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

In this paper, we study the granular equation of 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 equation of 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.

Another resolution of the configurational entropy paradox as applied to hard spheres.

Ozawa and Berthier [J. Chem. Phys. 146, 014502 (2017)] recently studied the configurational and vibrational entropies Sconf and Svib from the relation Stot = Sconf + Svib for polydisperse mixtures of spheres. They noticed that because the total entropy per particle Stot/N shall contain the mixing entropy per particle kBsmix and Svib/N shall not, the configurational entropy per particle Sconf/N shall diverge in the thermodynamic limit for continuous polydispersity due to the diverging smix. They also provided a resolution for this paradox and related problems-it relies on a careful redefining of Sconf and Svib. Here, we note that the relation Stot = Sconf + Svib is essentially a geometric relation in the phase space and shall hold without redefining Sconf and Svib. We also note that Stot/N diverges with N → ∞ with continuous polydispersity as well. The usual way to avoid this and other difficulties with Stot/N is to work with the excess entropy ΔStot (relative to the ideal gas of the same polydispersity). Speedy applied this approach to the relation above in his work [Mol. Phys. 95, 169 (1998)] and wrote this relation as ΔStot = Sconf + ΔSvib. This form has flaws as well because Svib/N does not contain the kBsmix term and the latter is introduced into ΔSvib/N instead. Here, we suggest that this relation shall actually be written as ΔStot = ΔcSconf + ΔvSvib, where Δ = Δc + Δv, while ΔcSconf = Sconf - kBNsmix and ΔvSvib=Svib-kBN1+lnVΛdN+UNkBT with N, V, T, U, d, and Λ standing for the number of particles, volume, temperature, internal energy, dimensionality, and de Broglie wavelength, respectively. In this form, all the terms per particle are always finite for N → ∞ and continuous when introducing a small polydispersity to a monodisperse system. We also suggest that the Adam-Gibbs and related relations shall in fact contain ΔcSconf/N instead of Sconf/N.

Analysis of packing microstructure and wall effects in a narrow-bore ultrahigh pressure liquid chromatography column using focused ion-beam scanning electron microscopy.

Column wall effects are well recognized as major limiting factor in achieving high separation efficiency in HPLC. This is especially important for modern analytical columns packed with small particles, where wall effects dominate the band broadening. Detailed knowledge about the packing microstructure of packed analytical columns has so far not been acquired. Here, we present the first three-dimensional reconstruction protocol for these columns utilizing focused ion-beam scanning electron microscopy (FIB-SEM) on a commercial 2.1mm inner diameter×50mm length narrow-bore analytical column packed with 1.7µm bridged-ethyl hybrid silica particles. Two sections from the packed bed are chosen for reconstruction by FIB-SEM: one from the bulk packing region of the column and one from its critical wall region. This allows quantification of structural differences between the wall region and the center of the bed due to effects induced by the hard, confining column wall. Consequences of these effects on local flow velocity in the column are analyzed with flow simulations utilizing the lattice-Boltzmann method. The reconstructions of the bed structures reveal significant structural differences in the wall region (extending radially over approximately 62 particle diameters) compared to the center of the column. It includes the local reduction of the external porosity by up to 10% and an increase of the mean particle diameter by up to 3%, resulting in a decrease of the local flow velocity by up to 23%. In addition, four (more ordered) layers of particles in the direct vicinity of the column wall induce local velocity fluctuations by up to a factor of three regarding the involved velocity amplitudes. These observations highlight the impact of radial variations in packing microstructure on band migration and column performance. This knowledge on morphological peculiarities of column wall effects helps guiding us towards further optimization of the packing process for analytical HPLC columns.

Chemical potential and entropy in monodisperse and polydisperse hard-sphere fluids using Widom's particle insertion method and a pore size distribution-based insertion probability.

We estimate the excess chemical potential Δµ and excess entropy per particle Δs of computer-generated, monodisperse and polydisperse, frictionless hard-sphere fluids. For this purpose, we utilize the Widom particle insertion method, which for hard-sphere systems relates Δµ to the probability to successfully (without intersections) insert a particle into a system. This insertion probability is evaluated directly for each configuration of hard spheres by extrapolating to infinity the pore radii (nearest-surface) distribution and integrating its tail. The estimates of Δµ and Δs are compared to (and comply well with) predictions from the Boublík-Mansoori-Carnahan-Starling-Leland equation of state. For polydisperse spheres, we employ log-normal particle radii distributions with polydispersities δ = 0.1, 0.2, and 0.3.

Upper bound on the Edwards entropy in frictional monodisperse hard-sphere packings.

We extend the Widom particle insertion method [B. Widom, J. Chem. Phys., 1963, 39, 2808-2812] to determine an upper bound sub on the Edwards entropy in frictional hard-sphere packings. sub corresponds to the logarithm of the number of mechanically stable configurations for a given volume fraction and boundary conditions. To accomplish this, we extend the method for estimating the particle insertion probability through the pore-size distribution in frictionless packings [V. Baranau, et al., Soft Matter, 2013, 9, 3361-3372] to the case of frictional particles. We use computer-generated and experimentally obtained three-dimensional sphere packings with volume fractions φ in the range 0.551-0.65. We find that sub has a maximum in the vicinity of the Random Loose Packing Limit φRLP = 0.55 and decreases then monotonically with increasing φ to reach a minimum at φ = 0.65. Further on, sub does not distinguish between real mechanical stability and packings in close proximity to mechanical stable configurations. The probability to find a given number of contacts for a particle inserted in a large enough pore does not depend on φ, but it decreases strongly with the contact number.

How to predict the ideal glass transition density in polydisperse hard-sphere packings.

The formula for the entropy s of the accessible volume of the phase space for frictionless hard spheres is combined with the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state for polydisperse three-dimensional packings to obtain an analytical expression for s as a function of packing density φ. Polydisperse hard-sphere packings with log-normal, Gaussian, and Pareto particle diameter distributions are generated to estimate their ideal glass transition densities φg. The accessible entropy s at φg is almost the same for all investigated particle diameter distributions. We denote this entropy as sg and can predict φg for an arbitrary particle diameter distribution through an equation s(φ) = sg. If the BMCSL equation of state is used for s(φ), then φg is found to depend only on the first three moments of a particle diameter distribution.

Computational investigation of longitudinal diffusion, eddy dispersion, and trans-particle mass transfer in bulk, random packings of core-shell particles with varied shell thickness and shell diffusion coefficient.

In recent years, chromatographic columns packed with core-shell particles have been widely used for efficient and fast separations at comparatively low operating pressure. However, the influence of the porous shell properties on the mass transfer kinetics in core-shell packings is still not fully understood. We report on results obtained with a modeling approach to simulate three-dimensional advective-diffusive transport in bulk random packings of monosized core-shell particles, covering a range of reduced mobile phase flow velocities from 0.5 up to 1000. The impact of the effective diffusivity of analyte molecules in the porous shell and the shell thickness on the resulting plate height was investigated. An extension of Giddings' theory of coupled eddy dispersion to account for retention of analyte molecules due to stagnant regions in porous shells with zero mobile phase flow velocity is presented. The plate height equation involving a modified eddy dispersion term excellently describes simulated data obtained for particle-packings with varied shell thickness and shell diffusion coefficient. It is confirmed that the model of trans-particle mass transfer resistance of core-shell particles by Kaczmarski and Guiochon [42] is applicable up to a constant factor. We analyze individual contributions to the plate height from different mass transfer mechanisms in dependence of the shell parameters. The simulations demonstrate that a reduction of plate height in packings of core-shell relative to fully porous particles arises mainly due to reduced trans-particle mass transfer resistance and transchannel eddy dispersion.

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).

Random-close packing limits for monodisperse and polydisperse hard spheres.

We investigate how the densities of inherent structures, which we refer to as the closest jammed configurations, are distributed for packings of 10(4) frictionless hard spheres. A computational algorithm is introduced to generate closest jammed configurations and determine corresponding densities. Closest jamming densities for monodisperse packings generated with high compression rates using Lubachevsky-Stillinger and force-biased algorithms are distributed in a narrow density range from φ = 0.634-0.636 to φ≈ 0.64; closest jamming densities for monodisperse packings generated with low compression rates converge to φ≈ 0.65 and grow rapidly when crystallization starts with very low compression rates. We interpret φ≈ 0.64 as the random-close packing (RCP) limit and φ≈ 0.65 as a lower bound of the glass close packing (GCP) limit, whereas φ = 0.634-0.636 is attributed to another characteristic (lowest typical, LT) density φLT. The three characteristic densities φLT, φRCP, and φGCP are determined for polydisperse packings with log-normal sphere radii distributions.