Jamming in granular polymers

Strain stiffening due to stretching of entangled particles in random packings of granular materials

Stress-strain relations for random packings of entangling chains under triaxial compression can exhibit strain stiffening and sustain stresses several orders-of-magnitude beyond typical granular materials. X-ray tomography reveals the transition to this strong strain stiffening occurs when chains are long enough to entangle an average of about one chain each, which results in system-filling clusters of entangled chains, similar to the Erdös-Rényi model for randomly connected graphs. The number of entanglements is nearly proportional to the area surrounded by entangling particles with an excluded volume effect, thus the existence of system-filling clusters of entanglements can be predicted assuming random particle positions and orientations with an excluded volume effect if the particle shapes in the packing are known. A tendency was found for chain links to stretch when the packing was strained. This suggests that the strength of these packings comes from stretching of the links of chains, but only when the system-filling network of entanglements provides constraints that prevents failure by shear banding, so that particles must be deformed to move further under strain. The slope of the stress-strain relation of a packing can be calculated from a mean-field model consisting of the product of the effective extensional modulus of the chain, packing fraction, probability of stretched links, and the ratio of strain of stretched links to packing strain. In this model, the increasing slope of the stress-strain curve is mainly due to the fraction of stretched links increasing with strain, and assuming the fraction of stretched links is proportional to strain results in a quadratic prediction for the stress-strain curve. The stress-strain model requires as input measurements of the ratio between local particle deformation and global average strain, and the probability of stretching for nonrigid particles, resulting in a quadratic curvature that agrees with experiments within the run-to-run variation (30%). This model for the stress-strain relation is shown to be generalizable to different shapes of entangling particles by applying it to staples, where the packing strength comes from the bending of staples instead of stretching links. The permanent plastic deformation of staples allows measuring statistical quantities from inspection of a poured-out sample after a triaxial compression, without the need for in situ imaging. Both the probability of staples bending and the average bend angle of the arms were found to increase with strain, and these inputs into the model result in a quadratic curvature of the stress-strain that agrees with experiments within the model uncertainties (37%).

Functional Molecular Granular Materials: Advances and Perspectives

Molecular granular materials (MGMs) are constructed with sub-nanoscale molecular clusters (MCs) as the building units and they have recently been observed to possess enriched functionalities distinct from granular materials of colloid nanoparticles. Herein, the birth and recent research advances in MGMs are summarized with the topics covering the precise synthesis of MC assemblies with target topologies, the hierarchical relaxation dynamics and tuneable viscoelasticity, impact-resistant capacity, and proton conductivity performance. The extremely small size of MC renders them two features: bulk diffusive dynamics with energy scale close to thermal fluctuation energy and the dominant volume fraction of surface structures. This finally leads to the hierarchical relaxation dynamics and broadly tuneable viscoelasticity of MGMs although the structural units are with small sizes and low M_w . Therefore, MGMs have been applied as impact resistant materials and proton conductors for the highly tuneable relaxation dynamics.

Directional clogging and phase separation for disk flow through periodic and diluted obstacle arrays

We model collective disk flow though a square array of obstacles as the flow direction is changed relative to the symmetry directions of the array. At lower disk densities there is no clogging for any driving direction, but as the disk density increases, the average disk velocity decreases and develops a drive angle dependence. For certain driving angles, the flow is reduced or drops to zero when the system forms a heterogeneous clogged state consisting of high density clogged regions coexisting with empty regions. The clogged states are fragile and can be unclogged by changing the driving angle. For large obstacle sizes, we find a uniform clogged state that is distinct from the collective clogging regime. Within the clogged phases, depinning transitions can occur as a function of increasing driving force, with intermittent motion appearing just above the depinning threshold. The clogging is robust against the random removal or dilution of the obstacle sites, and the disks are able to form system-spanning clogged clusters even under increasing dilution. If the dilution becomes too large, however, the clogging behavior is lost.

Two-stage athermal solidification of semiflexible polymers and fibers

We study how solidification of model freely rotating polymers under athermal quasistatic compression varies with their bond angle θ0. All systems undergo two discrete, first-order-like transitions: entanglement at φ = φE(θ0) followed by jamming at φ = φJ(θ0) ≃ (4/3 ± 1/12)φE(θ0). For φ < φE(θ0), systems are in a "gas" phase wherein all chains remain free to translate and reorient. For φE(θ0) ≤ φ ≤ φJ(θ0), systems are in a liquid-like phase wherein chains are entangled. In this phase, chains' rigid-body-like motion is blocked, yet they can still locally relax via dihedral rotations, and hence energy and pressure remain extremely small. The ability of dihedral relaxation mechanisms to accommodate further compression becomes exhausted, and systems rigidify, at φJ(θ0). At and slightly above φJ, the bulk moduli increase linearly with the pressure P rather than jumping discontinuously, indicating these systems solidify via rigidity percolation. The character of the energy and pressure increases above φJ(θ0) can be characterized via chains' effective aspect ratio αeff. Large-αeff (small-θ0) systems' jamming is bending-dominated and is similar to that observed in systems composed of straight fibers. Small-αeff (large-θ0) systems' jamming is dominated by the degree to which individual chains' dihedrals can collapse into compact, tetrahedron-like structures. For intermediate θ0, chains remain in highly disordered globule-like configurations throughout the compression process; jamming occurs when entangled globules can no longer even locally relax away from one another.

Effects of spherical confinement and backbone stiffness on flexible polymer jamming

We use molecular simulations to study jamming of a crumpled bead-spring model polymer in a finite container and compare to jamming of repulsive spheres. After proper constraint counting, the onset of rigidity is seen to occur isostatically as in the case of repulsive spheres. Despite this commonality, the presence of the curved container wall and polymer backbone bonds introduce new mechanical properties. Notably, these include additional bands in the vibrational density of states that reflect the material structure as well as oscillations in local contact number and density near the wall but with lower amplitude for polymers. Polymers have fewer boundary contacts, and this low-density surface layer strongly reduces the global bulk modulus. We further show that bulk-modulus dependence on backbone stiffness can be described by a model of stiffnesses in series and discuss potential experimental and biological applications.

Crossover from Jamming to Clogging Behaviours in Heterogeneous Environments

Jamming describes a transition from a flowing or liquid state to a solid or rigid state in a loose assembly of particles such as grains or bubbles. In contrast, clogging describes the ceasing of the flow of particulate matter through a bottleneck. It is not clear how to distinguish jamming from clogging, nor is it known whether they are distinct phenomena or fundamentally the same. We examine an assembly of disks moving through a random obstacle array and identify a transition from clogging to jamming behavior as the disk density increases. The clogging transition has characteristics of an absorbing phase transition, with the disks evolving into a heterogeneous phase-separated clogged state after a critical diverging transient time. In contrast, jamming is a rapid process in which the disks form a homogeneous motionless packing, with a rigidity length scale that diverges as the jamming density is approached.

Nonmonotonic dependence of polymer-glass mechanical response on chain bending stiffness

We investigate the mechanical properties of amorphous polymers by means of coarse-grained simulations and nonaffine lattice dynamics theory. A small increase of polymer chain bending stiffness leads first to softening of the material, while hardening happens only upon further strengthening of the backbones. This nonmonotonic variation of the storage modulus G^{'} with bending stiffness is caused by a competition between additional resistance to deformation offered by stiffer backbones and decreased density of the material due to a necessary decrease in monomer-monomer coordination. This counterintuitive finding suggests that the strength of polymer glasses may in some circumstances be enhanced by softening the bending of constituent chains.

Spontaneous symmetry breaking induced unidirectional rotation of a chain-grafted colloidal particle in the active bath

Exploiting the energy of randomly moving active agents such as bacteria is a fascinating way to power a microdevice. Here we show, by simulations, that a chain-grafted disk-like colloidal particle can rotate unidirectionally and hence output work when immersed in a thin film of active particle suspension. The collective spontaneous symmetry breaking of chain configurations is the origin of the unidirectional rotation. Long persistence time, large propelling force and/or small rotating friction are keys to sustaining the collective broken symmetry and realizing the rotation. In the rotating state, we find very simple linear relations, e.g. between the mean angular speed and the propelling force. The time-evolving asymmetry of chain configurations reveals that there are two types of non-rotating state. The basic phenomena are also observed in the macroscopic granular experiments, implying the generic nature of these phenomena. Our findings provide new insights into the collective spontaneous symmetry breaking in active systems with flexible objects and also open the way to conceive new soft/deformable microdevices.

Jamming of Semiflexible Polymers

We study jamming in model freely rotating polymers as a function of chain length N and bond angle θ_{0}. The volume fraction at jamming ϕ_{J}(θ_{0}) is minimal for rigid-rodlike chains (θ_{0}=0), and increases monotonically with increasing θ_{0}≤π/2. In contrast to flexible polymers, marginally jammed states of freely rotating polymers are highly hypostatic, even when bond and angle constraints are accounted for. Large-aspect-ratio (small θ_{0}) chains behave comparably to stiff fibers: resistance to large-scale bending plays a major role in their jamming phenomenology. Low-aspect-ratio (large θ_{0}) chains behave more like flexible polymers, but still jam at much lower densities due to the presence of frozen-in three-body correlations corresponding to the fixed bond angles. Long-chain systems jam at lower ϕ and are more hypostatic at jamming than short-chain systems. Implications of these findings for polymer solidification are discussed.

Optimizing packing fraction in granular media composed of overlapping spheres

What particle shape will generate the highest packing fraction when randomly poured into a container? In order to explore and navigate the enormous search space efficiently, we pair molecular dynamics simulations with artificial evolution. Arbitrary particle shape is represented by a set of overlapping spheres of varying diameter, enabling us to approximate smooth surfaces with a resolution proportional to the number of spheres included. We discover a family of planar triangular particles, whose packing fraction of ϕ ∼ 0.73 is among the highest experimental results for disordered packings of frictionless particles. We investigate how ϕ depends on the arrangement of spheres comprising an individual particle and on the smoothness of the surface. We validate the simulations with experiments using 3D-printed copies of the simplest member of the family, a planar particle consisting of three overlapping spheres with identical radius. Direct experimental comparison with 3D-printed aspherical ellipsoids demonstrates that the triangular particles pack exceedingly well not only in the limit of large system size but also when confined to small containers.

Celebrating Soft Matter's 10th Anniversary: toward jamming by design

In materials science, high performance is typically associated with regularity and order, while disorder and the presence of defects are assumed to lead to sub-optimal outcomes. This holds for traditional solids such as crystals as well as for many types of nanoscale devices. However, there are circumstances where disorder can be harnessed to achieve performance not possible with approaches based on regularity. Recent research has shown opportunities specifically for soft matter. There, the phenomenon of jamming leads to unique emergent behavior that enables disordered, amorphous systems to switch reversibly between solid-like rigidity and fluid-like plasticity. This makes it possible to envision materials that can change stiffness or even shape adaptively. We review some of the progress in this direction, discussing examples where jamming has been explored from micro to macro scales in colloidal systems, suspensions, granular-materials-enabled soft robotics, and architecture. We focus in particular on how the jammed aggregate state can be tailored by controlling particle level properties and discuss very recent ideas that provide an important first step toward actual design of specifically targeted jamming behavior.

Aspects of jamming in two-dimensional athermal frictionless systems

In this work we provide an overview of jamming transitions in two dimensional systems focusing on the limit of frictionless particle interactions in the absence of thermal fluctuations. We first discuss jamming in systems with short range repulsive interactions, where the onset of jamming occurs at a critical packing density and where certain quantities show a divergence indicative of critical behavior. We describe how aspects of the dynamics change as the jamming density is approached and how these dynamics can be explored using externally driven probes. Different particle shapes can produce jamming densities much lower than those observed for disk-shaped particles, and we show how jamming exhibits fragility for some shapes while for other shapes this is absent. Next we describe the effects of long range interactions and jamming behavior in systems such as charged colloids, vortices in type-II superconductors, and dislocations. We consider the effect of adding obstacles to frictionless jamming systems and discuss connections between this type of jamming and systems that exhibit depinning transitions. Finally, we discuss open questions such as whether the jamming transition in all these different systems can be described by the same or a small subset of universal behaviors, as well as future directions for studies of jamming transitions in two dimensional systems, such as jamming in self-driven or active matter systems.

Simple model for chain packing and crystallization of soft colloidal polymers

We study a simple bead-spring polymer model exhibiting competing crystallization and glass transitions. Constant-pressure molecular dynamics simulations are employed to study phase behavior and morphological order. For adequately slow quench rates, chain systems exhibit a first-order phase transition (crystallization) below a critical temperature T=T(cryst). We observe the formation of close-packed crystallites of FCC and/or HCP order, separated by domain walls, twin defects, and amorphous regions. Such crystal structures closely resemble the corresponding ordered morphologies of athermal polymer packings: fully flexible chains retain random-walk-like configurations in the crystalline state and do not form lamellae, while semiflexible chains do form lamellae. The model presented here is well suited to the modeling of granular and colloidal polymers, in particular for elucidating the factors that dictate the formation of specific ordered morphologies.

Segregation in mixtures of granular chains and spherical grains under vertical vibration

We experimentally investigate segregation behaviors of binary granular mixtures consisting of granular chains and spherical grains with different interstitial media under vertical vibrations. A quantitative criterion is proposed to locate the boundaries between different vibrating phases. The water-immersed granular mixture exhibits two interesting types of segregation behaviors: chain-on-top and sandwich patterns. However, the phenomenon of sandwich segregation is absent for the air-immersed mixture. The topological differences of phase diagrams between two different environments indicate that the interstitial fluid plays an important role on the granular demixing. Additionally, the phase behaviors of mixtures for the different chain lengths show a not significant discrepancy. Finally, the vibrating thickness ratio determining the phase boundary characterizes the mixing extent of the granular bed. The estimated ratios for various chain lengths exhibit a monotonically decreasing dependence, when the vibration frequency increases.

Rheology and shear band suppression in particle and chain mixtures

Using numerical simulations we consider an amorphous particle mixture which exhibits shear localization, and find that the addition of even a small fraction of chains strongly enhances the material strength, creating pronounced overshoot features in the stress-strain curves. The strengthening occurs in the case where the chains are initially perpendicular to the shear direction, leading to a suppression of the shear band. This also leads to stiffening effects that are typical of polymeric systems. For large strain, the chains migrate to the region where a shear band forms, resulting in a stress drop. For chains larger than the linear system size we find oscillatory behavior, which does not resemble polymeric systems since the second stress peak is larger than the first. Our results are also useful for providing insights into methods of controlling and strengthening granular materials against failure.

Spontaneous crystallization in athermal polymer packings

We review recent results from extensive simulations of the crystallization of athermal polymer packings. It is shown that above a certain packing density, and for sufficiently long simulations, all random assemblies of freely-jointed chains of tangent hard spheres of uniform size show a spontaneous transition into a crystalline phase. These polymer crystals adopt predominantly random hexagonal close packed morphologies. An analysis of the local environment around monomers based on the shape and size of the Voronoi polyhedra clearly shows that Voronoi cells become more spherical and more symmetric as the system transits to the ordered state. The change in the local environment leads to an increase in the monomer translational contribution to the entropy of the system, which acts as the driving force for the phase transition. A comparison of the crystallization of hard-sphere polymers and monomers highlights similarities and differences resulting from the constraints imposed by chain connectivity.

Calculations of the structure of basin volumes for mechanically stable packings

Experimental and computational model systems composed of frictionless particles in a fixed geometry have a finite number of distinct mechanically stable (MS) packings. The frequency of occurrence for each MS packing is highly variable and depends strongly on preparation protocol. Despite intense work, it is extremely difficult to predict a priori the MS packing probabilities. We describe a novel computational method for calculating the volume and other geometrical properties of the "basin of attraction" for each MS packing. The basin of attraction for an MS packing contains all initial conditions in configuration space that map to that MS packing using a given preparation protocol. We find that the basin is a highly complex structure. For a compressive-quench-from-zero-density protocol, we show the existence of a small core volume of the basin around each MS packing for which all points map to that MS packing. However, in contrast to previous studies for supercooled liquids, glasses, and over-compressed jammed systems, we find that the MS packing probabilities are very weakly correlated with this core volume. Instead, MS packing probabilities obtained from compression protocols that use initially dilute configurations and do not allow particle overlaps (i.e., those relevant to granular media) are determined by complex geometric features of the basin of attraction that are distant from the MS packing. In particular, we find that the shape of the average basin profile function S(l), which gives the probability for a point on a hyperspherical shell a distance l from a given MS packing to map back to that packing, can be described by a Γ distribution with a peak that increases as the system size increases and as the quench rate decreases. We find a simple model which predicts S(l) for the extreme cases of very slow and fast quench rates.

Entangled granular media

We study the geometrically induced cohesion of ensembles of granular "u particles" that mechanically entangle through particle interpenetration. We vary the length-to-width ratio l/w of the u particles and form them into freestanding vertical columns. In a laboratory experiment, we monitor the response of the columns to sinusoidal vibration (with peak acceleration Γ). Column collapse occurs in a characteristic time τ which follows the relation τ∝exp(Γ/Δ). Δ resembles an activation energy and is maximal at intermediate l/w. A simulation reveals that optimal strength results from competition between packing and entanglement.

Strain stiffening in random packings of entangled granular chains

Random packings of granular chains are presented as a model system to investigate the contribution of entanglements to strain stiffening. The chain packings are sheared in uniaxial compression experiments. For short chain lengths, these packings yield when the shear stress exceeds the scale of the confining pressure, similar to granular packings of unconnected particles. In contrast, packings of chains which are long enough to form loops exhibit strain stiffening, in which the effective stiffness of the material increases with strain, similar to many polymer materials. The latter packings can sustain stresses orders-of-magnitude greater than the confining pressure, and do not yield until the chain links break. X-ray tomography measurements reveal that the strain-stiffening packings contain system-spanning clusters of entangled chains.