Evolution of Voronoi/Delaunay Characterized Micro Structure with Transition from Loose to Dense Sphere Packing

Micro structures of equal sphere packing （ranging from loose to dense packing） generated numerically by discrete element method under different vibration conditions are characterized using Voronoi/Delaunay tessellation, which is applied on a wide range of packing densities. The analysis on micro properties such as the total perimeter, surface area, and the face number distribution of each Voronoi polyhedron, and the pore size distribution in each Voronoi/Delaunay subunit is systematically carried out. The results show that with the increasing density of sphere packing, the Voronoi//Delaunay pore size distribution is narrowed. That indicates large pores to be gradually substituted by small uniformed ones during densification. Meanwhile, the distributions of face number, total per/meter, and surface area of Voronoi polyhedra at high packing densities tend to be narrower and higher, which is in good agreement with those in random loose packing.     

DEM simulation of small particles clogging in the packing of large beads

The percolation of small particles through a periodic random loose packing of large beads is studied numerically with the Distinct Element Method. The representativity of periodic mono-sized sphere packing of varying system size was first studied by comparing their pore size distributions and tortuosities with those of a larger system, considered as an infinite medium. The results show that a periodic packing of size as low as 4-grain diameters gives a reasonable representation of the porous medium and allows reducing considerably the number of particles that has to be used in the simulations. The flow and clogging of small particles of varying concentrations and friction coefficients flowing through the former packing are then studied numerically. Results show that a steady state is rapidly reached where the mean velocity and mean vertical velocity of small particles are both constant. These mean velocities decrease with an increase in friction coefficient and in small particle concentration. The influence of the friction coefficient μ is much less marked for values of μ larger than or equal to 0.5. The distribution of small particles throughout the crossed packing becomes rapidly heterogeneous. Small particles concentrate in some pores where their velocity vanishes and where the density can reach values larger than the density of the random loose packing. The proportion of particles blocked in these pores varies linearly with concentration. Finally, the narrow throats of the porous medium responsible for blocking are identified and characterized for different values of the friction coefficient.     

Role of interparticle forces in the formation of random loose packing

We present a physical and numerical study of the settling of uniform spheres in liquids and show that interparticle forces play a critical role in forming the so-called random loose packing (RLP). Different packing conditions give different interparticle forces and, hence, different RLP. Two types of interparticle forces are identified: process dependent and process independent. The van der Waals force, as the major cohesive force in the present study, plays a critical role in effecting the process-dependent forces such as drag and lift forces. An equation is formulated to describe the relationship between the macroscopic packing fraction and microscopic interparticle forces in a packing. We argue there is no lowest packing fraction for a mechanically stable RLP; hence, the packing fractions of RLP can range from 0 to 0.64 depending on the cohesive and frictional conditions between particles.     

Jamming III: Characterizing randomness via the entropy of jammed matter

The nature of randomness in disordered packings of frictional and frictionless spheres is investigated using theory and simulations of identical spherical grains. The entropy of the packings is defined through the force and volume ensemble of jammed matter and this is shown to be difficult to calculate analytically. A mesoscopic ensemble of isostatic states is then utilized in an effort to predict the entropy through the definition of a volume function that is dependent on the coordination number. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter, and elucidating the phase diagram for jammed granular matter. Analytical calculations are compared to numerical simulations using volume fluctuation analysis and graph theoretical methods, with reasonable agreement. The entropy of the jammed system reveals that random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits. Ensemble calculations show that the entropy vanishes at random close packing (RCP), while numerical simulations show that a finite entropy remains in the microscopic states at RCP. The notion of a negative compactivity, which explores states with volume fractions below those achievable by existing simulation protocols, is also explored, expanding the equations of state. The mesoscopic theory reproduces the simulations results in shape well, though a difference in magnitude implies that the entire entropy of the packing may not be captured by the methods presented herein. We discuss possible extensions to the present mesoscopic approach describing packings from random loose packing (RLP) to RCP to the ordered branch of the equation of state in an effort to understand the entropy of jammed matter in the full range of densities from RLP to face-centered cubic (FCC) packing.     

Effects of N-Phosphoproline on the Phospholipid of Human Erythrocytes Membrane

The effects of phosphorylated proline on the packing and conformation of the membrane phospholipid have been studied. Generally it causes the polar parts of the phospholipid to become more mobile, the packing of the long chain of phospholipid to become more loose, the long chain to be in a lower trans/gauche ratio, and the end of the long chain to become more orderly. Any change in the level of phosphorylated proline affects the fluidity of the membrane phospholipid. Free proline and phosphorylated proline have different effects on the aliphatic chains of phospholipid.     

Fluctuation kinetics of the formation of nanoparticles

The method of stochastic simulations was applied to studying the kinetics of nucleation and growth of nanoparticles from solution in the presence of a ligand-yielding stabilizer. It was established that, at large stabilizer concentrations the process is not described within the framework of the law of mass action, being appreciably dependent on stochastic fluctuations in the number of free vacancies at the surface of the growing particle. This factor is responsible for the limitation of particle growth and for the formation of stable particles with a loose packing and dendrite-like structure. With decreasing stabilizer concentration, the kinetic characteristics of the process approach those typical of classical kinetics, whereas the packing density increases.     

Onset of mechanical stability in random packings of frictional spheres

Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction phiRLP of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure p and increasing interparticle friction coefficient mu. Using x-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit p-->0 a new lower bound, phiRLP=0.550+/-0.001.     

Jamming in two-dimensional packings

We investigate the existence of random close and random loose packing limits in two-dimensional packings of monodisperse hard disks. A statistical mechanics approach-based on several approximations to predict the probability distribution of volumes-suggests the existence of the limiting densities of the jammed packings according to their coordination number and compactivity. This result has implications for the understanding of disordered states in the disk packing problem as well as the existence of a putative glass transition in two-dimensional systems.     

Micromechanical simulation and analysis of one-dimensional vibratory sphere packing

We present a numerical method capable of reproducing the densification process from the so-called random loose to dense packing of uniform spheres under vertical vibration. The effects of vibration amplitude and frequency are quantified, and the random close packing is shown to be achieved only if both parameters are properly controlled. Two densification mechanisms are identified: pushing filling by which the contact between spheres is maintained and jumping filling by which the contact between particles is periodically broken. In general, pushing filling occurs when the vibration intensity is low and jumping filling becomes dominant when the vibration intensity is high.     

Simulation of random packing of spherical particles with different size distributions

A numerical model for a loose packing process of spherical particles is presented. The simulation model starts with randomly choosing a sphere according to a pregenerated continuous particle-size distribution, and then dropping the sphere into a dimension-specified box, and obtaining its final position by using dropping and rolling rules which are derived from a similar physical process of spheres dropping in the gravitational field to minimize its gravity potential. Effects of three different particle-size distributions on the packing structure were investigated. Analysis on the physical background of the powder-based manufacturing process is additionally applied to produce optimal packing parameters of bimodal and Gaussian distributions to improve the quality of the fabricated parts. The results showed that higher packing density can be obtained using bimodal size distribution with a particle-size ratio from 1.5 to 2.0 and the mixture composition around n ₂:n ₁=6:4. For particle size with a Gaussian distribution, the particle radii should be limited in a narrow range around 0.67 to 1.5.     

Theory of size-selective precipitation

A theory of size-selective precipitation of spherical nanoparticles is presented. The precipitate is assumed to be a random loose packing of spheres with low polydispersity (less than ≈30%). The interparticle interaction is treated in the Derjaguin approximation. The theory is used to calculate the polydispersity in the fractionated particle samples for various problem parameters. Our calculations show that, for typical particle concentrations, fractionation is effective for polydispersities above 20% and ineffective for polydispersities below 5%.     

Decoupling of ionic transport from segmental relaxation in polymer electrolytes

We present detailed studies of the relationship between ionic conductivity and segmental relaxation in polymer electrolytes. The analysis shows that the ionic conductivity can be decoupled from segmental dynamics and the strength of the decoupling correlates with the fragility but not with the glass transition temperature. These results call for a revision of the current picture of ionic transport in polymer electrolytes. We relate the observed decoupling phenomenon to frustration in packing of rigid polymers, where the loose local structure is also responsible for the increase in their fragility.     

Simulation of periodic mono-sized hard sphere systems under different vibration conditions and resulting compaction

This paper presents a simulation study of confined periodic mono-sized hard sphere systems under different vibration conditions and their influence on the final compaction. An initial random loose packing is submitted to a series of vibration cycles allowing to transform it into a “suspension” in which a given proportion of particles have the possibility to vibrate. This “suspension” is then let to settle down and brought to a new denser packing. Different random local rules are used for the simulation of the displacements of particles during vibration.Firstly, a symmetric vibration is applied in which particles attempt to perform vertical upwards and downwards displacements of equal length. Shocks between particles are simply simulated by random upwards or downwards displacements. It has already been shown that in these conditions, and when the whole system is vibrating, the final packing density can be related to the initial density and to the vibration amplitude (“suspension model”). We show here that for a periodic packing, this model can be extended to partial “suspensions” in which only a proportion of particles is vibrating. An excellent agreement is found between this model and the simulation results as long as the packing is disordered. For large vibration amplitudes, an order appears amongst the system allowing to reach high densities (up to 0.66).Then, the symmetric vibration is replaced by a random vibration in which particles attempt to perform alternatively random upwards and downwards displacements whose inclination with the vertical axis follows a normal distribution. It is shown that in these conditions, the vibration still allows transforming the initial packing into a partial “suspension” but the proportion of vibrating particles is lower than for the symmetric vibration previously used. However, the “suspension model” can be extended with a reasonable agreement to this kind of vibration.     

Random Loose Packing in Granular Matter

We introduce and simulate a two-dimensional Edwards-style model of granular matter at vanishing pressure. The model incorporates some of the effects of gravity and friction, and exhibits a random loose packing density whose standard deviation vanishes with increasing system size, a phenomenon that should be verifiable for real granular matter. Research of the second named author was supported in part by NSF Grant DMS-0700120.     

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 1960s. This problem finds applications spanning from the mathematician’s pencil, the processing of granular materials, the jamming and glass transitions, all the way to fruit packing in every grocery. There are presently numerous experiments showing that the loosest way to pack spheres gives a density of ∼55% (named random loose packing, RLP) while filling all the loose voids results in a maximum density of ∼63%-64% (named random close packing, RCP). While those values seem robustly true, to this date there is no well-accepted physical explanation or theoretical prediction for them. Here we develop a common framework for understanding the random packings of monodisperse hard spheres whose limits can be interpreted as the experimentally observed RLP and RCP. The reason for these limits arises from a statistical picture of jammed states in which the RCP can be interpreted as the ground state of the ensemble of jammed matter with zero compactivity, while the RLP arises in the infinite compactivity limit. We combine an extended statistical mechanics approach ‘a la Edwards’ (where the role traditionally played by the energy and temperature in thermal systems is substituted by the volume and compactivity) with a constraint on mechanical stability imposed by the isostatic condition. We show how such approaches can bring results that can be compared to experiments and allow for an exploitation of the statistical mechanics framework. The key result is the use of a relation between the local Voronoi volumes of the constituent grains (denoted the volume function) and the number of neighbors in contact that permits us to simply combine the two approaches to develop a theory of volume fluctuations in jammed matter. Ultimately, our results lead to a phase diagram that provides a unifying view of the disordered hard sphere packing problem and further sheds light on a diverse spectrum of data, including the RLP state. Theoretical results are well reproduced by numerical simulations that confirm the essential role played by friction in determining both the RLP and RCP limits. The RLP values depend on friction, explaining why varied experimental results can be obtained.     

Structural investigations of DNA-polycation complexes

The internal structure of DNA-polycation complexes is investigated by synchrotron small-angle X-ray scattering (SAXS). Hexagonal packing of DNA is observed for DNA complexed with poly-L-lysine (PL), poly-L-arginine (PA), spermine (Sp), and linear and branched polyethyleneimine (lPEI and bPEI, respectively). Variations in the internal spacings and degree of long-range ordering are dependent on both polycation type and concentration of added salt. With increasing concentration of monovalent salt, a discontinuous phase transition is observed from compact to loose bundles and finally to an isotropic network phase. This salt-induced melting transition was found to be universal for all polyplexes studied and is in quantitative agreement with a simple free energy model based solely on electrostatic and entropic contributions. Using the osmotic stress method, bulk modulus (K) is measured for PL-DNA and PA-DNA polyplexes at various salt concentrations. With increasing osmotic force, we show that the salt-induced melting transition is shifted and compression in the loose bundle regime is in qualitative agreement with our model.     

Unexpected Reorganization Effects of Cold-Crystallized Polylactide Stereocomplex

Cold crystallization of poly (l-lactide)/poly (d-lactide) blends at low temperatures results in the formation of a stereocomplex with loose intermolecular packing. Upon heating, it undergoes significant reorganization into a compact one with an extremely high melting point via a solid–solid transition. In contrast, the stereocomplex crystallized at high temperatures exhibits little reorganization and thus a relatively low melting point.     

Stress response function of a granular layer: Quantitative comparison between experiments and isotropic elasticity

We measured the vertical pressure response function of a layer of sand submitted to a localized normal force at its surface. We found that this response profile depends on the way the layer has been prepared: all profiles show a single centered peak whose width scales with the thickness of the layer, but a dense packing gives a wider peak than a loose one. We calculate the prediction of isotropic elastic theory in the presence of a bottom boundary and compare it to the data. We found that the theory gives the right scaling and the correct qualitative shape, but fails to really fit the data. Received 3 August 2001     

Entropy of jammed matter

We investigate the nature of randomness in disordered packings of frictional spheres. We calculate the entropy of 3D packings through the force and volume ensemble of jammed matter, a mesoscopic ensemble and numerical simulations using volume fluctuation analysis and graph theoretical methods. Equations of state are obtained relating entropy, volume fraction and compactivity characterizing the different states of jammed matter. At the mesoscopic level the entropy vanishes at random close packing, while the microscopic states contribute to a finite entropy. The entropy of the jammed system reveals that the random loose packings are more disordered than random close packings, allowing for an unambiguous interpretation of both limits.     

Synthesis, characterization and photodegradation study of mixed-phase titania hollow submicrospheres with rough surface

Titania hollow submicrospheres with mixed phase (anatase-brookite or anatase-rutile) were synthesized via the combination of hydrothermal treatment and calcination of submicrospheres consisting of a polystyrene core and an amorphous TiO₂ shell. After hydrothermal treatment, amorphous titania shell could be transformed to anatase-brookite shell consisting of loose packed titania nanocrystals, which could be further converted to anatase-brookite (below 700 °C) or anatase-rutile titania (700-800 °C) hollow spheres with rough surface via calcination. The loose packing of titania nanocrystals not only inhibited the transformation temperature from anatase to rutile, but also provided titania hollow submicrospheres with high photodegradation activity of Rhodamine B. The photocatalytic activity of titania hollow spheres increased firstly then decreased when the calcination temperature was varied in the range of 450-800 °C, while hollow spheres obtained via calcinating at 700 °C exhibited the highest photocatalytic activity, which was five times higher than that of counterpart without hydrothermal treatment.