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Register a new userDionysian Hard Sphere Packings are Mechanically Stable at Vanishingly Low Densities
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High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures. This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. Here, we demonstrate the existence of a packing of hard spheres that has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we term Dionysian, is the diametric opposite to the Apollonian sphere packing which completely and stably fills space. We create tools to evaluate the stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.
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The discrepancy in nucleation rate densities between simulated and experimental hard spheres remains staggering and unexplained. Suggestively, more strongly sedimenting colloidal suspensions of hard spheres nucleate much faster than weakly sedimenting systems. In this work we consider firstly the effect of sedimentation on the structure of colloidal hard spheres, by tuning the density mismatch between solvent and colloidal particles. In particular we investigate the effect on the degree of five fold symmetry present. Secondly we study the size of density fluctuations in these experimental systems in comparison to simulations. The density fluctuations are measured by assigning each particle a local density, which is related to the number of particles within a distance of 3.25 particle diameters. The standard deviation of these local densities gives an indication of the fluctuations present in the system. Five fold symmetry is suppressed by a factor of two when sedimentation is induced in our system. Density fluctuations are also increased by a factor of two in experiments compared to simulations. The change in five fold symmetry makes a difference to the expected nucleation rates, but we demonstrate that it is ultimately too small to resolve the discrepancy between experiment and simulation, while the fluctuations are shown to be an artefact of 3d particle tracking.
Descriptors that characterize the geometry and topology of the pore space of porous media are intimately linked to their transport properties. We quantify such descriptors, including pore-size functions and the critical pore radius $delta_c$, for four different models: maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, and inherent structures of the quantizer energy. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. Often, $delta_c$ is used as the key characteristic length scale that determines the fluid permeability $k$. A recent study [Torquato. Adv. Wat. Resour. 140, 103565 (2020)] suggested for porous media with a well-connected pore space an alternative estimate of $k$ based on the second moment of the pore size $langledelta^2rangle$. Here, we confirm that, for all porosities and all models considered, $delta_c^2$ is to a good approximation proportional to $langledelta^2rangle$. However, unlike $langledelta^2rangle$, the permeability estimate based on $delta_c^2$ does not predict the correct ranking of $k$ for our models. Thus, we confirm $langledelta^2rangle$ to be a promising candidate for convenient and reliable estimates of $k$ for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the $tau$ order metric. We find that (effectively) hyperuniform models tend to have lower values of $k$ than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.
There are a finite number of distinct mechanically stable (MS) packings in model granular systems composed of frictionless spherical grains. For typical packing-generation protocols employed in experimental and numerical studies, the probabilities with which the MS packings occur are highly nonuniform and depend strongly on parameters in the protocol. Despite intense work, it is extremely difficult to predict {it a priori} the MS packing probabilities, or even which MS packings will be the most versus the least probable. We describe a novel computational method for calculating the MS packing probabilities by directly measuring the volume of the MS packing `basin of attraction, which we define as the collection of initial points in configuration space at {it zero packing fraction} that map to a given MS packing by following a particular dynamics in the density landscape. We show that there is a small core region with volume $V^c_n$ surrounding each MS packing $n$ in configuration space in which all initial conditions map to a given MS packing. However, we find that the MS packing probabilities are very weakly correlated with core volumes. Instead, MS packing probabilities obtained using initially dilute configurations are determined by complex geometric features of the basin of attraction that are distant from the MS packing.
There is growing evidence that the flow of driven amorphous solids is not homogeneous, even if the macroscopic stress is constant across the system. Via event driven molecular dynamics simulations of a hard sphere glass, we provide the first direct evidence for a correlation between the fluctuations of the local volume-fraction and the fluctuations of the local shear rate. Higher shear rates do preferentially occur at regions of lower density and vice versa. The temporal behavior of fluctuations is governed by a characteristic time scale, which, when measured in units of strain, is independent of shear rate in the investigated range. Interestingly, the correlation volume is also roughly constant for the same range of shear rates. A possible connection between these two observations is discussed.
Packing spheres efficiently in large dimension $d$ is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be effectively constructed by this method, up to a packing fraction close to $7, d, 2^{-d}$. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.
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