Powder Technology 364 (2020) 49–59

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Shape effects on packing properties of bi-axial superellipsoids

Lufeng Liu a,b,⁎, Shuixiang Li b,⁎a Institute of Applied Physics and Computational Mathematics, Beijing 100094, Chinab Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

⁎ Corresponding authors at: Department ofMechanics aof Engineering, Peking University, Beijing 100871, China.

E-mail addresses: liulufeng@pku.edu.cn (L. Liu), lsx@p

https://doi.org/10.1016/j.powtec.2020.01.0400032-5910/© 2020 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 4 November 2019Received in revised form 11 December 2019Accepted 14 January 2020Available online 20 January 2020

Wegenerate themaximally random jammed (MRJ) packings andmaximally dense randompackings (MDRPs) ofbi-axially elongated superballs with different shape parameters. On the MRJ packing state, the packings of ellip-soids, general superellipsoids and cuboids are on different random degrees and the shape elongation effects onthe disordered-packing densities are not uniform. The effects of order degree and particle shape are compounded.However, we find uniform shape elongation effects on the MDRP state. Bi-axial elongation will improve therandom-packing densities. The packing densities reach the maximum when the aspect ratio is about 1.5 for allthe surface shape parameters and oblateness. The influence of order degree is eliminated on the MDRP state,which demonstrates the advantages of theMDRP state as a platform for comparing the packing density of disor-dered packings. Our work helps to explore the shape effects on disordered packings of non-spherical particles.

© 2020 Elsevier B.V. All rights reserved.

Keywords:SuperballShape effectBi-axial elongationMaximally dense random packingPacking density

1. Introduction

Packings of particles are often used to describe the structures andsome fundamental properties of many substances such as crystals, liq-uids, colloids, suspensions, particulate media, amorphous solids andglasses [1–8]. People have been pursuing the optimal particle shape,which leads to the highest disordered-packing density for decades.The random close packing density of equal spheres is about 0.64[9,10], which is a robust andwell-acknowledged value inmany particu-lar disordered states [10–14].Manymethods are applied to improve thedisordered-packing density above that of spheres. One way is the sizepolydispersity. The binary and polydisperse sphere packings havebeen widely studied since the 1930s by Furnas [15] and Westman[16], and still attracted many researchers in recent years [17–32]. Themaximal packing density of disordered binary sphere mixtures ap-proaches 0.87 with the small to large sphere size ratio closing to zero.Meanwhile, the packing density of disordered binary sphere mixturescan approach that of the densest known ordered packings for specificvalues of small to large sphere size ratio and the number fraction ofsmall spheres [24]. Moreover, people concentrated on the effects of par-ticle size distributions on the densities of randomclose packings of poly-disperse spheres [25–28].

The other way to improving the disordered-packing density is tovary the particle shape. The surface shape and aspect ratio are importantand independent shape parameters of a particle whichwere often stud-ied in the literature. The disordered-packing densities of superballs [33]

nd Engineering Science, College

ku.edu.cn (S. Li).

and spheropolyhedra [34] increase dramatically from 0.64 when theirsurface shapes slightly shift away from spheres, indicating that chang-ing the surface shape is an effectiveway to improve the packing density.The maximal packing density of the maximally dense random packings(MDRP) for superballs is about 0.72with the surface shape parameter tobe 0.7 or 2.0 [14,35]. Meanwhile, the disordered packings ofspherocylinders [36–45] and spheroids [14,39,46–53], which are elon-gated or compressed spheres, are also denser than that of spheres, indi-cating that elongating (compressing) the particles can also increase thedisordered-packing density. Themaximal packing density achieves 0.72with the aspect ratio to be 1.5 for both spherocylinders [43] and spher-oids [14,48,50]. The packing density of spheres is a local minimumamong all convex shapes below a certain asphericity, as discussed byJiao [54] and Kallus [55]. The disordered-packing density can alsobe improved via changing the surface shape and aspect ratio of particlessimultaneously. The maximal disordered-packing density ofsuperellipsoids, which are uniaxially compressed or elongated super-balls, can achieve 0.738 in a symmetrical packing density landscapeabout the surface shape parameter and aspect ratio [14]. Uniform anddecoupled effects of surface shape and aspect ratio on the random-packing densities of these superellipsoids were observed [14]. Delaney[50] and Zhao [53] also investigated the packing properties of thesesuperellipsoids under jammed or mechanically stable conditions,which demonstrated non-uniform particle shape effects on the packingproperties as a result of maintaining jamming or mechanically stable.

Moreover, Donev et al. [48] studied the maximal random jammed(MRJ) packings of mono-sized bi-axial ellipsoids. The ratios betweenthree semi-axes of bi-axial ellipsoids were defined as 1 : αβ : α, whereα ≥ 1 is the aspect ratio, and 0 ≤ β ≤ 1 is the oblateness or skewnesswith β = 0 corresponds to a prolate spheroid and β = 1 to an oblate

Fig. 1. The biaxial superellipsoids with different aspect ratio α and oblateness β. Thesurface shape parameter is p= 2.0.

Fig. 2. Configurations of the MRJ packings for biaxial superellipsoids with different aspectratio α and oblateness β. The surface shape parameter p is 1.0, 2.0 and +∞ in (a), (b) and(c), respectively.

50 L. Liu, S. Li / Powder Technology 364 (2020) 49–59

spheroid. The maximums of all the packing density versus aspect ratiocurves with different oblateness βwere obtained at α≈ 1.5, indicatinga uniform shape elongation effect on bi-axial ellipsoids. The maximalpacking density of ellipsoids is about 0.74 with α ≈ 1.5, β = 0.5 [48].Ciesla et al. investigated the random sequential adsorption (RSA) ofbi-axial ellipsoids [56] and cuboids [57] of axes length ratio of a : 1 : b.The packing density of spheres and cubes is not a maximum. Thehighest packing density is about 0.44 with axes ratios 0.7 : 1 : 1.6 for el-lipsoids [56] and 0.40with axes ratios 0.75 : 1 : 1.30 for cuboids [57]. Thelocal orientational orderwas observed for the RSA packings of elongatedcuboids [57]. Meanwhile, the fitted saturation density is sensitive to theterminal time [58] and generating a strictly saturated packing is still dif-ficult [58,59]. However, to our knowledge, the bi-axial shape elongationeffects on the randomclose packing (RCP) for cubes have been rarely in-vestigated, aswell as for other non-spherical particles, especially the su-perballs. Although the cuboids and elongated superballs are ubiquitousin nature and industry. Meanwhile, how the packing state, degree oforder and particle surface shape influence the bi-axial shape elongationeffects for general superballs are also unknown.

In this work, we study the biaxial elongation effects on the disor-dered packings of superballs, including cubes. We focus on the MRJpacking and MDRP mentioned above, both of which are the modifica-tions for the RCP proposed by Bernal [1]. Here, the MRJ packing is de-fined as the maximally random one among all jammed packings [2,11]and theMDRP is defined as themaximally dense one among all randompackings [45,60]. The MRJ packing must be jammed (mechanically sta-ble), and the randomness may be sacrificed to maintain jamming. Incontrast, the MDRP must be random, and the structure may not be me-chanically stable in order to keep random. For particles that are difficultto crystallize, the MDRPs are almost the same with the MRJ packings,such as the spheres [14,45], spheroids [14], spherocylinders [45], octa-hedra [60], and superellipsoids with small surface shape parameters[14]. However, for particles which are easy to crystallize, the packingdensities of MDRPs are lower than those of the MRJ packings, and theMDRPs are not mechanically stable but still keep random, for example,the cuboids [61], cylinders [62], and superellipsoids with large surfaceshape parameters [14]. Therefore, the differences between the MRJpackings and MDRPs of bi-axially elongated superballs should dependon the surface shape parameters, which has not been well studied andis the main topic of this work.

We generate theMRJ packings andMDRPs of bi-axial superellipsoidsvia the adaptive shrinking cell (ASC) algorithm [63,64] and the inverseMonte Carlo (IMC) method [14,61], respectively. We use the aspectratio α and oblateness βmentioned in Ref. [48] to represent the relativelengths of three axes. The ratios of three axes are set as 1 : αβ : α for theparticles. The influences of the aspect ratio and oblateness on the MRJpackings and MDRPs of bi-axial superellipsoids are systematically in-vestigated. We find uniform shape elongation effects on the MDRPs of

ellipsoids, superellipsoids and cuboids. However, the aspect ratio effectson the MRJ packings of these particles are diverse, further demonstrat-ing the advantages of the MRDP state in investigating the particle-shape effects on random packings. The rest of the paper is organizedas follows: In Section 3, we introduce the particle models and the pack-ing algorithms we use. The order parameters we used to evaluate therandomness of packing structures are also presented in this section. Fi-nally, the simulation results of the MRJ packings and MDRPs arediscussed in Section 4, and concluding remarks are provided inSection 4.

Fig. 3. The packing densityφ (a), the normalized local bond-orientational order parameterQ 6local (b), the global cubatic order parameter I4,x, I4,y, I4,z, S4 (c), and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local (d) of the MRJ packings for biaxial ellipsoids (p = 1.0).

51L. Liu, S. Li / Powder Technology 364 (2020) 49–59

2. Methodology

In this part, we introduce the particle models and packing algo-rithms used in this work. The order parameters applied to evaluatethe order degree of packing structures are also presented in this section.

2.1. The particle models and packing algorithms

The superellipsoid model [65] and the ideal polyhedron model [60]are used to describe the bi-axial superellipsoids. The superellipsoidmodel is a rich geometrical model and can be used to describe variouslyshaped particles, containing spheres, ellipsoids, superballs, cylinders,cuboids and octahedra. In this work, the surface shape of a bi-axialsuperellipsoid can be described as

xa

��� ���� �2pþ y

b

��� ���� �2pþ z

c

��� ���� �2p¼ 1:0 ð1Þ

where a, b and c are the semi-major axis lengths in the direction of x, y,and z axes, respectively. The p is the surface shape parameter determin-ing the sharpness of particles. The superellipsoid degenerates to an oc-tahedron, sphere and cube when the p is 0.5, 1.0 and +∞, respectively.

In this work, we mainly focus on the different shape effects on thepacking densities of MRJ packings and MDRPs for bi-axialsuperellipsoids. As mentioned in the fourth paragraph in the Introduc-tion part, according to our previous work in Ref. [14], the superballswith the shape parameter p ≤ 1.5, including regular octahedra, are diffi-cult to crystallize and the MDRPs are almost the same with the MRJpackings. The aspect ratio effects on these superballs with p ≤ 1.5 shouldbe in accordance with the spheres (p = 1.0). However, the superballswith the shape parameter p N 1.5 are easy to crystallize and the packingdensities ofMDRPs are lower than those of theMRJ packings. The aspectratio effects on the packing densities ofMRJ packings andMDRPs shouldbe also different. In order to decrease the computational cost withoutloss of generality, we just choose p = 1.0, 2.0 and +∞, representingthe ellipsoids, general superellipsoids and ideal cuboids, respectively,

Fig. 4. The packing densityφ (a), the normalized local bond-orientational order parameterQ6local (b), the global cubatic order parameter I4,x, I4,y, I4,z, S4 (c), and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local (d) of the MRJ packings for biaxial superellipsoids (p = 2.0).

52 L. Liu, S. Li / Powder Technology 364 (2020) 49–59

which are representative to investigate the bi-axial elongation effectson the packing densities of MRJ packings and MDRPs for bi-axialsuperellipsoids.

The values of a, b and c are set as a : b : c=1 : αβ : α, where α and βare the aspect ratio and oblateness mentioned in the introduction part.Fig. 1 shows the bi-axial superellipsoidswith different aspect ratioα andoblateness β. The surface shape parameter p is equal to 2.0. The axislength is a = 1.0. The b and c increase with the increase of β and α.More details about the superellipsoid model and the overlap detectionalgorithmweuse can be found in Ref. [14].Meanwhile, the ideal cuboidswith p = + ∞ are represented by the ideal polyhedron model men-tioned in Ref. [60].

We generate theMRJ packings andMDRPs of bi-axial superellipsoidsvia the adaptive shrinking cell (ASC) algorithm [63,64] and the inverseMonte Carlo (IMC) method [14,61], respectively. The ASC algorithm isa Monte Carlo (MC) compression process with random particle move-ments and allows the simultaneous change of different lattice vectorsof the simulation box at the same time [63,64]. It has been used to obtain

the densest-known packings [63,64,66] and the MRJ packings [54,67–69] of hard particles, as well as to investigate the phase behaviorand phase transition [66,68]. The IMC method is based on the ASCmethod with an order constraint to prevent the forming of orderedstructures in the packing [14,61,62]. The IMC method can be used togenerate the maximally dense packings under certain degrees of ordervia a preset value of order constraint. The final packings obtained withthe order constraint Opup = 0.5 are regarded as close approximationsto the ideal MDRPs. The total number of particles is set to be N = 200,which is large enough to ignore the system size effect under periodicalboundary conditions in all three directions [14,60,61]. More detailsabout the packing algorithms we applied can be found in Ref. [62].

2.2. The order parameters

In the IMC method, special local order parameters are required toevaluate and control the degrees of order in the packing configurations.Considering that the superellipsoids studied in this work are all

Fig. 5. The packing densityφ (a), the normalized local bond-orientational order parameterQ6local (b), the global cubatic order parameter I4,x, I4,y, I4,z, S4 (c), and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local (d) of the MRJ packings for biaxial cuboids (p= + ∞).

53L. Liu, S. Li / Powder Technology 364 (2020) 49–59

octahedral symmetric and the three axial directions are not equivalent,we use the normalized local bond-orientational order parameter Q6local

[14] to evaluate the bond-orientational order and the normalizedcubatic order parameters P4local,1, P4local,2, P4local,3 [62] and S4local [14] tosuppress the orientational orders in the x, y, z directions and their cou-pling orientational order, respectively. All the local order parametersdescribe the average order degrees of each particle with its n nearestneighbor particles and are normalized via the Monte Carlo tests[14,62]. The average bond-orientational and orientation correction ofparticles with n nearest neighbor particles can be defined as

Q6local;n ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π13

∑m¼6m¼−6

1N∑N

i¼11n∑n

j¼1Y6m θij;φij

� �� ����� ����2s

ð2Þ

P4local;k;n ¼ 1N

XNi¼1

18n

Xnj¼1

35 cos4θik;jk−30 cos2θik;jk þ 3� 24 35 ð3Þ

S4local;n ¼ 1N∑N

i¼1 maxk

114n

∑nj¼1∑

3l¼1 35 cos4θik;jl−30 cos2θik;jl þ 3

� � �ð4Þ

where N is the total number of particles in the packing system, n=1, 2,3…, 26 represents the number of particles that are closest to the ith par-ticle, k, l=1, 2, 3 are the three principal axes of particles, correspondingto x, y, z, respectively. Y6m(θ,φ) are the spherical harmonics. θij, φij arethe polar and azimuthal angles of the bond formed by particle i and j.cosθik;jl ¼ j u!ik∙ u

!jlj with u!ik the ith central particle's kth axis and u!jl

the lth axis of the jth neighbor particle of the ith central particle.According to the Monte Carlo tests in our previous work [14,61,62],

the probability distribution functions of the values of these local param-eters in the random state are in Gaussian distributions. The means andstandard deviations are related to the particle amounts N and the

Fig. 6. Configurations of theMDRPs for biaxial superellipsoidswith different aspect ratioαand oblateness β. The surface shape parameter p is 1.0, 2.0 and +∞ in (a), (b) and (c),respectively.

54 L. Liu, S. Li / Powder Technology 364 (2020) 49–59

number of neighbors taken into account, i.e.,

Qμ6local;n ¼ 0:98123=

ffiffiffiffiffiffiffiNn

p;Qσ

6local;n ¼ 0:19379=ffiffiffiffiffiffiffiNn

pð5Þ

Pμ4local;k;n ¼ 0:0; Pσ

4local;k;n ¼ 1= 3ffiffiffiffiffiffiffiNn

p� �ð6Þ

Sμ4local;n ¼ 0:30972=ffiffiffin

p; Sσ4local;n ¼ 0:38720=

ffiffiffiffiffiffiffiNn

pð7Þ

Finally, the local order parameters are normalized as

fOp;n ¼ Op;n−Opμ;nOpσ;n

���������� ð8Þ

and the normalized local order parameters are

Op ¼ maxn

fOp;njn ¼ 1;2;3…;26n o

ð9Þ

whereOp represents the order parametersQ6local, P4local,x, P4local,y, P4local,zand S4local.

Meanwhile, we also use the global cubatic order parameters I4, x, I4, y,I4, z [70] and S4 [71] to evaluate the global orientational order of the gen-erated packings of bi-axial superellipsoids. The order parameters are de-fined as

I4;k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π9∑m¼4

m¼−41N∑N

i¼1Y4m θik;φikð Þ���� ����2

sð10Þ

S4 ¼ maxn!

114N

∑Ni¼1∑

3k¼1 35 cos4θik; n!−30 cos2θik; n! þ 3

� �ð11Þ

where θik and φik are the polar and azimuthal angles of the ithcentral particle's kth axis u!ik, Y4m(θ,φ) are the spherical harmonics. cos

θik; n! ¼ j u!ik∙ n!jwith n!the unit vector forwhich S4 ismaximized. An ap-

proximate solution described in Ref. [71] is applied to get n! and S4. Allparticle axesuij are chosen as trial directors to get n!for themaximum S4.

The Q6local is larger than 80.0 in the Simple Cubic (SC), Body-Centered Cubic (BCC), Face-Centered Cubic (FCC) and HexagonalClose-Packed (HCP) packings with 216 particles. For a lattice packingswith all the particles owning the same directions, all the global andlocal cubatic order parameters I4,x, I4,y, I4,z, S4, P4local,n,x, P4local,n,y, P4local,n,z and S4local,n are equal to unity and the normalized local cubatic orderparameters P4local,x, P4local,y, P4local,z and S4local are larger than 170.0 for N=200. However, in a polycrystalline structure or a quasi-random pack-ing [13] with large amounts of local ordered clusters, the global cubaticorder parameters are small while the local cubatic order parameters arelarge. The packing with smaller order parameters will be more disor-dered. In an ideal random packing, all the order parameters should bezero.

3. Results and discussion

In this part, we study the MRJ packings and MDRPs of the bi-axialsuperellipsoids, the degrees of order and the shape elongation effectsare also systemically investigated.

3.1. The maximally random jammed packings

Asmentioned above, theMRJ packings of bi-axial superellipsoids aregenerated via the ASC algorithm. Here we choose p = 1.0, 2.0 and +∞,representing the ellipsoids, general superellipsoids and ideal cuboids,respectively. Some packing configurations are shown in Fig. 2 for differ-ent surface shape parameter p, aspect ratio α and oblateness β.

When p = 1.0, the superellipsoids are general ellipsoids. Fig. 3(a)shows the packing densities of different ellipsoid packings as functionsof the aspect ratio α for different oblateness β. The packing density ofspheres is a local minimum, demonstrating that changing the particleshape away from spheres should increase the disordered-packing den-sity. Meanwhile, the maximal packing densities are all obtained atα ≈ 1.5 for different β, which is consistent with the results obtainedby Donev et al. in Ref. [48]. The packing density curve for β= 0.5 is al-ways higher than the curveswith other β values.Meanwhile, the curvesfor β= 0.0 and 1.0 are very close, likewise the curves for β= 0.25 and0.75, which demonstrate a symmetric influence on the packing density

Fig. 7. The packing densityφ (a), the normalized local bond-orientational order parameterQ6local (b), the global cubatic order parameter I4,x, I4,y, I4,z, S4 (c), and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local (d) of the MDRPs for biaxial ellipsoids (p= 1.0).

55L. Liu, S. Li / Powder Technology 364 (2020) 49–59

for the oblateness β. The maximal packing density of ellipsoids isreached at α ≈ 1.5, β = 0.5 with the value to be about 0.72, which issmaller than 0.74 in Ref. [48]. This is because the compression rate Γ ap-plied in the ASC algorithm in our work is Γ= 1.0, leading to the fastestcompression process and the final packings are just MRJ-like packingswhich may be only locally jammed, as also discussed in Ref. [62].

As shown in Fig. 3 (c), all the packings are globally randomwith theglobal order parameters smaller than 0.1. However, small differences inthe values of local order parameters are observed in Fig. 3 (b) and (d).The normalized local bond-orientational order parameter Q6local reachthe maximal value for the packing of spheres as a result of the uniformlocal curvature of spheres. Moreover, the Q6local decreases with the in-crease of aspect ratio α and is not sensitive to the values of oblatenessβ. For very long or oblate ellipsoid packings, the normalized localcubatic order parameters are slightly higher with two or three particlesaligning or stacking together. Therefore, the sphere-like ellipsoid pack-ings show higher bond-orientational order for the particle centers andvery long or oblate ellipsoid packings show higher orientational orderfor the axes. However, the maximal value of the local bond-

orientational order parameter Q6local is around 20, and the maximalvalue of the normalized local cubatic order parameters is smallerthan 15, demonstrating that the degrees of local order for the ellipsoidpackings are not remarkable.

When p = 2.0, the superellipsoids demonstrate cuboid-like shapeswith rounded edges and corners. As shown in Fig. 4(a), the packing den-sity first increases and then decreases when the aspect ratio α increasesfrom 1.0 for different oblateness β. However, the optimal aspect ratioswhich correspond to the maximal packing density for different oblate-nessβ are diverse around 1.1 to 1.4. The effect of oblateness on thepack-ing densities of the MRJ packings with p = 2.0 is insensitive, which isdifferent from that for ellipsoids shown in Fig. 3(a). The trends of thelocal bond-orientational order parameter Q6local are almost the sameas those of ellipsoids shown in Fig. 3(b). The Q6local decrease when theaspect ratio α increases from 1.0 and is not sensitive to the values of ob-lateness β. The maximum of Q6local is about 17, obtained at α = 1.0 forthe packing of superballs with p=2.0. For theMRJ packing of superballswith p = 2.0, the global cubatic order parameters I4,x, I4,y and I4,z areabout 0.2 and the global cubatic order parameter S4 are about 0.3.

Fig. 8. The packing densityφ (a), the normalized local bond-orientational order parameterQ6local (b), the global cubatic order parameter I4,x, I4,y, I4,z, S4 (c), and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local (d) of the MDRPs for biaxial superellipsoids (p = 2.0).

56 L. Liu, S. Li / Powder Technology 364 (2020) 49–59

Meanwhile, the corresponding local cubatic order parameters P4local,x,P4local,y, P4local,z are around 25 and the S4local is about 50. Therefore, theMRJ packings of superballs with p = 2.0 demonstrate higher degreesof global and local cubatic order.

Fig. 4 (c) and (d) show the global and local cubatic order parametersfor the MRJ packings of superellipsoids with p = 2.0. All the globalcubatic order parameters decrease with when the aspect ratio α in-creases from 1.0 to 3.0, indicating that elongating the superballs withp = 2.0 will lower the global cubatic order in the MRJ packings. How-ever, when the aspect ratio α increases to 3.0 for β = 1.0, thesuperellipsoids are disk-like and form stacks of two to five particles inthe MRJ packings, resulting in a larger value of the uniaxial cubaticorder parameter P4local,x. The other local cubatic order parameters andall the global order parameters still decrease with the increase of α.Therefore, the equal elongation on the y and z axes can reduce the globalorientational order but increase the local orientational order. Elongatingwill decrease both the degrees of global and local cubatic order param-eters for all the other values of the oblateness β.

Finally, when the shape parameter p= +∞, the superballs are idealcubes, and the corresponding superellipsoids are cuboids. As shown in

Fig. 5 (a), the packing density of cubes is the maximum for all the cu-boids considered, as well as all the order parameters demonstrated inFig. 5 (b)-(d). In the MRJ packings of cubes, the packing density isabout 0.723, the normalized local bond-orientational order parameterQ6local is about 14, the global cubatic order parameters I4,x, I4,y, I4,z areabout 0.4, the S4 is about 0.6, the normalized local cubatic order param-eter P4local,x, P4local,y, P4local,z are about 60 and the S4local is about 110.Therefore, the MRJ packings of cubes are in much higher degrees ofglobal and local cubatic order than the packings of superballs withp = 2.0. Meanwhile, all these parameters decrease when the aspectratio α increases from 1.0 to 3.0, indicating that elongationwill decreasethe degrees of cubatic order in cuboid packings. All the parameters arenot sensitive to the oblateness β with large error bars.

In summary, the degrees of order in MRJ packings of superellipsoidswith different shape parameters are not consistent and shape elonga-tion effects on the packing densities are not uniform. The effects oforder degree and particle shape on the disordered-packing density arecompounded on the MRJ packing state, meaning that the MRJ statemay not be an appropriate platform for the purpose of studying the par-ticle shape effects on disordered-packing densities.

Fig. 9. The packing densityφ (a), the normalized local bond-orientational order parameterQ6local (b), the global cubatic order parameter I4,x,I4,y, I4, z, S4 (c), and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local (d) of the MDRPs for biaxial cuboids (p = + ∞).

57L. Liu, S. Li / Powder Technology 364 (2020) 49–59

3.2. The maximally dense random packings

The MDRPs of bi-axial superellipsoids with p = 1.0, 2.0 and +∞ aregenerated via the IMC algorithm and some packing configurations areshown in Fig. 6 for different values of aspect ratio α and oblateness β.Figs. 7, 8 and 9 demonstrate the packing density and order parametersfor theMDRPs of superellipsoids with p=1.0, 2.0 and+∞, respectively.The normalized local bond-orientational and cubatic order parametersapproach to 0.5, as the order constraint Opup in the IMC method is setto be 0.5, which is a very small value. Meanwhile, the global cubaticorder parameters are always lower than 0.15 for all the MDRPs we ob-tained, which indicates no apparent global cubatic order structures.Therefore, all the MDRPs are highly disordered.

When theaspect ratioα increases form1.0, thepackingdensity of theMDRPs always first increases and reaches themaximum at α≈ 1.5 andthendecreases for all the surface shapeparameterpandoblatenessβ, in-dicating a uniform shape elongation effect on the random-packing den-sity. The upper bound of the packing density is almost obtained at β=0.5 for all the aspect ratio α, consistent with the results obtained byDonev et al. in Ref. [48]. Meanwhile, the maximal packing densities of

ellipsoids, general superellipsoids with p= 2.0 and cuboids are about0.743, 0.741 and 0.661, respectively. The packing density of spheres,general superballs with p= 2.0 and cubes are about 0.642, 0.721 and0.643, respectively. The random-packing densities are improved by0.101,0.020,0.018viathebi-axialelongationofparticleshape.Therefore,the bi-axial elongation effects on the random-packing densities areweakenedwith the increase of the surface shape parameter p.

In summary, the MDRPs of all the superellipsoids we studied arehighly disordered without any obviously ordered structures. They areon the same random degree, and the shape elongation effects on therandom-packing densities of superballs are uniform. The influences oforder degree on the packing densities are eliminated on the MDRPstate, which demonstrates the advantages of the MDRP state as a plat-form for comparing the packing density of disordered packings.

4. Conclusions

We generate theMRJ packings andMDRPs of bi-axially elongated su-perballs with different surface shape parameters via the adaptive shrink-ing cell algorithm and the inverseMonte Carlomethod, respectively. The

58 L. Liu, S. Li / Powder Technology 364 (2020) 49–59

normalized local bond-orientational order parameter Q6local, the globalcubatic order parameter I4,x, I4,y, I4,z, S4, and the normalized local cubaticorder parameter P4local,x, P4local,y, P4local,z, S4local are used simultaneously toevaluate the degrees of order in different forms and sizes. The influencesof particle shapes on the disordered-packing densities under the twodis-ordered states are systematically investigated.

On the MRJ packing state, the packings of ellipsoids, generalsuperellipsoids with p = 2.0 and cuboids are on different degrees oforder, and the shape elongation effects on the disordered-packing den-sities are not uniform. The maximal packing density for ellipsoids is ob-tained when the aspect ratio is about 1.5 for all the oblateness, and theupper bound of the packing density is achieved when the oblateness is0.5 for all the aspect ratios. However, the maximal packing density forcuboids is obtained when the aspect ratio is 1.0 and the aspect ratioscorresponding to the maximal packing density for generalsuperellipsoids with p = 2.0 are diverse around 1.1 to 1.4. The effectsof oblateness on the packing densities of general superellipsoids withp = 2.0 and cuboids are insensitive. Moreover, the MRJ packings ofrod-like ellipsoids, general superballs with p = 2.0, disk-like generalsuperellipsoids, and ideal cubes demonstrate higher degrees of orderamong their corresponding particle shape families. The effects of orderdegree and particle shape on the disordered-packing density arecompounded on the MRJ packing state.

On the MDRP state, the packings of all the superellipsoids we studyare highly disordered without any obviously ordered structures. Theyare on the same random degree, and the shape elongation effects onthe random-packing densities of superballs are uniform. The packingdensities first increase and reach the maximum when the aspect ratiois about 1.5 and then decreases for all the surface shape parametersand oblateness. The upper bound of the packing density is obtainedwhen the oblateness is 0.5 for all the aspect ratios. Bi-axial elongationwill improve the random- packing densities and the effects are weak-ened with the increase of the surface shape parameter. The influencesof order degree on the packing densities are eliminated on the MDRPstate, which demonstrates the advantages of the MDRP state as a plat-form for comparing the packing density of disordered packings.

Our results suggest that 1.5 is a particular aspect ratio value whichcorresponds to the maximal random-packing density for superballswith different surface shapes. However, the critical aspect ratio valueof 1.5 is still not theoretically explained, even for the random packingof spheroids. Moreover, the aspect ratio and surface shape effects onthe other shaped superellipsoids, such as axisymmetric superellipsoid,is still not clear and will be investigated in our future work. Our workhelps to explore the shape effects on disordered packings of non-spherical particles.

Acknowledgments

This work was supported by the National Natural Science Founda-tion of China (Grant No. 11572004, U1630112 and 11972047), theChina Postdoctoral Science Foundation (Grant No. 2019TQ0040) andthe High-performance Computing Platform of Peking University.

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