Close Packing Limit Research Articles

SIMPLIFIED HARD-SPHERE AND HARD-SPHERE CHAIN EQUATIONS OF STATE FOR ENGINEERING APPLICATIONS

A simple hard-sphere equation of state is proposed. This hard-sphere equation is a ratio of second-order polynomials that meets the ideal gas and close-packed density limits. It predicts the compressibility of hard-sphere fluids at low and medium densities to within a degree of quality similar to the widely used Carnahan-Starling equation. In addition, the proposed equation performs better at high densities, particularly near the close-packed density. An expression is also derived to relate the site-site correlation function at contact for hard dimers with the site-site correlation function at contact for hard spheres. With this relationship, the thermodynamic perturbation-dimer theory (TPT-D) of hard-sphere chains is simplified. The new theory performs comparably with the TPT-D when the compressibility factors of hard-sphere chain fluids containing up to 201-mer are predicted, however, it has the advantages of both simplicity and accuracy. From a practical perspective, this theory can be used to construct equations of state for polymer solutions or fluid systems containing short- and long-chain molecules.

Elastic properties of the degenerate crystalline phase of two-dimensional hard dimers

Hard homonuclear dimers, each consisting of two identical and tangent hard disks, exhibit a thermodynamically stable, aperiodic solid phase, known as a degenerate crystal. The elastic properties of this phase are determined by the box fluctuation Monte Carlo method in the NpT ensemble. The computed elastic constants are well described by the free volume approximation and the coefficients of the leading singularities are estimated in the close packing limit. Three mechanisms decreasing Poisson’s ratio are indicated: one is based on increasing the pressure to temperature ratio, another comes from the constrains imposed on the disks forming a dimer, yet another, working in the close packing limit, arises from molecular motions. It is also shown that Poisson’s ratio increases with increasing concentration of vacancies.

Expansion evolution during foaming of amorphous metals

Amorphous Pd 43Ni 10Cu 27P 20 foam of 38, 49, and 70% porosity is produced by expanding a 25% porosity amorphous precursor in the supercooled liquid state in three isothermal stages of varying durations. Metallographic examination suggests that expansion evolves by bubble growth towards a limit at which bubbles become critically packed. This limit is found to be close to the limit of random close packing of spheres of 63.7%. Beyond this critical limit, bubbles tend to impinge and coalesce and expansion progresses by growth of bubble clusters via stretching of intracellular membranes. The expansion evolution is modeled by means of a dynamic treatment of over-damped growth of individual bubbles. The model captures the expansion evolution reasonably well up to porosities between 50–70%, hence verifying that a transition from a bubble growth mechanism to a cluster growth mechanism takes place at some intermediate porosity.

The nanogranular nature of C–S–H

Despite its ubiquitous presence as binding phase in all cementitious materials, the mechanical behavior of calcium–silicate–hydrates (C–S–H) is still an enigma that has deceived many decoding attempts from experimental and theoretical sides. In this paper, we propose and validate a new technique and experimental protocol to rationally assess the nanomechanical behavior of C–S–H based on a statistical analysis of hundreds of nanoindentation tests. By means of this grid indentation technique we identify in situ two structurally distinct but compositionally similar C–S–H phases heretofore hypothesized to exist as low density (LD) C–S–H and high density (HD) C–S–H, or outer and inner products. The main finding of this paper is that both phases exhibit a unique nanogranular behavior which is driven by particle-to-particle contact forces rather than by mineral properties. We argue that this nanomechanical blueprint of material invariant behavior of C–S–H is a consequence of the hydration reactions during which precipitating C–S–H nanoparticles percolate generating contact surfaces. As hydration proceeds, these nanoparticles pack closer to center on-average around two characteristic limit packing densities, the random packing limit ( η = 64 % ) and the ordered face-centered cubic (fcc) or hexagonal close-packed (hcp) packing limit ( η = 74 % ), forming a characteristic LD C–S–H and HD C–S–H phase.

Equation of state for hard-spheres and excess function calculations of binary liquid mixtures

Using our previously proposed matrix method, an equation of state for hard spheres is presented, which can reproduce the exact values of the first-eight virial coefficients. This equation meets both the low density and the close-packed limits and can predicts the first order fluid–solid phase transition of hard spheres. The results obtained show that the new equation of state can correlate the simulation data of compressibility factor up to high densities better than other equations of state. The new equation of state is extended to mixtures of hard spheres and excess functions of various binary liquid mixtures are calculated using the perturbation theory of Leonard–Henderson–Barker. The results are compared with existing theoretical and experimental data and with those calculated by other hard-sphere equations of state. It is seen that the results obtained by the new equation of state is quite satisfactory compared to other equations of state for the hard spheres and mixture of hard spheres.

Capturing shock waves in inelastic granular gases

Shock waves in granular gases generated by hitting an obstacle at rest are treated by means of a shock capturing scheme that approximates the Euler equations of granular gas dynamics with an equation of state (EOS), introduced by Goldshtein and Shapiro [J. Fluid Mech. 282 (1995) 75–114], that takes into account the inelastic collisions of granules. We include a sink term in the energy balance to account for dissipation of the granular motion by collisional inelasticity, proposed by Haff [J. Fluid Mech. 134 (1983) 401–430], and the gravity field added as source terms. We have computed the approximate solution to a one-dimensional granular gas falling on a plate under the acceleration of gravity until the close-packed limit.

A hard-sphere model on generalized Bethe lattices: statics

We analyse the phase diagram of a model of hard spheres of chemical radiusone, which is defined over a generalized Bethe lattice containing short loops. Wefind a liquid, two different crystalline, a glassy and an unusual crystalline glassyphase. Special attention is also paid to the close-packing limit in the glassy phase.All analytical results are cross-checked by numerical Monte Carlo simulations.

Monte Carlo simulation of two‐dimensional hard body systems with extreme values of the Poisson's ratio

AbstractMonte Carlo computer simulations with variable shape of the periodic box were performed for a few two‐dimensional, hard‐body models showing elastically isotropic solid phases. Two examples of homo‐molecular (i.e. consisting of identical molecules) systems of anisotropic molecules are discussed which form unusual solid phases, showing negative Poisson's ratio and know as auxetic ones, at densities and pressures higher than the elastically isotropic solid phases. Sharp extremes of the Poisson's ratio were observed for both systems at freezing of the orientational degrees of freedom of the molecules: a minimum was observed for the system for which the auxetic phase with frozen rotation is isotropic and a maximum was found for the other system. This indicates that isotropic auxetic phases can have auxetic precursors with much more negative Poisson's ratios than those characterizing the auxetic phases themselves. An example of a hetero‐molecular (i.e. mixture consisting of different molecules), elastically isotropic system is also discussed – the polydisperse hard disc system. The simulations revealed a maximum of the Poisson's ratio in the close packing limit at any nonzero polydispersity. Although no manifestation of any auxetic behaviour has been found in this system, the obtained results indicate other unusual effect – a jump (discontinuity) of the Poisson's ratio at close packing when the polydispersity tends to zero. The results obtained in this work demonstrate that the Poisson's ratio can play the role of a sensitive indicator of (at least) some structural changes in solids. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A model for the interior structure, evolution, and differentiation of Callisto

The recently measured dimensionless moment of inertia (MoI) factor for Callisto of 0.3549±0.0042 (Anderson et al., 2001, Icarus, 153, 157–161) poses a problem: its value cannot be explained by a model in which Callisto is completely differentiated into an ice shell above a rock shell and an iron core such as its neighboring satellite Ganymede nor can it be explained by a model of a homogeneous, undifferentiated ice–rock satellite. We show that Callisto may be incompletely differentiated into an outer ice–rock shell in which the volumetric rock concentration is close to the primordial one at the surface and decreases approximately linearly with depth, an ice mantle mostly depleted of rock, and an about 1800 km rock–ice core in which the rock concentration is close to the close-packing limit. The ice–rock shell thickness depends on uncertain rheology parameters and the heat flow and can be roughly 50 to 150 km thick. We show that if Callisto accreted from a mix of metal bearing rock and ice and if the average size of the rocks was of the order of meters to tens of meters, then Callisto may have experienced a gradual, but still incomplete unmixing of the two components. An ocean in Callisto at a depth of 100–200 km is difficult to obtain if the ice is pure H 2O and if the ice–rock lithosphere is 100 km or more thick; a water ocean is more plausible for ice contaminated by ammonia, methane or salts; or for pure H 2O at a depth of 400–600 km.

An exploding glass?

We draw analogies between self-similar, focusing dynamics in nonlinear partial differential equations (PDEs) and macroscopic dynamic features of the glass transition. In particular, we explore the divergence of the appropriate relaxation times in the case of hard spheres as the limit of random close packing is approached. We illustrate the analogy in the critical case, and suggest a description that can capture the onset of dynamic self-similarity in both phenomena.

Elastic properties of two-dimensional hard disks in the close-packing limit

Elastic constants and the Poisson ratio of defect-free hard-disk solid are determined by two independent methods: (1) analysis of the box side fluctuations in the N-p-T ensemble with variable box shape and (2) numerical differentiation (with respect to strain components) of the free energy computed in the N-V-T ensemble; N, p, V, and T denote the number of particles, the pressure, the volume, and the temperature, respectively. The efficiency of the applied methods is compared. It is shown that reasonable estimates of the elastic properties can be obtained by studying small systems in the N-p-T ensemble and that the singular behavior of the elastic constants near close packing is well described by the free volume approximation; the coefficients of the leading singularities are estimated.

Effect of charge reduction on shielding in dusty plasmas

The shielding of dust particles by each other in a dusty plasma is studied taking into account the effects of dust charging. It is shown that the assumption of a Boltzmann response for dust species is not appropriate under these circumstances. Further, it is shown that in the close-packing limit, dust grains screen each other by charge reduction, and an expression is obtained for this shielding scale length.

A General Expression for the Ordered-Packed Volume Fraction of Hard Spheres of Different Diameters

An expression is proposed to evaluate the ordered-packed volume fraction of hard spheres of unequal diameters in multicomponent mixtures. This value can be used with different mixing rules and with any one-component hard-sphere equation of state (EOS) meeting the correct close-packed limit. It is shown that the new method improves the results produced by conventional mixing rules for the individual radial distribution functions (RDFs) at contact value of hard-sphere mixtures. Particular attention is given to the case of the large diameter ratio especially when the larger spheres are at low concentration. Based on the observation that, in a binary mixture, the value of the RDF at contact value of the larger spheres approaches the value of the RDF at contact value of a one-component hard-sphere fluid, the proposed expression is used to correct the RDF at contact value of a pure hard-sphere fluid in the Santos et al. mixing rule. The EOS obtained by combining the Ghotbi−Vera one-component EOS with the correc...

Moderate-Reynolds-number flows in ordered and random arrays of spheres

Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is calculated as a function of the Reynolds number at solid volume fractions up to the close-packed limits of the arrays. At Reynolds numbers up to O(102), the non-dimensional drag force has a more complex dependence on the Reynolds number and the solid volume fraction than suggested by the well-known Ergun correlation, particularly at solid volume fractions smaller than those that can be achieved in physical experiments. However, good agreement is found between the simulations and Ergun's correlation at solid volume fractions approaching the close-packed limit. For ordered arrays, the drag force is further complicated by its dependence on the direction of the flow relative to the axes of the arrays, even though in the absence of fluid inertia the permeability is isotropic. Visualizations of the flows are used to help interpret the numerical results. For random arrays, the transition to unsteady flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are discussed.

The first effects of fluid inertia on flows in ordered and random arrays of spheres

Theory and lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is determined at small but finite Reynolds numbers, at solid volume fractions up to the close-packed limits of the arrays. For small solid volume fraction, the simulations are compared to theory, showing that the first inertial contribution to the drag force, when scaled with the Stokes drag force on a single sphere in an unbounded fluid, is proportional to the square of the Reynolds number. The simulations show that this scaling persists at solid volume fractions up to the close-packed limits of the arrays, and that the first inertial contribution to the drag force relative to the Stokes-flow drag force decreases with increasing solid volume fraction. The temporal evolution of the spatially averaged velocity and the drag force is examined when the fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. Theory for the short- and long-time behaviour is in good agreement with simulations, showing that the unsteady force is dominated by quasi-steady drag and added-mass forces. The short- and long-time added-mass coefficients are obtained from potential-flow and quasi-steady viscous-flow approximations, respectively.

Modeling and numerical simulation of particulate flows by the Eulerian–Lagrangian approach

In this paper we present an Eulerian–Lagrangian numerical simulation (LNS) scheme for particulate flows. The overall algorithm in the present approach is a variation of the scheme presented earlier. In this numerical scheme we solve the fluid phase continuity and momentum equations on an Eulerian grid. The particle motion is governed by Newton's law thus following the Lagrangian approach. Momentum exchange from the particle to fluid is modeled in the fluid phase momentum equation. Forces acting on the particles include drag from the fluid, body force and the interparticle force that prevents the particle volume fraction from exceeding the close-packing limit. There is freedom to use different models for these forces and to introduce other forces. In this paper we have used two types of interparticle forces. The effect of viscous stresses are included in the fluid phase equations. The volume fraction of the particles appear in the fluid phase continuity and momentum equations. The fluid and particle momentum equations are coupled in the solution procedure unlike an earlier approach. A finite volume method is used to solve these equations on an Eulerian grid. Particle positions are updated explicitly. This numerical scheme can handle a range of particle loadings and particle types. We solve the fluid phase continuity and momentum equations using a Chorin-type fractional-step method. The numerical scheme is verified by comparing results with test cases and experiments.

Direct laser fabrication of dense microlens arrays in semiconductor-doped glass

A direct laser writing technique in semiconductor-doped glasses is used to fabricate microlens arrays with fill factors exceeding the close-packing limit for circular microlenses. This is achieved by both structural and thermal interactions between adjacent lenses during fabrication, which result in the formation of microlenses shaped in accordance with the translational symmetry of the array. The effect of nearest-neighbor interaction on various time scales is studied and shown to be critical to the distortions observed.

Triangle Distribution and Equation of State for Classical Rigid Disks

The triangle distribution function f(3) for three mutual near neighbors in the plane describes basic aspects of short-range order and statistical thermodynamics in two-dimensional many-particle systems. This paper examines prospects for constructing a self-consistent calculation for the rigid-disk-system f(3). We present several identities obeyed by f(3). A rudimentary closure suggested by scaled-particle theory is introduced. In conjunction with three of the basic identities, this closure leads to an unique f(3) over the entire density range. The pressure equation of state exhibits qualitatively correct behaviors in both the low-density and the close-packed limits, but no intervening phase transition appears. We discuss extensions to improved disk closures, and to the three-dimensional rigid-sphere system.

Generalized density-functional theory: extended weighted density approaches.

A third-order density-functional theory is introduced by an approach that may be used to find density-functional theories to any higher-order accuracy provided only that known homogeneous state correlation functions are utilized as input. It is constructed from the required knowledge of a single weight function at each order. By way of application results are presented for the melting of classical hard spheres using functionals accurate to a third-order functional Taylor series in the homogeneous limit. Within the framework of the modified weighted density approximation, there is a uniform improvement in the solid phase-free energies, pressures and melting parameters, and further improvement also results when these functionals are optimized in a way that utilizes the close packing limit. The sensitivity of the results to existing and proposed models of the third-order direct correlation function is discussed.

Upper and lower bounds for the kinetics of hard-particle adsorption

We compare upper and lower bounds for the rate of the reversible, cooperative adsorption of hard particles from a reservoir at constant activity to a lattice surface where the only cooperative effect is excluded volume. The adsorption rate is proportional to the density of groups of unoccupied lattice sites: holes, large enough to accommodate a particle. The bounds on the rate of adsorption are then bounds on the density of holes. The upper bound for particles that are infinitely mobile on the surface is obtained from the equilibrium Mayer activity series for the pressure, while the lower bound is obtained from the extensive exact series calculated by Gan and Wang [J. Chem. Phys. 108, 3010 (1998)] for the case of irreversible random sequential adsorption where the particles are immobile once adsorbed. In all cases the bounds coincide at low densities. For the one-dimensional lattice with nearest-neighbor exclusion (where the bounds are known exactly) the upper and lower bounds are very close for all densities below the limit of random close packing as they are for the adsorption of hard dimers on two-dimensional lattices. Thus in these cases equilibrium statistical mechanics can give useful information about the kinetics of cooperative processes.