J.A. Elliott, A. Kelly and A.H. Windle- Recursive packing of dense particle mixtures

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J O U R N A L O F M A T E R I A L S S C I E N C E L E T T E R S 21, 2002, 1249 1251

Recursive packing of dense particle mixtures

J . A . E L L I O T T , A . K E L L Y , A . H . W I N D L E

Department of Materials Science and Metallurgy, University of Cambridge,Pembroke Street, Cambridge, CB2 3QZE-mail: [email protected]

The quest to obtain the densest possible randomlypacked arrangement of hard, geometric particles ofvarying shapes and sizes is an ongoing and physi-cally challenging problem, with very wide applica-

tions in sciences and engineering. The bulk densityof a packed assembly is characterized by the packingfraction, which is the ratio of the particle volume tothe total occupied volume of the system. We present a

straightforward recursive volume-filling model whichcan account for the maximum packing fractions attain-able using randomly ordered mixtures of particles ofdiscretely different sizes but similar shapes, and therelative proportions of each component required for op-

timum packing.We make use of existing experimental data on spher-

ical particles where the particle size is characterizedsimply by the diameter, and for which a wide varietyof sources are available. The model contains a single

parameter x which is the maximum packing fractionattainable for randomly packed particles of a single

size. For spheres, values of x between 0.6 and 0.64have been obtained in both experiments [18] and com-puter simulations [9, 10]. These are 1419% lower than

the closest-packed value of/3

2 = 0.7405, and thecause of this discrepancy is still not well understood[11, 12].

However, it is not our intention to justify a partic-ular value for x . Rather, by taking an experimentalvalue for this parameter, a simple recursive intersti-tial filling argument can predict the total packing frac-tion for mixtures of particles of different sizes, and ac-

counts closely for the existing experimental data onmixtures of spheres. Deviations from the model aredue to limits on the arrangements of particles withinthe interstices set by the relative sizes of the com-

ponents.In order to improve on the packing density for parti-

cles of a single size, it is necessary to fill the intersticesbetween the larger particles with smaller particles with-out disrupting the original packing. Such arrangements

are often called Apollonian packings, [13, 14] afterApollonius of Perga who studied the problem of recur-sively inscribed circles in two dimensions ca. 200 BCE.In fact, this is not the most efficient way to fill the

gaps between the larger particles because the smaller

particles do not fit the interstices well. Also, there arepractical difficulties in constructing such tight recursivepackings. In reality, the best that can be hoped for is tofill the interstices with the same efficiency as the larger

particles fill space.

In the case of unit spheres, the smallest possible holein a randomly packed structureis a triangular pore madeby three spheres in contact, which has a radius of ap-proximately 0.154. This leads to the condition that the

secondary spheres should be at least 6.46 times smallerthan the primary ones if they are to percolate throughthe packed assembly and fill all the interstices in it.Thisidea was suggested in passing for binary sphere mix-

tures by Yerazunis and co-workers, [15] and we havedeveloped it further as follows.

To illustrate our approach, let x be the packing frac-tion attained by randomly packing particles all of thesame size and shape. Then, the free volume remain-

ing is (1x). Suppose now that this free volume maybe packed to the same efficiency by a second popula-tion of particles of similar shape. The packing fractionobtained is then given by Equation 1:

Q = x + (1 x)x. (1)

Equation 1 requires not only a significant size differ-ence between the two sets of particles, but also that therelative volume fractions of the particles take particu-

lar values. The volume fractions for the densest totalpacking of the two sizes of particle will be:

c1 = x/Q =x

x + (1 x)x =1

2 x (2)

and c2 = (1x)x/Q=(1 x)x

x + (1 x)x =1 x2 x . (3)

So, if x = 0.625, the experimental value for spheresfound by McGeary, [5] then Q = 0.625+ 0.235 =0.859, and the volume fractions of two components asa proportion of the total packing fraction are c1= 0.727and c2= 0.273. We can now generalize this argu-ment to mixtures with more than two components as

follows.If we define QN as the maximum packing fraction

that can be achieved using an N-component mixture ofparticles of different sizes then by definition, Q1=x .So, assuming the interstices of each particle packing

are filled with the same efficiency as the packing ofparticles of a single size, we have:

QN+1 = QN+ (1 QN)x. (4)

Transforming this recursion relation into an explicit

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T AB L E I Comparison of predictions of volume-filling model with experimental data

c1 c2 c3 c4 QN

N Theory Expt. Theory Expt. Theory Expt. Theory Expt. Theory Expt.

1 1.000 1.000 0.625 0.625

2 0.727 0.723 0.273 0.277 0.859 0.840

3 0.660 0.670 0.247 0.230 0.093 0.100 0.947 0.900

4 0.638 0.607 0.239 0.230 0.090 0.102 0.034 0.061 0.980 0.951

Values of the relative proportions of spheres cn for maximum density packings of N-component mixtures calculated from the recursive filling model,

together with packing fractions, compared with experimental data of McGeary [5].

formula for QN, we obtain:

QN = 1 (1 x)N. (5)

This implies QN 1 as N independently of x ,provided x 1, which is the expected asymptotic limitfor a mixture with an infinite number of components.

In order to construct the recursively packed mixtureswith a finite number of components, the relative pro-

portions of each component cn , where c1 is proportionof the largest size component, can be calculated from

the ratio of the partial packing fraction qn of componentn to the total packing fraction QN. The total packingfraction is related to the partial packing fractions ofeach component by Equation 6, and the values ofcn byEquation 7.

QN =N

n=1qn. (6)

cn

=qn/QN. (7)

The values ofqn must satisfy a recursion relation:

qn+1 = (1 qn)x (8)

which can be transformed into an explicit equation for

qn :

qn = x(1 x)n. (9)

Substituting Equations 5 and 9 into Equation 7 yieldsan explicit expression for cn :

cn = qn/QN =x(1 x)n

1 (1 x)N (10)

which can be used to calculate the appropriate propor-tions of particles with which to construct an maximallydenserandomly packed N-component particle mixture.

The predicted relative proportions for binary, ternaryand quaternary mixtures of spheres, calculated assum-ingx = 0.625, are compared with the experimental dataof McGeary [5] in the first four columns of Table I. The

relative sizes of the components are not specified in our

model; it is assumed that the size ratio of successivecomponents is sufficiently large so that random packingis achieved within all interstices. Although the calcula-tions are generally in good agreement with experiment,

there are some discrepancies in the relative amounts

of the smaller components dueto practicallimitson par-ticle size. This issue can be made clear using the model

in the following way.From Equation 5 we canderivethe inversion formula:

x = 1 (1 QN)1/N (11)

So, given QN we can infer x from the successive or-ders of experimental data in addition to the first, and usethis to assess the quality of our model. Specifically, we

can test the assumption that x remains constant at eachfilling iteration. The results for McGearys data [5] aregiven in Table II, and it is found that the value of x ob-tained varies somewhat with N. In general, the valuesbecome smaller as N becomes larger. This is due to the

practical limitations in obtaining monodisperse sphereswith which to construct a packed assembly whose re-lative sizes are sufficiently different that each succes-sive level of interstices is maximally filled. The size

ratios for the sphere packings of McGeary are given inTable III.

In general, a size ratio of at least 10 : 1 is needed be-tween each component for the packings to be indepen-dent, as assumed in the model. As noted by McGeary,

[5] this is just above the percolation limit dictated by thetriangular pore size. If the size ratio is not sufficientlylarge then the packings will be less dense due to finitesize effects at the boundaries of the interstices. So, ifthe largest particles are of the order of centimeters, then

the smallest must be of the order of microns for a quate-rnary mixture. This explains why, in practice, it is rareto find particle mixes with more than four size grades

TA BL E II Application of Equation 8 to infer value for x from

McGearys [5] maximally dense N-component sphere mixtures

N QN x

1 0.625 0.625

2 0.840 0.600

3 0.900 0.536

4 0.951 0.530

TABL E III Size ratios of McGearys [5] maximally dense

N-component sphere mixtures

N Size ratio(s)

1 1

2 1 : 77

3 1 : 7 : 77

4 1 : 7 : 38 : 316

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of components, as it is not possible to fill the intersticesmore efficiently using additional finer grades due to theinfluence of non-geometric forces.

It is worth noting that we have not assumed anythingabout the shapes of the particles involved in the recur-

sive filling model. Thus, the arguments presented couldapply equally to similar angular particles which packrandomly with a different value of x to spheres. How-

ever, it would be necessary to define some mean sizeparameter for the particles, and it might be expectedthat the boundary effects in the interstices would havea different, possibly more severe, effect on the nestedpacking of particles which are not sufficiently differentin size.

We finish by reiterating our main conclusion that itis possible to predict quantitatively both the maximumpacking fraction and relative proportions of compo-nents in a mixture of particles of similar shape un-

der the assumption of random packing by using asimple recursive procedure, provided that the relativesizes of each component are such that they pack inde-

pendently.

References1. A . E . R . W E S T M A N and H . R . H U G I L L , J. Amer. Ceram.

Soc. 13 (1930) 767.

2. C . C . F U R N A S , Ind. Eng. Chem. 23 (1931) 1052.3. O . K . R I C E , J. Chem. Phys. 12 (1944) 1.4. G . D . S C O T T , Nature 188 (1960) 908.5. R . K . M C G E A R Y , J. Amer. Ceram. Soc. 44 (1961) 513.6. R . R U T G E R S , Nature 193 (1962) 465.7. J . D . B E R N A L , Proc. R. Soc. London 280 (1964) 299.8. G . D . S C O T T and D . M . K I L G O U R , Brit. J. Appl. Phys.

(J. Phys. D) 2 (1969) 863.9. D . J . A D A M S and A . J . M A T H E S O N , J. Chem. Phys. 56

(1972) 1989.

10. W . M . V I S S C H E R and M . B O L S T E R L I ,Nature 239 (1972)504.11. K . G O T O H and J . L . F I N N E Y , ibid. 252 (1974) 202.12. S . T O R Q U A T O , T . M . T R U S K E T T and P . G .

D E B E N E D E T T I , Phys. Rev. Lett. 84 (2000) 2064.

13. B . B . M A N D E L B R O T , The Fractal Geometry of Nature(Freeman and Company, New York, 1983).

14. S . V . A N I S H C H I K and N . N . M E D V E D E V , Phys. Rev. Lett.75 (1995) 4314.

15. S . Y E R A Z U N I S , S . W . C O R N E L L and B . W I N T N E R ,Nature 207 (1965) 835.

Received 18 March

and accepted 22 April 2002

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J.A. Elliott, A. Kelly and A.H. Windle- Recursive packing of dense particle mixtures