Close Packing Limit Research Articles

Elasticity of two-dimensional crystals of polydisperse hard disks near close packing: Surprising behavior of the Poisson's ratio

The equation of state, elastic constants, and Poisson's ratio of a crystalline two-dimensional polydisperse hard disk system were determined in the close packing limit. Monte Carlo simulations in the NpT ensemble with variable shape of the periodic box reveal that the pressure and elastic constants grow with increasing polydispersity. The equation of state and the bulk modulus are well described by the free volume approximation. The latter approximation fails, however, for the shear modulus. The simulations also show that the introduction of any amount of size polydispersity in the hard disk systems causes a discontinuous "jump" of the Poisson's ratio in the close packing limit from the value ν(δ=0) = 0.1308(22), obtained for equidiameter hard disks, to ν(δ>0) ≈ 1, estimated for the polydisperse disks.

Packing Spheres Tightly: Influence of Mechanical Stability on Close-Packed Sphere Structures

Many experiments and simulations of packings of monodisperse hard spheres report a dominance of the face-centered cubic structure in the hexagonally close-packed limit, even though it has no significant energetic or entropic gain over other close-packed configurations. Combining simulations and experiments, we demonstrate that a simple mechanical instability which occurs during the packing process may play an important role in selecting the face-centered cubic structure over other close-packed alternatives. Our argument is supported by detailed quantitative analyses of key configurations in sphere packings and highlights the importance of the packing dynamics. The proposed mechanism is elementary and should therefore play a role in a wide range of sphere systems.

Jammed spheres: Minkowski tensors reveal onset of local crystallinity

The local structure of disordered jammed packings of monodisperse spheres without friction, generated by the Lubachevsky-Stillinger algorithm, is studied for packing fractions above and below 64%. The structural similarity of the particle environments to fcc or hcp crystalline packings (local crystallinity) is quantified by order metrics based on rank-four Minkowski tensors. We find a critical packing fraction φ(c)≈0.649, distinctly higher than previously reported values for the contested random close packing limit. At φ(c), the probability of finding local crystalline configurations first becomes finite and, for larger packing fractions, increases by several orders of magnitude. This provides quantitative evidence of an abrupt onset of local crystallinity at φ(c). We demonstrate that the identification of local crystallinity by the frequently used local bond-orientational order metric q(6) produces false positives and thus conceals the abrupt onset of local crystallinity. Since the critical packing fraction is significantly above results from mean-field analysis of the mechanical contacts for frictionless spheres, it is suggested that dynamic arrest due to isostaticity and the alleged geometric phase transition in the Edwards framework may be disconnected phenomena.

Short-ranged pair distribution function for concentrated suspensions of soft particles

Many rheological properties of concentrated suspensions of soft particles jammed together beyond the random close packing limit for hard-spheres are determined by the contacts between the particles and are computable from the short range value of the pair distribution function. It is a challenge to compute the structure of these amorphous suspensions because they are athermal, non-equilibrium materials. Here we develop a microscopic theory to predict the short ranged pair distribution function with a transport or conservation of mass equation for the stationary state that includes a concentration dependent mean force that captures the effect of the bulk suspension on the pair interaction. The resulting distribution function is singular as the concentration approaches random close packing as expected from the theory for hard spheres. The pair distribution function and elastic properties that can be calculated from it are validated by computer simulations and experiments with concentrated suspensions of particles with varying soft potentials.

Characteristics of Pickering Emulsion Gels Formed by Droplet Bridging

We experimentally characterize the microstructure and rheology of a carefully designed mixture of immiscible fluids and near-neutral-wetting colloidal particles. Particle bridging across two fluid interfaces provides a route to highly stable gel-like emulsions at volume fractions of the dispersed phase well below the random close-packing limit for spheres. We investigate the microstructural origins of this behavior by confocal microscopy and reveal a percolating network of colloidal particles that serves as a cohesive scaffold, bridging together droplets of the dispersed phase. Remarkably, the mixture's salient rheological characteristics are governed predominantly by the solids loading and can be tailored irrespective of the droplet volume fraction. The identification of this rheological hallmark could provide a means toward the improved design of modern products that utilize solid-stabilized interfaces.

A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions

Convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level. In this paper, we are concerned with solving continuity equations that govern the evolution of volume fractions in Eulerian models of disperse two-phase flows. An implicit Galerkin finite element approximation is equipped with a flux limiter for the convective terms. The fully multidimensional limiting strategy is based on a flux-corrected transport (FCT) algorithm. This nonlinear high-resolution scheme satisfies a discrete maximum principle for divergence-free velocities and ensures positivity preservation for arbitrary velocity fields. To enforce an upper bound that corresponds to the maximum-packing limit, an FCT-like overshoot limiter is applied to the converged convective fluxes at the end of each time step. This postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model. Numerical results for 2D implosion problems illustrate the performance of the proposed limiting procedure.

Inter-membrane adhesion mediated by mobile linkers: Effect of receptor shortage

Giant unilamellar vesicles (GUVs) adhering to supported bilayers were used as a model system to mimic ligand–receptor mediated cell-cell adhesion. We present the effect of varying the concentration of receptors (neutravidin on the bilayer) and ligands (biotin on the vesicle) on GUV adhesion and the organization of receptors in the adhesion zone. At high concentrations of both ligands and receptors, the adhesion is strong, all the available membrane is adhered and receptors are accumulated under the adhered membrane up to the geometrical limit of close packing. At low concentrations of receptors (<0.5%), and an arbitrary concentration of ligands (≥0.1%), adhesion does not proceed to completion: the membrane is only partially bound and parts of it still fluctuate. The receptors tend to accumulate under the adhered membrane but the filling is partial. Receptors get jammed and form clusters with fractal like shapes along the rim of the adhered vesicle in such a way that the annular cluster prevents further filling of the adhesion disc. We characterize the filling in terms of a compaction factor and the final concentration. Interestingly, the closing of the ring of jammed clusters switches the interior of the adhesion disc from one thermodynamic ensemble to another. In the new ensemble the receptors sealed within the adhesion disc are mobile but their number is fixed. Under such conditions, the usually strong neutravidin/biotin bond is weak. The incomplete adhesion state can be attributed to a combination of the effects of diffusion, jamming and the competition of enthalpy and entropy on bond formation. The formation of jammed receptor clusters reported here represents a new mechanism that influences membrane adhesion.

Concentration Dependent Diffusion of Self-Propelled Rods

We examine the persistent random motion of self-propelled rods (SPR) as a function of the area fraction phi and study the effect of steric interactions on their diffusion properties. SPR of length l and width w are fabricated with a spherocylindrical head attached to a beaded chain tail, and show directed motion on a vibrated substrate. The mean square displacement (MSD) on the substrate grows linearly with time t for phi<w/l, before displaying caging as phi is increased, and stops well below the close packing limit. Direction autocorrelations decay progressively slower with phi. We describe the observed decrease of SPR propagation speed c(phi) with a tube model. Further, MSD parallel to the SPR collapse with tau=l/c(phi) and scales as (lt/tau);{2}, and MSD in the perpendicular direction grows progressively slower than l{2}t/tau with phi, consistent with dynamics inside a thinning tube.

Synthesis, characterization, and remodeling of weight-bearing allograft bone/polyurethane composites in the rabbit

The process of bone healing requires the restoration of both anatomy and physiology, and there is a recognized need for innovative biomaterials that facilitate remodeling throughout this complex process. While porous scaffolds with a high degree of interconnectivity are known to accelerate cellular infiltration and new bone formation, the presence of pores significantly diminishes the initial mechanical properties of the materials, rendering them largely unsuitable for load-bearing applications. In this study, a family of non-porous composites has been fabricated by reactive compression molding of mineralized allograft bone particles (MBPs) with a biodegradable polyurethane (PUR) binder, which is synthesized from a polyester polyol and a lysine-derived polyisocyanate. At volume fractions exceeding the random close-packing limit, the particulated allograft component presented a nearly continuous osteoconductive pathway for cells into the interior of the implant. By varying the molecular weight of the polyol and manipulating the surface chemistry of the MBP via surface demineralization, compressive modulus and strength values of 3–6 GPa and 107–172 MPa were achieved, respectively. When implanted in bilateral femoral condyle plug defects in New Zealand White rabbits, MBP/PUR composites exhibited resorption of the allograft and polymer components, extensive cellular infiltration deep into the interior of the implant, and new bone formation at 6 weeks. While later in vivo timepoints are necessary to determine the ultimate fate of the MBP/PUR composites, these observations suggest that allograft bone/polymer composites have potential for future development as weight-bearing devices for orthopedic applications.

Irreversible deposition of extended objects with diffusional relaxation on discrete substrates

Random sequential adsorption with diffusional relaxation of extended objects both on a one-dimensional and planar triangular lattice is studied numerically by means of Monte Carlo simulations. We focus our attention on the behavior of the coverage θ(t) as a function of time. Our results indicate that the lattice dimensionality plays an important role in the present model. For deposition of k-mers on 1D lattice with diffusional relaxation, we found that the growth of the coverage θ(t) above the jamming limit to the closest packing limit θCPL is described by the pattern θCPL-θ(t) ∝Eβ[-(t/τ)β], where Eβ denotes the Mittag-Leffler function of order β∈(0,1). In the case of deposition of extended lattice shapes in 2D, we found that after the initial “jamming", a stretched exponential growth of the coverage θ(t) towards the closest packing limit θCPL occurs, i.e., θCPL - θ(t) ∝exp [-(t/τ)β]. For both cases we observe that: (i) dependence of the relaxation time τ on the diffusion probability Pdif is consistent with a simple power-law, i.e., τ∝Pdif-δ; (ii) parameter β depends on the object size in 1D and on the particle shape in 2D.

New virial equation of state for hard-disk fluids

Although many equations of state of hard-disk fluids have been proposed, none is capable of reproducing the currently calculated or estimated values of the first eighteen virial coefficients at the same time as giving very good accuracy when compared with computer simulation values for the compressibility factor over the whole fluid range. A new virial-based expression is here proposed which achieves these aims. For that, we use the fact that the currently accepted estimated values for the highest virial coefficients behave linearly with their order, and also that virial coefficients must have a limiting behaviour that permits the closest packing limit in the compressibility factor to be also adequately reproduced.

Adsorption, desorption, and diffusion ofk-mers on a one-dimensional lattice

Kinetics of the deposition process of k -mers in the presence of desorption or/and diffusional relaxation of particles is studied by Monte Carlo method on a one-dimensional lattice. For reversible deposition of k-mers, we find that after the initial "jamming," a stretched exponential growth of the coverage theta(t) toward the steady-state value theta(eq) occurs, i.e., theta(eq)-theta(t) is proportional to exp[-(t/tau)(beta)]. The characteristic time scale tau is found to decrease with desorption probability P(des) according to a power law, tau is proportional to P(des)(-gamma), with the same exponent gamma=1.22+/-0.04 for all k-mers. For irreversible deposition with diffusional relaxation, the growth of the coverage theta(t) above the jamming limit to the closest packing limit (CPL) theta(CPL) is described by the pattern theta(CPL)-theta(t) is proportional to E(beta)[-(t/tau)(beta)], where E(beta) denotes the Mittag-Leffler function of order beta(0,1) . Similarly to the reversible case, we found that the dependence of the relaxation time tau on the diffusion probability P(dif) is consistent again with a simple power-law, i.e., tau is proportional to P(dif)(-delta). When adsorption, desorption, and diffusion occur simultaneously, coverage always reaches an equilibrium value theta(eq), which depends only on the desorption/adsorption probability ratio. The presence of diffusion only hastens the approach to the equilibrium state, so that the stretched exponential function gives a very accurate description of the deposition kinetics of these processes in the whole range above the jamming limit.

Dynamics of dense sheared granular flows. Part 1. Structure and diffusion

Shear flows of inelastic spheres in three dimensions in the volume fraction range 0.4–0.64 are analysed using event-driven simulations. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution en (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution et (negative of the ratio of post- and pre-collisional velocities perpendicular to the line joining the centres). Here, we have considered both et = +1 and et = en (rough particles) and et = −1 (smooth particles), and the normal coefficient of restitution en was varied in the range 0.6–0.98. Care was taken to avoid inelastic collapse and ensure there are no particle overlaps during the simulation. First, we studied the ordering in the system by examining the icosahedral order parameter Q6 in three dimensions and the planar order parameter q6 in the plane perpendicular to the gradient direction. It was found that for shear flows of sufficiently large size, the system continues to be in the random state, with Q6 and q6 close to 0, even for volume fractions between φ = 0.5 and φ = 0.6; in contrast, for a system of elastic particles in the absence of shear, the system orders (crystallizes) at φ = 0.49. This indicates that the shear flow prevents ordering in a system of sufficiently large size. In a shear flow of inelastic particles, the strain rate and the temperature are related through the energy balance equation, and all time scales can be non-dimensionalized by the inverse of the strain rate. Therefore, the dynamics of the system are determined only by the volume fraction and the coefficients of restitution. The variation of the collision frequency with volume fraction and coefficient of restitution was examined. It was found, by plotting the inverse of the collision frequency as a function of volume fraction, that the collision frequency at constant strain rate diverges at a volume fraction φad (volume fraction for arrested dynamics) which is lower than the random close-packing volume fraction 0.64 in the absence of shear. The volume fraction φad decreases as the coefficient of restitution is decreased from en = 1; φad has a minimum of about 0.585 for coefficient of restitution en in the range 0.6–0.8 for rough particles and is slightly larger for smooth particles. It is found that the dissipation rate and all components of the stress diverge proportional to the collision frequency in the close-packing limit. The qualitative behaviour of the increase in the stress and dissipation rate are well captured by results derived from kinetic theory, but the quantitative agreement is lacking even if the collision frequency obtained from simulations is used to calculate the pair correlation function used in the theory.

Generation of porous particle structures using the void expansion method

The newly developed “void expansion method” allows for an efficient generation of porous packings of spherical particles over a wide range of volume fractions using the discrete element method. Particles are randomly placed under addition of much smaller “void-particles”. Then, the void-particle radius is increased repeatedly, thereby rearranging the structural particles until formation of a dense particle packing. The structural particles’ mean coordination number was used to characterize the evolving microstructures. At some void radius, a transition from an initially low to a higher mean coordination number is found, which was used to characterize the influence of the various simulation parameters. For structural and void-particle stiffnesses of the same order of magnitude, the transition is found at constant total volume fraction slightly below the random close packing limit. For decreasing void-particle stiffness the transition is shifted towards a smaller void-particle radius and becomes smoother.

Granular temperature in a liquid fluidized bed as revealed by diffusing wave spectroscopy

We report granular temperature and solid fraction fields for a thin rectangular bed (20×200 mm cross-section and 500 mm high) of glass particles (mean diameter of 165 μm and density of 2500 kg/m 3) fluidized by water for superficial velocities ranging from 0.05 U t , which is approximately double the minimum fluidization velocity, to 0.49 U t , where U t is the particle terminal velocity estimated by fitting the Richardson–Zaki correlation to the bed expansion data. At superficial velocities below 0.336 U t , the solid fraction and granular temperature are uniform throughout the bed. At higher superficial velocities, the solid fraction tends to decrease with height above the distributor, whilst the granular temperature first increases to a maximum before decaying towards the top of the bed. Correlation of the mean granular temperature with the mean solid fraction and the local granular temperature with the local solid fraction both suggest that the granular temperature in the liquid fluidized bed can be described solely in terms of the solid fraction. The granular temperature increases monotonically with solid fraction to a maximum at φ≈0.18 where it then decreases monotonically as φ approaches the close-packed limit.

Morphological Transformations of Native Petroleum Emulsions. I. Viscosity Studies

Emulsions of water in as-recovered native crude oils of diverse geographical origin evidently possess some common morphological features. At low volume fractions varphi of water, the viscosity behavior of emulsions is governed by the presence of flocculated clusters of water droplets, whereas characteristic tight gels, composed of visually monodisperse small droplets, are responsible for the viscosity anomaly at varphi approximately 0.4-0.5. Once formed, small-droplet gel domains apparently retain their structural integrity at higher varphi, incorporating/stabilizing new portions of water as larger-sized droplets. The maximum hold-up of disperse water evidently is the close-packing limit of varphi approximately 0.74. At higher water contents (up to varphi approximately 0.83), no inversion to O/W morphology takes place, but additional water emerges as a separate phase. The onset of stratified flow (W/O emulsion gel + free water) is the cause of the observed viscosity decrease, contrary to the conventional interpretation of the viscosity maximum as a reliable indicator of the emulsion inversion point.

Structural transitions in granular packs: statistical mechanics and statistical geometry investigations

We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common neighbours. This geometrical analysis reveals sharp changes in the network's clustering occurring at the packing fractions (fraction of volume occupied by the spheres respect to the total volume, $\rho$) corresponding to the so called Random Loose Packing limit (RLP, $\rho \sim 0.555$) and Random Close Packing limit (RCP, $\rho \sim 0.645$). At these packing fractions we also observe abrupt changes in the fluctuations of the portion of free volume around each sphere. We analyze such fluctuations by means of a statistical mechanics approach and we show that these anomalies are associated to sharp variations in a generalized thermodynamical variable which is the analogous for these a-thermal systems to the specific heat in thermal systems.

Relations between Dewetting of Polymer Thin Films and Phase-Separation of Encompassed Quantum Dots

Incorporation of semiconductor quantum dots (QDs) at high densities in polymer thin films is promising for the development of nanoscale sensors, light-emitting diodes, and photovoltaic devices. However, inhomogeneous distribution and aggregation of QDs in polymers limit the applications of QD−polymer composites. We investigated the origin of the aggregation of CdSe-ZnS QDs in polybutadiene (PB) thin films and its relations with the dewetting of polymer thin films on inorganic surfaces. Microscale dewetting of PB thin films on glass surfaces resulted in the formation of honeycomb structures of PB, and nanoscale dewetting on the QD surface resulted in phase-separation and aggregation of QDs. The formation of the honeycomb structures of PB is attributed to phase-separation between PB and glass during the dewetting on the glass surface, and the aggregation of QDs is attributed to the phase-separation between PB and QDs during the dewetting on the QD surface. The honeycomb structures of PB are characterized using optical microscopy and atomic force microscopy (AFM) imaging, and the phase-separation and aggregation of QDs are characterized using transmission electron microscopy (TEM) imaging and inter-QD Förster resonance energy-transfer (FRET). FRET is confirmed from photoluminescence (PL) spectral shifts and decreases of PL intensity and lifetime of QDs. Without the phase-separation and aggregation of QDs, inter-QD FRET is unexpected under the selected densities of QDs; the calculated inter-QD distance (>60 nm) for a homogeneous distribution of QDs is beyond the Förster distance (<10 nm). The phase-separation and aggregation of QDs in PB thin films are further characterized based on a decrease of FRET from QD to rhodamine 590 (R590) with an increase in the density of QDs. The aggregation of QDs occurs because of the rejection of QDs by PB molecules during the release of recoiling energy, which is distributed among entangled polymer chains and between polymer chains and QDs in the thin films. In the current work, we characterized the nanoscale dewetting of PB on the surface of QDs, the phase-separation and aggregation of QDs in PB thin films, and the limit of close-packing of QDs in polymer thin films without aggregation that are valuable during the construction of QD and polymer-based thin-film devices.

Relationships between the single particle barrier hopping theory and thermodynamic, disordered media, elastic, and jamming models of glassy systems

The predictions of the ultralocal limit of the activated hopping theory of highly viscous simple fluids and colloidal suspensions [K. S. Schweizer and G. Yatsenko, J. Chem. Phys. 127, 164505 (2007), preceding paper] for the relaxation time and effective activation barrier are compared with those of diverse alternative theoretical approaches and computer simulation. A nonlinear connection between the barrier height and excess pressure as empirically suggested by simulations of polydisperse repulsive force fluids is identified. In the dense normal and weakly dynamical precursor regime, where entropic barriers of hard spheres are nonexistent or of order the thermal energy, agreement with an excess entropy ansatz is found. In the random close packing or jamming limit, the barrier hopping theory predicts an essential singularity stronger than the free volume model, which is in agreement with the simplest entropic droplet nucleation and replica field theoretic approaches. Upon further technical simplification of the theory, close connections with renormalization group and nonperturbative memory function based studies of activated transport of a Brownian particle in a disordered medium can been identified. Several analytic arguments suggest a qualitative consistency between the barrier hopping theory and solid-state elastic models based on the high frequency shear modulus and a molecular-sized apparent activation volume. Implications of the analysis for the often high degeneracy of conflicting explanations of glassy dynamics are discussed.

Rheology of Highly Concentrated Emulsions – Concentration and Droplet Size Dependencies

Abstract The concentration and size dependencies of elastic properties of highly concentrated w/o emulsions were studied. The range of weight concentration of the disperse phase was 90 - 96%, the range of the average droplet size was 16 - 20 mm, and the droplet size distribution remained unchanged. The disperse phase consists of droplets of over-cooled concentrated aqueous solutions of inorganic salts. The concentration range being studied lies above the limit of maximal close packing, j &gt; jm. The droplet size distribution is fairly wide and the shape of droplets is polygonal. These factors alone determine possible new rheological effects, such as the elasticity and visco-plastic behaviour of emulsions, as well as the observed form of concentration and size dependencies of rheological properties of emulsions. The complete flow curves were measured for these fairly new emulsion systems. It emerged that they were similar to the entire concentration and droplet size ranges being studied. The concentration dependencies of the yield stress and storage modules corresponded to the Princen-Kiss theory with critical volume concentration approximately 0.71 - 0.74. However, this theory describes the size dependence of elastic modules incorrectly. A new model is proposed, which correctly describes the dependencies of elastic modules on both determining parameters - those of concentration and droplet size.