Liu, Andrea J
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Publication Elasticity and Response in Nearly Isostatic Periodic Lattices(2009-11-13) Souslov, Anton; Liu, Andrea J.; Lubensky, Tom C.The square and kagome lattices with nearest-neighbor springs of spring constant k are isostatic with a number of zero-frequency modes that scale with the perimeter. We analytically study the approach to this isostatic limit as the spring constant k' for the next-nearest-neighbor bond vanishes. We identify a characteristic frequency ω* ~ √(k') and length ω* ~ √(k/k' ) for both lattices. The shear modules C44 = k' of the square lattice vanishes with k', but that for the kagome lattice does not.Publication Heat Transport in Model Jammed Solids(2010-02-03) Xu, Ning; Vitelli, Vincenzo; Liu, Andrea J; Wyart, Matthieu; Nagel, Sidney RWe calculate numerically the normal modes of vibrations in three-dimensional jammed packings of soft spheres as a function of the packing fraction and obtain the energy diffusivity, a spectral measure of transport that controls sound propagation and thermal conductivity. The crossover frequency between weak and strong phonon scattering is controlled by the coordination and shifts to zero as the system is decompressed toward the critical packing fraction at which rigidity is lost. We present a scaling analysis that relates the packing fraction dependence of the crossover frequency to the anomalous scaling of the shear modulus with compression. Below the crossover, the diffusivity displays a power-law divergence with inverse frequency consistent with Rayleigh law, which suggests that the vibrational modes are primarily transverse waves, weakly scattered by disorder. Above it, a large number of modes appear whose diffusivity plateaus at a nearly constant value before dropping to zero above the localization frequency. The thermal conductivity of a marginally jammed solid just above the rigidity threshold is calculated and related to the one measured experimentally at room temperature for most glasses.Publication Universal Jamming Phase Diagram in the Hard-Sphere Limit(2011-03-21) Haxton, Thomas K; Liu, Andrea J.; Schmiedeberg, MichaelWe present a new formulation of the jamming phase diagram for a class of glass-forming fluids consisting of spheres interacting via finite-ranged repulsions at temperature T , packing fraction φ or pressure p, and applied shear stress Σ. We argue that the natural choice of axes for the phase diagram are the dimensionless quantities T/pσ3, pσ3/ϵ, and Σ/p, where T is the temperature, p is the pressure, Σ is the stress, σ is the sphere diameter, ϵ is the interaction energy scale, and m is the sphere mass. We demonstrate that the phase diagram is universal at low pσ3/ϵ; at low pressure, observables such as the relaxation time are insensitive to details of the interaction potential and collapse onto the values for hard spheres, provided the observables are nondimensionalized by the pressure. We determine the shape of the jamming surface in the jamming phase diagram, organize previous results in relation to the jamming phase diagram, and discuss the significance of various limits.Publication Finite-Size Scaling at the Jamming Transition(2012-08-27) Goodrich, Carl P; Liu, Andrea J; Nagel, Sidney RWe present an analysis of finite-size effects in jammed packings of N soft, frictionless spheres at zero temperature. There is a 1/N correction to the discrete jump in the contact number at the transition so that jammed packings exist only above isostaticity. As a result, the canonical power-law scalings of the contact number and elastic moduli break down at low pressure. These quantities exhibit scaling collapse with a nontrivial scaling function, demonstrating that the jamming transition can be considered a phase transition. Scaling is achieved as a function of N in both two and three dimensions, indicating an upper critical dimension of 2.Publication Vibrational Modes Identify Soft Spots in a Sheared Disordered Packing(2011-08-31) Manning, M. Lisa; Liu, Andrea JWe analyze low-frequency vibrational modes in a two-dimensional, zero-temperature, quasistatically sheared model glass to identify a population of structural ‘‘soft spots’’ where particle rearrangements are initiated. The population of spots evolves slowly compared to the interval between particle rearrangements, and the soft spots are structurally different from the rest of the system. Our results suggest that disordered solids flow via localized rearrangements that tend to occur at soft spots, which are analogous to dislocations in crystalline solids.Publication Low-Frequency Vibrations of Soft Colloidal Glasses(2010-07-09) Chen, Ke; Ellenbroek, Wouter G; Zhang, Zexin; Yunker, Peter J; Chen, Daniel T.N.; Henkes, Silke; Brito, Carolina; Dauchot, Oliver; Liu, Andrea J; Van Saarloos, Wim; Yodh, Arjun GWe conduct experiments on two-dimensional packings of colloidal thermosensitive hydrogel particles whose packing fraction can be tuned above the jamming transition by varying the temperature. By measuring displacement correlations between particles, we extract the vibrational properties of a corresponding ‘‘shadow’’ system with the same configuration and interactions, but for which the dynamics of the particles are undamped. The vibrational properties are very similar to those predicted for zerotemperature sphere packings and found in atomic and molecular glasses; there is a boson peak at low frequency that shifts to higher frequency as the system is compressed above the jamming transition.Publication Temperature-Pressure Scaling for Air-Fluidized Grains near Jamming(2012-03-30) Daniels, L. J.; Haxton, T. K.; Liu, Andrea J; Xu, N.; Durian, Douglas J.We present experiments on a monolayer of air-fluidized beads in which a jamming transition is approached by increasing pressure, increasing packing fraction, and decreasing kinetic energy. This is accomplished, along with a noninvasive measurement of pressure, by tilting the system and examining behavior versus depth. We construct an equation of state and analyze relaxation time versus effective temperature. By making time and effective temperature dimensionless using factors of pressure, bead size, and bead mass, we obtain a good collapse of the data but to a functional form that differs from that of thermal hard-sphere systems. The relaxation time appears to diverge only as the effective temperature to pressure ratio goes to zero.Publication Equivalence of Glass Transition and Colloidal Glass Transition in the Hard-Sphere Limit(2009-12-10) Xu, Ning; Haxton, Thomas K.; Liu, Andrea J.; Nagel, Sidney R.We show that the slowing of the dynamics in simulations of several model glass-forming liquids is equivalent to the hard-sphere glass transition in the low-pressure limit. In this limit, we find universal behavior of the relaxation time by collapsing molecular-dynamics data for all systems studied onto a single curve as a function of T/p, the ratio of the temperature to the pressure. At higher pressures, there are deviations from this universal behavior that depend on the interparticle potential, implying that additional physical processes must enter into the dynamics of glass formation.Publication Branching, Capping, and Severing in Dynamic Actin Structures(2007-08-07) Gopinathan, Anjay; Lee, Kun-Chun; Schwarz, Jennifer; Liu, Andrea JBranched actin networks at the leading edge of a crawling cell evolve via protein-regulated processes such as polymerization, depolymerization, capping, branching, and severing. A formulation of these processes is presented and analyzed to study steady-state network morphology. In bulk, we identify several scaling regimes in severing and branching protein concentrations and find that the coupling between severing and branching is optimally exploited for conditions in vivo. Near the leading edge, we find qualitative agreement with the in vivo morphology.Publication Excess Vibrational Modes and the Boson Peak in Model Glasses(2007-04-24) Xu, Ning; Liu, Andrea J; Wyart, Matthieu; Nagel, Sidney R.The excess low-frequency normal modes for two widely used models of glasses are studied at zero temperature. The onset frequencies for the anomalous modes for both systems agree well with predictions of a variational argument, which is based on analyzing the vibrational energy originating from the excess contacts per particle over the minimum number needed for mechanical stability. Even though both glasses studied have a high coordination number, most of the additional contacts can be considered to be weak.