Jamming, glass transition, and entropy in monodisperse and polydisperse hard-sphere packings
This thesis is dedicated to the investigation of properties of computer-generated monodisperse and polydisperse three-dimensional hard-sphere packings, frictional and frictionless. For frictionless packings, we (i) assess their total (fluid) entropy in a wide range of packing densities (solid volum...
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|Zusammenfassung:||This thesis is dedicated to the investigation of properties of computer-generated monodisperse and polydisperse three-dimensional hard-sphere packings, frictional and frictionless. For frictionless packings, we (i) assess their total (fluid) entropy in a wide range of packing densities (solid volume fractions), (ii) investigate the structure of their phase space, (iii) and estimate several characteristic densities (the J-point, the ideal glass transition density, and the ideal glass density). For frictional packings, we estimate the Edwards entropy in a wide range of densities. We utilize the Lubachevsky–Stillinger, Jodrey–Tory, and force-biased packing generation algorithms. We always generate packings of 10000 particles in cubic boxes with periodic boundary conditions. For estimation of the Edwards entropy, we also use experimentally produced and reconstructed packings of fluidized beds. In polydisperse cases, we use the log-normal, Pareto, and Gaussian particle diameter distributions with polydispersities (relative radii standard deviations) from 0.05 (5%) to 0.3 (30%) in steps of 0.05. This work consists of six chapters, each corresponding to a published paper. In the first chapter, we introduce a method to estimate the probability to insert a particle in a packing (insertion probability) through the so-called pore-size (nearest neighbour) distribution. Under certain assumptions about the structure of the phase space, we link this probability to the (total) entropy of packings. In this chapter, we use only frictionless monodisperse hard-sphere packings. We conclude that the two characteristic particle volume fractions (or densities, φ) often associated with the Random Close Packing limit, φ ≈ 0.64 and φ ≈ 0.65, may refer to two distinct phenomena: the J-point and the Glass Close Packing limit (the ideal glass density), respectively. In the second chapter, we investigate the behaviour of jamming densities of frictionless polydisperse packings produced with different packing generation times. Packings produced quickly are structurally closer to Poisson packings and jam at the J-point (φ ≈ 0.64 for monodisperse packings). Jamming densities (inherent structure densities) of packings with sufficient polydispersity that were produced slowly approach the glass close packing (GCP) limit. Monodisperse packings overcome the GCP limit (φ ≈ 0.65) because they can incorporate crystalline regions. Their jamming densities eventually approach the face-centered cubic (FCC) / hexagonal close packing (HCP) crystal density φ = π/(3 √2) ≈ 0.74. These results support the premise that φ ≈ 0.64 and φ ≈ 0.65 in the monodisperse case may refer to the J-point and the GCP limit, respectively. Frictionless random jammed packings can be produced with any density in-between. In the third chapter, we add one more intermediate step to the procedure from the second chapter. We take the unjammed (initial) packings in a wide range of densities from the second chapter, equilibrate them, and only then jam (search for their inherent structures). Thus, we investigate the structure of their phase space. We determine the J-point, ideal glass transition density, and ideal glass density. We once again recover φ ≈ 0.64 as the J-point and φ ≈ 0.65 as the GCP limit for monodisperse packings. The ideal glass transition density for monodisperse packings is estimated at φ ≈ 0.585. In the fourth chapter, we demonstrate that the excess entropies of the polydisperse hard-sphere fluid at our estimates of the ideal glass transition densities do not significantly depend on the particle size distribution. This suggests a simple procedure to estimate the ideal glass transition density for an arbitrary particle size distribution by solving an equation, which requires that the excess fluid entropy shall equal to some universal value characteristic of the ideal glass transition density. Excess entropies for an arbitrary particle size distribution and density can be computed through equations of state, for example the Boublík–Mansoori–Carnahan–Starling–Leland (BMCSL) equation. In the fifth chapter, we improve the procedure from the first chapter. We retain the insertion probability estimation from the pore-size distribution, but switch from the initial assumptions about the structure of the phase space to a more advanced Widom particle insertion method, which for hard spheres links the insertion probability to the excess chemical potential. With the chemical potential at hand, we can estimate the excess fluid entropy, which complies well with theoretical predictions from the BMCSL equation of state. In the sixth chapter, we extend the Widom particle insertion method from the fifth chapter as well as the insertion probability estimation method from the first chapter to determine the upper bound on the Edwards entropy per particle in monodisperse frictional packings. The Edwards entropy counts the number of mechanically stable configurations at a given density (density interval). We demonstrate that the Edwards entropy estimate is maximum at the Random Loose Packing (RLP) limit (φ ≈ 0.55) and decreases with density increase. In this chapter, we accompany computer-generated packings with experimentally produced and reconstructed ones. Overall, this study extends the understanding of the glass transition, jamming, and the Edwards entropy behavior in the system of hard spheres. The results can help comprehend these phenomena in more complex molecular, colloidal, and granular systems.|