A soft departure from jamming: the compaction of deformable granular matter under high pressures.

The high-pressure compaction of three dimensional granular packings is simulated using a bonded particle model (BPM) to capture linear elastic deformation. In the model, grains are represented by a collection of point particles connected by bonds. A simple multibody interaction is introduced to control Poisson's ratio and the arrangement of particles on the surface of a grain is varied to model both high- and low-frictional grains. At low pressures, the growth in packing fraction and coordination number follow the expected behavior near jamming and exhibit friction dependence. As the pressure increases, deviations from the low-pressure power-law scaling emerge after the packing fraction grows by approximately 0.1 and results from simulations with different friction coefficients converge. These results are compared to predictions from traditional discrete element method simulations which, depending on the definition of packing fraction and coordination number, may only differ by a factor of two. As grains deform under compaction, the average volumetric strain and asphericity, a measure of the change in the shape of grains, are found to grow as power laws and depend heavily on the Poisson's ratio of the constituent solid. Larger Poisson's ratios are associated with less volumetric strain and more asphericity and the apparent power-law exponent of the asphericity may vary. The elastic properties of the packed grains are also calculated as a function of packing fraction. In particular, we find the Poisson's ratio near jamming is 1/2 but decreases to around 1/4 before rising again as systems densify.

Fractal dimensions of jammed packings with power-law particle size distributions in two and three dimensions.

Static structure factors are computed for large-scale, mechanically stable, jammed packings of frictionless spheres (three dimensions) and disks (two dimensions) with broad, power-law size dispersity characterized by the exponent -ß. The static structure factor exhibits diverging power-law behavior for small wave numbers, allowing us to identify a structural fractal dimension d_{f}. In three dimensions, d_{f}≈2.0 for 2.5≤ß≤3.8, such that each of the structure factors can be collapsed onto a universal curve. In two dimensions, we instead find 1.0≲d_{f}≲1.34 for 2.1≤ß≤2.9. Furthermore, we show that the fractal behavior persists when rattler particles are removed, indicating that the long-wavelength structural properties of the packings are controlled by the large particle backbone conferring mechanical rigidity to the system. A numerical scheme for computing structure factors for triclinic unit cells is presented and employed to analyze the jammed packings.

Molecular dynamics simulations of binary sphere mixtures.

Explicit simulations of fluid mixtures of highly size-dispersed particles are constrained by numerical challenges associated with identifying pair-interaction neighbors. Recent algorithmic developments have ameliorated these difficulties to an extent, permitting more efficient simulations of systems with many large and small particles of disperse sizes. We leverage these capabilities to perform molecular dynamics simulations of binary sphere mixtures with elastically stiff particles approaching the hard sphere limit and particle size ratios of up to 50, approaching the colloidal limit. The systems considered consist of 500 large particles and up to nearly 3.6×10^{6} small particles with total particle volume fractions up to 0.51. Our simulations confirm qualitative predictions for correlations between large particles previously obtained analytically and for simulations employing effective depletion interactions, but also reveal additional insights into the near-contact structure that result from the explicit treatment of the small particle solvent. No spontaneous crystal nucleation was observed during the simulations, suggesting that nucleation rates in the fluid-solid coexistence region are too small to observe crystal nucleation for feasible simulation system sizes and timescales.

Large-scale frictionless jamming with power-law particle size distributions.

Due to significant computational expense, discrete element method simulations of jammed packings of size-dispersed spheres with size ratios greater than 1:10 have remained elusive, limiting the correspondence between simulations and real-world granular materials with large size dispersity. Invoking a recently developed neighbor binning algorithm, we generate mechanically stable jammed packings of frictionless spheres with power-law size distributions containing up to nearly 4 000 000 particles with size ratios up to 1:100. By systematically varying the width and exponent of the underlying power laws, we analyze the role of particle size distributions on the structure of jammed packings. The densest packings are obtained for size distributions that balance the relative abundance of large-large and small-small particle contacts. Although the proportion of rattler particles and mean coordination number strongly depend on the size distribution, the mean coordination of nonrattler particles attains the frictionless isostatic value of six in all cases. The size distribution of nonrattler particles that participate in the load-bearing network exhibits no dependence on the width of the total particle size distribution beyond a critical particle size for low-magnitude exponent power laws. This signifies that only particles with sizes greater than the critical particle size contribute to the mechanical stability. However, for high-magnitude exponent power laws, all particle sizes participate in the mechanical stability of the packing.

Green's function method for dynamic contact calculations.

Resolving atomic scale details while capturing long-range elastic deformation is the principal difficulty when solving contact mechanics problems with computer simulations. Fully atomistic simulations must consider large blocks of atoms to support long-wavelength deformation modes, meaning that most atoms are far removed from the region of interest. Building on earlier methods that used elastic surface Green's functions to compute static substrate deformation, we present a numerically efficient dynamic Green's function technique to treat realistic, time-evolving, elastic solids. Our method solves substrate dynamics in reciprocal space and utilizes precomputed Green's functions that exactly reproduce elastic interactions without retaining the atomic degrees of freedom in the bulk. We invoke physical insights to determine the necessary number of explicit substrate layers required to capture the attenuation of subsurface waves as a function of surface wave vector. We observe that truncating substrate dynamics at depths that fall as a power of wave vector allows us to accurately model wave propagation without implementing arbitrary damping. The framework we have developed substantially accelerates molecular dynamics simulations of large elastic substrates. We apply the method to single asperity contact, impact, and sliding friction problems and present our preliminary findings.

Sliding Friction of Amorphous Asperities on Crystalline Substrates: Scaling with Contact Radius and Substrate Thickness.

Disorder in the contact between an amorphous slider and a crystalline substrate leads to a cancellation of lateral forces. Atomically flat, rigid surfaces exhibit structural superlubricity, with the frictional stress in circular contacts of radius a vanishing as 1/a. The inclusion of elasticity allows relative motion of domains on the surface in response to the random interfacial forces. The competition between disorder and elastic deformation is predicted to limit structural superlubricity and produce a constant frictional stress for a larger than a characteristic domain size λ that depends on the ratio of the shear modulus G to the magnitude of interfacial shear stresses τ0. Extensive simulations of a flat, amorphous punch sliding on a crystalline substrate with different system sizes and G/τ0 are used to test scaling predictions and determine unknown prefactors that are needed for quantitative analysis. For bulk systems, we find an exponential decrease of the large a frictional stress and 1/λ with increasing G/τ0. For thin free-standing films, the stress and 1/λ are inversely proportional to G/τ0. These results may help explain the size-dependent friction of nanoparticles and plate-like materials used as solid lubricants.

Effect of Roughness and Elasticity on Interactions between Charged Colloidal Spheres.

The effects of realistic roughness and elasticity on the interactions between charged silica spheres are studied as a function of surface potential, screening length, interfacial energy, and roughness. The repulsive force Frep that must be overcome to bring charged spheres into contact is relatively insensitive to elasticity unless spheres are hundreds of times softer than silica. Frep is also insensitive to roughness and interfacial energy. In contrast, roughness has a large effect on the binding energy of spheres and the force Fsep to separate them. Both are lowered by 1 to 2 orders of magnitude by the measured surface roughness of less than 1 nm on 1 µm silica spheres. The reason is that interactions between rigid spheres are dominated by the highest surface peaks rather than the entire spherical surface. Elasticity can increase the pull-off force of rough spheres by a factor of 2 or more because additional surface area can be brought into contact. The implications of these results for shear-thickening transitions are discussed.