PHYSICAL REVIEW E 98, 042903 (2018)

Coupling effects of particle size and shape on improving the densityof disordered polydisperse packings

Ye Yuan, Lufeng Liu, Yuzhou Zhuang, Weiwei Jin, and Shuixiang Li*

Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

(Received 5 May 2018; revised manuscript received 3 September 2018; published 10 October 2018)

It is well established that the packing density (volume fraction) of the random close packed (RCP) state of con-gruent 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 tononspherical shapes (e.g., superballs or ellipsoids). In this paper, we numerically investigate the coupling effectsof particle size and shape on improving the density of disordered polydisperse particle packings in a quantitativemanner. A previously introduced concept of “equivalent diameter” (De), which encodes information of both theparticle volume and shape, is reexamined and utilized to quantify the effective size of a nonspherical particle inthe 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 (reciprocalof ϕc) can be expressed as a linear combination of the specific volume ek for each component k (particles withidentical shape), weighted by its corresponding volume fraction Xk in the mixture, i.e., e = ∑

kXkek . In thiscase, 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 linearpart 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-dependentmodification to improve the accuracy from ∼0.01 to ∼0.005. Especially for nearly spherical particles, finc isonly 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 resultsalso have implications for the rational design of granular materials and model glass formers.

DOI: 10.1103/PhysRevE.98.042903

I. INTRODUCTION

Disordered packing has served as an elegant model forunderstanding the physics of those complex phenomena inliquids, glasses, and colloidal and granular materials [1–4].In particular, disordered packings of monodisperse spheresgenerated by different methods possess a common densityvalue of ∼0.64 consistent with the well-acknowledged ran-dom close packing (RCP) [5–9]. Apparently, this criticaldensity ϕc is affected by both the system size distribution andindividual particle shape. Therefore, it is of great interest toquantitatively study how these two factors function.

The most simplified model for tackling this issue ispolydisperse disk packing in two dimensions [10–12].It was found that ϕc was positively correlated withthe size span in general. Similar phenomena were ob-served in polygon packings [13,14]. However, monodispersesystems of most two-dimensional shapes tend to form par-tially crystallized patterns including disks [10,15] and othershapes [16–18]. Thus, a slight size dispersity may even re-duce the packing density for certain particle shapes in twodimensions. This is counterintuitive and impractical for aquantitative approach because of the lack of a prototype ϕc

*Corresponding author: lsx@pku.edu.cn

of monodisperse systems in two dimensions. By contrast,polydisperse sphere packings in three dimensions are alwaysdenser than 0.64 as reported in many prior studies [19–24].Therefore, we mainly discuss three-dimensional (3D) poly-disperse systems of different particle shapes in this paper.Note that the term “polydisperse” can also include discretedistributions such as binary or ternary referring to the context.

It was reported that the density increment from 0.64 inpolydisperse sphere packings is positively correlated withpolydispersity δ defined as δ =

√〈(d − 〈d〉)2〉/〈d〉, where d

is the particle diameter and 〈·〉 denotes the average overthe entire packing system [20,23,25–28]. Specifically, spherepackings of log-normal size distribution possess ϕc ∼ 0.73as δ ∼ 0.6, which are much higher than 0.64 [25]. Insteadof being restricted to a certain distribution form, researcherstried to empirically predict ϕc of general polydisperse spherepackings including both discrete and continuous distributions[23,29–31]. Parameter δ alone is apparently not sufficientto characterize the packing behavior. This was also revealedby studies of size span and distribution shapes in two di-mensions [12]. Note that mean-field approaches were pro-posed to deal with polydisperse sphere packings [32–34],yet they cannot be easily extended to nonspherical particle(NSP) shapes. The knowledge of polydisperse sphere packingconstitutes the basis for further studies with the NSP effectsencoded.

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YUAN, LIU, ZHUANG, JIN, AND LI PHYSICAL REVIEW E 98, 042903 (2018)

The packing properties of various 3D NSPs have beenmuch discussed recently, including ellipsoids [16,35], sphe-rocylinders [36–38], polyhedra [39–42], superballs [17], andsuperellipsoids [43]. It was discovered that nearly sphericalshapes possess ϕc higher than 0.64 [44,45], increasing totypically ∼0.7 and even ∼0.74 for certain ellipsoids [16].Therefore, it is intriguing to explore the coupling effectsof both particle size and shape polydispersity on improvingϕc of disordered packings. To understand the properties ofsuch “generalized polydisperse packings” quantitatively, wecan immediately address two questions: (1) how the ϕc of adisordered packing of mixed shapes is correlated with ϕc ofeach component (shape), and (2) how size dispersity furtheraffects such a system.

Most existing studies approached this issue from limitedperspectives. First, we need sufficient but not weak polydis-persity [18,46] of both size and shape to investigate a widerange of ϕc. In several studies mentioned above, shape poly-dispersity was introduced by randomly deviating individualparticles from ideal pentagons [13,14], which can only dealwith a fraction of shapes. Binary disordered packings of NSPs,namely, spherocylinders [47] and irregular polyhedra [48],were also studied as simple forms of polydisperse packings.Nevertheless, one should look into a mixture with a broadsize range and distinct shapes to study the correlations amongthem, which is the main highlight in this paper.

Remarkably, prior researchers proposed the concept of“equivalent diameter” De to evaluate the “size” of an NSP[49,50] and further predict ϕc of a generalized polydispersepacking [51], which highly correlates with the topics of thispaper. However, NSPs discussed in their works are mainlycylinders with various aspect ratios, which only represent partof the shape effect. Their packing states are also ambiguousand vary owing to different packing methods. In spite of theselimitations, Refs. [49,51] offered the fundamental rationaleto examine the effects of size dispersity and individual shapeseparately. Briefly speaking, the RCP states of different shapeswill not be disturbed by each other on the condition that theyhave identical De. Then, the entire packing can be regarded asa superposition of different RCP subsystems of correspondingshapes. If De of particles varies, the size dispersity effect isthen introduced similarly as in polydisperse sphere systems.We inherit these main ideas and provide detailed explanationsfor them in Sec. II. It is worth noting that there is no rigoroustheoretical proof for these arguments, yet they are stronglyvalidated as demonstrated.

In this paper, we conduct an elaborate numerical investi-gation on the coupling effects of both the particle size andshape polydispersity in disordered packings using a fast-compressing Monte Carlo algorithm [40]. A superellipsoidparticle model is applied in the simulations, which can rep-resent a wide spectrum of NSPs. Here, we emphasize thatthis numerical research meets the conditions of broad sizedistribution, universal particle shapes, and uniform packingstate typically as RCP. These conditions are crucial for thefollowing quantitative approaches. The concept of equivalentdiameter De is reexamined and utilized to represent the sizesof NSPs in mixed-shape packings.

In such a mixture with particles of identical De, the overallspecific volume e (reciprocal of ϕc) can be expressed as a

linear combination of the specific volume ek for each shape,weighted by its volume fraction Xk , i.e., e = ∑

kXkek . In thiscase, the mixture can be considered as a superposition of RCPpackings of each shape, as implied by the Voronoi tessellation[52] and contact number analysis. When De of particlesvaries, the overall packing density can be decomposed asϕc = ϕL + finc, where ϕL is the linear part determined bythe superposition law, i.e., ϕL = 1/

∑kXkek , and finc is a cor-

rection part owing to the size polydispersity. We empiricallyestimate finc using two distribution parameters, and apply ashape-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. Gener-alized polydisperse packings even with a moderate size ratio(∼4) can achieve a relatively high density ϕc ∼ 0.8, whichrequires a size ratio of ∼7 in binary sphere packings [53].Our results also have implications for the rational design ofgranular materials and model glass formers.

This paper is organized as follows. In Sec. II, we explainseveral basic definitions of this paper. The particle model andpacking algorithm are briefly introduced in Sec. III. Then,we discuss the equivalent diameter of NSPs and provide anumerical test of binary-shaped packings in Sec. IV. Next, thestructural peculiarities of the linear superposition of mixed-shape packings are demonstrated in Sec. V A. We furtherstudy an empirical method to predict ϕc in Sec. V B and applyit to generalized polydisperse packings in Sec. V C. Finally,we summarize our paper and point out its significance andpossible applications in Sec. VI.

II. BASIC DEFINITIONS

In this section, we explain several basic definitions todepict a packing. The entire packing space (specifically, theperiodic packing cell in this paper) can be tessellated into acellular network with each cell containing exactly one parti-cle, typically a Voronoi tessellation in a monodisperse spherepacking, for instance. This treatment can be also extendedinto NSPs using special techniques [52]. Then, the localpacking density for a certain particle is provided by ϕL =Vp/Vc, where Vp and Vc denote the volumes of this particleand its local cell, respectively. For a monodisperse RCPpacking, the overall density is expressed as ϕc = NVp/Vps ,where N denotes the particle number, and the volume of thepacking space is expressed as Vps = ∑

iVc,i with i loopsfor all particles. After a simple transformation, we obtainVc = Vps/N = Vp/ϕc = Vpec, where ec denotes the specificvolume (reciprocal of ϕc).

In a mixed-shape packing, ϕc of each component shouldcontribute to the overall ϕc. We note that the size dispersityalways promotes ϕc above 0.64 for sphere packings. Such anequal-sized ground state also exists for a mixed-shape systemwith the knowledge of De. Here, we set aside the specificapproach to determine De for a certain NSP, which will beexplained later. In a mixed-shape packing with identical De,the mixture is simply composed of several RCP subsystems ofcorresponding shapes. If this claim holds, some elegant resultsproposed in early literature [51] can be straightforwardlyderived as follows.

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Considering a packing in this critical state, Vps =∑kNkVc,k , where k is the number of loops for components

of different shapes, and Nk is the particle number of eachshape. This equation holds because the RCP states of dif-ferent shapes are not disturbed, nor are the correspondingmonodisperse ones. The overall specific volume is givenby eL = Vps/

∑kNkVp,k = ∑

kNkVp,kek/∑

kNkVp,k , and ap-parently Xk = NkVp,k/

∑kNkVp,k is the volume fraction of

component k. Therefore, we obtain the equation

eL =∑

k

Xkek (1)

for mixed-shape systems where the label L indicates the linearsuperposition case. Equation (1) shows the simplest correla-tion form between the bulk property and its components in amixed-shape packing. From this, we can expect a rather largeϕc if NSPs are imported into the packings.

Here, we review the definition and rationale of the con-ception of De [49,51]. First, we define DV of a NSP as thespherical diameter with equal volume. For disordered binarypackings of a sphere and specific NSP, there exists a criticalstate when Eq. (1) holds with the size ratio between these twoshapes tuned. At this state, De is defined as the diameter of thesphere, such that there is no size dispersity between the sphereand NSP in the system by definition. Then a shape-dependentdimensionless factor can be defined as αc = DV /De for thiscritical state. If this NSP is degenerated to a sphere, Eq. (1)holds only if the whole system is a monodisperse one and thusαc is 1 for the sphere obviously.

Actually, one can choose a shape other than a sphere asthe reference, and it will provide the same results owing toits transferability. Currently, αc of each NSP is a posteriorparameter directly based on the condition of Eq. (1) with thehelp of extensive packing simulations [49]. This is discussedin Sec. IV. With the knowledge of αc after a calibrationsimulation of the sphere and a certain NSP, De of each particleis calculated by DV /αc (DV can distinguish different sizeswith the same shape), which represents its corresponding size.We emphasize that all of these arguments about De are basedon physical intuition and not rigorous theoretical proof, yetthey are valid in practice as demonstrated in this paper.

Typically, ϕc of a polydisperse packing is larger thanϕL = 1/eL owing to the varied De of different particles,which is in accordance with the physical intuition that sizedispersity always promotes the density. A so-called modifiedlinear packing model proposed in Ref. [23] disposes of thisnonlinear deviation from Eq. (1) by empirically quantifyingthe contributions of all particle pairs with size disparities. Thiscan be described as an accumulative approach. In addition tothis approach, a polydisperse-sized system can be describedby only several controlling distribution parameters includingδ. Inspired by the results of the polydisperse sphere packings[31], we investigate an empirical method to estimate thedeviation between ϕL and ϕc by utilizing size distributionparameters. Specifically, we decompose ϕc into two parts,formulated as

ϕc = ϕL + finc (2)

where ϕL denotes the linear part and finc denotes the in-cremental part determined by these distribution parameters.

Since De can represent the size of the NSPs, we might expectthat the dependence of finc on the distribution parametersfor polydisperse sphere systems can be simply extended toa generalized polydisperse packing. A detailed empirical esti-mation method will be thoroughly discussed in this paper.

III. MODEL AND ALGORITHM

In this paper, we utilize a superellipsoid defined as∣∣∣∣x

a

∣∣∣∣2p

+∣∣∣∣y

b

∣∣∣∣2p

+∣∣∣∣z

c

∣∣∣∣2p

= 1 (3)

as our particle model, where a, b, and c are the semimajor axislengths, and p denotes the shape parameter. This degeneratesto an ellipsoid as p = 1 and approaches cubic shapes asp increases. We can study a large number of NSP shapesevolving smoothly from a sphere and reproduce typical NSPs,namely, ellipsoids and superballs, with this unified model.Here, we consider limited shapes with a = b, and define theaspect ratio as w = c/a.

To generate disordered packings of NSPs, we implementa fast-compressing Monte Carlo algorithm of a hard particlemodel. Details of the algorithm can be found in Ref. [40],and here we briefly explain this algorithm. During the com-pression process, the periodic boundary shrinks once eachparticle moves T times on average, where T is a presetconstant and the compression rate can be defined as 1/T.To suppress crystallization, we should initially set T smallenough (T = 1 in this paper) to pull the system out of theequilibrium liquid branch. This process also ensures that thereis no phase separation between particles of different sizes. Asthe interparticle gap becomes small, the compression shouldbe slow (T reset as 500) to ensure jamming. This is becauseparticles are resettled in sequence during the Monte Carloprocess, which theoretically can only ensure local jammingespecially when the compression rate is rather high.

By implementing this slowdown procedure, precise jam-ming (collective jamming) can be better approached from theaspects of both density and structure. The magnitudes of bothtranslational and rotational trial movement are self-adaptiveto keep the acceptance ratio at ∼0.35. Furthermore, particlesare grouped into different bins based on their similaritiesin size and shape in polydisperse systems, and self-adaptivemovements are carried out separately for particles in differentbins.

In this paper, the system size (particle amount) is 500 forbinary and ternary packings, while in polydisperse cases itis 1000. All of the results discussed in this paper are theaverage of five to ten independent replications. To verify therationality of this algorithm, we generate disordered packingsof binary spheres with size ratios up to 3, as shown in Fig. 1.In addition, the same algorithm is utilized in monodispersepackings of superellipsoids of various shapes, as shown inFig. 2. The results shown in Figs. 1(a) and 2(a) are in generalagreement with those of previous studies [25,32,43,53,54].The packing structures illustrated are apparently disordered.Here, we note that rattlers, floating particles in the voids,become non-negligible as the particle size ratio enlarges,especially when small particles account for a small part ofthe overall volume. This issue should be managed carefully

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FIG. 1. (a) Packing density of binary spheres vs volume fractionof small spheres XS obtained in this paper (solid lines), comparedwith several existing results in Refs. [25,53,54]. Labels indicatecertain size ratios r in different works. Two typical packings of r = 2and 3 are displayed in (b) and (c), respectively.

in polydisperse-sized packings because it makes the jammingcondition problematical in this algorithm. A detailed discus-sion of the rattlers in binary sphere packings can be found inRefs. [53,55].

IV. EQUIVALENT DIAMETER

The conception of equivalent diameter is proposed to con-vert NSP shapes into the sizes of spheres, which makes itpossible to compare sizes of different NSPs [49–51]. For adisordered binary packing composed of a sphere and a specificNSP, there should exist a special size for the NSP such thatthe entire system can be seen as a simply superposed mixtureof these two RCP states. In Eq. (2), the linear part remainsconstant if we just tune the size ratio of the binary packingwhile fixing the volume fraction. Thus, ϕc of the linear casemust be located at the minimum in this protocol since finc ispositively correlated with the size ratio. This feature providesthe approach to obtain αc of certain NSPs [49].

In Fig. 3(a), we show part of the results for 50:50 vol-ume fraction binary packings of a sphere and certain NSPs,including superballs with varied p and ellipsoids with variedw. These are labeled by filled and hollow dots, respectively.As α (defined as the particle volume ratio of an NSP over asphere) is varied, ϕc shows a clear minimum for all of theseNSPs. In this case, De of a certain NSP is equal to DV of thesphere, which gives exactly α = αc. Then, we apply a cubiccurve fitting to obtain both αc and ϕc of the linear combination

FIG. 2. (a) Packing density vs aspect ratio w for monodispersesuperellipsoids of varied p, compared with results in Refs. [16,43].Typical monodisperse packing structures of superellipsoids (p =0.75, w = 0.5) and ellipsoids (w = 1.5) are shown in (b) and (c),respectively.

cases. We check this method using 30:70 binary mixtures forseveral NSPs, which provides nearly the same results.

For a certain particle shape, sphericity is defined as theratio between the spherical surface area of equal volumeand its own. We plot αc versus sphericity in Fig. 3(b) afterperforming simulations for numerous NSPs and fitting thedata. Starting from the sphere (at bottom right corner), αc

of all of these NSPs first increases significantly to the peak∼1.4 with sphericity ∼0.95 and then decreases. In general,any NSP considered in this paper should possess a largerparticle volume than a sphere if these two have identicalDe. Then, De is applied to represent the particle size in amixed-shape packing in Sec. V. We reorganized the data of thesuperellipsoids in the inset of Fig. 3(b), where the x axis is setas w. Interestingly, this shares similar features with Fig. 2(a)when the shape is close to a sphere, which means that theaspect ratio influences both ϕc and αc in an analogous manner.

To intuitionally illustrate the peculiarity of αc, we performa complete simulation of the binary packings composed ofa sphere and superball with p = 1.5. As shown in Fig. 4(a),there is a clear linear relationship between the overall specificvolume and the volume fraction of the sphere when α is about1.1–1.2. This is in excellent agreement with the obtainedαc = 1.18 of this kind of superball [see the inset of Fig. 3(b)].The left and right end points indicate the monodisperse ϕc

of the superball (∼0.71, p = 1.5) and sphere (∼0.64), re-spectively. As the size ratio deviates from this critical value,nonlinearity emerges and activates the incremental part of finc.

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FIG. 3. (a) Packing density of 50:50 volume fraction binarymixtures of a sphere and certain NSPs, including different superballs(dashed lines) and ellipsoids (solid lines). We enlarged the lengthscale of NSPs by α times to find the minimum via cubic curvefitting. (b) Obtained αc of NSPs vs their sphericity. Note that thesphere is labeled with a black diamond as a special case in the figure.Reorganized data with aspect ratios w are shown in the inset of (b).

Undoubtedly, finc is positively correlated with the size ratio ifthe particle volume fraction is fixed in binary systems.

Since the minimum ϕc of the 50:50 volume fraction binarymixtures as in Fig. 2(a) are known via data fitting, ϕc or thespecific volume equivalent of these NSPs can be predictedwith the relationship epre = 2ebi − esp, where ebi is the inverseof fitted ϕc, and esp is the specific volume of the RCP of amonodisperse sphere (1/0.64 = 1.5625). Therefore, anothersimple verification of the linear case is to compare epre withthe results from a direct simulation of monodisperse NSPpackings. In Fig. 4(b), we find that most of the data lieclose to the identity line (dashed line), which validates ourargument. Note that superellipsoids with relatively large p

(green diamonds and also one black cube in Fig. 4(b)) deviatefrom this line since these particles tend to form ordered struc-tures in a monodisperse simulation, which produces denserpackings [43].

FIG. 4. (a) Results of binary packings of a sphere and superball(p = 1.5). We plot the specific volume of mixtures vs volumefraction of spheres Xsphere. The size ratio is up to 3 in interpretation ofequivalent diameter, including cases of both a larger sphere (dashedlines) and smaller sphere (solid lines). Note that the same colormeans the identical size ratio. (b) Predicted specific volume epre ofmonodisperse NSPs vs simulation results esimu. The dashed line isthe identity line.

V. POLYDISPERSE PACKINGS

With the knowledge of ϕc and αc for the NSPs studiedabove, we can further investigate Eq. (2) quantitatively. Whenwe study the shape polydispersity, the linear interpolationmethod is applied to obtain the values of ϕc and αc for anyNSPs in the same class, namely, superballs or ellipsoids, basedon the existing data. First, we discuss the linear superpositioncase in Sec. V A, and then look into more complicated situa-tions in Secs. V B and V C. Note that a detailed explanation ofthe constitution of polydisperse packings in this paper appearsin the Appendix.

A. Linear superposition

We generate ternary or polydisperse shape packings withidentical De for each particle, which leads to finc = 0.

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FIG. 5. (a) Packing density predicted by Eq. (1) vs simulationresults. We consider packings of both ternary and polydisperseparticle shapes with identical De. Specifically, ter1, ter2, and ter3are three ternary systems; ps1 and ps2 are polydisperse superballpackings; and pe1 and pe2 are polydisperse ellipsoid packings (seethe Appendix for further details). The dashed line is the identity line.(b) Ternary packing of a sphere and two kinds of superballs (p =0.75 and 2). (c) Polydisperse ellipsoid packing with lnw uniformlydistributed within (ln0.5, ln2). Colors are varied among differentparticle shapes.

Then, in these systems, the overall specific volume shouldbe exactly eL as determined by Eq. (1). As displayed inFig. 5(a), the simulation results are in perfect agreement withthe linear combination predictions. Specifically, we show aternary packing and a polydisperse packing in Figs. 5(b) and5(c), respectively. Different shapes are well mixed and are inhighly disordered states.

The lack of a density increment indicates that the entiresystem can be seen as the superposition of different RCPsubsystems. Here, we reveal this character by inspecting thelocal packing structures. As explained in Sec. II, Eq. (1) canbe directly derived with the concept of local volume, whichcan be determined structurally in dense packings. Therefore,we utilize the point set Voronoi tessellation technique to dealwith NSP packings, and then study the distribution of localVoronoi cell volumes. Once the cell volume is normalized bythe corresponding particle volume, denoted as v0, it directlyequals the specific volume locally. Then, we can compare thedistribution of v0 for a certain particle shape within differentpackings. This provides elaborate microscopic informationabout local structures.

In Fig. 6(a), we show this distribution of three monodis-perse systems (solid lines), two binary systems (dashed lines),

FIG. 6. (a) Distribution of local Voronoi cell volume v0 normal-ized by particle volume. We concentrate on three particle shapes: asphere (labeled as sp) and two kinds of superballs (p = 0.85 and1.5), in monodisperse (solid lines), ternary (dash-dotted lines), andbinary (dashed lines) packings. Three arrows above mark the specificvolume of monodisperse RCP for these shapes. (b) Average contactnumber in binary packings of a sphere and two kinds of superballs ofp = 0.85 (solid lines) and 1.5 (dashed lines) with different volumeratios α. Two vertical dotted lines mark αc of these two superballs atabout 1.05 and 1.18, respectively. Three horizontal dotted lines markthe contact number of these three shapes in ternary packings. Forcritical cases at αc, average contact numbers of these three shapesin binary mixtures (joint points of dashed and solid lines) are about6.09, 6.50, and 7.73, respectively, in general accordance with dottedlines. They are also close to monodisperse counterparts in this paper,yet smaller than results in Ref. [17], not shown here.

and one ternary system (dash-dotted line), which contain asphere and two types of superballs with p = 0.85 and 1.5. Par-ticles in these systems have identical De. Note that the threearrows above mark the specific volume of the monodisperseRCP for these shapes. It can be observed that the distributionof all three particles in binary and ternary systems generallyagrees with the monodisperse counterparts, which stronglysuggests that their local structures are similar as in the RCP.

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We also apply this analysis to systems with size disparitiesand discover that the distribution of v0 for each component inpolydisperse systems deviates from that of the monodisperseones.

Additionally, we count the average contact number of eachcomponent in these systems and plot Fig. 6(b). As the volumeratio α changes in the binary packings, the average contactnumber of the sphere or superball varies monotonously. Whenthe two shapes are of equal size (at αc), their contact numbers[joint points of vertical dotted lines and curves in Fig. 6(b)]generally agree with the results in the ternary-shaped packings(horizonal dotted lines). The exact contact numbers of thesetwo kinds of superballs observed in binary or ternary packingsas ∼6.5 and 7.7 are close to those in the monodisperse ones(6.68 and 7.25). Note that the values given in this paper aresmaller than those presented in Ref. [17], probably owing tothe different contact tolerances. Both observations of the localcell volume and contact number strongly indicate that thelinear case is indeed composed of different RCP subsystems.

B. Polydisperse packings of single shape

The quantitative property of finc in Eq. (2) is of greatimportance because it may offer a predictive approach for thegeneralized polydisperse packings of NSPs. For this purpose,we investigate ternary- and polydisperse-sized packings, con-sidering both cases of a single particle shape in this sectionand multiple ones in Sec. V C. We generate ternary pack-ings including size ratios of 0.8:1:1.25, 0.7:1:1.4, 0.6:1:1.5,and 0.5:1:2. Meanwhile, the continuous distribution for thepolydisperse-sized systems takes the form P (d ) = Ad−3, d ∈[dmin, dmax], where d is the particle size represented by De,and A is the normalized constant with dmin set as 1 and dmax

up to 5 in practice. Note that the volume fractions of eachsmall-sized interval are identical in this distribution.

For cases of a single particle shape, part of our simulationresults is shown in Fig. 6. In these systems, the densityincrement is simply equal to ϕ−ϕ0, where ϕ0 is the foregonedensity of the corresponding monodisperse NSP. Figures 7(a)and 7(b) illustrate a polydisperse packing of superballs (p =1.5) and a ternary packing of ellipsoids (w = 2), respectively.We find that particles of different sizes are well mixed in thesesystems. The relationship between the incremental densityand δ is plotted in Fig. 7(c), including both ternary-sized (opensymbols) and polydisperse-sized (closed symbols) packings.This conveys two features: first, the dependency of ϕ−ϕ0

on the distribution form is qualitatively similar for differentparticle shapes; moreover, there is no clear one-to-one corre-spondence between ϕ−ϕ0 and δ since a bandwidth exists forall of these data.

Following Ref. [31], we utilize an empirical formula to fitfinc as below:

finc = (k1δ + k2δ2S )/k3 (4)

where the skewness S = 〈(d−〈d〉)3〉/〈(d−〈d〉)2〉3/2, k1 andk2 are two empirical constants, and k3 is a modificationcoefficient depending on the particle shapes. The property ofS is discussed in detail in Ref. [31]. In general, it can largelyinfluence finc as the magnitude of δ becomes considerable(∼0.4). This is illustrated by the spread of data points in

FIG. 7. (a) Polydisperse-sized packing of a superball (p = 1.5).(b) Ternary-sized packing of an ellipsoid (w = 2). Colors are variedamong different particle sizes. (c) Polydispersity δ vs incrementaldensity of five different single-shape systems. We consider bothternary-sized (open symbols) and polydisperse-sized (closed sym-bols) packings.

Fig. 7(c). In Eq. (4), the former part in the bracket is directlycorrelated with the size distribution. The factor k3 reflects theshape driven modification about finc, which is 1 for a sphere.We obtain k1 = 0.07376 and k2 = 0.06031 by fitting the dataof the ternary and polydisperse sphere packings. Note thatthe values of these two constants are slightly different fromthose in Ref. [31]. This is because the range of ϕc of thesphere packings in this paper is wider, up to ∼0.73, as shownin Fig. 7(c). Then, for other polydisperse-sized systems ofa single NSP, k3 is fitted with the obtained k1 and k2. Weshow the calculated finc using Eq. (4) in comparison withthe simulation results in Fig. 8(a). The prediction performsfairly well even if the systems possess a large polydispersity(δ ∼ 0.5). In the inset of Fig. 8(a), the dependency of k3 onthe specific volume of a single NSP system is demonstrated,from which we can empirically depict this relationship usingthe formula below:

k3 = 1 + 25.4(1.4825 − ec )2 (5)

where ec is for the monodisperse NSP considered if ec <

1.4825, and k3 = 1 if ec > 1.4825. For a given NSP, k3 isalways larger than 1 and is positively correlated with itsmonodisperse ϕc (reciprocal of ec). This feature implies that alarge ϕc indicates a low density incremental potential becausethe upper density limit caused by sufficient size polydispersityis fixed as 1. Provided that particles are close to the sphere ina polydisperse system, k3 becomes nearly constant as 1, and

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FIG. 8. (a) Prediction of density increment finc via Eq. (4)vs simulation results for polydisperse packings (including ternarycases) of single particle shape. We consider various shapes includ-ing spheres, superballs (different p), and ellipsoids (different w).The inset of (a) shows the relationship between obtained k3 andcorresponding specific volume of a monodisperse NSP, which isfitted using Eq. (5). (b) Same plot of two series of ternary packings,including ternary1 composed of a sphere and two kinds of ellipsoids(w = 0.5 and 2), and ternary2 composed of a sphere and two kindsof superballs (p = 0.85 and 1.5). In the inset of (b), we show theresult of Eq. (4) assuming DV instead of De is utilized with k3 = 1(triangles). The dashed line indicates the identity line.

finc depends only on the system size distribution. This meansthat the overall density can be simply decomposed into twoparts ϕL and finc, solely determined by the shape and sizedistribution, respectively. In practice, we find that the absolutedensity error is at most ∼0.01 even when k3 is fixed as 1 in oursamples, which can be tolerated in some cases.

C. Generalized polydisperse packings

To implement this modification process directly into shapepolydisperse systems, we simply assume k3 = ∑

iXik3,i ,where i loops over all of the components, and k3,i denotesits corresponding coefficient. We generate ternary packings

FIG. 9. (a) finc vs simulation results for three polydisperse sys-tems: a sphere (cubes) and two generalized polydisperse systems ofsuperballs (circles) and ellipsoids (triangles). We also plot obtaineddensity vs δ for these packings in the inset of (a). Two typicalgeneralized polydisperse packings of superballs and ellipsoids areillustrated in (b) and (c), respectively. Colors are varied amongdifferent particle shapes.

with each component set as a certain particle shape and predicttheir density increments, plotted in Fig. 8(b). Note that ϕL iscalculated using Eq. (1). Suppose we utilize DV instead ofDe to study the polydisperse systems. The prediction result ofEq. (4) is plotted in the inset of Fig. 8(b), which obviouslydeviates the identity line. This observation again verifies thatthe definition of particle size in disordered packings withshape polydispersity is De rather than DV , as introduced inRefs. [49,51].

Finally, we investigate the generalized polydisperse pack-ings of NSPs, which include both shape and size polydisper-sity. We generate packings with a continuous size distributiondescribed above, and meanwhile randomize the shape parame-ters, namely, p for superballs and w for ellipsoids. In Fig. 9(a),we show a plot similar to that in Fig. 8 for these two cases incomparison with polydisperse sphere packings. It is observedthat finc is slightly larger than the real density increment in allcases because small NSPs cannot pack the voids as efficientlyas spheres do. In the inset, the relationship between ϕc and δ

is plotted, and we emphasize that such dependency is onlycorrect for this unique size and shape distribution form asdiscussed above. Apparently, the packings of the NSPs aremuch denser than those of the spheres, and are inherentlydriven by their shapes.

Here, we draw some overall comments on this densityprediction framework. As shown in Eq. (4), this process isnot based on certain distribution forms but can deal withboth discrete and continuous polydispersity. The empirical

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formula guarantees the smoothness of the prediction, whichis confused in some packing models. In Eq. (2), the density isdecomposed into a linear part ϕL and an incremental part finc.Approximately, finc is weakly correlated with particle shapes,as mentioned above. If we apply the precise modificationin Eq. (4), the absolute error between the prediction andsimulation is almost within 0.005, even as the systems showrather significant polydispersity as δ ∼ 0.5. Moreover, NSPscan largely promote the packing density as shown in the insetof Fig. 9(a). A large number of NSPs studied in this paper pos-sess a monodisperse ϕc ∼ 0.7, which is significantly denserthan that of the sphere. Then, one can produce polydispersepackings with both a rather high density of ∼0.8 and moderatesize ratio if they are tactfully designed, considering the resultsin Fig. 8. Again, we emphasize that the basis of our studies onthe packings with both size and shape polydispersity is priorknowledge of monodisperse systems.

VI. DISCUSSION

In this paper, we numerically investigate the couplingeffects of both the particle size and shape on densifying thepolydisperse packings using a particle model of a superellip-soid. A flexible fast-compressing Monte Carlo algorithm isutilized to generate a large number of disordered packingsincluding monodisperse, binary, ternary, and polydisperseones. Our simulations meet the fundamental conditions ofbroad size distribution, universal particle shapes, and uniformpacking state for RCPs. These are crucial for a valid approachto this issue.

In highly disordered packings of mixed shapes, the linearsuperposition law in Eq. (1) holds if all particles possessidentical “sizes.” Logically, we show that the earlier intro-duced concept of equivalent diameter De for NSPs worksperfectly to represent a single particle “size” in polydisperseshape systems. By studying the binary packings of spheresand certain NSPs, we numerically measure αc of differentNSPs (a factor solely determined by the particle shape asDe = DV /αc) as the basis for further studies on generalizedpolydisperse packings. Specifically, αc of an ellipsoid of revo-lution with w = 0.5 or 2 is about 1.3 from our simulations.Then, in a binary packing composed of this ellipsoid andsphere, the particle volume of the ellipsoid should be about2.2 times the volume of the sphere in order to ensure the linearsuperposition feature, which shows a considerable difference.

A clear observation of the linear cases is reported indifferent systems of binary, ternary, and even polydisperseshapes in this paper, which was rarely reported in previousworks to our knowledge. The rationale behind this featureis that the overall mixture can be simply considered as asuperposition of RCP packings of each shape. The linearcombination form of a specific volume in Eq. (1) is just themacroscopic embodiment of this nature. We perform a preciselocal analysis on different packings in linear cases using aset Voronoi tessellation technique to verify this statement. Wefind that for each particle shape the distribution of its Voronoicell volume in binary and ternary shape packings is nearly thesame as those in the monodisperse counterparts. In addition,the property of the average contact number of certain shapesis exactly the same. These observations strongly validate the

arguments about the linear superposition explained above.Owing to the rather large monodisperse ϕc of typical NSPs,ϕc of a sphere-rich polydisperse shape packing can be ∼0.7as controlled by Eq. (1) even without any size dispersity. Wenote that the observations here may not simply apply to specialNSPs which tend to crystallize, as shown in Fig. 4(b).

Then, we look into systems with size polydispersity, i.e.,where De of the particles varies. As in Eq. (2), the overall ϕc

is decomposed into two parts, namely, the linear part ϕL (asexplained above) and the increment part finc. Different fromthe linear packing model characterized as an accumulative ap-proach, we try to estimate finc directly with the information ofthe system size distribution. Here, we emphasize again that theterm “size” in polydisperse shape packings is in the meaningof De. An empirical formula as in Eq. (4) is utilized to evaluatefinc, which depends not only on polydispersity δ but also onskewness S and a modification coefficient k3 with Eq. (5).

We perform extensive simulations of polydisperse-sizedpackings for different particle shapes as our samples to studythis density prediction model. This works well for superballand ellipsoid packings with both size and shape polydisper-sity. In particular, finc mainly depends on the size distributionif the particle shapes are nearly spherical (k3 ≈ 1). We canstraightforwardly interpret this approximation that ϕL and finc

reflect the effects of NSP constitution and size dispersity,respectively, which are nearly decoupled in this case. Withthe shape-driven modification in Eq. (5), we improve theprecision of the density prediction method in this paper withan absolute error from ∼0.01 to ∼0.005, which is quiteacceptable in potential applications. Note that the predictionmethod for generalized polydisperse packings discussed inthis paper does not rely on a certain distribution but can dealwith any given discrete or continuous polydispersity. Overall,the coupling effect of the particle size and shape on densifyingpolydisperse packings can be evaluated via this empiricalmodel.

Since finc is just weakly correlated with the particle shapes,ϕc of a polydisperse-sized packing with a single NSP shapecan be seen as a simple shift from the corresponding poly-disperse sphere system, estimated by the difference betweenthe monodisperse ϕc of this NSP and 0.64 for the sphere.Considering that the vast majority of NSPs studied in thispaper possess a monodisperse ϕc ∼ 0.7, which is much higherthan 0.64, one can produce polydisperse packings with arather high density of ∼0.8 and even at a moderate size ratioof ∼4, which requires a size ratio of ∼7 in binary spherepackings. Therefore, it is intriguing to utilize the couplingeffect of size and shape polydispersity to improve ϕc ofdisordered packings, which provides an alternative approachfrom the well-studied size polydispersity for material design.Here, we can provide an example to clarify this point.

For an originally monodisperse packing, we can estimatehow ϕc will change if all the particle shapes smoothly evolvefrom its prototype under certain distributions while maintain-ing their volume. This shape change will first affect ϕL in acertain manner. More subtly, it will introduce size dispersitysince De of each particle is deviated. Consequently, we mayobtain complex results based on the specific particle shape andthe distribution form, which can be optimized even withoutexhaustive simulations. We believe that a precise prediction

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framework in this paper for generalized polydispersity hasimplications for the rational design of granular materials andmodel glass formers.

It is worth noting that the set Voronoi tessellation techniqueis extremely helpful in local structural analyses of NSP pack-ings. The local origin of the density increment driven by thesize dispersity might be discovered by evaluating the informa-tion of local cells. Moreover, particles with different shapesquite probably behave qualitatively alike yet quantitativelydistinctively in this analysis, because we find that a shape-driven modification is crucial to improving the predictionprecision in this paper. These interesting ideas may contributeto understanding some complex phenomena in NSP packings,and they will be discussed in future work.

ACKNOWLEDGMENTS

This work was supported by the National Natural Sci-ence Foundation of China (Grant No. 11572004 and No.U1630112), the Science Challenge Project (Grant No.TZ2016002), and the High-Performance Computing Platformof Peking University.

APPENDIX

In this Appendix, we explain the constitution of our poly-disperse systems in detail. For the systems with identical De

in Sec. V A, we study the packings of ternary and polydisperseshapes. The particle number (N1, N2, N3) is (200, 150, 150),(200, 200, 100), or (300, 100, 100) for the ternary packings.In polydisperse-shaped systems, the shape parameters arelinearly or uniformly distributed in the given intervals. Weset variables as 1/p for superballs and lnw for ellipsoidssince it is reasonable that the shape distribution on eitherside of the sphere has equal weight. For a linear distribu-tion, both one-side distributions and two-side distributions areconsidered. As shown in Fig. 10(a), the choice of the solid anddashed slash can make up three distributions. tc is set as 1 forsuperballs and 0 (ln1) for ellipsoids, which always refers to asphere.

This kind of linear distribution always produces a sphere-dominative system. The uniform distribution is plotted in

FIG. 10. Schematic plots of (a) linear and (b) uniformdistributions.

Fig. 10(b). Once the distribution is provided, we just needto randomize the particle shapes and tune their sizes asdetermined by αc, which can be interpolated from knowndata as shown in the inset of Fig. 3(b). By changing theup and down limits of the distribution, we generate differentshape polydispersities, as shown in Fig. 5. Specifically, ter1 iscomposed of a sphere, ellipsoid (w = 0.5), and superellipsoid(p = 0.75, w = 1.5); ter2 is composed of a sphere, ellipsoid(w = 1.5), and superball (p = 2); ter3 is composed of asphere and two kinds of superballs (p = 0.75 and 2); ps1 andps2 are polydisperse superball packings of uniform and lineardistributions, respectively; and pe1 and pe2 are polydisperseellipsoid packings of uniform and linear distributions, respec-tively.

For ternary-sized systems in Secs. V B and V C, we con-sider size ratios of 0.8:1:1.25, 0.7:1:1.4, 0.6:1:1.5, and 0.5:1:2.We set each size group as certain particle shapes, as shown inFig. 8(b). As explained in the caption of Fig. 8, ternary1 iscomposed of a sphere and two kinds of ellipsoids (w = 0.5and 2), and ternary2 is composed of a sphere and two kinds ofsuperballs (p = 0.85 and 1.5). Distribution function P (d ) =Ad−3, d ∈ [dmin, dmax] is used to study the polydisperse-sizedpackings of both single (Sec. V B) and mixed (Sec. V C)shapes, where d is represented by De. We change dmax from2 to 5 to obtain systems of different size polydispersity. Forthe generalized polydisperse packings of NSPs in Sec. V C,we directly generated uniformly distributed shape parame-ters p ∼ U (0.75, 1)(1, 2) and w ∼ U (0.5, 1)(1, 2) with equalprobability on two sides of 1. The corresponding results areshown in Fig. 9.

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