Random Close Packing Limit Research Articles

Estimating angoricity and granular equationsof state for monodisperse packings of frictional hard spheres in a wide range of densities.

In this paper, we study the granular equationof state (EOS) for computer-generated three-dimensional mechanically stable packings of frictional monodisperse particles over a wide range of densities (packing fractions), φ=0.56-0.72. As a statistical physics framework, we utilize the statistical ensemble for granular matter, specifically the "angoricity" ensemble, where the compressional component Σ_{p} of the force-moment tensor serves as granular energy and angoricity A_{p} is the corresponding granular "temperature." We demonstrate that the systems under study conform well to this statistical description, and the simple equationof state Σ_{p}=2.8NA_{p} holds very well, where N is the number of particles. We show that granular temperature exhibits a rapid drop around the random-close packing (RCP) limit φ≈0.64-0.65, and, hence, one can say that granular packings "freeze" at the RCP limit. We repeat these calculation for shear angoricity A_{sh} and shear component Σ_{sh} of the force-moment tensor and obtain a similar EOS, Σ_{sh}=0.85NA_{sh}. Additionally, we measure the so-called keramicity, an inverse temperature variable corresponding to the determinant of the force-moment tensor, while pressure angoricity corresponds to its trace. We show that inverse keramicity κ^{-1} and angoricity A_{p} conform to an EOS 1/A_{p}Σ_{p}/N+0.11κ(Σ_{p}/N)^{3}=1.2, whose form is predicted by mean-field theory. Finally, we demonstrate that the alternative statistical ensemble where Voronoi volumes serve as granular energy (and so-called compactivity serves as temperature) does not describe the systems under study well.

Ethyl cellulose nanoparticles as stabilizers for Pickering emulsions

Pickering emulsions stabilized by ethyl cellulose nanoparticles have recently received –great attention for their remarkable stability and numerous industrial applications. De- spite this, the exact stabilization mechanism of such Pickering emulsions is still not fully understood. Both the stabilization of the emulsion by particle adsorption at the inter- face and through network formation in the continuous phase (leading to a yield stress) have been suggested. In this work we study soybean oil-in-water emulsions stabilized by ethyl cellulose nanoparticles and find, by the use of confocal microscopy and interfa- cial tension measurements, that the main stabilization mechanism of this nanoparticle- stabilized emulsions is the adsorption of the particles at the interface, instead of forming a network in the continuous phase. At the same time, oscillatory rheology measurements reveal that the emulsions exhibit a yield stress well below the random close-packing limit for hard spheres, suggesting short-range interactions between the droplets caused by the presence of the particles at the interface. The presence of the particles at the interface in combination with the observed rheological behavior of an attractive emulsion gives a strong indication for a particle-bridged stabilized emulsions.

Long Wavelength Thermal Density Fluctuations in Molecular and Polymer Glass-Forming Liquids: Experimental and Theoretical Analysis under Isobaric Conditions.

We establish via an in-depth analysis of experimental data that the dimensionless compressibility (proportional to the dimensionless amplitude of long wavelength thermal density fluctuations) of one-component normal and supercooled liquids of chemically complex nonpolar and weakly polar molecules and polymers follows extremely well a surprisingly simple and general temperature dependence over an exceptionally wide range of pressures and temperatures. A theoretical basis for this behavior is shown to exist in the venerable van der Waals model and its more modern interpretations. Although associated hydrogen-bonding (and to a lesser degree strongly polar) liquids display modestly more complex behavior, rather simple temperature and pressure dependences are also discovered. A new approach to collapse the temperature- and pressure-dependent dimensionless compressibility data onto a master curve is formulated that differs from the empirical thermodynamic scaling approach. As a practical matter, we also find that the dimensionless compressibility scales well as an inverse power law with temperature with an exponent that is system dependent and decreases with pressure. At very high pressures and low temperatures, the thermal liquid behavior appears to approach (but not reach) a repulsion-dominated random close packing limit. All these findings are relevant to our recent theoretical work on the problem of activated relaxation and vitrification of supercooled molecular and polymeric liquids.

Modeling meso- and microstructure in materials patterned with acoustic focusing

We conduct numerical simulations of acoustic focusing in dense suspensions to map the design space of acoustically patterned materials and understand the relationships between input parameters, structural features, and functional properties. We develop closed-form expressions for acoustic forces on particles, enabling rapid simulation of thousands of particles, and find excellent agreement with experimentally focused patterns over a range of conditions. We map the geometrical and microstructural features of focused particle patterns and their dependence on processing parameters. We find that mesostructural geometrical features (focused line height, width, and profile shape) can be controlled reliably over a broad range by modulating input parameters, and that while microstructural features are less readily modulated via input parameters, they are well-suited for various transport properties in functional materials. Notably, packing density nears the random close packing limit at 0.64, and particle contact density shows anisotropy favoring particle contacts along the focused lines. These results guide process design for controlling the properties of patterned materials, and outline the property ranges accessible via acoustic focusing. Additionally, we discuss the dependence of material functionalities, particularly electrical, thermal, and ionic transport properties, on the meso- and micro-structural features of patterned composite materials in the context of acoustic focusing.

Relaxation times, jamming densities, and ideal glass transition densities for hard spheres in a wide range of polydispersities

We computationally study the behavior of asymptotic alpha-relaxation times τα as well as jamming densities for equilibrated frictionless polydisperse hard spheres in wide ranges of particle volume fractions φ. Log-normal particle radii distribution (r) with polydispersities δ=⟨Δr2⟩/⟨r⟩=0.1−0.3 in steps of 0.05 is used. We discover that τα(φ) can be fitted well with the Vogel–Fulcher–Tammann (VFT) form. Through the VFT fits, we estimate positions of the ideal glass transition densities φg. For each equilibrated configuration, we calculate equilibrium kinetic pressure Z. Equilibrium pressures can be well described by the Boublík–Mansoori–Carnahan–Starling–Leland fluid equation of state. For each equilibrated configuration, a jammed particle configuration, which is the closest one in the configuration space, is determined. We measure jamming densities φEJ of these configurations and present plots φEJ(φ) for all polydispersities. We demonstrate that the lines τα(φ), φEJ(φ), and Z(φ), as well as values φg, depend significantly on δ. These results show that φg is, in general, distinct from the random close packing limit (φEJ at φ = 0). We plan to use these data in the future to estimate glass equations of state and the configurational entropy for these hard-sphere systems.

Looking for a compact semi empirical equation of state for hard spheres and the possibility of a glassy transition

The equation of state of a hard sphere fluid at high density should exhibit a simple pole at the random close packing limit. Here we show that trying to obtain a compact semi-empirical equation of state simultaneously compatible with that asymptotic behaviour and with the known virial coefficients raises an analytical difficulty which can be solved if a glassy transition occurs in the disordered metastable phase at a density intermediate between the freezing point and the random close packing limit. The comparison of the estimated value for the transition point with earlier estimations suggests a more subtle situation.

Compressive resistance of granular-scale microgels: From loose to dense packing

Microgels are deformable and compressible particles that can be packed to concentrations that exceed the random close packing limit of hard spheres. For reaching high packing levels, one has to overcome the resistance to compression of the system. This resistance potentially originates from many different phenomena (thermal agitation effects, surface interactions, microgel deformation, interpenetration, water expulsion) that depend on the microgel properties (size, ionic charge, structure, softness). Here, we investigate granular-scale dextran-based microgels with different native water contents. The resistance to compression of the suspensions is measured through the variation of the osmotic pressure with packing concentration. In parallel, we characterize the structure of the packings in terms of polymer heterogeneity, microgel deformation, and average size using confocal microscopy. We find that all microgel suspensions resist compression in the same manner; however, the mechanisms involved clearly depend on the actual degree of compression. In the loose packing regime, the resistance originates mainly from the resistance of the microgels to their own deformation, with no or negligible deswelling; the osmotic pressure rises abruptly with concentration in analogy to compressed emulsion droplets. In the second and dense packing regime, the microgels necessarily have to expel water to withstand compression. The resistance of the packing is then similar to that of a continuous gel of the same polymer. Importantly, we find that structural macro-voids are still observable in these systems; the presence of which needs to be taken into account when modeling the osmotic resistance.

Influence of emulsification process on the properties of Pickering emulsions stabilized by layered double hydroxide particles.

This paper reports the influence of emulsification process on the packing of layered double hydroxide (LDH) particles at the aqueous/oil phase interface and the properties of the resulting Pickering emulsions. Emulsions prepared by ultrasonication display superior long-term stability and gel-like characteristics at the dispersed phase volume fraction well below the random close packing limit, whereas emulsions with same compositions prepared by vortex mixing show some extent of sedimentation and liquid-like behaviors. Rheological measurements demonstrate that the zero-shear elastic modulus and yield stress of gel-like emulsions exhibit power-law dependences on particle concentration and independence on aqueous/oil phase ratio. The microstructural origin of this behavior is investigated by optical microscopy, revealing the droplets become strongly adhesive and a heterogeneous percolating network is formed among neighboring droplets. Fluorescent confocal microscopy measurements further confirm that the droplet adhesion is due to particle layers bridging opposite interfaces. In contrast, homogeneous, isolated, and densely packed droplets are present in emulsions prepared by vortex mixing, which results in these systems being dominantly viscous like the suspending fluid. This study shows that the emulsification process can be used as a trigger to modify long-term stability and rheology of solid-stabilized multiphase mixtures, which greatly expands their potential technological applications.

Pressure-driven suspension flow near jamming.

We report here magnetic resonance imaging measurements performed on suspensions with a bulk solid volume fraction (ϕ_{0}) up to 0.55 flowing in a pipe. We visualize and quantify spatial distributions of ϕ and velocity across the pipe at different axial positions. For dense suspensions (ϕ_{0}>0.5), we found a different behavior compared to the known cases of lower ϕ_{0}. Our experimental results demonstrate compaction within the jammed region (characterized by a zero macroscopic shear rate) from the jamming limit ϕ_{m}≈0.58 at its outer boundary to the random close packing limit ϕ_{rcp}≈0.64 at the center. Additionally, we show that ϕ and velocity profiles can be fairly well captured by a frictional rheology accounting for both further compaction of jammed regions as well as normal stress differences.

Confined flow of suspensions modelled by a frictional rheology

AbstractWe investigate in detail the problem of confined pressure-driven laminar flow of neutrally buoyant non-Brownian suspensions using a frictional rheology based on the recent proposal of Boyer et al. (Phys. Rev. Lett., vol. 107 (18), 2011, 188301). The friction coefficient (shear stress over particle normal stress) and solid volume fraction are taken as functions of the dimensionless viscous number $I$ defined as the ratio between the fluid shear stress and the particle normal stress. We clarify the contributions of the contact and hydrodynamic interactions on the evolution of the friction coefficient between the dilute and dense regimes reducing the phenomenological constitutive description to three physical parameters. We also propose an extension of this constitutive framework from the flowing regime (bounded by the maximum flowing solid volume fraction) to the fully jammed state (the random close packing limit). We obtain an analytical solution of the fully developed flow in channel and pipe for the frictional suspension rheology. The result can be transposed to dry granular flow upon appropriate redefinition of the dimensionless number $I$. The predictions are in excellent agreement with available experimental results for neutrally buoyant suspensions, when using the values of the constitutive parameters obtained independently from stress-controlled rheological measurements. In particular, the frictional rheology correctly predicts the transition from Poiseuille to plug flow and the associated particles migration with the increase of the entrance solid volume fraction. We also numerically solve for the axial development of the flow from the inlet of the channel/pipe toward the fully developed state. The available experimental data are in good agreement with our numerical predictions, when using an accepted phenomenological description of the relative phase slip obtained independently from batch-settlement experiments. The solution of the axial development of the flow notably provides a quantitative estimation of the entrance length effect in a pipe for suspensions when the continuum assumption is valid. Practically, the latter requires that the predicted width of the central (jammed) plug is wider than one particle diameter. A simple analytical expression for development length, inversely proportional to the gap-averaged diffusivity of a frictional suspension, is shown to encapsulate the numerical solution in the entire range of flow conditions from dilute to dense.

On the jamming phase diagram for frictionless hard-sphere packings

We computer-generated monodisperse and polydisperse frictionless hard-sphere packings of 10(4) particles with log-normal particle diameter distributions in a wide range of packing densities φ (for monodisperse packings φ = 0.46-0.72). We equilibrated these packings and searched for their inherent structures, which for hard spheres we refer to as closest jammed configurations. We found that the closest jamming densities φ(J) for equilibrated packings with initial densities φ ≤ 0.52 are located near the random close packing limit φ(RCP); the available phase space is dominated by basins of attraction that we associate with liquid. φ(RCP) depends on the polydispersity and is ∼ 0.64 for monodisperse packings. For φ > 0.52, φ(J) increases with φ; the available phase space is dominated by basins of attraction that we associate with glass. When φ reaches the ideal glass transition density φ(g), φ(J) reaches the ideal glass density (the glass close packing limit) φ(GCP), so that the available phase space is dominated at φ(g) by the basin of attraction of the ideal glass. For packings with sphere diameter standard deviation σ = 0.1, φ(GCP) ≈ 0.655 and φ(g) ≈ 0.59. For monodisperse and slightly polydisperse packings, crystallization is superimposed on these processes: it starts at the melting transition density φ(m) and ends at the crystallization offset density φ(off). For monodisperse packings, φ(m) ≈ 0.54 and φ(off) ≈ 0.61. We verified that the results for polydisperse packings are independent of the generation protocol for φ ≤ φ(g).

Stability and convergence analysis of a dynamics-based collective method for random sphere packing

The simulation of granular materials requires an initial overlap-free packing of spherical particles at high volume packing fractions. In previous work, a dynamics-based collective approach, the Quasi-Dynamics Method (QDM), has been proposed to generate densely distributed spheres in an enclosed container. However, the stability and efficiency of the QDM were not fully addressed. In this paper, the algorithm is reformulated with two control parameters and the impact of these parameters on the algorithm performance is investigated. First, theoretical analyses and numerical verifications for extreme 1-D/2-D packing systems are conducted and the range of the control parameters in which the algorithm is convergent is analytically determined. Then, the analysis is extended to 3-D packing systems and the estimation of the parameter range is verified numerically. Finally, the algorithm is applied to modeling a cylindrical 3-D packing system and the convergence performance at different volume packing fractions is studied. Results show that the QDM is highly efficient in packing spheres at volume packing fractions that are close to the random close packing limit.

Dense colloidal fluids form denser amorphous sediments

We relate, by simple analytical centrifugation experiments, the density of colloidal fluids with the nature of their randomly packed solid sediments. We demonstrate that the most dilute fluids of colloidal hard spheres form loosely packed sediments, where the volume fraction of the particles approaches in frictional systems the random loose packing limit, ϕRLP = 0.55. The dense fluids of the same spheres form denser sediments, approaching the so-called random close packing limit, ϕRCP = 0.64. Our experiments, where particle sedimentation in a centrifuge is sufficiently rapid to avoid crystallization, demonstrate that the density of the sediments varies monotonically with the volume fraction of the initial suspension. We reproduce our experimental data by simple computer simulations, where structural reorganizations are prohibited, such that the rate of sedimentation is irrelevant. This suggests that in colloidal systems, where viscous forces dominate, the structure of randomly close-packed and randomly loose-packed sediments is determined by the well-known structure of the initial fluids of simple hard spheres, provided that the crystallization is fully suppressed.

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.

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.

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.

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.

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.