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Slides for a colloquium: Solidity without Elasticity. Given at Indiana University by TW January 10, 2007. (pdf 1.6 megabytes)
Granular convergence as an iterated local mapAnna Movsheva and Thomas A. WittenThis speculative model aims to understand the pattern selection that occurs when a granular medium is repeatedly sheared back and forth. The process terminates in a final state in which the shearing produces an exact repetition of numerous microstates. The convergence to this final state requires a large number of cycles. Our approach following our earlier work, offers a generic framework to understand such convergence phenomena based on two features of the granular system: determinism and irreversibility. It frames the passage from one microstate to the next as a deterministic map over all microstates. We limit this map by requiring it to be compatible with extensive behavior of the pack and with local causality. Using an ad-hoc way of imposing locality, we show that tighter constraints on locality result in shorter convergence times, with little dependence on system size.
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Cyclic annealing as an iterated random mapMuhittin Mungan, Thomas A. WittenDisordered magnets, martensitic mixed crystals, and glassy solids can be irreversibly deformed by subjecting them to external deformation. The deformation produces a smooth, reversible response punctuated by abrupt relaxation "glitches". Under appropriate repeated forward and reverse deformation producing multiple glitches, a strict repetition of a single sequence of microscopic configurations often emerges. We exhibit these features by describing the evolution of the system configuration from glitch to glitch as a mapping of states into one-another. A map U controls forward deformation; a second map D controls reverse deformation. Iteration of a given sequence of forward and reverse maps, e.g. DDDDUUU necessarily produces a convergence to a fixed cyclic repetition of states covering multiple glitches. The repetition may have a period of more than one strain cycle, as recently observed in simulations. Using numerical sampling, we characterize the convergence properties of four types of random maps implementing successive physical restrictions. The most restrictive is the much-studied Preisach model. These maps show only the most qualitative resemblance to annealing simulations. However, they suggest further properties needed for a realistic mapping scheme.
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Geometrical frustration yields fiber formation in self-assemblyMartin Lenz and Thomas A. WittenControlling the self-assembly of supramolecular structures is vital for living cells, and a central challenge for engineering at the nano- and microscales. Nevertheless, even particles without optimized shapes can robustly form well-defined morphologies. This is the case in numerous medical conditions where normally soluble proteins aggregate into fibers. Beyond the diversity of molecular mechanisms involved, we propose that fibers generically arise from the aggregation of irregular particles with short-range interactions. Using a minimal model of ill-fitting, sticky particles, we demonstrate robust fiber formation for a variety of particle shapes and aggregation conditions. Geometrical frustration plays a crucial role in this process, and accounts for the range of parameters in which fibers form as well as for their metastable character. Nature Physics (2017), doi:10.1038/nphys4184 . Interactive Supplement Reports and comments: | |
Signal transmissibility in marginal granular materialsMatthew Pinson and Thomas A. WittenWe examine the 'transmissibility' of a simulated two-dimensional pack of frictionless disks formed by confining dilute disks in a shrinking, periodic box to the point of mechanical stability. Two opposite boundaries are then removed, thus allowing a set of free motions. Small free displacements on one boundary then induce proportional displacements on the opposite boundary. Transmissibility is the ability to distinguish different perturbations by their distant responses. We assess transmissibility by successively identifying free orthonormal modes of motion that have the smallest distant responses. The last modes to be identified in this 'pessimistic' basis are the most transmissive. The transmitted amplitudes of these most transmissive modes fall off exponentially with mode number. Similar exponential falloff is seen in a simple elastic medium, though the responsible modes differ greatly in structure in the two systems. Thus the marginal pack's transmissibility is qualitatively similar to that of a simple elastic medium. We compare our results with recent findings based on the projection of the space of free motion onto interior sites. Journal of Physics: Condensed Matter, Volume 28, Number 49 (2016) | |
Force propagation in isostatic granular packsNathan KrapfIn which we probe the anomalous ray-like force propagation expected seen in some marginally jammed packs. We extend the study to isotropic marginal jamming systems developed by Nagel, Liu and collaborators. By altering the boundary conditions so that all forces are absorbed on one unique boundary, we apply point forces on individual beads and investigate the spatial distribution of the resulting stress. By studying variants of these packs plus a set of regular lattices, we find that ray-like force propagation is observed when the distribution of contact directions is anisotropic. For the isotropic packs of Nagel and Liu, only diffuse stress propagation like that of an elastic solid is observed, even very close to the jamming threshold, where many other features differ strongly from conventional elastic solids. | |
Compressing nearly hard sphere fluids increases glass fragilityLudovic Berthier and T. A. Witten
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Effects of compression on the vibrational modes of marginally jammed solidsMatthieu Wyart, Leonardo E. Silbert, Sidney R. Nagel, Thomas A. WittenGlasses have a large excess of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature is a necessary consequence of the weak connectivity of the solid, and that the frequency of modes in excess is very sensitive to the pressure. We analyze in particular two systems whose density D(w) of vibrational modes of angular frequency w display scaling behaviors with the packing fraction: (i) simulations of jammed packings of particles interacting through finite-range, purely repulsive potentials, comprised of weakly compressed spheres at zero temperature and (ii) a system with the same network of contacts, but where the force between any particles in contact (and therefore the total pressure) is set to zero. We account in the two cases for the observed a) convergence of D(w) toward a non-zero constant as w goes to 0, b) appearance of a low-frequency cutoff w*, and c) power-law increase of w* with compression. Differences between these two systems occur at lower frequency. The density of states of the modified system displays an abrupt plateau that appears at w*, below which we expect the system to behave as a normal, continuous, elastic body. In the unmodified system, the pressure lowers the frequency of the modes in excess. The requirement of stability despite the destabilizing effect of pressure yields a lower bound on the number of extra contact per particle dz: dz > p^(1/2), which generalizes the Maxwell criterion for rigidity when pressure is present. This scaling behavior is observed in the simulations. We finally discuss how the cooling procedure can affect the microscopic structure and the density of normal modes. | |
Geometric origin of excess low-frequency vibrational modes in amorphous solids Matthieu Wyart, Sidney R. Nagel, T.A. Witten Glasses have a large excess of low-frequency vibrational modes in comparison with crystalline solids. We show that such a feature is a necessary consequence of the geometry generic to a marginally connected solid. In particular, we analyze the density of states of a recently simulated system, comprised of weakly compressed spheres at zero temperature. We account for the observed a) constancy of the density of modes with frequency, b) appearance of a low-frequency cutoff, and c) power-law increase of this cutoff with compression. We predict a length scale below which the boundary conditions strongly affect the system. | |
Robust propagation direction of stresses in a minimal granular packing D. A. Head, A. V. Tkachenko, T. A. Witten By employing the adaptive network simulation method, we demonstrate that the ensemble-averaged stress caused by a local force for packings of frictionless rigid beads is concentrated along rays whose slope is consistent with unity: forces propagate along lines at 45 degrees to the horizontal or vertical. This slope is shown to be independent of polydispersity or the degree to which the system is sheared. Further confirmation of this result comes from fitting the components of the stress tensor to the null stress constitutive equation. The magnitude of the response is also shown to fall off with the -1/2 power of distance. We argue that our findings are a natural consequence of a system that preserves its volume under small perturbations. | |
Stress Propagation through Frictionless Granular Material | |
Stress in frictionless granular material: adaptive network simulations |