We revisit the concept of minimal rigidity as applied to soft repulsive, frictionless sphere packings in two-dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman's theorem in two-dimensions. It constrains the global, average coordination number of the graph, for example. However, minimal rigidity does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local moves via the Henneberg construction such that these graphs fall under the jurisdiction of correlated percolation. We then probe how jamming graphs destabilize, or become unjammed, by deleting a bond and computing the resulting rigid cluster distribution. We also study how the system restabilizes with the addition of new contacts and how a jamming graph with extra/redundant contacts destabilizes. The latter endeavor allows us to probe a disc packing in the rigid phase and uncover a potentially new diverging lengthscale associated with the random deletion of contacts as compared to the study of cut-out (or frozen in) subsystems.
The dynamical properties of a dense horizontally vibrated bidisperse granular monolayer are experimentally investigated. The quench protocol produces states with a frozen structure of the assembly, but the remaining degrees of freedom associated with contact dynamics control the appearance of macroscopic rigidity. We provide decisive experimental evidence that this transition is a critical phenomenon, with increasingly collective and heterogeneous rearrangements occurring at length scales much smaller than the grains' diameter, presumably reflecting the contact force network fluctuations. Dyna…
We investigate the rigidity transition associated with shear jamming in frictionless, as well as frictional, disk packings in the quasi-static regime and at low shear rates. For frictionless disks, the transition is under quasistatic shear is discontinuous, with an instantaneous emergence of a system spanning rigid cluster at the jamming transition. For frictional systems, the transition appears continuous for finite shear rates, but becomes sharper for lower shear rates. In the quasi-static limit, it is discontinuous as in the frictionless case. Thus, our results show that the rigidity transi…
We study the nature of the frictional jamming transition within the framework of rigidity percolation theory. Slowly sheared frictional packings are decomposed into rigid clusters and floppy regions with a generalization of the pebble game including frictional contacts. We discover a second-order transition controlled by the emergence of a system-spanning rigid cluster accompanied by a critical cluster size distribution. Rigid clusters also correlate with common measures of rigidity. We contrast this result with frictionless jamming, where the rigid cluster size distribution is noncritical.
We discuss a microscopic scheme to compute the rigidity of glasses or the plateau modulus of supercooled liquids by twisting replicated liquids. We first summarize the method in the case of harmonic glasses with analytic potentials. Then we discuss how it can be extended to the case of repulsive contact systems : the hard sphere glass and related systems with repulsive contact potentials which enable the jamming transition at zero temperature. For the repulsive contact systems we find entropic rigidity which behaves similarly as the pressure in the low temperature limit: it is proportional to …
We study the effect of thermalization on the rigidity of a randomly packed soft repulsive sphere system around the jamming point by analyzing the shear-modulus using the cloned liquid theory with the 1 step replica symmetry breaking ansatz and molecular dynamics simulations. Contrarily to the usual harmonic picture for solids, we found that the thermalized jamming system is anomalously softer than at zero temperature such that the shear-modulus becomes as small as the pressure down to vanishingly low temperatures.
The jamming of bi-disperse soft core disks is considered, using a variety of different protocols to produce the jammed state. In agreement with other works, we find that cooling and compression can lead to a broad range of jamming packing fractions $ϕ_J$, depending on cooling rate and initial configuration; the larger the degree of big particle clustering in the initial configuration, the larger will be the value of $ϕ_J$. In contrast, we find that shearing disrupts particle clustering, leading to a much narrower range of $ϕ_J$ as the shear strain rate varies. In the limit of vanishingly small…
Recent studies of athermal systems such as dry grains and dense, non-Brownian suspensions have shown that shear can lead to solidification through the process of shear jamming in grains and discontinuous shear thickening in suspensions. The similarities observed between these two distinct phenomena suggest that the physical processes leading to shear-induced rigidity in athermal materials are universal. We present a non-equilibrium statistical mechanics model, which exhibits the phenomenology of these shear-driven transitions: shear jamming and discontinuous shear thickening in different regio…
The jamming transition characterizes athermal systems of particles interacting via finite range repulsive potentials, and occurs on increasing the density when particles cannot avoid making contacts with those of their first coordination shell. We have recently shown [M. Pica Ciamarra and P. Sollich, arXiv:1209.3334] that the same systems are also characterized by a series of jamming crossovers. These occur at higher volume fractions as particles are forced to make contact with those of subsequent coordination shells. At finite temperature, the crossovers give rise to dynamic and thermodynamic…
Solids are distinguished from fluids by their ability to resist shear. In traditional solids, the resistance to shear is associated with the emergence of broken translational symmetry as exhibited by a non-uniform density pattern, which results from either minimizing the energy cost or maximizing the entropy or both. In this work, we focus on a class of systems, where this paradigm is challenged. We show that shear-driven jamming in dry granular materials is a collective process controlled solely by the constraints of mechanical equilibrium. We argue that these constraints lead to a broken tra…
By analyzing the displacement statistics of an assembly of horizontally vibrated bidisperse frictional grains in the vicinity of the jamming transition experimentally studied before, we establish that their superdiffusive motion is a genuine Levy flight, but with `jump' size very small compared to the diameter of the grains. The vibration induces a broad distribution of jumps that are random in time, but correlated in space, and that can be interpreted as micro-crack events at all scales. As the volume fraction departs from the critical jamming density, this distribution is truncated at a smal…
Growth in confined spaces can drive cellular populations through a jamming transition from a fluid-like state to a solid-like state. Experiments have found that jammed budding yeast populations can build up extreme compressive pressures (over 1MPa), which in turn feed back onto cellular physiology by slowing or even stalling cell growth. Using extensive numerical simulations, we investigate how this feedback impacts the mechanical properties of model jammed cellular populations. We find that feedback directs growth toward poorly-coordinated regions, resulting in an excess number of cell-cell c…
We experimentally study the vicinity of the Jamming transition by investigating the statics and the dynamics of the contact network of an horizontally shaken bi-disperse packing of photo-elastic discs. Compressing the packing very slowly, while maintaining a mechanical excitation, yields a granular glass, namely a frozen structure of vibrating grains. In this glass phase, we observe a remarkable dynamics of the contact network, which exhibits strong dynamical heterogeneities. Such heterogeneities are maximum at a packing fraction $ϕ^*$, \emph{distinct} and smaller than the jamming packing frac…
We numerically study the jamming transition in particulate systems with attraction by investigating their mechanical response at zero temperature. We find three regimes of mechanical behavior separated by two critical transitions--connectivity and rigidity percolation. The transitions belong to different universality classes than their lattice counterparts, due to force balance constraints. We also find that these transitions are unchanged at low temperatures and resemble gelation transitions in experiments on colloidal and silica gels.
A jammed packing of frictionless spheres at zero temperature is perfectly specified by the network of contact forces from which mechanical properties can be derived. However, we can alternatively consider a packing as a geometric structure, characterized by a Voronoi tessellation which encodes the local environment around each particle. We find that this local environment characterizes systems both above and below jamming and changes markedly at the transition. A variety of order parameters derived from this tessellation carry signatures of the jamming transition, complete with scaling exponen…
Jamming is a fundamental transition that governs the mechanical behavior of particulate media, including sand, foam and dense suspensions but also biological tissues: Upon compression, particulate media can change from freely flowing to a disordered solid. Jamming has previously been conceived as a bulk phenomenon involving particle motions in fixed geometries. In a diverse class of soft materials, however, solidification can take place in a deformable geometry, such as on the surface of a fluid droplet or in the formation of a bijel. In these systems the nature and dynamics of jamming remains…
We comprehensively study mechanical and vibrational properties of dimer packings in three-dimensional space with particular attention on critical scaling behaviors near the jamming transition. First, we confirm the dependence of the packing fraction at the transition on the aspect ratio, the isostatic contact number at the transition, and the scaling dependence of the excess contact number on the excess density. Second, we study the elastic moduli, bulk and shear moduli, and establish power-law scaling of them. Finally, we study the vibrational density of states and its characteristic frequenc…
Aqueous foams are an important model system that displays coarsening dynamics. Coarsening in dispersions and foams is well understood in the dilute and dry limits, where the gas fraction tends to zero and one, respectively. However, foams are known to undergo a jamming transition from a fluid-like to a solid-like state at an intermediate gas fraction,$ϕ_c$. Much less is known about coarsening dynamics in wet foams near jamming, and the link to mechanical response, if any, remains poorly understood. Here, we probe coarsening and mechanical response using numerical simulations of a variant of th…
In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, $P(s)$, is continuous and shows a power-law decay. For systems with short-range attractions, however, $P(s)$ appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing ``discontinuity'' does not mean that the transition is discontinuous. In fact, it signifies the coexis…
We numerically simulate mechanically stable packings of soft-core, frictionless, bidisperse disks in two dimensions, above the jamming packing fraction $ϕ_J$. For configurations with a fixed isotropic global stress tensor, we compute the averages, variances, and correlations of conserved quantities (stress $Γ_{\cal C}$, force-tile area $A_{\cal C}$, Voronoi volume $V_{\cal C}$, number of particles $N_{\cal C}$, and number of small particles $N_{s{\cal C}}$) on compact subclusters of particles ${\cal C}$, as a function of the cluster size and the global system stress. We find several significan…
The suppression of density fluctuations at different length scales is the hallmark of hyperuniformity. However, its existence and significance in jammed solids is still a matter of debate. We explore the presence of this hidden order in a manybody interacting model known to exhibit a rigidity transition, and find that in contrary to exisiting speculations, density fluctuations in the rigid phase are only suppressed up to a finite lengthscale. This length scale grows and diverges at the critical point of the rigidity transition, such that the system is hyperuniform in the fluid phase. This sugg…
Granular matter at the jamming transition is poised on the brink of mechanical stability, and hence it is possible that these random systems have topologically protected surface phonons. Studying two model systems for jammed matter, we find states that exhibit distinct mechanical topological classes, protected surface modes, and ubiquitous Weyl points. The detailed statistics of the boundary modes enable tests of a standard understanding of the detailed features of the jamming transition, and show that parts of this argument are invalid.
Jamming is a geometric phase transition occurring in dense particle systems in the absence of temperature. We use computer simulations to analyse the effect of thermal fluctuations on several signatures of the transition. We show that scaling laws for bulk and shear moduli only become relevant when thermal fluctuations are extremely small, and propose their relative ratio as a quantitative signature of jamming criticality. Despite the nonequilibrium nature of the transition, we find that thermally induced fluctuations and mechanical responses obey equilibrium fluctuation-dissipation relations …
Mechanically stable sphere packings are generated in three-dimensional space using the discrete element method, which span a wide range in structural order, ranging from fully amorphous to quasi-ordered structures, as characterized by the bond orientational order parameter. As the packing pressure, $p$, varies from the marginally rigid limit at the jamming transition ($p \approx 0$) to that of more robust systems ($p \gg 0$), the coordination number, $z$, follows a familiar scaling relation with pressure, namely, $Δz = z - z_c \sim p^{1/2}$, where $z_c = 2d = 6$ ($d=3$ is the spatial dimension…
We investigate experimentally the mechanical response of a monolayer of bi-disperse frictional grains to an inhomogeneous shear perturbation across the jamming transition. We inflate an intruder inside the packing and use photo-elasticity and tracking techniques to measure the induced shear strain and stresses at the grain scale. We quantify experimentally the constitutive relations for strain amplitudes as low as 0.001 and for a range of packing fractions within 2% variation around the jamming transition. At the transition strong nonlinear effects set in : both the shear modulus and the dilat…
We introduce two new concepts, frictional rigidity percolation and minimal rigidity proliferation, to help identify the nature of the frictional jamming transition as well as significantly broaden the scope of rigidity percolation. For frictional rigidity percolation, we construct rigid clusters in two different lattice models using a $(3,3)$ pebble game, while taking into account contacts below and at the Coulomb threshold. The first lattice is a honeycomb lattice with next-nearest neighbors, the second, a hierarchical lattice. For both, we generally find a continuous rigidity transition. Our…
Rigidity regulates the integrity and function of many physical and biological systems. This is the first of two papers on the origin of rigidity, wherein we propose that "energetic rigidity," in which all non-trivial deformations raise the energy of a structure, is a more useful notion of rigidity in practice than two more commonly used rigidity tests: Maxwell-Calladine constraint counting (first-order rigidity) and second-order rigidity. We find that constraint counting robustly predicts energetic rigidity only when the system has no states of self stress. When the system has states of self s…
In 2005, Wyart et al. (Europhys. Lett., 72 (2005) 486) showed that the low frequency vibrational properties of jammed amorphous sphere packings can be understood in terms of a length scale, called l*, that diverges as the system becomes marginally unstable. Despite the tremendous success of this theory, it has been difficult to connect the counting argument that defines l* to other length scales that diverge near the jamming transition. We present an alternate derivation of l* based on the onset of rigidity. This phenomenological approach reveals the physical mechanism underlying the length sc…
In this paper, we present a theory to efficiently deal with mechanical properties of heterogeneous polymer chain in free space and the central problem is to evaluate the diffusion equation and orientation-orientation correlation function, under the condition of varying persistence length (a measure of bending rigidity) along the chain contour. Additionally, we give the specifically experimental method to measure the variable persistence length to examine our theory. In order to verify the theoretical predictions, we also performed large numbers of Brownian dynamics simulations based on the Gen…
Athermal packings of soft repulsive spheres exhibit a sharp jamming transition in the thermodynamic limit. Upon further compression, various structural and mechanical properties display clean power-law behavior over many decades in pressure. As with any phase transition, the rounding of such behavior in finite systems close to the transition plays an important role in understanding the nature of the transition itself. The situation for jamming is surprisingly rich: the assumption that jammed packings are isotropic is only strictly true in the large-size limit, and finite-size has a profound ef…
We study the influence of cell-level mechanical heterogeneity in epithelial tissues using a vertex-based model. Heterogeneity in single cell stiffness is introduced as a quenched random variable in the preferred shape index($p_0$) for each cell. We uncovered a crossover scaling for the tissue shear modulus, suggesting that tissue collective rigidity is controlled by a single parameter $f_r$, which accounts for the fraction of rigid cells. Interestingly, the rigidity onset occurs at $f_r=0.21$, far below the contact percolation threshold of rigid cells. Due to the separation of rigidity and con…
Recent progress on studies of the nanoscale mechanical responses in disordered systems has highlighted a strong degree of heterogeneity in the elastic moduli. In this contribution, using computer simulations, we study the elastic heterogeneities in athermal amorphous solids, composed of isotropic, static, sphere packings, near the jamming transition. We employ techniques, based on linear response methods, that are amenable to experimentation. We find that the local elastic moduli are randomly distributed in space and are described by Gaussian probability distributions, thereby lacking any sign…
The origin of rigidity in disordered materials is an outstanding open problem in statistical physics. Previously, a class of 2D cellular models has been shown to undergo a rigidity transition controlled by a mechanical parameter that specifies cell shapes. Here, we generalize this model to 3D and find a rigidity transition that is similarly controlled by the preferred surface area: the model is solid-like below a dimensionless surface area of $s_0^\ast\approx5.413$, and fluid-like above this value. We demonstrate that, unlike jamming in soft spheres, residual stresses are necessary to create r…
We show how rigidity emerges in experiments of sheared frictional granular materials by using generalizations of two methods for identifying rigid structures. Both approaches, the force-based dynamical matrix and the topology-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a contact number of $z=2.4\pm 0.1$, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a sponge-like morphology. We also find that the pressure within rigid structures always exceeds the pressur…
Crossover motifs are integral components for designing DNA based nanostructures and nanomechanical devices due to their enhanced rigidity compared to the normal B-DNA. Although the structural rigidity of the double helix B-DNA has been investigated extensively using both experimental and theoretical tools, to date there is no quantitative information about structural rigidity and the mechanical strength of parallel crossover DNA motifs. We have used fully atomistic molecular dynamics simulations in explicit solvent to get the force-extension curve of parallel DNA nanostructures to characterize…
Mechanotransduction, the biological response to mechanical stress, is often initiated by the activation of mechanosensitive (MS) proteins upon mechanically induced deformations of the cell membrane. A current challenge to fully understand this process is to predict how lipid bilayers deform upon application of mechanical stress. In this context, it is now well established that anionic lipids influence the function of many proteins. Here, we test the hypothesize that anionic lipids could indirectly modulate MS proteins by alteration of the lipid bilayer mechanical properties. Using all-atom mol…
This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus $B=0$ and shear modulus $G>0$, with either a generalized untwisted kagome lattice with $B>0$ and $G>0$ or with a honeycomb lattice with $B>0$ and $G=0$. These lattices exhibit jamming-like critical end-points at which $B$, $G$, or both $B$ and $G$ jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of …
Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of $k$ choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally…
We investigate morphologies of proliferating cellular tissues using a newly developed numerical simulation model for mechanical cell division and migration in 2D. The model is applied to a bimodal mixture consisting of stiff cells with a low growth potential and soft cells with a high growth potential; cancer cells are typically considered to be softer than healthy cells. In an even mixture, the soft cells develop into a tissue matrix and the stiff cells into a dendrite-like network structure. When soft cells are placed inside a tissue consisting of stiff cells (to model cancer growth), the so…
Tissue mechanical properties such as rigidity and fluidity, and changes in these properties driven by jamming-unjamming transitions (UJT), have come under recent highlight as mechanical markers of health and disease in various biological processes including cancer. However, most analysis of these mechanical properties and UJT have sidestepped the effect of cellular death and division in these systems. Cellular apoptosis (programmed cell death) and mitosis (cell division) can drive significant changes in tissue properties. The balance between the two is crucial in maintaining tissue function, a…
We study theoretically and numerically the microscopic cause of the mechanical stability of hard sphere glasses near their maximum packing. We show that, after coarse-graining over time, the hard sphere interaction can be described by an effective potential which is exactly logarithmic at the random close packing $ϕ_c$. This allows to define normal modes, and to apply recent results valid for elastic networks: mechanical stability is a non-local property of the packing geometry, and is characterized by some length scale $l^*$ which diverges at $ϕ_c$ [1, 2]. We compute the scaling of the bulk a…
The rigid to floppy transitions and the associated intermediate phase in glasses are studied in the case where the local structure is not fully determined from the macroscopic concentration. The approach uses size increasing cluster approximations and constraint counting algorithms. It is shown that the location and the width of the intermediate phase and the corresponding structural, mechanical and energetical properties of the network depend crucially on the way local structures are selected at a given concentration. The broadening of the intermediate phase is obtained for networks combining…
A statistical mechanical distance constraint model (DCM) is presented that explicitly accounts for network rigidity among constraints present within a system. Constraints are characterized by local microscopic free energy functions. Topological re-arrangements of thermally fluctuating constraints are permitted. The partition function is obtained by combining microscopic free energies of individual constraints using network rigidity as an underlying long-range mechanical interaction -- giving a quantitative explanation for the non-additivity in component entropies exhibited in molecular systems…
