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(Color online) Three typical packing states of tetrahedra. (a) The fully random sate: positions and orientations of particles are random and no particles form clusters. (b) The quasirandom state: particles assemble into clusters and these clusters are packed randomly. (c) The ordered state: particles are orientationally or positionally ordered.
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The disordered packings of tetrahedra often show no obvious macroscopic orientational or positional order for a wide range of packing densities, and it has been found that the local order in particle clusters is the main order form of tetrahedron packings. Therefore, a cluster analysis is carried out to investigate the local structures and properti...
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... fully random packing is defined as a packing in which the particles have no positional, orientational or local orders. Figure 3 illustrates three typical states, i.e., the fully random, quasirandom, and ordered state of tetrahedral particles. The quasirandom state shows no obvious positional or orientational order, but all particles in this packing form wagon wheels. ...
Citations
... There are also a number of computational [27][28][29] and experimental 30,31 works concerning random tetrahedron packing. Jin et al. 32 examined disordered packings of tetrahedra and performed cluster analysis. They found out that two special types of clusters are dominant, i.e. dimer and 5-unit wagon wheels. ...
There are a number of exceptional examples indicating the unique position of tetrahedral symmetry in the vast landscape of different spatial organization pathways which can be sampled by matter. This work shows that the design and analysis of relatively simple tetrahedron clusters can lead to the formulation of a new type of dendritic structure together with unique periodic frameworks resembling clathrates and foams. A simple sequential protocol leading from regular tetrahedron clusters to more complex structural motifs can be employed to determine interesting repetitive building units. Accordingly, four different hierarchical superstructures are introduced, in which the dominant population of nodes is based on tetrahedral symmetry. The introduced architectures could be of particular interest for the field of regenerative medicine and metamaterial engineering.
... Here, we focus on superballs [46,47] (w 1 = w 2 = 1) and ellipsoids [10,48,49] (p = 1, including prolates with w = w 2 > w 1 = 1, oblates with w = w 1 < w 2 = 1, self-dual ellipsoids with w 1 w 2 = 1, and other general ellipsoids with two different aspect ratios). For polyhedral particles, we investigate tetrahedron [29,[50][51][52][53], cube [52], as well as different types of cuboids [54] with aspect ratios defined via the three side lengths similar to those of a superellipsoid. The size of a particle is quantified by D v , which is the diameter of a sphere with equal volume to the particle. ...
... For polyhedra, a certain contact can be classified into distinct types of contact topologies formed by vertexes, edges, and facets, which constrain different numbers of degrees of freedom, namely 3 for a facet-facet contact, 2 for an edgefacet contact, and 1 for the rest. Considering the summed constraint number C (see Appendix A) for each particle on average, previous studies showed that C ≈ z 0 iso = 12 in jammed packings of frictionless polyhedra [29,45,53,64]. In our model, the constraint number for one contact between two polyhedra can be explicitly obtained without any geometric threshold. ...
... The fractions of different types of contact topologies as a function of μ for these polyhedra are given in Appendix D, Fig. 7(a). More importantly, in contrast with z J , C indeed approaches the isostatic value z 0 iso = 12 as μ → 0 as a result of facet-facet and edge-facet contacts [29,45,53,64]. For large μ, C z J ≈ 4 since the simple contacts dominate. ...
Understanding disordered particle packings is of great significance from both theoretical and engineering perspectives. Establishing a quantitative relationship between nonspherical particle shape and disordered packing properties is generally challenging, due to the complex geometry and topology. Here we resolve this issue by numerically investigating disordered jammed packings of various frictional congruent nonspherical particles, including superellipsoids and polyhedra, over a wide range of friction coefficients. We discover several universal packing characteristics across different particle shapes and frictions. In the infinite friction limit, the coordination numbers for all shapes approach the identical lower bound for jamming. The resulting “random loose packing” (RLP) state possesses minimal structural correlations, with the packing fraction as a simple monotonic decreasing function of the orientation-averaged excluded volume for different particle shapes. Packings with finite friction can then be understood via a perturbative approach based on RLP. The nature of RLP can be illuminated by the percolation transition of contacting particle network during the quasistatic densification process. Moreover, the large-scale density fluctuations for all jammed frictional packings are also strongly suppressed, broadening the previous claim of hyperuniformity for the frictionless ones.
... All cement particles in the fresh state of cement paste can be thought as a random packing system. There are many models that can be employed to simulate the random packing behavior of particles, such as the random sequential addition (RSA) model [25,58], adaptive shrinking cell scheme [55,59], discrete element method [60], molecular dynamic method [61], and relaxation iteration model [62]. In this study, the RSA model is used to simulate the state of cement particles randomly suspended in a cubic cell as a representative volume element (RVE). ...
A microstructure-guided diffusivity model of cement paste is devised to explore the dependence of the relative diffusivity on the microstructure evolutions of cement paste from fresh state via non-spherical cement particles hydration to hardened state. The microstructure-guided diffusivity model contains three main components: (1) the microstructure of fresh cement paste is generated to simulate the initial state of non-spherical cement particles in water; (2) a continuum-based hydration model of non-spherical cement particles (HYD-NSP) is proposed to describe the evolutions of various phases (hydration products and pores) from fresh state to hardened state; (3) a random walk model is implemented to the microstructures of hydrated cement paste digitalized at the resolution of 0.1 μm/voxel for the determination of relative diffusivity of cement paste. Although this study takes five kinds of Platonic particles and sphere as an introductory example, this microstructure-guided model is readily applicable to other complex microstructures induced by the hydration of non-spherical cement particles. Finally, we utilize the framework to evaluate the influence of the geometrical shape of cement particles on the hydrated microstructure and relative diffusivity of cement paste. Results shed light on that the relative diffusivity increases with the increase of sphericity as a shape descriptor of cement particle. This is due to the specific surface area decreasing with the increase of sphericity, resulting in less hydration products and more porosity, which ultimately increases the relative diffusivity.
... [21][22][23][24] Also, φ J shows complicated dependency on individual particle shape, revealed by studies using various NSP models [25][26][27] ellipsoids, 11,28,29 superellipsoids, [30][31][32][33][34] and polyhedra. [35][36][37][38] As the particle shape is deviated from sphere, φ J will first increase and then decrease with growing asphericity. 34,39 One may naturally pose a question of both theoretical and practical significance: Can we estimate the packing properties for coupled size and shape polydispersity based on the existing results of merely polydisperse-sized or monodisperse NSP systems? ...
We numerically investigate disordered jammed packings with both size and shape polydispersity, using frictionless superellipsoidal particles. We implement the set Voronoi tessellation technique to evaluate the local specific volume, i.e.,...
... For tetrahedral particles, the reason is because of the particularity of the particle shape, and the packing densification of tetrahedra is contributed by forming local dense clusters (e.g. dimer and wagon wheel clusters) [38,41,42]. While for binary mixture of tetrahedral and spherical particles with similar size ratio, the incorporation of spherical particles will destroy the local dense packing structure formed by tetrahedral particles, thus reducing the packing density of the mixture. ...
The random packings of binary tetrahedron-sphere mixtures were numerically reproduced by DEM simulations. The influences of particle shape (characterized by eccentricity ζ and height ratio η), particle size, and composition on the packing density of binary tetrahedron-sphere mixtures were systematically investigated. The properties of
equivalent packing diameter are identified by both macroscopic and microscopic parameters. The results show that the equivalent packing diameter of tetrahedra is independent of the particle shape (ζ and η) deviation. The minimal specific volume variations ΔV are negative for each case with the size ratio ranging from 0.5 to 1 caused by the shape particularity of tetrahedra. The overall mean coordination number (CN) obtained at r =1 (here, r=ds/dte, ds and dte are respectively sphere diameter and equivalent volume sphere diameter of tetrahedral
particles) is almost identical and does not changewith the variation of composition. The mean stress analysis shows that the equivalent packing diameter is not only universal for the microstructure parameters, but also for the mechanical properties of tetrahedron-sphere binary packings.
... In addition to F-F contacts which were also mentioned as dimer structure in the packing of regular tetrahedral particles [30,42,48]. Another structure called wagon wheel structure also played an important role in the packing densification of regular tetrahedral particles [17,42,48,49]. The evolution of the number of wagon wheel structures with the eccentricity and the height ratio is shown in Fig. 18. ...
Packings of different mono-sized tetrahedral particles under 3D vibrations were studied by physical experiments and DEM simulations. The effects of vibration conditions and particle shape on the packing
densification were comprehensively investigated and optimized. Corresponding characteristic microscopic properties such as coordination number (CN), particle contact type, radial distribution function (RDF), and particle orientations were numerically characterized and analyzed. The results show that the DEM model can be well validated by physical experiments. Microscopic analysis indicates that the minimum mean CN appears for tetrahedral particles with regular shape. The RDF shows that as the shape deviates from regular tetrahedral particles, the frequency of face-face, vertex-face and edge-edge contacts all decreases while that of edge-face contact increases. The cluster evolutions demonstrate that the reduction or disappearance of two important local clusters (dimer and wagon wheel structures) is one of the main reasons for the decrease of packing density of irregular tetrahedral particles.
... Neudecker et al. [26] investigated the effects of shaking conditions on the packing structures of tetrahedron particles. On the other hand, multi-sphere model of relaxation algorithm [27][28][29] and the adaptive-shrinking cell method [30][31][32] have been used to investigated generate a random tetrahedron packing. Zhou et al. [33] generated the random packing of tetrahedron particles using DEM and investigated the effects of friction, height ratio and eccentricity on packing structures. ...
A CFD-DEM modelling of the densification of uniform tetrahedron particles under air impact was conducted. The effect of air impact on packing structures such as packing density, coordination number (CN), radial distribution function (RDF) were analysed, and the densification mechanisms were discussed based on the analysis of local structure. The void distributions in the packing were analysed using a sphere filling method. The packings after air impact had smaller pores due to the reduction of void with normalized sphere diameter from 0.15 to 0.35. It was observed that the final dense packing had less contacts because of the change of non-stable contacts to more stable face-face contacts. Both dimer and wagon wheel structures increased at the final packing and the increase in the wagon-wheel clusters was more significant than the dimer clusters. The transfer of the dimer structure to the wagon wheel structure during was analogue to the nucleation and growth of crystals. The normal forces distribution exhibited exponential decays for both initial and final packing, and the normal force had a non-linear increase with depth in packings before air impact, while had a linear increase with depth in the packings after air impact.
... In this work, we do not differentiate these two concepts and just use the term RCP. Prior researches have discussed the RCPs of various non-spherical particles in three dimensions, including rods [10][11][12], ellipsoids [13], polyhedra [14][15][16][17], and superellipsoids [18][19][20][21][22] using different packing methods. Common knowledge is that all nearly-spherical particles pack more densely than spheres, typically~0.7 and even~0.74 ...
... Other two Platonic solids, namely dodecahedron (Ψ = 0.910) and icosahedron (Ψ = 0.939), are more spherical and have less facet effect, which are not included in this work. Previous numerical studies reported the φ c of ideal tetrahedra [15,16], cubes [15] and octahedra [14,15] arẽ 0.64,~0.73 and~0.69, ...
... The densest packing density of SPC shows a monotonic trend from 1 to 0.74 (crystalline density of sphere) [32], while the φ c of SPC is still unknown. Moreover, the dimer defined as two neighboring particles sharing a joint facet was found to be the dominant local structural pattern in dense tetrahedral packings [16,[33][34][35]. How this particular structure evolves as the s varied for SPT, SPC and SPO is still unknown. ...
It is of both theoretical and engineering significance to understand the random packings of non-spherical particles. However, apart from the well-discussed aspect ratio, studies on particle shapes continuously evolved from sphere to ideal polyhedra are still lacking, which represents the roundness effect. In this work, we investigate two packing states, namely random close packing (RCP) and saturated random packing (SRP), of spheropolyhedra (SPP), including three shape families, namely spherotetrahedron (SPT), spherocube (SPC), and spherooctahedron (SPO). We observe a common density peak phenomenon of these two states for all the families with respect to sphericity. Specifically, the RCP densities can reach ~0.746, ~0.750, and ~0.731 for the SPT, SPC, and SPO respectively, comparable with the crystalline density ~0.74 for spheres. Density peaks of the SRP locate at the sphericity ~0.96 for all the families. Additionally, the local structural analysis reveals the complex dependencies of order parameters on the roundness, including positional order q 6 and facet alignment Δ. The SRP states are more random in particle position than orientation. The dimer clusters formed by particles sharing common facets are also explored. We find that the facet number of a single particle is positively correlated with the q 6 yet negatively correlated with the Δ and the cluster ratio for all the polyhedron-like shapes at the RCP. Furthermore, the mechanism of excluded volume can explain the density peak of both the RCP and SRP for all the families and even partly reproduce the general trend of the RCP density for SPT.
... The packing properties of various 3D NSPs have been much discussed recently, including ellipsoids [16,35], sphe- rocylinders [36][37][38], polyhedra [39][40][41][42], superballs [17], and superellipsoids [43]. It was discovered that nearly spherical shapes possess ϕ c higher than 0.64 [44,45], increasing to typically ∼0.7 and even ∼0.74 for certain ellipsoids [16]. ...
It is well established that the packing density (volume fraction) of the random close packed (RCP) state of congruent three-dimensional spheres, i.e., φc∼0.64, can be improved by introducing particle size polydispersity. In addition, the RCP density φc can also be increased by perturbing the particle shape from a perfect sphere to nonspherical shapes (e.g., superballs or ellipsoids). In this paper, we numerically investigate the coupling effects of particle size and shape on improving the density of disordered polydisperse particle packings in a quantitative manner. A previously introduced concept of “equivalent diameter” (De), which encodes information of both the particle volume and shape, is reexamined and utilized to quantify the effective size of a nonspherical particle in the disordered packing. In a highly disordered packing of mixed shapes (i.e., polydispersity in particle shapes) with particles of identical De, i.e., no size dispersity effects, we find that the overall specific volume e (reciprocal of φc) can be expressed as a linear combination of the specific volume ek for each component k (particles with identical shape), weighted by its corresponding volume fraction Xk in the mixture, i.e., e=∑kXkek. In this case, the mixed-shape packing can be considered as a superposition of RCP packings of each component (shape) as implied by a set Voronoi tessellation and contact number analysis. When size polydispersity is added, i.e., De of particles varies, the overall packing density can be decomposed as φc=φL+finc, where φL is the linear part determined by the superposition law, i.e., φL=1/∑kXkek, and finc is the incremental part owing to the size polydispersity. We empirically estimate finc using two distribution parameters, and apply a shape-dependent modification to improve the accuracy from ∼0.01 to ∼0.005. Especially for nearly spherical particles, finc is only weakly coupled with the particle shape. Generalized polydisperse packings even with a moderate size ratio (∼4) can achieve a relatively high density φc∼0.8 compared with polydisperse sphere packings. Our results also have implications for the rational design of granular materials and model glass formers.
... To comprehensively characterize the contact configuration between icosahedral particles, the variation of the contact types in the different packing structures was analyzed. There are four main types of contacts in packing structures: face to face (F-F), face to edge (F-E), face to vertex (F-V), and edge to edge (E-E), as shown in Fig. 5(a) (Jaoshvili et al., 2010;Jiao & Torquato, 2011;Jin, Lu, Liu, & Li, 2015;Neudecker, Ulrich, Herminghaus, & Schröter, 2013). Fig. 5(b) shows evolution of the percentages of the four contact types with the packing density. ...
Packing densification of monosized regular icosahedral particles under three-dimensional mechanical vibration has been simulated by the discrete element method (DEM). The effects of the vibration conditions and container size on packing densification were systematically investigated. In addition to the macroscale properties (packing density and porosity), the microscale properties, such as the coordination number (CN), radial distribution function (RDF), particle contact type, particle orientation distribution, and stresses/forces, in random loose packing (RLP) and random close packing (RCP) were also characterized and analyzed. The results show that transformation of icosahedral particle packing from RLP to RCP can be realized by properly controlling the vibration conditions. The maximum random packing density without the wall effect reaches 0.7078. Microscale property analysis shows that the average CN increases after vibration. The RDF curves contain two clear peaks for RLP and three for RCP. From RLP to RCP, the probability of face to face contact between two particles increases, while the probabilities of edge to edge, edge to face, and face to vertex contact decrease. The orientation correlation functions indicate the randomness of the vibrated packing structure. In addition, more uniform force and stress distributions are observed within the dense packing structure. © 2018 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences
