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Register a new userVibrations in jammed solids: Beyond linear response
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We propose a `phase diagram for particulate systems that interact via purely repulsive contact forces, such as granular media and colloidal suspensions. We identify and characterize two distinct classes of behavior as a function of the input kinetic energy per degree of freedom $T_0$ and packing fraction deviation above and below jamming onset $Delta phi=phi - phi_J$ using numerical simulations of purely repulsive frictionless disks. Iso-coordinated solids (ICS) only occur above jamming for $Delta phi > Delta phi_c(T_0)$; they possess average coordination number equal to the isostatic value ($< z> = z_{rm iso}$) required for mechanically stable packings. ICS display harmonic vibrational response, where the density of vibrational modes from the Fourier transform of the velocity autocorrelation function is a set of sharp peaks at eigenfrequencies $omega_k^d$ of the dynamical matrix evaluated at $T_0=0$. Hypo-coordinated solids (HCS) occur both above and below jamming onset within the region defined by $Delta phi > Delta phi^*_-(T_0)$, $Delta phi < Delta phi^*_+(T_0)$, and $Delta phi > Delta phi_{cb}(T_0)$. In this region, the network of interparticle contacts fluctuates with $< z> approx z_{rm iso}/2$, but cage-breaking particle rearrangements do not occur. The HCS vibrational response is nonharmonic, {it i.e} the density of vibrational modes $D(omega)$ is not a collection of sharp peaks at $omega_k^d$, and its precise form depends on the measurement method. For $Delta phi > Delta phi_{cb}(T_0)$ and $Delta phi < Delta phi^*_{-}(T_0)$, the system behaves as a hard-particle liquid.
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Penrose tilings form lattices, exhibiting 5-fold symmetry and isotropic elasticity, with inhomogeneous coordination much like that of the force networks in jammed systems. Under periodic boundary conditions, their average coordination is exactly four. We study the elastic and vibrational properties of rational approximants to these lattices as a function of unit-cell size $N_S$ and find that they have of order $sqrt{N_S}$ zero modes and states of self stress and yet all their elastic moduli vanish. In their generic form obtained by randomizing site positions, their elastic and vibrational properties are similar to those of particulate systems at jamming with a nonzero bulk modulus, vanishing shear modulus, and a flat density of states.
Using a system of repulsive, soft particles as a model for a jammed solid, we analyze its force network as characterized by the magnitude of the contact force between two particles, the local contact angle subtended between three particles, and the local coordination number. In particular, we measure the local contact angle distribution as a function of the magnitude of the local contact force. We find the suppression of small contact angles for locally larger contact forces, suggesting the existence of chain-like correlations in the locally larger contact forces. We couple this information with a coordination number-spin state mapping to arrive at a Potts spin model with frustration and correlated disorder to draw a potential connection between jammed solids (no quenched disorder) and spin glasses (quenched disorder). We use this connection to measure chaos due to marginality in the jammed system. In addition, we present the replica solution of the one-dimensional, long-range Potts glass as a potential toy building block for a jammed solid, where a sea of weakly interacting spins provide for long-range interactions along a chain-like backbone of more strongly interacting spins.
Memory encoding by cyclic shear is a reliable process to store information in jammed solids, yet its underlying mechanism and its connection to the amorphous structure are not fully understood. When a jammed sphere packing is repeatedly sheared with cycles of the same strain amplitude, it optimizes its mechanical response to the cyclic driving and stores a memory of it. We study memory by cyclic shear training as a function of the underlying stability of the amorphous structure in marginally stable and highly stable packings, the latter produced by minimizing the potential energy using both positional and radial degrees of freedom. We find that jammed solids need to be marginally stable in order to store a memory by cyclic shear. In particular, highly stable packings store memories only after overcoming brittle yielding and the cyclic shear training takes place in the shear band, a region which we show to be marginally stable.
Contact breaking and Hertzian interactions between grains can both give rise to nonlinear vibrational response of static granular packings. We perform molecular dynamics simulations at constant energy in 2D of frictionless bidisperse disks that interact via Hertzian spring potentials as a function of energy and measure directly the vibrational response from the Fourier transform of the velocity autocorrelation function. We compare the measured vibrational response of static packings near jamming onset to that obtained from the eigenvalues of the dynamical matrix to determine the temperature above which the linear response breaks down. We compare packings that interact via single-sided (purely repulsive) and double-sided Hertzian spring interactions to disentangle the effects of the shape of the potential from contact breaking. Our studies show that while Hertzian interactions lead to weak nonlinearities in the vibrational behavior (e.g. the generation of harmonics of the eigenfrequencies of the dynamical matrix), the vibrational response of static packings with Hertzian contact interactions is dominated by contact breaking as found for systems with repulsive linear spring interactions.
It is well known that jammed soft materials will flow if sheared above their yield stress - think mayonnaise spread on bread - but a complete microscopic description of this seemingly sim- ple process has yet to emerge. What remains elusive is a microscopic framework that explains the macroscopic flow, derived from a 3-D spatially resolved analysis of the dynamics of the droplets or particles that compose the soft material. By combining confocal-rheology experiments on compressed emulsions and numerical simulations, we unravel that the primary microscopic mechanisms for flow are strongly influenced by the rate of the imposed deformation. When shearing fast, small coordinated clusters of droplets move collectively as in a conga line, while at low rates the flow emerges from bursts of droplet rearrangements, correlated over large domains. These regions exhibit complex spatio-temporal correlation patterns that reflect the long range elasticity embedded in the jammed material. These results identify the three-dimensional structure of microscopic rearrangements within sheared soft solids, revealing that the characteristic shape and dynamics of these structures are strongly determined by the rate of the external shear.
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