Equation of state and jamming density for equivalent

bi- and polydisperse, smooth, elastic sphere systems

V. Ogarko, S. Luding

Multi Scale Mechanics (MSM), TS, CTW, UTwente,PO Box 217, 7500 AE Enschede, Netherlands;

v.ogarko@utwente.nl, s.luding@utwente.nl

Abstract

We study bi- and polydisperse mixtures of hard particle fluids. Size ratiosbetween 1 and 100 are studied for different size distributions. Simulationresults are compared with previously found analytical equations of state bylooking at the compressibility factor, Z, and agreement is found with muchbetter than 1% deviation in the fluid regime. A slightly improved empiricalcorrection to Z is proposed.

When the density is further increased, the behavior of Z changes andthere is a close relationship between many-component mixtures and theirbinary, two-component equivalents (where our contribution is to define theterm “equivalent”). We determine the size ratios for which the liquid-solidtransition exhibits crystalline, amorphous or mixed system structure.

Near the jamming density, Z is independent of the size distribution andfollows a −1 power law as function of the difference from the jamming density(Z →∞). In this limit, Z depends only on one free parameter, the jammingdensity itself, as reported for several different size distributions with a widerange of widths.

Keywords: equation of state, jamming, polydisperse, colloids, glasses

1. Introduction

The hard-sphere model can be applied with some success for variousphysical phenomena and systems, like e.g., disorder-order transitions, theglass transition, complex dynamical behavior and simple gases and liquids[1, 2, 3, 4, 5]. Kinetic theory describes the behavior of such particle systems,

Preprint submitted to The Journal of Chemical Physics October 22, 2011

assuming that the particles are infinitely rigid and collisions are instanta-neous [1, 6], like in the hard-sphere model. In this paper we study the highdensity limit, where the caging-effect, that is, particles captured by theirneighbors, becomes important in the dynamics of the global system and thusa free-volume theory needs to be formulated [7, 8, 9, 10, 11, 12, 13]. Thereexists no satisfactory theory, to our knowledge, which is valid for the in-termediate densities where the system changes from the disordered to theordered state. However, various theoretical approaches were proposed in thelast decades, see Refs. [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and refer-ences therein. In this study systems of particles of many different sizes areinvestigated in the high-density limit using theory and simulations. Fluidand jammed configurations of hard sphere mixtures are examined for varioussize distributions and compression rates. It is possible to predict the jam-ming density of many-component mixtures of spheres based on the jammingdensity of two-component mixtures. Additionally, we give accurate data forthe jamming density as a function of size polydispersity, which is importantfor e.g. experiments on non-monosized colloidal or granular systems.

In section 2, the theoretical ideas are presented, which are needed toanalyze the numerical results presented in section 3, before the results aresummarized in section 4.

2. Theory

We consider a thermalized mixture of N elastic smooth spheres, homo-geneously placed in three-dimensional (3D) systems of volume V . Spheresare located at positions ri with velocities vi, radii ai and masses mi. Thekinetic energy Ek = (1/2)

∑Ni=1 miv

2i is dependent on time via the particle

velocities vi. For rigid spheres, the total energy is given by E = Ek, whereasfor soft spheres the potential energy also has to be considered. The temper-ature in equilibrium hard-sphere systems is not a relevant parameter, sinceit does not affect the equilibrium configuration [25], it only scales time. Asa consequence, there is only one independent thermodynamic state variable,which can be either the dimensionless pressure (so-called compressibility fac-tor) Z = PV/NkBT or the density (volume fraction) ν = 4π

∑a3

i /(3V ),related through the equation of state (EOS), where kBT = 2E/(3N).

2

2.1. Bi-disperse systems

Let us consider a binary mixture of spheres with radii a1 and a2, with N1

and N2 the number of particles of each kind, and N = N1 + N2. Thus themixture can be classified by only two parameters, n1 = N1/N and the sizeratio R = a1/a2. The total volume fraction ν = ν1 + ν2 is the last relevantsystem parameter, since the partial volume fractions ν1,2 = 4N1,2πa3

1,2/(3V )can be expressed [21] in terms of n1 and R, using the dimensionless moments

Ak = n1 + (1− n1)R−k =

⟨ak

⟩/ak

1, (1)

where⟨ak

⟩= n1a

k1 + n2a

k2. With this, one has ν1 = n1ν/A3 and ν2 =

(1− n1)ν/(R3A3).A calculation in style of Jenkins and Mancini, see Refs. [26, 27], leads

to the partial translational pressures pti = 2niE/(3V ) for species i and to

the collisional pressures pcij = πN1N2gijaij(1 + rij)T/(3V 2) with the particle

correlation function gij evaluated at contact, and aij = ai + aj. In thefollowing, for simplification the inter-species restitution coefficients rij areset to unity (elastic case), r11 = r12 = r22 = 1. The particle correlationfunctions gij from Ref. [27], are here expressed in terms of A2,3, R and ν:

g11 =1

1− ν+

3ν A2

A3

2(1− ν)2+

(ν A2

A3)2

2(1− ν)3, (2)

g22 =1

1− ν+

3ν A2

RA3

2(1− ν)2+

(ν A2

RA3)2

2(1− ν)3, (3)

g12 =1

1− ν+

3ν A2

(1+R)A3

(1− ν)2+

2(ν A2

(1+R)A3)2

(1− ν)3. (4)

Thus, the global pressure in the mixture is:

pm = pt1 + pt

2 + pc11 + 2pc

12 + pc22

=2E

3V

[1 +

ν

2 〈a3〉(g11a311n

21 + 2g12a

312n1n2 + g22a

322n2)

]

=2E

3V[1 + 4νgO(ν)] .

(5)

Like done for 2D systems in [21], the effective correlation function gO(ν) canbe expressed in term of the dimensionless moments:

O1 =〈a〉 〈a2〉〈a3〉 and O2 =

〈a2〉3〈a3〉2 (6)

3

so that (see Appendix A)

gO(ν) =(1− ν)2 + 3O1(1− ν) + O2(3− ν)ν

4(1− ν)3. (7)

Then the equation of state for mono- and bi-disperse systems reads:

Z = 1 + 4νgO(ν). (8)

Note that in the monodisperse case all gij and gO(ν) are identical to thepair distribution function at contact gCS(ν) = (1 − ν/2)/(1 − ν)3, proposedby Carnahan and Starling (CS) [28], since R = 1, Ak = O1,2 = 1 (usingthe relation ZCS = 1 + 4νgCS [29]). The Carnahan-Starling pair distributionfunction is quite accurate at low and moderate volume fractions, but doesnot show the reported divergence due to excluded volume effects at the closepacking volume fraction [30, 31, 32].

2.2. Polydisperse systems

In the case of a polydisperse mixture (N → ∞) characterized by a sizedistribution function f(a),

⟨ak

⟩ ≡ ∫akf(a)da is the k-th moment of the size

distribution. By just using these moments, the function gO(ν) from Eq. (7)is well defined also for multi-component systems of spheres. It turns out thatthe equation of state (8) coincides with the well-known Boublık, Mansoori,Carnahan, Starling, and Leland (BMCSL) equation of state proposed formixtures [29, 33]:

ZBMCSL =1

1− ν+ O1

3ν

(1− ν)2+ O2

ν2(3− ν)

(1− ν)3. (9)

There are several modified equations of state for multi-component mixturesin the literature, which require the knowledge of the equation of state fora one-component system [18]. First, we consider Santos’ equation of state,based on the Carnahan-Starling equation of state:

ZSCS = ZBMCSL + (O1 −O2)ν3

(1− ν)3. (10)

Second, following Santos’ procedure based on the Carnahan-Starling-Kolafa(CSK) equation of state, ZCSK = [1 + ν + ν2 − 2ν3(1 + ν)/3]/(1 − ν)3 [34],one gets

ZSCSK = ZSCS + (O1 + O2)ν3(1− 2ν)

6(1− ν)3. (11)

4

Third, Boublık extended the CSK equation of state to mixtures [35], yielding:

ZBCSK = ZBMCSL + O2ν3(1− 2ν)

3(1− ν)3. (12)

Although we could consider the extensions of other equations of state, forthe sake of simplicity we will restrict our analysis to the above mentionedequations of state.

2.3. Discussion

It is interesting to observe that all the equations of state considered insection 2.2 are functions of the particle size distribution only through itsdimensionless moments O1 and O2. Therefore, we study and discuss theproperties of O1 and O2. It can be shown that 0 < O1 ≤ 1 and O2

1 ≤O2 ≤ O1 for any size distribution (see Appendix B). This property makes itconvenient to characterize a size distribution by a point on the O1O2 plane.

In the bi-disperse case the system of equations (6) can be solved an-alytically in terms of the variables n1 and R, yielding a unique solution:nbi

1 (O1, O2) and Rbi(O1, O2) (see Appendix C). This means that for anygiven polydisperse system we can construct an “equivalent” bi-disperse sys-tem, which has the same compressibility factor (in the fluid regime) at thesame volume fraction. We use this in section 3 to check how the compress-ibility factor and the jamming density of polydisperse systems are related tothose of bi-disperse ones, constructed using the relation above.

3. Comparison with simulations

Since we are interested in the behavior of rigid particles, we use an event-driven molecular dynamics algorithm as the primary tool for our investiga-tions [25, 36]. The system consists of a cubic cell of volume V , with periodicboundary conditions, which are imposed to simulate an infinite system, i.e.,a statistically homogeneous medium. The compressibility factor is calculatedfrom the total exchanged momentum in all interparticle collisions during acertain short time period ∆t, given by 400 events per particle. By grow-ing the size of the particles linearly in time with a growing rate Γ one canchange the volume fraction (for details see Appendix D). Over time the ad-ditional energy created during collisions would accelerate the particles, butthis is avoided by periodic rescaling of the average particle velocity to hold

5

the mean temperature constant [36]. If the growing is sufficiently slow, thesystem will be in approximate equilibrium during the densification and onecan rather efficiently gather quasi-equilibrium data as a function of density[23, 37]. The number of particles N used in most simulations is 4096, if notexplicitly stated otherwise. In order to improve the performance of neigh-borhood search the ideas of a multilevel contact detection algorithm [38] canbe used, but were not implemented here.

3.1. Particle size distributions

The following types of particle size distributions are used: (i) uniformsize distribution; (ii) uniform volume distribution, i.e., the probability dis-tribution of the volume of the particles is constant; (iii) systems constructedby mapping the aforementioned polydisperse systems to bi-disperse ones (seesection 2.3 for details).

For a given size distribution, we denote the ratio between the maximumand the minimum particle radius as ω 1. Therefore, ω = 1 corresponds tothe monodisperse case, i.e., all sizes are equal, and in our convention ω ≥ 1.Further in the paper when we use ω for bi-disperse systems we mean theextreme size ratio of the equivalent polydisperse system, while for the truesize ratio we use Rbi. Analytical expressions for O1 and O2 as function ofω for uniform size and uniform volume distributions are shown in AppendixE.

The sizes of particles of polydisperse systems are drawn from the particlesize distribution function using the systematic sampling approach [39, 40].It guarantees a more evenly spread sample, i.e., there will always be largeparticles, even if they are rare, like in the case of the uniform volume distri-bution.

3.2. Equation of state in the fluid regime

In Fig. 1 we show the compressibility factor Z from simulations scaled bythe BMCSL, SCS, SCSK and BCSK equations of state for different densitiesand for different size distributions. Note, that in the monodisperse case(O1 = O2 = 1) ZCS = ZBMCSL = ZSCS and ZSCSK = ZBCSK. We observe

1The use of the size ratio ω excludes continuous distributions like log-normal, for whichone has to use O1 and O2 for classification, see Appendix E. The size-distributions usedhere, due to their sharp edges with well defined ω, can be obtained by sieving from wider,smooth continuously distributed realistic distributions.

6

0.99

0.995

1

1.005

1.01

1.015

0 0.1 0.2 0.3 0.4 0.5 0.6

⟨ Z /

Zα

⟩

ν

α=CSα=BCSK

α=OL

(a)

0.99

0.995

1

1.005

1.01

1.015

0 0.1 0.2 0.3 0.4 0.5 0.6⟨ Z

/ Z

α ⟩

ν

α=BMCSLα=BCSK

α=SCSα=SCSK

α=OL

(b)

0.99

0.995

1

1.005

1.01

1.015

0 0.1 0.2 0.3 0.4 0.5 0.6

⟨ Z /

Zα

⟩

ν

α=BMCSLα=BCSK

α=SCSα=SCSK

α=OL

(c)

0.99

0.995

1

1.005

1.01

1.015

0 0.1 0.2 0.3 0.4 0.5 0.6

⟨ Z /

Zα

⟩

ν

α=BMCSLα=BCSK

α=SCSα=SCSK

α=OL

(d)

Figure 1: (Color online) The quality factor, i.e., the numerical Z scaled by the theoreticalpredictions for different densities and for different size distributions (a) monodisperse, andpolydisperse with (b) uniform size, ω = 5, (c) uniform size, ω = 100, and (d) uniformvolume, ω = 4. The growth rate for all data is Γ = 16 × 10−6. The error bars indicatethe standard deviation of the quality factor within an averaging bin. The error bars areshown only for the BMCSL EOS since in the other cases they have the same trend andmagnitude.

7

for monodisperse systems that the deviation of the compressibility factorfrom all considered theories is below 0.3% (a) in the volume fraction regionof 0 < ν < 0.54. The deviation is well below 1% in the volume fractionregion of 0 < ν ≤ 0.6 for polydisperse systems: for uniform size, with ω = 5(b), uniform size, with ω = 100 (c), and uniform volume, with ω = 4 (d),distributions, respectively. This indicates that the number of particles is largeenough to realistically represent the respective size distributions to suppresspossible effects of finiteness of the system. The uniform volume distributiondoes not allow to properly realize wider distributions, due to the small samplesize, N = 4096.

Based on all these results, we propose a more accurate equation of statefor the fluid regime, which is simply the arithmetic average of Santos’ andBoublik’s extensions:

ZOL =1

2(ZSCS + ZBCSK). (13)

Equation (13) is more accurate at intermediate volume fractions 0.20− 0.60and is almost identical to the other forms for lower volume fractions. For allsystems, with ν ≤ 0.60, presented in Fig. 1, ZOL performs better than 0.2%except for the monodisperse case data, for ν > 0.54, due to crystallization.

Our simulations also confirm that for ν < 0.54 the agreement in thecompressibility factor between the considered polydisperse systems and theirbi-disperse equivalents (by compressibility factor, see Appendix C) is oforder of 0.1% for ω ≤ 50 (data not shown).

In the following we show what happens for higher volume fractions.

3.3. How much disorder is necessary to avoid order?

According to Skoge et al. [25], monodisperse hard-sphere systems undergoa first-order fluid-solid phase transition, characterized by a melting point, i.e.,the density at which the crystal thermodynamically begins to melt, and afreezing point, i.e., the density at which the fluid thermodynamically begins tofreeze. In this subsection, we investigate how much polydispersity is neededto avoid partial crystallization, complementing an earlier study in 2D [22].

Since the compressibility factor increases very rapidly at densities higherthan the freezing point, we plot the estimated jamming density φJ = ν/(1−d/Z) instead of the compressibility factor, inspired by [25], as shown in Fig.2, since φJ(Z →∞) = ν. The estimated jamming density φJ here is derived

8

from the free-volume EOS for a d-dimensional system [10]:

Zfv =d

1− ν/φJ

. (14)

In Fig. 2, φJ is plotted against ν for different systems with uniform sizedistribution. Data agree perfectly with the fluid theory before the particlesystem changes from the fluid state to the state of coexistence of fluid andsolid, as indicated by a sharp increase (jump) above the fluid prediction,see Fig. 2(a). The jumps are due to partial crystallization observed for sizeratios 1 ≤ ω . 1.2 but not for larger ω. It must be noted that for thesystem with ω = 1.2, partial crystallization is observed not for every run,as shown in Fig. 2(a) for two runs with random initial velocities, i.e., fromsix runs we found that four systems exhibit partial crystallization, while twosystems do not. This indicates that near the critical size ratio, where or-dering effects disappear, also the history and fluctuations of the system playan important role. In the monodisperse case the freezing density is veryclose to 0.54, which is in good agreement with data from Refs. [23, 25]. Thelocation of the freezing point, νf , is shifted to higher density with increas-ing polydispersity. Taken from a few representative simulations, values areνf = 0.540, 0.548, 0.556, 0.557, 0.565, 0.573 for ω = 1, 1.1, 1.12, 1.14, 1.16, 1.2,respectively, fluctuating with different runs (±0.005). The latter trend re-verses (decreasing νf ) for another type of crystallization which we observe inthe bi-disperse systems with (relatively) large size ratios, see Fig. 3. This isdiscussed in more detail in sections 3.4 and 3.5. In contrast, there is no strongdependence of the (final) jamming density on ω for partially crystallized sys-tems, since crystallization is a stochastic nucleation process, i.e., differentruns of the same system with random initial particle velocities will lead todifferent jamming densities (data not shown). The trends in Fig. 2(b) showthat the larger ω, the larger the density at which disordered systems (notcrystallized) deviate from the fluid theory (the sharp increase of φJ relativeto the fluid theory turns to a smooth decrease). In addition, for such systemsthe jamming density is increasing with ω, which is discussed in more detailin section 3.5. Finally, all curves end at the identity line φJ = ν at densitieswell below the perfect crystal EOS [41], for which φJ = ν ≈ 0.74 would beexpected (for mono-disperse systems).

We checked (data not shown) that the jumps in φJ vanish around ω ≈ 1.2(Rbi ≈ 1.1) also for other size distributions, which are defined in subsection3.1. We only remark that the bi-disperse systems which are equivalent (by

9

compressibility factor) to systems with uniform volume distribution showsome fine structure around the jumps, which we do not discuss here for thesake of brevity.

In Fig. 4, systems with uniform size distribution shown for different sizeratios ω. It can be seen how the ordering disappears as the size ratio increases.

In Fig. 5 bi-disperse systems corresponding to uniform size distributionsare shown for different size ratios, i.e., with (relatively) small size ratios justbelow the crystallization ratio, Fig. 5(a), just above, Fig. 5(b), and with(relatively) large size ratios just below and above partial (single species)crystallization in Figs. 5(c) and 5(d), respectively. Note that changing ofthe polydispersity from 10% (a) to 11% (b) changes the physical state of thesystem a lot, so that we can see the difference even by the eye, which is quiteremarkable.

In section 3.5 we investigate the order-disorder transition for many dif-ferent size distributions in more detail using another criterion to measurepartial crystallization.

10

0.64

0.66

0.68

0.7

0.72

0.74

0.5 0.55 0.6 0.65 0.7 0.75

Est

imat

ed φ

J

ν

11.121.16

1.11.14

1.21.221.2*

ν = φJCS EOS

Crystal EOS

(a)

0.64

0.66

0.68

0.7

0.72

0.74

0.5 0.55 0.6 0.65 0.7 0.75

Est

imat

ed φ

J

ν

2010

532

1.4ν = φJ

BMCSL EOSCrystal EOS

(b)

Figure 2: (Color online) The estimated jamming density φJ as a function of volumefraction ν for systems with uniform size distribution. Shown are systems of 4096 sphereswith various size ratios ω. In (a) ω, given in the inset, corresponds to decreasing φJ (top-to-bottom), and in (b) increasing ω corresponds to bottom-to-top. Also plotted are thefluid theory (BMCSL EOS) and the approximation for the crystal phase [41]. The usedgrowth rate here is Γ = 8 × 10−6. For ω = 1.2 data for two different runs are shown,marked with 1.2 and 1.2∗.

11

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.52 0.54 0.56 0.58 0.6 0.62

Est

imat

ed φ

J

ν

2.40 (6)2.53 (7)2.63 (8)2.79 (10)3.19 (20)3.35 (30)3.49 (50)3.72 (103)

Figure 3: (Color online) The estimated jamming density φJ as a function of volumefraction ν for bi-disperse systems corresponding to uniform size distribution. Increasingω corresponds to bottom-to-top, with growth rate Γ = 16× 10−6. The freezing density isdecreasing with increasing size ratio. The BMCSL EOS are shown by dash-dotted lines.

12

(a) (b)

(c) (d)

Figure 4: (Color online) Particle systems with uniform size distribution with different sizeratios at densities very close to the jamming transition. Size ratios are ω = 1 (a), ω = 1.12(b), ω = 1.18 (c), and ω = 1.22 (d). The order-disorder transition can be clearly seen asthe size ratio increases. Color is by relative size, i.e., yellow (light) corresponds to smallparticles, and blue (dark) corresponds to big ones. The used growth rate to reach theseconfigurations was Γ = 8× 10−6.

13

(a) (b)

(c) (d)

Figure 5: Particle systems with bi-disperse size distribution corresponding to a uniformsize distribution, i.e., n1 = 1/2, with different size ratios at densities very close to thejamming transition. Size ratios are Rbi ≈ 1.100 (ω = 1.18) (a), Rbi ≈ 1.111 (ω = 1.2) (b),Rbi ≈ 2.060 (ω = 4) (c), and Rbi ≈ 2.404 (ω = 6) (d). The order-disorder transition canbe clearly seen. Note that in (d) also some signs of segregation / clusterization of smallparticles can be seen, though investigation of this is beyond the scope of this paper.

14

3.4. Bi-disperse versus polydisperse

The aim of this subsection is to compare the estimated jamming den-sity φJ of polydisperse systems and their bi-disperse equivalents at volumefractions where fluid theory is not valid, i.e., in their “glassy” states or in co-existence of fluid and solid phases. Figure 6 shows φJ as a function of volumefraction for a few systems with uniform size distribution and their bi-disperseequivalents, 6(a), and for systems with uniform volume distribution and theirbi-disperse equivalents, 6(b).

While for small size ratios, ω . 1.2, ordering / crystallization occurs (seeFig. 2), for moderate size ratios, 1.2 . ω . 4, the estimated jamming densityof the considered polydisperse systems and their respective bi-disperse equiv-alents are within 1%, where the bi-disperse data are slightly below. Thus, onecan reduce mixtures to two-component (binary) systems, which have similarphysics, even when the system densities are above the fluid-regime.

The equivalency of two- and many-component mixtures breaks down inthe case of uniform size distribution as polydispersity (width) increases, e.g.,for ω = 6 (Rbi ≈ 2.4, nbi

1 = 1/2), due to partial crystallization of the bigparticles in the bi-disperse system, as it can also be seen in Fig. 5(d).

In the case of bi-disperse distributions corresponding to uniform volumedistributions, we did not observe strong evidence of partial crystallization inthe 1.18 ≤ ω ≤ 10 (1.10 . Rbi . 3.89) range, even though a bit of ordering,see Fig. 6(b), seems to happen. This is because the bi-disperse systems havenbi

1 ¿ 1/2, reflecting that less volume is occupied by large particles.In the inset of Fig. 6(b) the zoomed data for ω = 8 are plotted, showing

that the (final) jamming density can change (jump) also very close to theidentity line φJ = ν, indicating considerable re-structuring even very closeto jamming.

In the next subsection we determine more precisely the range of size ratiosfor which systems show crystallization and / or partial single-species ordering.

15

0.64

0.66

0.68

0.7

0.72

0.74

0.5 0.55 0.6 0.65 0.7 0.75

Est

imat

ed φ

J

ν

1.22 (1.12)

2 (1.48)

3 (1.81)

4 (2.06)

6 (2.40)

ν = φJBMCSL EOSCrystal EOS

(a)

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.5 0.55 0.6 0.65 0.7 0.75

Est

imat

ed φ

J

ν

1.2 (1.11)

2 (1.49)

3 (1.89)

4 (2.24)

5 (2.56)

6 (2.85)

7 (3.13)

8 (3.39) 10 (3.89)

ν = φJBMCSL EOSCrystal EOS

0.735

0.736

0.737

0.738

0.732 0.734 0.736

(b)

Figure 6: (Color online) The estimated jamming density φJ as a function of volume fractionν for systems with (a) uniform size distribution and their bi-disperse equivalents, usingΓ = 8× 10−6, and for systems with (b) uniform volume distribution and their bi-disperseequivalents, using Γ = 16× 10−6. Size ratios ω and Rbi are displayed in the inset, wherethe latter is given in brackets. Also plotted are the fluid theory (BMCSL EOS) and theapproximation for the crystal phase [41]. Data for polydisperse systems in (b) are shownfor ω ≤ 4. In the inset of (b) the zoomed data for ω = 8 are shown.

16

3.5. Towards the jamming density

In this subsection, we investigate the maximum density νmax as a functionof polydispersity. By maximum density we mean the density obtained afterlong-term (slow) compression, i.e., further compression for a long time periodabout 4× 105 events per particle does not increase the density, possibly dueto numeric limits. The average compressibility factor of the final (jammed)systems is typically Z ≥ 1013. (In the next subsection we will also verify thatthe (related) difference between obtained densities νmax and the true jammingdensities, i.e., the densities where the compressibility factor becomes infinity,is at most 10−13.)

In the amorphous regime φJ decreases (Z increases) relative to the fluidprediction, due to excluded volume effects. In Figs. 2 and 6, ordering / crys-tallization was characterized by an increase of φJ above the fluid prediction(i.e. a decrease of Z below). Now, we also relate this to the change of behaviorof νmax, which is consistent with [42].

Figure 7 shows the maximum density νmax as a function of the inversesize ratio ω−1 for different size distributions and for different compressionrates. The size ratio ω corresponds to polydisperse systems, while equivalentbi-disperse ones are constructed using formulas (C.1) and (C.2). Firstly, wesee that for small polydispersity ω−1 & 0.85, i.e., ω ≤ 1.18 (Rbi . 1.100), allthe considered systems display (partial) crystallization or ordering effects,characterized by a jump (discontinuous increase) in νmax with ω−1. Notethat the bi-disperse systems corresponding to uniform volume distributionsdo not crystalize for ω = 1.18 (checked for three different runs), but forslightly smaller ω ≤ 1.17 (Rbi . 1.095). Partial crystallization of this type(small ω) can be seen by the eye in Fig. 5(b).

In the 1.22 ≤ ω ≤ 4 range (partial) crystallization is not observed forall considered systems. Furthermore, the maximum density of polydispersesystems is just slightly larger (within 0.6%) than for their bi-disperse equiva-lents. Note that in the case of uniform volume distributions, larger ω cannotbe properly realized for too small N = 4096, so we do not show results forω > 5.

For larger ω, i.e., ω & 5 (ω−1 . 0.2), bi-disperse systems corresponding touniform size distributions undergo partial crystallization of the large species(Rbi & 2.25), which also can be seen by the eye in Fig. 5(d). This confirmsresults from Ref. [43], where the structure of binary hard sphere mixtures isstudied in more detail.

17

As already evident in Fig. 6, the system with uniform volume distributionand their bi-disperse equivalents pack “better” and reach much larger νmax,see Fig. 7(b), when compared to 7(a), even without apparent partial (species)crystallization [44].

In contrast to their bi-disperse counterparts, polydisperse systems withuniform size distribution do not show any signs of crystallization for ω ≥1.22, and the maximum density converges with increasing polydispersity toνmax ≈ 0.6828 ± 0.0004, seemingly not changing much even for extremelywide distributions. In addition, νmax can be fitted by a function of ω:

νmax(ω) = c0 − c2(3ω−2 − 2ω−3), (15)

where c0 = νmax(ω À 1) = 0.6828 is the maximal density for uniform sizedistributions and c2 = 0.0327517 is the density range. The deviation is within±0.3% for all data ω & 1.2, and ν ′max ≡ 0 for ω−1 → 1. The minimal density,by extrapolation of νmax(ω → 1), is close to the random close packing density,φRCP ≈ 0.64, corresponding to the monodisperse system (ω = 1).

Finally, the maximum density of systems with uniform volume distribu-tion and their bi-disperse equivalents can be fitted by a linear function of ωin the range 1.2 ≤ ω ≤ 5:

ναmax(ω) = cα

0 + cα1 ω, (16)

where cpoly0 = 0.634744, cpoly

1 = 0.016413, and cbi0 = 0.635278, cbi

1 = 0.0157049,with deviation within ±0.3%. Extrapolation of Eqs. (15) and (16) to ω = 1gives νmax(1) ≈ 0.650 ± 0.001, which is very close to the empirical value forrandom close packing [45, 46, 47, 48, 49], and can be approached for ex-tremely fast compression, i.e. Γ & 16 × 10−4 [42, 47], as studied in moredetail elsewhere [50].

18

0.65

0.66

0.67

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ν max

ω−1

US1

US2

BUS1

BUS2

Eq. (15)

0.664

0.666

0.668

0.67

10−7 10−6 10−5 10−4

ν max

Γ

US

BUS

(a)

0.65

0.66

0.67

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ν max

ω−1

UV2BUV2

Eq. (16)

(b)

Figure 7: (Color online) The maximum density νmax as a function of the inverse size ratioω−1 for different size distributions and for different compression rates, in the inset subscript1 corresponds to the reference growth rate Γ = 8 × 10−6, and subscript 2 corresponds toΓ = 16 × 10−6, i.e. two times faster. Size distributions are (a) uniform size distribution(US) and their bi-disperse equivalents (BUS), or (b) uniform volume (UV) and their bi-disperse equivalents (BUV). The size ratio ω corresponds to the polydisperse systems,while bi-disperse ones are constructed using Appendix C. Results from Ref. [51] obtainedby compression of soft frictionless particles with a uniform size distribution are shown ascrosses in (a). In the inset of (a), νmax is plotted for different growth rates Γ for theuniform size distribution (ω = 2) and their bi-disperse equivalent (Rbi ≈ 1.48).

19

3.6. Different growth rates

The effect of using different growth rates Γ on the maximum density isshown in the inset of Fig. 7(a) for two size distributions. We observe that themaximum density slightly increases (within 1%) with decreasing the growthrate. There are no signs of saturation for Γ → 0, which indicates a quite in-teresting slow dynamics, leaving plenty of questions open for further research.The effect of using various expansion rates for mono-disperse systems is morecomprehensively studied in Refs. [25, 47], so that we do not display more ofour data.

3.7. Super dense limit

In this subsection we show that sufficiently close to the jamming den-sity, the compressibility factor is independent of the size distribution anddepends on only one parameter, the jamming density itself. Figure 8 showsthe compressibility factor scaled by equation (14) for systems with uniformsize distributions.

The jamming densities φJ used in Fig. 8 are determined using Eq. (14)by mapping the diverging compressibility factor to finite values, as suggestedby Torquato [52]. It must be noted that all φJ differ from νmax obtained insimulations by at most 10−13, which confirms that we are close to the infinitepressure limit. We see from Fig. 8 perfect agreement with the free volumetheory for all considered systems at the densities (φJ − ν) . 10−8. Onlyone free parameter is required for the free volume theory, φJ, even thoughin other studies two fitting parameters are used [22, 42]. In order to capturethe deviation of the compressibility factor from the free volume theory as thedensity difference increases, theories which take into account the geometryand structure of the free volume should be developed. The point where datacollapse with the free-volume theory is the point where cages are establishedand do not change any more [50].

It must be noted that the results of Fig. 8 are obtained using numericaldouble type (8 bytes). We used also long double type (16 bytes) for a fewsimulations and (while the data in Fig. 8 did not change) could reach perfectagreement with the free-volume theory Eq. (14) even for 10−15 . (φJ− ν) .10−8, i.e., over seven orders of magnitude, which is quite remarkable.

Finally, we remark that even though the free volume theory presented inEq. (14) is designed for monodisperse systems, we observe a very good agree-ment for polydisperse systems as well (see Fig. 8). We also confirmed Eq.(14) for two-dimensional systems, and for systems with uniform volume and

20

bi-disperse size distributions (data not shown), and for different compressionrates.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10-12 10-10 10-8 10-6 10-4 10-2 1

Z /

Zfv

φJ - ν

1.21.6

247

20

0.8

1

1.2

1.4

10-6 10-4 10-2 1

Figure 8: (Color online) The compressibility factor scaled by the free volume equationof state (14) in the limit of diverging pressure for systems with uniform size distributionfor different size ratios, increasing ω corresponds to top-to-bottom, with growth rate Γ =16× 10−6. When partial crystallization happens we see some fine structures as shown inthe inset for the monodisperse system (ω = 1).

21

4. Summary and Conclusions

In this study we have shown analytically that for any given polydispersesystem a bi-disperse system can be constructed with the same compressibil-ity factor, Z, in the fluid regime, using Eqs. (C.1) and (C.2). In order toconfirm this result simulation data of the measured compressibility factorare compared to previously found equations of state and an agreement wasfound with less than 1% deviation. Based on the simulation data a slightlymore accurate equation of state (13) was proposed for the fluid regime.

Surprisingly, for densities higher than the freezing density, beyond thefluid regime, our simulations show that a polydisperse system and its bi-disperse equivalent have similar estimated jamming density (within 1%) aslong as (partial) crystallization does not happen.

We have observed three distinct types of liquid-solid organization / behaviorin the bi-disperse systems with equal number of small and big particles, de-pending on the size ratio Rbi. First, for Rbi . 1.1 the crystalline phaseformed at high densities consists of a crystal made of both small and bigparticles; second, in the range 1.1 . Rbi . 2.25 we do not observe any signsof ordering effects, i.e., an amorphous solid is formed; finally, at relativelylarge size ratios Rbi & 2.40, we observe an ordered structure that consists ofa crystal of predominately large particles. We confirm this using two differ-ent criteria to quantify partial crystallization, i.e., the compressibility factorand the (final) jamming density. Note that above results are obtained using(relatively) slow density growth rates Γ = 8× 10−6 and Γ = 16× 10−6, whilefor extremely fast compression it is possible to avoid partial crystallization.For the polydisperse systems tested, we find partial crystallization only for(relatively) low size ratios ω, i.e., ω . 1.2, but not for larger ω.

Sufficiently close to the jamming density the compressibility factor isindependent of the size distribution and compression rate and depends onlyon one free parameter, the jamming density itself. The free volume equationof state (14) is in very good agreement with our simulation results for allconsidered systems in two- and three-dimensions.

The result with potential for applications is the fact that polydisperse sys-tems can be replaced [53] by their bi-disperse equivalents and their jamming(packing) density can be predicted from the moments O1 and O2. Further-more, the results of this study can be used for better understanding the equa-tion of state for mixtures of fluids and thus, to predict mixing / segregation.Finally, our predictions concerning the size ratios for which crystallization

22

happens can be useful for experiments on colloids and granular media, toeither avoid or establish ordering effects.

Acknowledgments

The authors would like to thank Martin van der Hoef, Sebastian Gonzalez,Nicolas Rivas and Fatih Goncu for helpful discussions, and Monica Skoge forconsultations about the Event-Driven code. This research is supported by theDutch Technology Foundation STW, which is the applied science division ofNWO, and the Technology Programme of the Ministry of Economic Affairs,project Nr. STW-MUST 10120.

Appendix A. Derivation of gO(ν)

In order to express the compressibility factor in the 3D mixtures, usingdimensionless moments, plug Eqs. (2), (3) and (4) into the expression inbrackets of Eq. (5):

g11a311n

21 + 2g12a

312n1n2 + g22a

322n2

≡ 1

1− ν(a3

11n21 + 2a3

12n1n2 + a322n2)

+3ν A2

A3

2(1− ν)2(a3

11n21 +

4

1 + Ra3

12n1n2 +1

Ra3

22n22)

+(ν A2

A3)2

2(1− ν)3(a3

11n21 +

8

(1 + R)2a3

12n1n2 +1

R2a3

22n22)

≡ 6 〈a〉 〈a2〉+ 2 〈a3〉1− ν

+6ν(〈a〉 〈a2〉 〈a3〉+ 〈a2〉3)

(1− ν)2 〈a3〉 +4ν2〈a2〉3

(1− ν)3 〈a3〉 .(A.1)

23

Using the expression above, and inserting Eqs. (6), one obtains:

Z − 1 = (1 + r)ν

4 〈a3〉(g11a311n

21 + 2g12a

312n1n2 + g22a

322n2)

= (1 + r)ν3 〈a〉 〈a2〉 〈a3〉 (1− ν) + 〈a3〉2(1− ν)2 + 〈a2〉3(3− ν)ν

2〈a3〉2(1− ν)3

= (1 + r)2ν(1 + 3O1) + ν(3O2 − 3O1 − 2) + ν2(1−O2)

4(1− ν)3

= (1 + r)2νgO(ν),

(A.2)

where r is the restitution coefficient.

Appendix B.

In order to show that O1 ≤ 1, find the extrema of g(ai) = NO1 =∑ai

∑a2

i /∑

a3i , i ∈ [1, N ] on the interval ai ∈ (0,∞). For this solve the

system of k = 1, ..., N equations:

∂g(ai)

∂ak

=

[2ak

∑ai +

∑a2

i∑a3

i

− 3a2k

∑ai

∑a2

i

(∑

a3i )

2

]= 0. (B.1)

Solving the quadratic equation (B.1) for ak yields the system:

ak =

∑ai

∑a3

i ± (3∑

ai(∑

a2i )

2∑

a3i + (

∑ai

∑a3

i )2)1/2

3∑

ai

∑a2

i

for any k ∈ [1, N ].

(B.2)

It has a solution ai = a = const. It can be checked that this extremum isthe local maximum gmax = N . Therefore the local maximum of O1 is equalto unity, which corresponds to the monodisperse case.

By analogy, considering the extrema of the function f(ai) = O2/O1, itcan be shown that O2 ≤ O1.

Finally, to show that O21 ≤ O2, note that O2

1/O2 = A, where A =〈a〉2/ 〈a2〉 ≤ 1. This yields O2

1 ≤ O2, where equality is reached only inthe monodisperse case (O1 = O2 = 1).

24

Appendix C.

In order to find a bi-disperse system that has the same dimensionlessmoments O1 and O2 as a given polydisperse one, solve analytically the systemof equations (6) in terms of the variables n1 and R. This yields a uniquesolution (O1 6= O2):

nbi1 =

1

2+

3O1O2 −O2 − 2O31

2λ, (C.1)

Rbi =2O3

1 + 2O22 + O2 − 4O1O2 −O2

1O2 + (1−O1)λ

2(O2 −O1)(O21 −O2)

, (C.2)

where λ =√

O2

√4O3

1 + O2 − 3O1(2 + O1)O2 + 4O22 and nbi

1 correspondshere to the number fraction of large particles, i.e., Rbi ≥ 1. Inserting thesevalues into Eq. (8) leads also to the same compressibility factor Zbi(nbi

1 , Rbi) =Zpoly(O1, O2).

Appendix D.

In order to maintain the size distribution during the growing process, theradius ai of the particle i changes with time as:

dai

dt= Γ

ai

amax(t), (D.1)

where amax(t) is the radius of the largest particle (which depends on time)and Γ is the growth rate with units of velocity. In this convention, we haveamax(t) = Γt and amin(t) = Γω−1t, where amin(t) is the radius of the smallestparticle and ω is constant, as desired.

Appendix E.

Assume a polydisperse distribution of particle radii with probability f(a)dato find the radius a between radii a and a + da, and with

∫∞0

f(a)da =1. Using the definition of the k-th moment of the size distribution f(a),⟨ak

⟩ ≡ ∫akf(a)da, it is straightforward to calculate dimensionless moments

O1 = 〈a〉 〈a2〉 / 〈a3〉 and O2 = 〈a2〉3/〈a3〉2 for any given polydisperse sizedistribution.

25

For the uniform size distribution, with ω = amax/amin, we obtain:

O1 = 1− 2

3

ω20

1 + ω20

, O2 =1

27

(3 + ω20)

3

(1 + ω20)

2, (E.1)

where ω0 = (ω−1)/(ω+1), and 2ω0 〈a〉 is the width of size distribution func-tion f(a). It must be noted, that plugging Eq. (E.3) into Eq. (C.1) yieldsnbi

1 = 1/2. Therefore, a bi-disperse system which is equivalent (by compress-ibility factor) to a polydisperse system with uniform size distribution has thesame number of small and large particles for any ω.

For the uniform volume distribution, we have (ω > 1):

O1 =2ω

ω2 − 1ln ω, O2 =

2ω2

(ω − 1)3(ω + 1)ln3 ω. (E.2)

For numerical values see Appendix F.The moments of the log-normal distribution (not used in simulations) can

be computed from the moment generating function of the normal distribution[54]. If X has the log-normal distribution with parameters µ (mean) and σ(standard deviation) then E(Xn) = exp[nµ + (1/2)n2σ2], n ∈ N. Therefore,we obtain:

O1 = exp(−2σ2), O2 = exp(−3σ2). (E.3)

26

Appendix F.

Simulations Figs. ω O1 O2 Rbi nbi1

Uniform size 1, 2, 3, 1.0 1.0 1.0 1.0distribution 6(a), 7(a), 8 1.2 0.99453 0.99182 1.111

1.4 0.98198 0.97322 1.2132 0.93333 0.90337 1.4774 0.82353 0.75958 2.0605 0.79487 0.72542 2.251 1/210 0.73267 0.65659 2.79020 0.69992 0.62355 3.18750 0.67999 0.60467 3.491100 0.67333 0.59858 3.607

∞ 2/3 16/27 2+√

3Uniform volume 1(d), 6(b), 7(b) 1.0 1.0 1.0 1.0 1/2distribution 1.2 0.99448 0.99173 1.111 0.4528

1.4 0.98138 0.97217 1.214 0.41342 0.92419 0.88807 1.493 0.32704 0.73936 0.63151 2.241 0.18985 0.67060 0.54283 2.559 0.15597 0.56756 0.41788 3.133 0.11398 0.52811 0.37283 3.398 0.100010 0.46517 0.30448 3.896 0.0801100 0.09211 0.01993 17.83 0.0058∞ 0 0 – –

Table F.1: Given are size ratios ω and dimensionless moments O1 and O2 for a fewpolydisperse systems and Rbi with nbi

1 for their bi-disperse equivalents.

27

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