as metastability or resistance to applied forces, emerge from the underlying energy landscape. We probe the energy landscape of a granular material by modeling a jammed 2D packing of bidisperse disks as a collection of overlapping circles, defining an energy penalty based on the amount of overlap. We propose a systematic approach to mapping out the transition pathways from energy minimizer to saddle point to minimizer forming a network of transition pathways. Our goal is to relate observable phenomena like a granular material’s rearrangements preceding failure events to dynamics on the network representation of the system.

Disordered networks are used widely to study heterogeneous material failure. These structures are inherent to many systems, such as rigid foams or granular materials. In particular, the latter exhibit highly heterogeneous force chain networks that appear to control the response of such media to external perturbations. To characterize these networks, we focus on their mechanical stability. We study the uniaxial response of networks with geometry derived from the force chains observed in granular experiments. We perform experiments on samples created by laser-cutting these networks from acrylic sheets. We find that the mean degree of the network is a control parameter of the failure behavior, which ranges from ductile to brittle. We develop a test to study the failure locations. We use a measure taken from network-science tools and capable of identifying likely failure locations in the samples. It consists in comparing the relative importance of the beams of the lattice by studying their geodesic edge betweenness centrality.

We analyse climate dynamics from a complex network approach. This leads to an inverse problem: Is there a backbone-like structure underlying the climate system? For this we propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system. This approach enables us to uncover relations to global circulation patterns in oceans and atmosphere. In particular we identify global teleconnection patterns which have a predictive power for strong rainfall events.