Powder Technology 351 (2019) 186–194

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Random-packing properties of spheropolyhedra

Ye Yuan, Lufeng Liu, Wei Deng, Shuixiang Li ⁎Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

⁎ Corresponding author.E-mail address: lsx@pku.edu.cn (S. Li).

https://doi.org/10.1016/j.powtec.2019.04.0180032-5910/© 2019 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 6 December 2018Received in revised form 28 March 2019Accepted 8 April 2019Available online 10 April 2019

It is of both theoretical and engineering significance to understand the random packings of non-spherical parti-cles. However, apart from the well-discussed aspect ratio, studies on particle shapes continuously evolved fromsphere to ideal polyhedra are still lacking,which represents the roundness effect. In thiswork,we investigate twopacking states, namely random close packing (RCP) and saturated random packing (SRP), of spheropolyhedra(SPP), including three shape families, namely spherotetrahedron (SPT), spherocube (SPC), andspherooctahedron (SPO). We observe a common density peak phenomenon of these two states for all the fam-ilies with respect to sphericity. Specifically, the RCP densities can reach ~0.746, ~0.750, and ~0.731 for the SPT,SPC, and SPO respectively, comparablewith the crystalline density ~0.74 for spheres. Density peaks of the SRP lo-cate at the sphericity ~0.96 for all the families. Additionally, the local structural analysis reveals the complex de-pendencies of order parameters on the roundness, including positional order q6 and facet alignment Δ. The SRPstates aremore random in particle position than orientation. The dimer clusters formed by particles sharing com-mon facets are also explored.Wefind that the facet number of a single particle is positively correlatedwith the q6yet negatively correlated with the Δ and the cluster ratio for all the polyhedron-like shapes at the RCP. Further-more, the mechanism of excluded volume can explain the density peak of both the RCP and SRP for all the fam-ilies and even partly reproduce the general trend of the RCP density for SPT.

© 2019 Elsevier B.V. All rights reserved.

Keywords:SpheropolyhedraRandom close packingSaturated random packingStructural analysisExcluded volume

1. Introduction

Understanding disordered particulate systems is of great signifi-cance for both theoretical researches [1,2] and industrial applications[3]. An important prototype issue in this community is the extensionof the concept of random close packing (RCP) originally for spheres[4,5] towards non-spherical shapes [6,7]. As its literal meaning, RCP isdefined as the densest random packing obtained experimentally. Forspheres, it coincides with the physically critical state of point J, whichcan be regarded as the loosest packing maintainingmechanically stablefor frictionless particles, whose packing density φc is ~0.64 [8,9]. In thiswork, we do not differentiate these two concepts and just use the termRCP. Prior researches have discussed the RCPs of various non-sphericalparticles in three dimensions, including rods [10–12], ellipsoids [13],polyhedra [14–17], and superellipsoids [18–22] using different packingmethods. Common knowledge is that all nearly-spherical particles packmore densely than spheres, typically ~0.7 and even ~0.74 for a certainellipsoid. This feature was also predicted by both the mean-field theory[6] and the perturbative calculation [23]. Furthermore, the φc reaches apeak value as asphericity (or aspect ratio) becomes moderate forspherocylinders (SCY) and ellipsoids. Recently, it was reported that an-other shape parameter p for superellipsoids can affect the random

packings similarly as the aspect ratio [21]. p is an independent shape pa-rameter besides the aspect ratio which quantifies the similarity of sur-face geometry with spheres (p = 1) or ideal polyhedra (p = 0.5 or ∞)[18,24]. Therefore, we are interested in the effect of particle surface ge-ometry on the packing densities and specific structures apart from theaspect ratio.

A comprehensive parameter for a certain non-spherical shape is thesphericity Ψ defined as the ratio between the surface area of an equal-volume sphere and its own, intowhich both the aspect ratio and surfacegeometry are encoded. Researchers even proposed empirical methodsto predict the φc of arbitrary shapes using Ψ that the relationship be-tween these two variables demonstrated a general trend, not single-valued function [25,26]. Flattening or elongating a shape will reduceitsΨ. An ideal tetrahedron possesses theΨ of ~0.67, themost noticeableasphericity for shape without axial bias. Note that the Ψ of a SCY withaspect ratio 5 is ~0.69 comparable with that of an ideal tetrahedron.Thus, one can considerably reduceΨ from 1 to ~0.7 by changing solelythe surface geometry regardless of the aspect ratio, onwhich systematicresearches are lacking to our knowledge.

To tackle this issue, the above-mentionedmodel of superellipsoids ismathematically elegant yet may have numerical problems of the con-tact detection typically when approaching polyhedra. In this work, weutilize the spheropolyhedron (SPP) model [7,27–30] with kernels ofthree Platonic solids, namely the spherotetrahedron (SPT), spherocube(SPC) and spherooctahedron (SPO), to study the particle shape effect

187Y. Yuan et al. / Powder Technology 351 (2019) 186–194

without aspect ratio. This model based on the three representative Pla-tonic solids has no special numerical difficulties. A specific SPP possessesa certain proportion of facet surface, which is different from asuperellipsoid. For these SPP, we define a roundness parameter s,depicting the continuous evolving shapes from the sphere (s = 1) tothe ideal polyhedra (s = 0). The concept of roundness is regularlydiscussed in geological literature [31] and in this work, its detailed def-inition is quite compendious as explained in the following section.Othertwo Platonic solids, namely dodecahedron (Ψ = 0.910) and icosahe-dron (Ψ = 0.939), are more spherical and have less facet effect, whichare not included in this work. Previous numerical studies reported theφc of ideal tetrahedra [15,16], cubes [15] and octahedra [14,15] are~0.64, ~0.73 and ~0.69, respectively. A recent simulation suggestedthat the φc of SPT and SPC with small roundness are higher than thoseof ideal tetrahedra and cubes respectively [30]. Therefore, it is reason-able to suggest the φc of SPT has a peak with a particular s, similar tothe behavior of the reported densest packings of SPT [29]. The densestpacking density of SPC shows amonotonic trend from1 to 0.74 (crystal-line density of sphere) [32], while the φc of SPC is still unknown. More-over, the dimer defined as two neighboring particles sharing a jointfacet was found to be the dominant local structural pattern in dense tet-rahedral packings [16,33–35]. How this particular structure evolves asthe s varied for SPT, SPC and SPO is still unknown. Therefore, the depen-dency of both theφc and local structural pattern on the s for SPP remainsto be resolved in this work, which is of both theoretical and engineeringinterest.

Additionally, the saturated random packing (SRP) obtained from therandom sequential adsorption (RSA) is another much-discussed statebesides the RCP in surface science and two-phase media [36–40]. Theprocess of RSA is briefly explained as follows. Within a large space, par-ticles are randomly positioned in sequence and one insertion action isonly accepted if it introduces no overlaps with existing particles. Thenafter sufficient trials, a saturated packing density φs is obtained sinceany new insertion can never succeed. Since there are no collective par-ticle movements during the RSA process, the obtained SRP are appar-ently looser than the RCP (the φs of spheres is ~0.38) and notmechanically stable. It was discovered that the φs of spheroids showedsimilar peak behaviors as the φc [37]. Meanwhile, the local structuresat the φs should be more random than those at the φc because there isa compromise between themechanical stability and the random degreein the neighboring environment. Thus, a comparative study on both theRCP and SRP of different SPP is helpful for a better understanding ofnon-spherical shape effect on random packings.

In this paper, we numerically investigate both the φc and φs of thethree families of SPP, namely the SPT, SPC, and SPO, evolving fromspheres to ideal polyhedra. The main findings are listed below:

(1) We observe upper convex curves about the φc or φs versus the sfor all the families considered. Yet, the φc or φs reach their peaksat different s due to the distinct local geometries. We emphasizethat the highest φc of these SPP are all comparable to the crystal-line packing density of spheres ~0.74. Specifically, the φc of SPT,SPC and SPO can reach ~0.746, ~0.750 and ~0.731with the partic-ular s as ~0.12, 0.2–0.4, and ~0.21 respectively. This observationillustrates that surface geometry can solely promote the disor-dered packing density comparable to the aspect ratio effect.

(2) We investigate the contact geometry and two order parameters,namely the bond orientation parameter q6 [41] andfacet alignment parameter Δ [42], to depict the local packingstructures. We find that the contact number Z does not changemonotonously with the s because a facet alignment can offermore geometrical constraints than a normal point contact. As aresult, the average constraint number C varies from 12 to 6 forall the three families, in good agreement with the isostatic num-ber. Order parameters show strong dependencies on the s forboth RCP and SRP of each family and the q6 remains ~0.3 nearly

constantly for SRP as a special case. We also consider the clusterratio defined as the proportion of particles involved in dimers.Overall, the facet number is positively correlated with the q6and negatively correlated with the Δ and cluster ratio for all thepolyhedron-like shapes at the φc.

(3) Since the above structural analysis cannot reproduce the densitypeak of theφc orφs, we try to approach this issue considering thecompetition between the excluded volume Vex and Z [13]. As theasphericity begins to increase orΨ deviates from 1 equivalently,the Zwill dramatically increase as discussed above, also theoret-ically predicted [6], which leads to the first stage of increasingφc.Then the Z becomes less dominant than the Vex for shapes withmoderate asphericity, reducing the packing density in turn. Wecarefully study the random contact equation previously pro-posed for long rods [10] to extend it to the SPP in this work.We find that the peak location of the φc or φs can be roughly es-timated using the C or Z respectively, while the specific values ofpacking density deviate significantly. Aftermodifying the originalmodel with Ψ, we can generally predict the φc of SPT and SCY.

This work thoroughly investigates the surface geometry of non-spherical shapes alternative from the well-studied aspect ratio usingthe SPP model. Besides those particular characters of different SPP dis-covered in this work, the universal phenomenon about density peak isqualitatively explained for both the surface geometry and aspect ratioeffects. These discussions contribute to both fundamental understand-ing and industrial application about disordered packings of non-spherical particles.

2. Model and method

In this paper, the spheropolyhedra (SPP) model [7] is utilized to de-pict the smooth evolution of surface geometry from sphere to threekinds of Platonic solids, namely the tetrahedron, cube, and octahedron.The SPP can be generated by sweeping the profile of an ideal polyhe-dron using a sphere shown in Fig. 1. The edge length of the kernel poly-hedron is L and the diameter of the sweeping sphere is D. Then theroundness parameter is defined as.

s ¼ D= Lþ Dð Þ ð1Þ

which varies from 0 (ideal polyhedra) to 1 (sphere). Shapes of identicalkernel polyhedrawith various s are called as a family of SPP, particularlythe SPT, SPC, and SPO in this work. In Fig. 1, typical SPT, SPC, and SPO aredisplayed with s = 0.12, 0.4 and 0.214 respectively. Both the particlevolume Vp and surface Ap can be analytically calculated for these SPP,which determines sphericity.

Ψ ¼ 4πð Þ1=3 3Vp� �2=3

=Ap ð2Þ

The lower bounds ofΨ for the SPT, SPC, and SPO (in the cases of idealpolyhedra) are 0.671, 0.806 and 0.846 respectively. Additionally, wealso partly consider the SCY whose kernel can be seen as a segmentand the Ψ of the SCY with aspect ratio 5 is 0.694.

We numerically generate disordered packings of a variety of SPPusing the two hard-particle packing algorithms, i.e. the fast-compressing Monte Carlo [14] and the RSA [38]. Both algorithms needto deal with the contact detection problems between two neighboringSPP. Firstly, we need to judgewhether there exists any overlap betweenthe two kernel polyhedra. If not, we compute theminimumdistance be-tween them and compare itwith the summation of the two correspond-ing D. Apparently, these two SPP separate if the distance is larger thanthe summation. Moreover, the real distance of two SPP is defined asthe kernel distance minus this summation in an overlap-free packing,which is always larger than 0. One of the advantages of a hard-particle

Fig. 1. (a) SPT with s = 0.12 (b) SPC with s = 0.4 (c) SPO with s = 0.214. Their kernelpolyhedra are specially marked for viewing.

188 Y. Yuan et al. / Powder Technology 351 (2019) 186–194

packing algorithm is that there is no need to consider the contact-forcemodel of non-spherical particles, which might be complicated to someextent in the discrete element method. Thus, this treatment is propersince we are not interested in the mechanical properties here.

In the fast-compressing Monte Carlo algorithm, we compress a di-lute overlap-free system to a dense disordered state, namely the RCP.Details of this Monte Carlo algorithm can be found in Ref. [14] andhere we just briefly explain it. During the compressing process, thecubic periodic boundary isotropically shrinks (with all the particle posi-tions rescaled) once each particlemoves Tc times on average,where Tc isa pre-set constant dominantly controlling the obtained packings. In thiswork, we set Tc =1 to suppress order forms, which generates the RCPs.All the trials of particle movement and boundary shrinking can only beaccepted if introducing no overlap. The magnitudes of particle move-ment trial are self-adaptive to keep the acceptance ratio as ~0.35. The se-quential boundary shrinking increases the packing density along withthe decrease of trial magnitudes. This Monte Carlo process terminateswhen the magnitudes of both translational and rotational trials are suf-ficiently small, producing the RCP state whose density is defined as φc.Typical RCPs of different shapes are shown in Fig. 2 with particles col-ored by the local order parameters.

We also study the SRP following themethod in Ref. [38]. The volumeof cubic periodic space Vs is initially set 10,000 times as large as the in-dividual particle volume Vp. One insertion iteration is defined as

generating a particle of both random position and orientation. Anewly inserted particle will be accepted if it does not overlap with allthe existing ones. We track the immediate density φ(t) where t is nor-malizedwith the iteration step ni as t= niVp/Vs. The probability of a suc-cessful insertion becomes infinitesimal approaching saturation. Thesaturated packing density φs is obtained via data fitting with the equa-tion [38,39].

φ tð Þ ¼ φs–kt−α ð3Þ

where φs, k and α are the fitted constants. This law holds not distantfrom the saturation, thuswe truncate thedatawith t N 1000 to eliminatethe influence of initial condition.

In the studies about the RCP of SPP, the particle number is 500. Weutilize 250:250 binary particles of size ratio 1.4 to suppress order de-gree in the systems of SPC. As for the SPT and SPO, we find thatmonodisperse and binary packings lead to similar results and thenthe monodisperse systems are discussed for simplicity. For the SRP,the finally obtained packings typically possess 3000–4000 monodis-perse particles. The reported statistics in this work are the averagesover 10–15 independent replications.

3. Packing densities

The results of theφc andφs are shown in Fig. 3,where solid or dashedlines mark the RCP or SRP states respectively. The s andΨ are the shapeparameters. For a certain SPP, the φs is apparently much lower than theφc, since there is no collectivemovement for particles in the RSA processto form a dense local packing. For ideal polyhedra, theφc=0.657, 0.730and 0.685 for tetrahedron, cube, and octahedron respectively, close tothose reported values in the literature [14–16]. Additionally, the φs =0.36 for the cube which generally agrees with a recent result [40]. Asthe s or Ψ approaches 1, the φc and φs become ~0.64 and ~0.38, typicaldensities of the RCP and SRP of spheres [43], respectively.

Furthermore, we can clearly inspect the evolution of φc and φs as svaries from 0 (ideal polyhedra) to 1 (sphere) for each family in Fig. 3(a). All these curves are in general upper convex, while they reachtheir peaks at different s. This counter-intuitive observation verifiesthat the round corner effect can promote the disordered packing den-sity, similar to the case in the densest packing of SPT [29]. Even theSPC with moderate roundness can pack more densely than the idealcube at the RCP states, since the round corner may act as the bufferwhen two facets are not in perfect alignment leading to denser localstructures. Specifically, the φc of SPT, SPC and SPO can reach theirpeaks ~0.746, ~0.750 and ~0.731 with particular s as ~0.12, 0.2–0.4and ~0.21 respectively. It is worth noting that theφc of SPT rises remark-ably as s is small compared with SPC and SPO, which persists even bi-nary sized particles are used. Thus, the peak location of SPT s ~ 0.12 islower than the others. Those shapes displayed in Fig. 1 are just aroundthese special s. We emphasize that these peak densities are comparableto the crystalline density of sphere ~0.74, as well as the φc of a specialellipsoid [13]. The corresponding peaks of φs locate at s ~ 0.45, ~0.6and ~0.38, relatively more spherical. Yet, their trends withΨ are similarnear the polyhedral shapes in Fig. 3(b). The relationship betweenφs andΨ for the three SPP shows certain universality with the peaks nearly atΨ ~ 0.96, while only weaker universality can be observed for s fromFig. 3(a).

The relationship betweenφc versusΨ shown in Fig. 3(b) needs care-ful discussion. For sphere-like shapes with Ψ N 0.95, the data points ofdifferent families of SPP shapes cannot be distinguished from eachother. Within this region, an empirical density prediction formula canbe easily proposed, not discussed here. It was theoretically shown thatthe φc of nearly spherical particles is higher than 0.64 [6,23]. Recently,such universal law about theφc and shape parameter was also reportedin two-dimension systems [44]. However, this universality breaks downfor moderately non-spherical particles, as discussed in the previous

Fig. 2. RCP structures of the SPT (s ~ (a) 0.12 and (b) 0.65), SPC (s ~ (c) 0.2 and (d) 0.7) and SPO (s ~ (e) 0.21 and (f) 0.49). We set a color map from green to red to demonstrate thefacet alignment order Δ and bond orientation order q6 for the individual particle in (a)(c)(e) and (b)(d)(f) respectively, the color of the particles with higher order degree closes tored. 1/8 corner of the packing is erased for visualization.

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literature [25,26]. In the following section of this work, we perform anintensive structural analysis and then try to explain such upper convexfeatures and universal relationships.

4. Structural properties

We carefully investigate the contact topology of all the obtained RCPstates, while any contact is rigorously invalid for the SRP. Direct contactnumber (Z) counting will offer misleading results because differenttypes of contact of facets shapes can constrain different degrees of free-dom (DOF). A facet-facet (FF) contact will offer three constraints andthis number is two or one for an edge-facet (EF) contact or the else(point contact, P), respectively. The concept of isostaticity [2] will clarifythis point as briefly explained here. Since the RCP can be regarded as the

loosest packings maintaining mechanically stable, the Z offered by thepacking network should be the minimum, just to stabilize all the parti-cles. For a certain particle, we sum up all the constraints offered by itscontact neighboring particles to obtain its constraint number. Consider-ing Newton's third law, we obtain the equation of average constraintnumber C/2 = df, where df is the DOF of an individual particle. Obvi-ously, the df is 3 for sphere and 6 for general non-spherical shapesdiscussed in this work.

A brief explanation of the numerical treatment of contact type clas-sification is provided as follows, similar to previous studies [16,45],which needs a series of judgment in sequence. If the distance betweentwo particles is smaller than a pre-set threshold ε (in the unit of the di-ameter of the bounding sphere), they are judged as in contact. Next, thiscontact is classified as an FF, EF, or the else in sequence [16]. We record

Fig. 3. Packing densities of the RCP (solid lines) and SRP (dashed lines) versus(a) roundness s or (b) sphericity for all the SPP. Fig. 4. (a) Contact number Z (solid lines) or constraint number C (dashed lines) versus

roundness s for all the SPP. (b) The ratio of FF (solid lines) or P (dashed lines) typewithin the contacts versus s for all the SPP.

190 Y. Yuan et al. / Powder Technology 351 (2019) 186–194

these quantities with the ε varied from 0.003 to 0.006 and obtain theexact results by the linear extrapolation to ε = 0. The statistics aboutcontact topology are demonstrated in Fig. 4. We sum up all of the con-straints offered by different contact types as C shown in Fig. 4(a),which varies almost monotonously from ~12 to ~6 for all the families(dashed lines), in good agreement with the prediction. Yet, the trendsof Z (solid lines) are distinct. Overall, the Z dramatically increases as svaries from 1 to ~0.7 and then behaves differently. To be specific, the Zof SPOmonotonously increases from 6 to 7.8 deviating from the sphere.The SPT and SPC showmore complex phenomena in Z, ending upwith 8and 6 respectively. Fig. 4(b) shows a general trend of increasing FF anddecreasing P contacts approaching polyhedra. This is quite reasonablebecause it coincides with the change of facet proportion in the wholesurface area. Note that there exists a less evident peak of FF near s = 0for SPT, reproducing the special trend about the φc of SPT in Fig. 4(a).The SPC typically have a larger proportion of FF with less Z, leading tosimilar C as the others. We clarify that the definition of FF is based onthe limited touching of two neighboring facets, distinct from a totalfacet alignment to be discussed. Thus, the RCPs discussed here are gen-erally disordered even with a finite amount of FF shown in Fig. 4(b).

Since these families of SPP evolve smoothly from sphere to idealpolyhedra, we can quantify their local order degree using proper orderparameters, demonstrated in Fig. 2. Moreover, the information ofthese order parameters helps to distinguish the RCP from the SRP.First, we compute the conventional local bond orientation parameter

q6 for an individual particle to quantify the positional order degree. Itis defined as [41,46].

q6 ¼ 4π13

∑6m¼−6

1nb

∑nbi¼1Y6m αi;βið Þ

��������2

ð4Þ

where Y6m(αi, βi) is the spherical harmony functionwithαi andβi as thepolar and azimuthal angles of a neighboring bond between central par-ticle and particle i. Note that i loops over the nearest nb = 12 particles(in the meaning of center distance) according to the crystalline struc-tures of the sphere. Then we obtain the system q6 averaging over allthe particles, displayed in Fig. 5(a) for both the RCP and SRP. It is discov-ered that the q6 of the SRP states (dashed lines) keeps almost constant~0.3 for all the shapes except for a part of SPT with small s, and itshows a generally increasing trend as s becomes larger. A small rise upcan be observed as s just deviates from 1, which illustrates that asmall shape deviation from sphere can even promote the q6 order atthe RCP. This is due to such deviation will introduce more neighboringparticles compared to the value 6. For those polyhedron-like SPP, theq6 approaches quite distinct values basically determined by the uniquepolyhedral shape. In this region, a nontrivial behavior of SPT can be ob-served that the q6 of RCP is even lower than SRP since the tetrahedralshape is incompatible with q6.

Fig. 5. Two order parameters, namely (a) the bond orientation parameter q6 and(b) facet alignment parameter Δ, of both RCP (solid lines) and SRP (dashed lines) versuss for all the SPP.

Fig. 6. (a) Cluster ratio defined by two thresholds 0.9 (solid lines) and 0.6 (dashed lines)versus s for all the SPP. Particles involved in dimers in the RCP of (b) SPT (s ~ 0.02) and(c) SPC (s ~ 0.05) are visualized, with the threshold set as 0.9.

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For those polyhedron-like SPP, q6 cannot directly depict the domi-nate order pattern, namely the facet alignment. We define a localfacet alignment parameter [42] for the individual particle as

Δ ¼ 1nf

∑n f

i¼1 maxp;q ið Þ γpq ið Þ� �

ð5Þ

where γpq(i) denotes the proportion of the projection of facet q(i) of par-ticle i on facet p of the central particle (or the opposite order andwe usethe large one) to the average facet area of both facets. Label i loops overthe nearest nf particles, where nf denotes the facet number of the centralparticle. We also obtain the system averagedΔ, shown in Fig. 5(b). Bothstates of the RCP and SRP share a common trend of correlation betweenincreasing Δ and decreasing s. For a certain family, the RCP curve is al-ways above the SRP just because of the denser local structures. The Δof SPT shows a sharp increase in accordance with similar behavior inFig. 3(a).

For tetrahedral particles, it was reported that the dimer cluster,formed by two neighboring particles sharing a joint facet, worked asthe dominant structural pattern in dense packings [16,33,35].We definea joint facet using the item inside the bracket in Eq. (5). If maxp(i),q(j)

(γp(i)q(j)) is close to 1 for two contacting particles i and j, they are judgedto share a joint facet. The relationship of the relative ratio of particles in-volved in such dimer clusters and the s at RCP is plotted in Fig. 6(a). Weset different thresholds for the judgments about joint facet with 0.9(solid lines) and 0.6 (dashed lines) shown in Fig. 6(a), and find similartrends for a specific family. A lower threshold will further count inthose facet touching particle pairs with a twisting orientation. Forthreshold 0.9, only those systems of polyhedron-like SPT and SPC

Fig. 7. (a) ηC/Vex (dashed lines) and (b) ηZ/Vex (dashed lines) in comparison with (a) φc

(solid lines) and (b) φs (solid lines) of SPT, SPC and SPO, respectively. η = 0.853 in(a) and 0.512 in (b).

192 Y. Yuan et al. / Powder Technology 351 (2019) 186–194

possess considerable dimer particles, accounting for about 0.2–0.4. InFig. 6(b) and (c), we visualize those dimer particles with threshold 0.9in the RCP of SPT (s ~ 0.022) and SPC (s ~ 0.05), respectively. A strikingrise is observed only for polyhedron-like SPT. Note that the transitionoccurs at s ~ 0.1, close to the peak φc. This illustrates that the increasingdimer proportion for s close to 0 is not positively correlated or even op-posing the decreasing φc in Fig. 3(a), which can be understood that thefacet proportion of the surface will also largely affect the local packingstructures.

Due to the distinct geometry of SPT, SPC, and SPO, they show abun-dant characters in the local packing structures of both RCP and SRP.Some specific phenomena are detailly pointed out as above. Then, wecan make some general remarks here. Both states of the RCP and SRPare not totally random, driven by distinct physics. By definition, theRCP is mechanically stable quantified by the C not Z, and concurrentlyloose enough which is roughly correlated with the Z as well. Therefore,a good compromise can be achieved by introducing a part of complexcontact types to ensure C = 2df, which inevitably promotes the localorder degree to some extent. The obtained Z still show a generally in-creasing trend until the s is decreased to ~0.7. On the other hand, as asystem becomes saturated in the RSA process, the hard particle interac-tionwill show certain facet alignment bias [47]. From Fig. 5 we find thatSRP states aremore random in position than orientation.We can simplyconclude that the facet number is positively correlated with the q6 andnegatively correlated with the Δ and the cluster ratio for all thepolyhedron-like shapes. However, all these structural quantities fail todirectly reproduce the peak phenomena in Fig. 3.

5. Excluded volume

The density peak features of different shape families seem to be uni-versal even with uncertain specific peak locations, as shown in Fig. 3(b) for SPP. Results of SCY [11,12] and ellipsoids [13] can also validatethis universality. A qualitative explanation about this can be simplymade via considering the effects of Z and excluded volume Vex [13]. Asthe particle shapedeviates froma sphere, there isfirstly a sharp increasein the Z to maintain mechanical stability since the additional rotationaldegrees of freedomneed to be constrained. Also, the excluded volume ispositively related to the asphericity. Since Z cannot increase unlimit-edly, the excluded volume effect will dominate over Z for considerableasphericity, which reduces the φc [13]. Specifically, a concise randomcontact equation [10,11] for long rods is formulated as.

φcVex ¼ Z ð6Þ

where Vex denotes the orientation averaged excluded volume normal-ized by the particle volume. It was claimed that this relationship canonly be applied to those systems where the particle contact points arenearly uncorrelated with each other, typically for the long rods [10]with C ~ 10. This equation was also successfully extended to thenonconvex U-shaped particles composed of three long rods [48].

Here we perform a similar analysis for SPP, trying to reproduce thepeak features in the curves of φc or φs versusΨ. We replace the originalright-hand item with ηZ or ηC, where η denotes a modification coeffi-cient considering the case of a sphere. Thus, η = 0.853 or 0.512 for φc

or φs respectively. The Vex is numerically obtained for all the SPP [49].In Fig. 7(a), we find that the estimation using C deviates significantlyfrom φc, as well as Z for φs in Fig. 7(b). However, the peak locations ofa certain family can be roughly predicted, especially for peaks of φs atΨ ~ 0.96. Thus, such treatment can only qualitatively catch the rationaleof the competition between C or Z and Vex, giving rise to the ubiquitouspeak phenomena. Eq. (6) in the current form cannot be extended to theshapes of SPP discussed in this work to predict the packing densities.

Typically, solving about φc is muchmore computationally expensivethanφs. Herewe propose an empiricalmethod to organize the results inthis work. We straightforward look into the term φcVex to check its

dependency on the particle shapes alternatively, not the contact infor-mation. Some additional data about the SCY within aspect ratio 5 arealso analyzed, plotted in Fig. 8(a). Note that polyhedron-like SPC andSPO deviate from the fitted straight line, which cannot be explainedby theΨ solely. We can empirically obtain the following equation.

φcVex ¼ 20:3–14:8Ψ ð7Þ

which correlates the right-hand term with the Ψ independent of theshape families. We emphasize that one can directly predict the φc viaΨ for nearly spherical shapesΨ N 0.95 shown in Fig. 3(b). Yet, a moder-ate asphericity is themain concern here. After back substitutions,φc andthe calculated ones for different shape families are compared in Fig. 8(b). The identity line marks the relatively good prediction results forthe SPT (black cubes) and SCY (green diamonds) than the others. Thissuggests that SPC and SPO show relativelymore probable contact corre-lation than SPT and SCY, which violates the precondition of the randomcontact model. This effect cannot be simply understood via theΨ sincedifferent levels of prediction deviation are discovered for families of SPPwith similarΨ.

Fig. 8. (a) Ψ of SPT, SPC, SPO and SCY versus their corresponding φcVex. The dashed lineshows the fitting Eq. (7). (b) The prediction density using Eq. (7) in comparison withthe simulated φc. The dashed line shows the identity line.

193Y. Yuan et al. / Powder Technology 351 (2019) 186–194

The extensive and careful discussion above is enlightening and herewe can clarify several observations. Firstly, the peak of φs can bereproduced by Eq. (6) using Z, a quantity that even does not exist atφs. Considering the unsatisfying results about φc, it illustrates that therelative positional randomness at φs suits the random contact modelwhich requires sufficient decorrelation among the contacts or neighbor-ing structures [10]. Moreover, Eq. (7) can well describe SPT and SCYwith moderate Ψ in Fig. 7(b), reflecting a certain universality of shapeeffect, specifically the surface geometry and the aspect ratio for SPTand SCY respectively. Vex encodes certain information neglected by asole parameter Ψ, which determines φc. Recalling the previous re-searches [26], a tough problem is to predict the φc of a given non-spherical shape using several parameters not Ψ only, which has notbeen perfectly solved by existing theory [6].

6. Conclusion

In this work, we numerically investigate the disordered states ofboth the RCP and SRP for three families of SPP, namely SPT, SPC, andSPO. We observe common peak phenomena in the relationships be-tween φc or φs and roundness s (defined in Eq. (1)) for each shape fam-ily. This is in accordancewith the previous researches in which theφc ofpolyhedron-like SPP are found to increase as the s monotonously [30]

and the densest packings of SPT also show a peak at a small s [29]. Spe-cifically, theφc of SPT, SPC and SPO can reach ~0.746, ~0.750 and ~0.731with particular s as ~0.12, 0.2–0.4, and ~0.21 respectively. This observa-tion illustrates that the surface geometry can solely promote the disor-dered packing density comparable to the aspect ratio effect found inellipsoids [13]. From Fig. 3(a), it is discovered that the φc of SPT risesup remarkably at small s, which might explain the mismatch betweenprevious simulation [15,16] and experiment [34] since the smallround corner of a real tetrahedron dice can lead to a misleadingoverestimated φc. For a certain shape, the φs is apparently lower thanthe φc owing to their distinct physics. Ψ ~ 0.96 is found to be the com-mon peak location for the φs of all the families.

Theoretically, it is the C, not Z affecting the mechanical stability of apacking system, verified by the results in Fig. 4(a). The minimum re-quirement to ensure stability is given by C = 2df (isostaticity) wheredf denotes the DOF of an individual particle, which is 3 for a sphereand 6 for a general non-spherical shape. By definition, the RCP isthe loosest packingmaintainingmechanically stable for frictionless par-ticles. From a geometrical aspect, the Z is positively correlated with thedensity. By introducing a certain proportion of complex facet-basedcontacts which can offer more than one constraint, a compromise isachieved satisfying the requirements of RCP. Note that the Zwill still in-crease overall with the asphericity. This mechanism explains the in-creasing facet alignment order Δ for the RCP of SPP evolving fromsphere towards ideal polyhedra, as shown in Fig. 5(b). The SRP showssimilar yet less considerable order due to the hard particle interaction,different from the mechanism of RCP. The positional order quantifiedby q6 can distinguish the RCP from the SRP, which almost keeps a con-stant q6 ~ 0.3 displayed in Fig. 5(a).We also directly study those clustersformed by the neighboring particles sharing a joint facet, which is dom-inating in the dense packings of tetrahedron [16,33,35]. It is simply con-cluded that the facet number of a single particle is positively correlatedwith the q6 and negatively correlatedwith theΔ and the cluster ratio forall the polyhedron-like shapes.

The relationship between the φc and certain shape parameter forvarious non-spherical shapes (including SPP in this work, and ellipsoids[13], SCY [11,12], superellipsoids [19] as well) demonstrates a ubiqui-tous upper convex curve as in Fig. 3, which is universal to some extent.However, the above structural analysis cannot reproduce this behavior.Alternatively, such universality can be regarded as a result of the com-petition between Z and Vex [13]. As the asphericity begins to increaseor the Ψ deviates from 1 equivalently, the Z will dramatically increaseas discussed above, also theoretically predicted [6], which leads tothe first stage of increasing φc. Also, the excluded volume is positivelyrelated to the asphericity. Since Z cannot increase unlimitedly, the ex-cluded volume effect will be dominating over Z for considerableasphericity, reducing the packing density in turn. We carefully discussthe previously proposed random contact equation [10], formulated asEq. (6), to verify this broad claim. We find that the peak location of φc

or φs can be roughly estimated using the C or Z respectively, while thespecific values of density deviate significantly, shown in Fig. 7. The pre-condition of Eq. (6) is that the contact positions are decorrelated witheach other typically for long rods, which is probably violated for mostSPP discussed in this work. We rewrite the equation as Eq. (7) withthe contact replaced by the Ψ related item, to deal with differentshape families including SCY. We can generally predict the φc of SPTand SCY shown in Fig. 8(b). This part of analysis based on the Vex pro-vides a qualitative explanation on the peak of φc or φs and partly suc-ceeds to unify the shape effects of surface geometry and aspect ratioon φc.

There are several interesting points that can be further investigatedabout the competition between Vex and Z in the upcoming researches.Firstly, we should take more non-spherical shapes into consideration,whichmight need other general shape descriptors besidesΨ. Moreover,packing density cannot be simply correlated with Z due to the complexcontact topology. Thus, proper formula for the right-hand item in

194 Y. Yuan et al. / Powder Technology 351 (2019) 186–194

Eqs. (6) or (7) should be discussed if it indeed works. Overall, we stillface the problem about a density prediction method, typically of theφc, for general non-spherical shapes.

Acknowledgements

Thisworkwas supportedby theNationalNatural ScienceFoundationof China (Grant Nos. 11272010, 11572004 and U1630112), the ScienceChallenge Project (Grant No. TZ2016002), and the High-PerformanceComputing Platform of Peking University.

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