On the caging number of two- and three-dimensional hard spheres

Tetrahedral colloidal clusters from random parking of bidisperse spheres

Using experiments and simulations, we investigate the clusters that form when colloidal spheres stick irreversibly to--or "park" on--smaller spheres. We use either oppositely charged particles or particles labeled with complementary DNA sequences, and we vary the ratio α of large to small sphere radii. Once bound, the large spheres cannot rearrange, and thus the clusters do not form dense or symmetric packings. Nevertheless, this stochastic aggregation process yields a remarkably narrow distribution of clusters with nearly 90% tetrahedra at α = 2.45. The high yield of tetrahedra, which reaches 100% in simulations at α = 2.41, arises not simply because of packing constraints, but also because of the existence of a long-time lower bound that we call the "minimum parking" number. We derive this lower bound from solutions to the classic mathematical problem of spherical covering, and we show that there is a critical size ratio α(c) = (1 + sqrt[2]) ≈ 2.41, close to the observed point of maximum yield, where the lower bound equals the upper bound set by packing constraints. The emergence of a critical value in a random aggregation process offers a robust method to assemble uniform clusters for a variety of applications, including metamaterials.

Snowballs, quantum solvation and coordination: lead ions inside small helium droplets

Ab initio calculations are used to construct an analytical many-body potential for Pb(2+)He(n) and Pb(+)He(n) clusters which accounts for non pairwise additive interactions. The potential surface reproduces the global minima for cluster sizes ranging from n = 1 to n = 16 obtained from explicit ab initio calculations and found in a previous search for ultrahigh coordination numbers. Ground state energies and structures obtained by accurate diffusion quantum Monte Carlo calculations are used to investigate if quantum effects qualitatively affect the formation of coordination shells. For Pb(2+) doped clusters a first solvation shell is closed at n = 12 and gradually softened by additional helium atoms which start to form a distinct second shell only at n = 16. Spin-orbit coupling profoundly influences the structure of Pb(+)He(n) clusters and causes a gradual structural evolution without pronounced solvation shells.

Use of the cage formation probability for obtaining approximate phase diagrams

In this work, we introduce the idea of cage formation probability, defined by considering the angular space needed by a particle in order to leave a cage given an average distance to its neighbors. Considering extreme fluctuations, two phases appear as a function of the number of neighbors and their distances to a central one: Solid and fluid. This allows us to construct an approximated phase diagram based on a geometrical approach. As an example, we apply this probability concept to hard disks in two dimensions and hard spheres in three dimensions. The results are compared with numerical simulations using a Monte Carlo method.

Geometrical cluster ensemble analysis of random sphere packings

We introduce a geometric analysis of random sphere packings based on the ensemble averaging of hard-sphere clusters generated via local rules including a nonoverlap constraint for hard spheres. Our cluster ensemble analysis matches well with computer simulations and experimental data on random hard-sphere packing with respect to volume fractions and radial distribution functions. To model loose as well as dense sphere packings various ensemble averages are investigated, obtained by varying the generation rules for clusters. Essential findings are a lower bound on volume fraction for random loose packing that is surprisingly close to the freezing volume fraction for hard spheres and, for random close packing, the observation of an unexpected split peak in the distribution of volume fractions for the local configurations. Our ensemble analysis highlights the importance of collective and global effects in random sphere packings by comparing clusters generated via local rules to random sphere packings and clusters that include collective effects.

Micromechanical behavior of adhesive granular silica layers: Structure deformation

We studied the mechanical behavior of packed layers of 1-mum-sized silica particles immersed in liquids, upon indentation with a 10-mum glass sphere, attached to the cantilever of an atomic force microscope (AFM). Simultaneously, a confocal scanning laser microscope (CSLM) was used to study the deformations in the material. Our liquids consisted of (nearly) refractive-index-matching water-DMSO mixtures. Particle layers were formed by sedimentation in normal gravity. In the absence of (added) electrolyte, the collective behavior of the layer is reminiscent of that of a simple liquid. Crystal-like structures were observed, with the individual particles showing positional fluctuations. Carefully adding 2 wt % LiCl to this system leads to the formation of a weakly aggregated network, in which the crystal-like order gets lost and the particles lose their mobility. On indenting into these aggregated layers, the CSLM recordings showed imprints that closely resembled the size and shape of the indenter. A more accurate inspection of the structural changes was allowed after localizing all silica particles in three dimensions. Calculated local concentrations and coordination numbers showed that even at the level of these highly local quantities, no deformation gradients could be observed in the vicinity of the probe. Particle image velocimetry analysis suggested that deformation occurs mostly in the lateral directions. On pulling the indenter out, adhesion between the silica particles and the glass indenter became manifest via a distortion of the initially spherical dent and lower coordination numbers under the dent. Together all these behaviors indicate that the aggregated layers behave like yield-stress materials, which are solidlike up to a critical stress and liquidlike above it. The results of this study also illustrate the potential of the AFM-CSLM combination to study the detailed 3D deformation in other types of systems, like granular packings or more open particle networks.