Local structure controls the nonaffine shear and bulk moduli of disordered solids

Effects of local incompressibility on the rheology of composite biopolymer networks

Fibrous networks such as collagen are common in biological systems. Recent theoretical and experimental efforts have shed light on the mechanics of single component networks. Most real biopolymer networks, however, are composites made of elements with different rigidity. For instance, the extracellular matrix in mammalian tissues consists of stiff collagen fibers in a background matrix of flexible polymers such as hyaluronic acid (HA). The interplay between different biopolymer components in such composite networks remains unclear. In this work, we use 2D coarse-grained models to study the nonlinear strain-stiffening behavior of composites. We introduce a local volume constraint to model the incompressibility of HA. We also perform rheology experiments on composites of collagen with HA. Theoretically and experimentally, we demonstrate that the linear shear modulus of composite networks can be increased by approximately an order of magnitude above the corresponding moduli of the pure components. Our model shows that this synergistic effect can be understood in terms of the local incompressibility of HA, which acts to suppress density fluctuations of the collagen matrix with which it is entangled.

Jamming on convex deformable surfaces

Jamming is a fundamental transition that governs the behavior of particulate media, including sand, foams and dense suspensions. Upon compression, such media change from freely flowing to a disordered, marginally stable solid that exhibits non-Hookean elasticity. While the jamming process is well established for fixed geometries, the nature and dynamics of jamming for a diverse class of soft materials and deformable substrates, including emulsions and biological matter, remains unknown. Here we propose a new scenario, metric jamming, where rigidification occurs on a surface that has been deformed from its ground state. Unlike classical jamming processes that exhibit discrete mechanical transitions, surprisingly we find that metric jammed states possess mechanical properties continuously tunable between those of classically jammed and conventional elastic media. The compact and curved geometry significantly alters the vibrational spectra of the structures relative to jamming in flat Euclidean space, and metric jammed systems also possess new types of vibrational mode that couple particle and shape degrees of freedom. Our work provides a theoretical framework that unifies our understanding of solidification processes that take place on deformable media and lays the groundwork to exploit jamming for the control and stabilization of shape in self-assembly processes.

Designing the pressure-dependent shear modulus using tessellated granular metamaterials

Jammed packings of granular materials display complex mechanical response. For example, the ensemble-averaged shear modulus 〈G〉 increases as a power law in pressure p for static packings of soft spherical particles that can rearrange during compression. We seek to design granular materials with shear moduli that can either increase or decrease with pressure without particle rearrangements even in the large-system limit. To do this, we construct tessellated granular metamaterials by joining multiple particle-filled cells together. We focus on cells that contain a small number of bidisperse disks in two dimensions. We first study the mechanical properties of individual disk-filled cells with three types of boundaries: periodic boundary conditions (PBC), fixed-length walls (FXW), and flexible walls (FLW). Hypostatic jammed packings are found for cells with FLW, but not in cells with PBC and FXW, and they are stabilized by quartic modes of the dynamical matrix. The shear modulus of a single cell depends linearly on p. We find that the slope of the shear modulus with pressure λ_{c}<0 for all packings in single cells with PBC where the number of particles per cell N≥6. In contrast, single cells with FXW and FLW can possess λ_{c}>0, as well as λ_{c}<0, for N≤16. We show that we can force the mechanical properties of multicell granular metamaterials to possess those of single cells by constraining the end points of the outer walls and enforcing an affine shear response. These studies demonstrate that tessellated granular metamaterials provide a platform for the design of soft materials with specified mechanical properties.

Universal scaling for disordered viscoelastic matter near the onset of rigidity

The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.

Local symmetry predictors of mechanical stability in glasses

The mechanical properties of crystals are controlled by the translational symmetry of their structures. But for glasses with a disordered structure, the link between the symmetry of local particle arrangements and stability is not well established. In this contribution, we provide experimental verification that the centrosymmetry of nearest-neighbor polyhedra in a glass strongly correlates with the local mechanical stability. We examine the distribution of local stability and local centrosymmetry in a glass during aging and deformation using microbeam x-ray scattering. These measurements reveal the underlying relationship between particle-level structure and larger-scale behavior and demonstrate that spatially connected, coordinated local transformations to lower symmetry structures are fundamental to these phenomena. While glassy structures lack obvious global symmetry breaking, local structural symmetry is a critical factor in predicting stability.

Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory

We combine hydrodynamic and field theoretic methods to develop a general theory of phonons as Goldstone bosons in crystals, glasses, and liquids based on nonaffine displacements and the consequent Goldstone phase relaxation. We relate the conservation, or lack thereof, of specific higher-form currents with properties of the underlying deformation field-nonaffinity-which dictates how molecules move under an applied stress or deformation. In particular, the single-valuedness of the deformation field is associated with conservation of higher-form charges that count the number of topological defects. Our formalism predicts, from first principles, the presence of propagating shear waves above a critical wave vector in liquids, thus giving a formal derivation of the phenomenon in terms of fundamental symmetries. The same picture provides also a theoretical explanation of the corresponding "positive sound dispersion" phenomenon for longitudinal sound. Importantly, accordingly to our theory, the main collective relaxation timescale of a liquid or a glass (known as the α relaxation for the latter) is given by the phase relaxation time, which is not necessarily related to the Maxwell time. Finally, we build a nonequilibrium effective action using the in-in formalism defined on the Schwinger-Keldysh contour, that further supports the emerging picture. In summary, our work suggests that the fundamental difference between solids, fluids, and glasses has to be identified with the associated generalized higher-form global symmetries and their topological structure, and that the Burgers vector for the displacement fields serves as a suitable topological order parameter distinguishing the solid (ordered) phase and the amorphous ones (fluids, glasses).

Stress hyperuniformity and transient oscillatory-exponential correlation decay as signatures of strength vs fragility in glasses

We examine and compare the local stress autocorrelation in the inherent states of a fragile and a strong glass: the Kob-Andersen (KA) binary mixture and the Beest-Kramer-Santen model of silica. For both systems, local (domain-averaged) stress fluctuations asymptotically reach the normal inverse-volume decay in the large domain limit; accordingly, the real-space stress autocorrelation presents long-range power law tails. However, in the case of silica, local stress fluctuations display a high degree of hyperuniformity, i.e., their asymptotic (normal) decay is disproportionately smaller than their bond level amplitude. This property causes the asymptotic power law tails of the real-space stress autocorrelation to be swamped, up to very large distances (several nanometers), by an intermediate oscillatory-exponential decay regime. Similar contributions exist in the KA stress autocorrelation, but they never can be considered as dominating the power law decay and fully disappear when stress is coarse-grained beyond one interatomic distance. Our observations document that the relevance of power-law stress correlation may constitute a key discriminating feature between strong and fragile glasses. Meanwhile, they highlight that the notion of local stress in atomistic systems involves by necessity a choice of observation (coarse-graining) scale, the relevant value of which depends, in principle, on both the model and the phenomenon studied.

Mechanical response of packings of nonspherical particles: A case study of two-dimensional packings of circulo-lines

We investigate the mechanical response of jammed packings of circulo-lines in two spatial dimensions, interacting via purely repulsive, linear spring forces, as a function of pressure P during athermal, quasistatic isotropic compression. The surface of a circulo-line is defined as the collection of points that is equidistant to a line; circulo-lines are composed of a rectangular central shaft with two semicircular end caps. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power law, 〈G(P)〉∼P^{β}, with β∼0.5, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of 〈G(P)〉 over the same range of pressure for aspect ratios R≳1.2. However, the power-law scaling exponent β∼0.8-0.9 is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose 〈G〉 into separate contributions from geometrical families, G_{f}, and from changes in the interparticle contact network, G_{r}, such that 〈G〉=〈G_{f}〉+〈G_{r}〉. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase and decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution 〈G_{f}〉 is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent for 〈G(P)〉.

Translational and rotational critical-like behaviors in the glass transition of colloidal ellipsoid monolayers

Critical-like behaviors have been found in translational degrees of freedom near the glass transition of spherical particle systems mainly with local polycrystalline structures, but it is not clear if criticality exists in more general glassy systems composed of nonspherical particles without crystalline structures. Here, through experiments and simulations, we show critical-like behaviors in both translational and rotational degrees of freedom in monolayers of monodisperse colloidal ellipsoids in the absence of crystalline orders. We find rich features of the Ising-like criticality in structure and slow dynamics at the ideal glass transition point ϕ₀, showing the thermodynamic nature of glass transition at ϕ₀ A dynamic criticality is found at the mode-coupling critical point ϕ_c for the fast-moving clusters whose critical exponents increase linearly with fragility, reflecting a dynamic glass transition. These results cast light on the glass transition and explain the mystery that the dynamic correlation lengths diverge at two different temperatures.

Characterizing nonaffinity upon decompression of soft-sphere packings

Athermal elastic moduli of soft-sphere packings are known to exhibit universal scaling properties near the unjamming point, most notably the vanishing of the shear-to-bulk moduli ratio G/B upon decompression. Interestingly, the smallness of G/B stems from the large nonaffinity of deformation-induced displacements under shear strains, compared to insignificant nonaffinity of displacements under compressive strains. In this work, we show using numerical simulations that the relative weights of the affine and nonaffine contributions to the bulk modulus, and their dependence on the proximity to the unjamming point, can differ qualitatively between different models that feature the same generic unjamming phenomenology. In canonical models of unjamming, we observe that the ratio of the nonaffine to total bulk moduli B_{na}/B approaches a constant upon decompression, while in other, less well-studied models, it vanishes. We show that the vanishing of B_{na}/B in noncanonical models stems from the emergence of an invariance of net (zero) forces on the constituent particles to compressive strains at the onset of unjamming. We provide a theoretical scaling analysis that fully explains our numerical observations, and allows us to predict the scaling behavior of B_{na}/B upon unjamming, given the functional form of the pairwise interaction potential.

Can liquid foams and emulsions be modeled as packings of soft elastic particles?

When two immersed bubbles are pushed against each other, a facet is formed at their contact, leading to an increase of interfacial energy and hence a repulsive interaction force. Foams (and concentrated emulsions) in mechanical equilibrium may thus be modeled as an assembly of soft elastic interacting particles. Such a model has been used in many studies of their structure and mechanical properties, in particular near the jamming transition (or wet limit) where the contact forces are so small that bubbles remain roughly spherical. We review analytical ab initio models and simulations, based on the equilibration of pressure and surface tension forces or, equivalently, minimization of interfacial energy. Two-body interaction behavior dominates asymptotically at packing fractions approaching the jamming transition, but the interaction is intrinsically anharmonic and cannot be captured by a power law. This phenomenon was first identified by D. Morse and T. Witten: we offer a detailed analysis and transparent derivation of their classic result. For packing fractions well above the jamming transition point, the coupling among contacts mediated by bubble volume conservation has a significant impact on the macroscopic elastic response of foam. This effect is captured by a many-body interaction law, derived from first principles. Applications are explored in two and three dimensions, as are future directions for this kind of theory.

Favored local structures in amorphous colloidal packings measured by microbeam X-ray diffraction

Local structure and symmetry are keys to understanding how a material is formed and the properties it subsequently exhibits. This applies to both crystals and amorphous and glassy materials. In the case of amorphous materials, strong links between processing and history, structure and properties have yet to be made because measuring amorphous structure remains a significant challenge. Here, we demonstrate a method to quantify proportions of the bond-orientational order of nearest neighbor clusters [Steinhardt, et al. (1983) Phys Rev B 28:784-805] in colloidal packings by statistically analyzing the angular correlations in an ensemble of scanning transmission microbeam small-angle X-ray scattering (μSAXS) patterns. We show that local order can be modulated by tuning the potential between monodisperse, spherical colloidal silica particles using salt and surfactant additives and that more pronounced order is obtained by centrifugation than sedimentation. The order in the centrifuged glasses reflects the ground state order in the dispersion at lower packing fractions. This diffraction-based method can be applied to amorphous systems across decades in length scale to connect structure to behavior in disordered systems with a range of particle interactions.

Advances and challenges in the rheology of concentrated emulsions and nanoemulsions

We review advances that have been made in the rheology of concentrated emulsions and nanoemulsions, which can serve as model soft materials that have highly tunable viscoelastic properties at droplet volume fractions near and above the glass transition and jamming point. As revealed by experiments, simulations, and theoretical models, interfacial and positional structures of droplets can depend on the applied flow history and osmotic pressure that an emulsion has experienced, thereby influencing its key rheological properties such as viscoelastic moduli, yield stress and strain, and flow behavior. We emphasize studies of monodisperse droplets, since these have led to breakthroughs in the fundamental understanding of dispersed soft matter. This review also covers the rheological properties of attractive emulsions, which can exhibit a dominant elasticity even at droplet volume fractions far below maximal random jamming of hard spheres.