Simulation of the coordination number of random sphere packing

Given article presents a generalized equation for calculating the average coordination number from the density of a random sphere packing, supplemented by a dependence on the threshold value of the interparticle distance in two- and three-dimensional spaces. It is shown that the calculation of the average coordination numbers according to the proposed equation gives an unambiguous correspondence between the simulated, calculated and experimental data for threshold values of more than 1.02 particle diameters. An explanation of the weak dependence of the average coordinate number on the packing density for small threshold values of the interparticle distance is given in this wor

Predicting the Drug Release Kinetics of Matrix Tablets

In this paper we develop two mathematical models to predict the release kinetics of a water soluble drug from a polymer/excipient matrix tablet. The first of our models consists of a random walk on a weighted graph, where the vertices of the graph represent particles of drug, excipient and polymer, respectively. The graph itself is the contact graph of a multidisperse random sphere packing. The second model describes the dissolution and the subsequent diffusion of the active drug out of a porous matrix using a system of partial differential equations. The predictions of both models show good qualitative agreement with experimental release curves. The models will provide tools for designing better controlled release devices.Comment: 17 pages, 7 figures; Elaborated at the first Workshop on the Application of Mathematics to Problems in Biomedicine, December 17-19, 2007 at the University of Otago in Dunedin, New Zealan

Birthday Inequalities, Repulsion, and Hard Spheres

We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. We give three different applications of the birthday inequality in statistical physics and combinatorics: we prove lower bounds on the free energy of the hard sphere model and upper bounds on the number of independent sets and matchings of a given size in d-regular graphs. The birthday inequality is implied by a repulsion inequality: the expected volume of the union of spheres of radius r around n randomly placed centers increases if we condition on the event that the centers are at pairwise distance greater than r. Surprisingly we show that the repulsion inequality is not true in general, and in particular that it fails in 24-dimensional Euclidean space: conditioning on the pairwise repulsion of centers of 24-dimensional spheres can decrease the expected volume of their union

Compaction of anisotropic granular materials : experiments and simulations

We present both experimental and numerical investigations of compaction in granular materials composed of rods. As a function of the aspect ratio of the particles, we have observed large variations of the asymptotic packing volume fraction in vertical tubes. The relevant parameter is the ratio between the rod length $\ell$ and the tube diameter $D$. Even the compaction dynamics remains unchanged for various particle lengths, a 3d/2d phase transition for grain orientations is observed for $\ell/D = 1$. A toy model for the compaction of needles on a lattice is also proposed. This toy model gives a complementary view of our experimental results and leads to behaviors similar to experimental ones.Comment: 5 pages, 10 figure

Image-Based Pore-Scale Modeling of Inertial Flow in Porous Media and Propped Fractures

Non-Darcy flow is often observed near wellbores and in hydraulic fractures where relatively high velocities occur. Quantifying additional pressure drop caused by non-Darcy flow and fundamentally understanding the pore-scale inertial flow is important to oil and gas production in hydraulic fractures. Image-based pore-scale modeling is a powerful approach to obtain macroscopic transport properties of porous media, which are traditionally obtained from experiments and understand the relationship between fluid dynamics with complex pore geometries. In image-based modeling, flow simulations are conducted based on pore structures of real porous media from X-ray computed tomographic images. Rigorous pore-scale finite element modeling using unstructured mesh is developed and implemented in proppant fractures. The macroscopic parameters permeability and non-Darcy coefficient are obtained from simulations. The inertial effects on microscopic velocity fields are also discussed. The pore-scale network modeling of non-Darcy flow is also developed based on simulation results from rigorous model (FEM). Network modeling is an appealing approach to study porous media. Because of the approximation introduced in both pore structures and fluid dynamics, network modeling requires much smaller computational cost than rigorous model and can increase the computational domain size by orders of magnitude. The network is validated by comparing pore-scale flowrate distribution calculated from network and FEM. Throat flowrates and hydraulic conductance values in pore structures with a range of geometries are compared to assess whether network modeling can capture the shifts in flow pattern due to inertial effects. This provides insights about predicting hydraulic conductance using the tortuosity of flow paths,which is a significant factor for inertial flow as well as other network pore and throat geometric parameters

Controlling the Short-Range Order and Packing Densities of Many-Particle Systems

Questions surrounding the spatial disposition of particles in various condensed-matter systems continue to pose many theoretical challenges. This paper explores the geometric availability of amorphous many-particle configurations that conform to a given pair correlation function g(r). Such a study is required to observe the basic constraints of non-negativity for g(r) as well as for its structure factor S(k). The hard sphere case receives special attention, to help identify what qualitative features play significant roles in determining upper limits to maximum amorphous packing densities. For that purpose, a five-parameter test family of g's has been considered, which incorporates the known features of core exclusion, contact pairs, and damped oscillatory short-range order beyond contact. Numerical optimization over this five-parameter set produces a maximum-packing value for the fraction of covered volume, and about 5.8 for the mean contact number, both of which are within the range of previous experimental and simulational packing results. However, the corresponding maximum-density g(r) and S(k) display some unexpected characteristics. A byproduct of our investigation is a lower bound on the maximum density for random sphere packings in $d$ dimensions, which is sharper than a well-known lower bound for regular lattice packings for d >= 3.Comment: Appeared in Journal of Physical Chemistry B, vol. 106, 8354 (2002). Note Errata for the journal article concerning typographical errors in Eq. (11) can be found at http://cherrypit.princeton.edu/papers.html However, the current draft on Cond-Mat (posted on August 8, 2002) is correct