Optimizing packing fraction in granular media composed of overlapping spheres

Optimal three-dimensional particle shapes for maximally dense saturated packing

Saturated packing is a random packing state of particles widely applied in investigating the physicochemical properties of granular materials. Optimizing particle shape to maximize packing density is a crucial challenge in saturated packing research. The known optimal three-dimensional shape is an ellipsoid with a saturated packing density of 0.437 72(51). In this work, we generate saturated packings of three-dimensional asymmetric shapes, including spherocylinders, cones, and tetrahedra, via the random sequential adsorption algorithm and investigate their packing properties. Results show that the optimal shape of asymmetric spherocylinders gives the maximum density of 0.4338(9), while cones achieve a higher value of 0.4398(10). Interestingly, tetrahedra exhibit two distinct optimal shapes with significantly high densities of 0.4789(19) and 0.4769(18), which surpass all previous results in saturated packing. The study of adsorption kinetics reveals that the two optimal shapes of tetrahedra demonstrate notably higher degrees of freedom and faster growth rates of the particle number. The analysis of packing structures via the density pair-correlation function shows that the two optimal shapes of tetrahedra possess faster transitions from local to global packing densities.

Contact networks and force transmission in aggregates of hexapod-shaped particles

Hexapods, consisting of three mutually orthogonal arms, have been utilized as a representative nonconvex shape to demonstrate the impact of interlocking on the strength properties of granular materials. Nevertheless, the microstructural characteristics of hexapod packings, which underlie their strength, have remained insufficiently characterized. We use particle dynamics simulations to build isotropically-packed aggregates of hexapods and we analyze the effects of aspect ratio and interparticle friction on the microstructure and force transmission. We find that the packing fraction is an unmonotonic function of aspect ratio due to competition between steric exclusions and interlocking. Interestingly, the contact coordination number declines considerably with friction coefficient, showing the stronger effect of friction on the stability of hexapod packings as compared with sphere packings. The pair distribution functions show that local ordering due to steric exclusions disappears beyond the aspect ratio 3 and the hexapods touch their second neighbors. Remarkably, hexapods of aspect ratio 3 tend to align with their neighbors and form locally ordered structures, implying a contact coordination number which is highly sensitive to the confining pressure. We also show that the probability density function of forces between hexapods is similar to that of sphere packings but with broadening exponential fall-off of strong forces as aspect ratio increases. Finally, the elastic bulk modulus of the aggregates is found to increase considerably with aspect ratio as a consequence of the rapid increase of contact density and the number of contacts with second neighbors.

Machine learning approaches for the optimization of packing densities in granular matter

The fundamental question of how densely granular matter can pack and how this density depends on the shape of the constituent particles has been a longstanding scientific problem. Previous work has mainly focused on empirical approaches based on simulations or mean-field theory to investigate the effect of shape variation on the resulting packing densities, focusing on a small set of pre-defined shapes like dimers, ellipsoids, and spherocylinders. Here we discuss how machine learning methods can support the search for optimally dense packing shapes in a high-dimensional shape space. We apply dimensional reduction and regression techniques based on random forests and neural networks to find novel dense packing shapes by numerical optimization. Moreover, an investigation of the regression function in the dimensionally reduced shape representation allows us to identify directions in the packing density landscape that lead to a strongly non-monotonic variation of the packing density. The predictions obtained by machine learning are compared with packing simulations. Our approach can be more widely applied to optimize the properties of granular matter by varying the shape of its constituent particles.

Optimal shapes of disk assembly in saturated random packings

Particle morphology is one of the most significant factors influencing the packing structures of granular materials. With certain targeted properties or optimization criteria, inverse packing problems have drawn extensive attention in terms of their adaptability to many material design tasks. An important question hard to answer is which particle shape, especially within given shape families, forms the densest (loosest) random packing? In this paper, we address this issue for the disk assembly model in two dimensions with an infinite variety of shapes, which are simulated in the random sequential adsorption process to suppress crystallization. Via a unique shape representation method, particle shapes are transformed into genotype sequences in the continuous shape space where we utilize the genetic algorithm as an efficient shape optimizer. Specifically, we consider three representative species of disk assembly, i.e., congruent tangent disks, incongruent tangent disks, and congruent overlapping disks, and carry out shape optimization on their packing densities in the saturated random state. We numerically search optimal shapes in the three species with a variable number of constituent disks which yield the maximal and minimal packing densities. We obtain an isosceles circulo-triangle and an unclosed ring for the maximal and minimal packing density in saturated random packings, respectively. The perfect sno-cone and isosceles circulo-triangle are also specifically investigated which give remarkably high packing densities of around 0.6, much denser than those of ellipses. This study is beneficial for guiding the design of particle shapes as well as the inverse design of granular materials.

Efficient DEM simulations of railway ballast using simple particle shapes

For complex shaped materials, computational efficiency and accuracy of DEM models are usually opposing requirements. In the literature, DEM models of railway ballast often use very complex and computationally demanding particle shapes in combination with very simple contact laws. In contrast, this study suggests efficient DEM models for railway ballast using simple particle shapes together with a contact law including more physical effects. In previous works of the authors, shape descriptors, calculated in a shape analysis of two types of ballast, were used to construct simple particle shapes (clumps of three spheres). Using such a shape in DEM simulations of compression and direct shear tests, accurate results were achieved only when the contact law included additional physical effects e.g. edge breakage. A parametrisation strategy was developed for this contact law comparing DEM simulations with the measurements. Now, all the constructed simple particle shapes are parametrised allowing to study their suitability and relating their shape descriptors to those of railway ballast. The most suitable particle shapes consist of non-overlapping spheres, thus have a high interlocking potential, and have lowest sphericity and highest convexity values. In a micromechanical analysis of the four best performing shapes, three shapes show similar behaviour on the bulk and the micro-scale, while one shape differs clearly on the micro-scale. This analysis shows, which shapes can be expected to produce similar results in DEM simulations of other tests/load cases. The presented approach is a step towards both efficient and accurate DEM modelling of railway ballast.

DEM modelling of railway ballast using the Conical Damage Model: a comprehensive parametrisation strategy

Despite ongoing research, the parametrisation of a DEM model is a challenging task, as it depends strongly on the particle shape representation used, particle-particle contact law and the simulated applications: for railway ballast e.g. lab tests or track conditions. The authors previously modelled railway ballast with a DEM model using a simple particle shape. The DEM model was parametrised, by trial-and-error, to compression and direct shear test results. A good agreement between DEM model and experimental results was achieved only when the Conical Damage Model (CDM) was used as the contact law. Compared to the well-known linear-spring Cundall-Strack law or the Hertz-Mindlin law, this contact law takes into account additional physical effects (e.g. edge breakage) occurring in the experiment. Little is known on the influence of the CDM model parameters on the simulation results or on possible parameter ambiguities. This lack of knowledge hinders a reliable and efficient parametrisation of DEM models using different particle shapes. Both points are addressed in this work in detail by investigating a DEM model for railway ballast using one simple particle shape. Suggestions for a parametrisation strategy of reduced computational effort are formulated and tested using a second particle shape. In future works, the newly presented parametrisation strategy can help to calibrate different DEM models and to study the influence of particle shape.

Structural analysis of disordered dimer packings

Jammed disordered packings of non-spherical particles show significant variation in the packing density as a function of particle shape for a given packing protocol. Rotationally symmetric elongated shapes such as ellipsoids, spherocylinders, and dimers, e.g., pack significantly denser than spheres over a narrow range of aspect ratios, exhibiting a characteristic peak at aspect ratios of α_max ≈ 1.4-1.5. However, the structural features that underlie this non-monotonic behaviour in the packing density are unknown. Here, we study disordered packings of frictionless dimers in three dimensions generated by a gravitational pouring protocol in LAMMPS. Focusing on the characteristics of contacts as well as orientational and translational order metrics, we identify a number of structural features that accompany the formation of maximally dense packings as the dimer aspect ratio α is varied from the spherical limit. Our results highlight that dimer packings undergo significant structural changes as α increases up to α_max manifest in the reorganisation of the contact configurations between neighbouring dimers, increasing nematic order, and decreasing local translational order. Remarkably, for α > α_max our metrics remain largely unchanged, indicating that the peak in the packing density is related to the interplay of structural rearrangements for α < α_max and subsequent excluded volume effects with unchanged structure for α > α_max.

Excluded area of superellipse sector particles

Superellipse sector particles (SeSPs) are segments of superelliptical curves that form a tunable set of hard-particle shapes for granular and colloidal systems. SeSPs allow for continuous parametrization of corner sharpness, aspect ratio, and particle curvature; rods, circles, rectangles, and staples are examples of shapes SeSPs can model. We investigate the space of allowable (nonoverlapping) configurations of two SeSPs, which depends on both the center-of-mass separation and relative orientation. Radial correlation plots of the allowed configurations reveal circular regions centered at each of the particle's two end points that indicate configurations of mutually entangled particle interactions. Simultaneous entanglement with both end points is geometrically impossible; the overlap of these two regions therefore represents an excluded area in which no particles can be placed regardless of orientation. The regions' distinct boundaries indicate a translational frustration with implications for the dynamics of particle rearrangements (e.g., under shear). Representing translational and rotational degrees of freedom as a hypervolume, we find a topological change that suggests geometric frustration arises from a phase transition in this space. The excluded area is a straightforward integration over excluded states; for arbitrary relative orientation this decreases sigmoidally with increasing opening aperture, with sharper SeSP corners resulting in a sharper decrease. Together, this work offers a path towards a unified theory for particle shape control of bulk material properties.

Simple particle shapes for DEM simulations of railway ballast: influence of shape descriptors on packing behaviour

ABSTRACT

In any DEM simulation, the chosen particle shape will greatly influence the simulated material behaviour. For a specific material, e.g. railway ballast, it remains an open question how to model the particle shape, such that DEM simulations are computationally efficient and simulation results are in good accordance with measurements. While DEM shape modelling for railway ballast is well addressed in the literature, approaches mainly aim at approximating the stones' actual shape, resulting in rather complex and thus inefficient particle shapes. In contrast, very simple DEM shapes will be constructed, clumps of three spheres, which aim to approximate shape descriptors of the considered ballast material. In DEM simulations of the packing behaviour, a set of clump shapes is identified, which can pack at porosities observed at track sites, as well as in lab tests. The relation between particle shape (descriptors) and obtained packing (characteristic) is investigated in a correlation analysis. The simulated packing's porosity is strongly correlated to four shape descriptors, which are also strongly correlated among each other. Thus, to derive simple shape models of a given particle shape, matching one of these shape descriptors, might be a good first step to bring simulated porosities closer to measured ones. The conducted correlation analysis also shows that packing's coordination number and isotropic fabric are correlated to more shape descriptors, making it more difficult to estimate the effect of particle shape on these quantities.

Densest versus jammed packings of bent-core trimers

We identify putatively maximally dense packings of tangent-sphere trimers with fixed bond angles (θ=θ_{0}), and contrast them to the disordered jammed states they form under quasistatic and dynamic athermal compression. Incommensurability of θ_{0} with three-dimensional (3D) close packing does not by itself inhibit formation of dense 3D crystals; all θ_{0} allow formation of crystals with ϕ_{max}(θ_{0})>0.97ϕ_{cp}. Trimers are always able to arrange into periodic structures composed of close-packed bilayers or trilayers of triangular-lattice planes, separated by "gap layers" that accommodate the incommensurability. All systems have ϕ_{J} significantly below the monomeric value, indicating that trimers' quenched bond-length and bond-angle constraints always act to promote jamming. ϕ_{J} varies strongly with θ_{0}; straight (θ_{0}=0) trimers minimize ϕ_{J} while closed (θ_{0}=120^{∘}) trimers maximize it. Marginally jammed states of trimers with lower ϕ_{J}(θ_{0}) exhibit quantifiably greater disorder, and the lower ϕ_{J} for small θ_{0} is apparently caused by trimers' decreasing effective configurational freedom as they approach linearity.

Random sequential adsorption of cuboids

The subject of this study was random sequential adsorption of cuboids of axes length ratio of a : 1 : b for a ∈ [0.3, 1.0] and b ∈ [1.0, 2.0], and the aim of this study was to find a shape that provides the highest packing fraction. The obtained results show that the densest packing fraction is 0.401 87 ± 0.000 97 and is reached for axes ratios near cuboids of 0.75:1:1.30. Kinetics of packing growth was also studied, and it was observed that its power-law character seems not to be governed by the number of cuboid degrees of freedom. The microstructural properties of obtained packings were studied in terms of density correlation function and propagation of orientational ordering.

Anisotropic particles strengthen granular pillars under compression

We probe the effects of particle shape on the global and local behavior of a two-dimensional granular pillar, acting as a proxy for a disordered solid, under uniaxial compression. This geometry allows for direct measurement of global material response, as well as tracking of all individual particle trajectories. In general, drawing connections between local structure and local dynamics can be challenging in amorphous materials due to lower precision of atomic positions, so this study aims to elucidate such connections. We vary local interactions by using three different particle shapes: discrete circular grains (monomers), pairs of grains bonded together (dimers), and groups of three bonded in a triangle (trimers). We find that dimers substantially strengthen the pillar and the degree of this effect is determined by orientational order in the initial condition. In addition, while the three particle shapes form void regions at distinct rates, we find that anisotropies in the local amorphous structure remain robust through the definition of a metric that quantifies packing anisotropy. Finally, we highlight connections between local deformation rates and local structure.

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles

Random sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage ∼t^{-ν} as t→∞. However, the exact value of the exponent ν is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi's seminal "car parking problem"), where ν=1 can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general nonspherical particles, ν=1/(d+d[over ˜]), where d denotes the dimension of the domain, and d[over ˜] the number of orientational degrees of freedom of a particle. Here, we solve this long-standing problem analytically for the d=1 case-the "Paris car parking problem." We prove, in particular, that the scaling exponent depends on the particle shape, contrary to the original conjecture and, remarkably, falls into two universality classes: (i) ν=1/(1+d[over ˜]/2) for shapes with a smooth contact distance, e.g., ellipsoids, and (ii) ν=1/(1+d[over ˜]) for shapes with a singular contact distance, e.g., spherocylinders and polyhedra. The exact solution explains, in particular, why many empirically observed scalings fall in between these two limits.