Liu, Andrea J
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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 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 Direct Determination of the Size of Basins of Attraction of Jammed Solids(2011-06-17) Xu, Ning; Frenkel, Daan; Liu, Andrea JWe propose a free-energy-based Monte Carlo method to measure the volume of potential-energy basins in configuration space. Using this approach we can estimate the number of distinct potential-energy minima, even when this number is much too large to be sampled directly. We validate our approach by comparing our results with the direct enumeration of distinct jammed states in small packings of frictionless spheres. We find that the entropy of distinct packings is extensive and that the entropy of distinct hard-sphere packings must have a maximum as a function of packing fraction.Publication Measurement of Correlations between Low-Frequency Vibrational Modes and Particle Rearrangements in Quasi-Two-Dimensional Colloidal Glasses(2011-08-31) Chen, Ke; Manning, M. L.; Yunker, Peter J.; Ellenbroek, Wouter G.; Liu, Andrea J; Zhang, Zexin; Yodh, Arjun G.We investigate correlations between low-frequency vibrational modes and rearrangements in two-dimensional colloidal glasses composed of thermosensitive microgel particles, which readily permit variation of the sample packing fraction. At each packing fraction, the particle displacement covariance matrix is measured and used to extract the vibrational spectrum of the "shadow" colloidal glass (i.e., the particle network with the same geometry and interactions as the sample colloid but absent damping). Rearrangements are induced by successive, small reductions in the packing fraction. The experimental results suggest that low-frequency quasilocalized phonon modes in colloidal glasses, i.e., modes that present low energy barriers for system rearrangements, are spatially correlated with rearrangements in this thermal system.Publication Energy Transport in Jammed Sphere Packings(2009-01-21) Xu, Ning; Vitelli, Vincenzo; Wyart, Matthieu; Liu, Andrea J.; Nagel, Sidney R.We calculate the normal modes of vibration in jammed sphere packings to obtain the energy diffusivity, a spectral measure of transport. At the boson peak frequency, we find an Ioffe-Regel crossover from a diffusivity that drops rapidly with frequency to one that is nearly frequency independent. This crossover frequency shifts to zero as the system is decompressed towards the jamming transition, providing unambiguous evidence of a regime in frequency of nearly constant diffusivity. Such a regime, postulated to exist in glasses to explain the temperature dependence of the thermal conductivity, therefore appears to arise from properties of the jamming transition.Publication Why is Random Close Packing Reproducible?(2007-10-09) Kamien, Randal D.; Liu, Andrea JWe link the thermodynamics of colloidal suspensions to the statistics of regular and random packings. Random close packing has defied a rigorous definition yet, in three dimensions, there is near universal agreement on the volume fraction at which it occurs.We conjecture that the common value of фrcp ≈ 0.64 arises from a divergence in the rate at which accessible states disappear.We relate this rate to the equation of state for a hard-sphere fluid on a metastable, noncrystalline branch.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 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 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 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.