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 2 1, 2 0 0 2, 1249 – 1251
Recursive packing of dense particle mixtures
J. A. ELLIOTT, A. KELLY, A. H. WINDLE
Department of Materials Science and Metallurgy, University of Cambridge,
Pembroke Street, Cambridge, CB2 3QZ
E-mail: [email protected]
The quest to obtain the densest possible randomly
packed arrangement of hard, geometric particles of
varying shapes and sizes is an ongoing and physically challenging problem, with very wide applications in sciences and engineering. The bulk density
of a packed assembly is characterized by the packing
fraction, which is the ratio of the particle volume to
the total occupied volume of the system. We present a
straightforward recursive volume-filling model which
can account for the maximum packing fractions attainable using randomly ordered mixtures of particles of
discretely different sizes but similar shapes, and the
relative proportions of each component required for optimum packing.
We make use of existing experimental data on spherical particles where the particle size is characterized
simply by the diameter, and for which a wide variety
of sources are available. The model contains a single
parameter x which is the maximum packing fraction
attainable for randomly packed particles of a single
size. For spheres, values of x between 0.6 and 0.64
have been obtained in both experiments [1–8] and computer simulations [9, 10]. These are
√ 14–19% lower than
the closest-packed value of π/3 2 ∼
= 0.7405, and the
cause of this discrepancy is still not well understood
[11, 12].
However, it is not our intention to justify a particular value for x. Rather, by taking an experimental
value for this parameter, a simple recursive interstitial filling argument can predict the total packing fraction for mixtures of particles of different sizes, and accounts closely for the existing experimental data on
mixtures of spheres. Deviations from the model are
due to limits on the arrangements of particles within
the interstices set by the relative sizes of the components.
In order to improve on the packing density for particles of a single size, it is necessary to fill the interstices
between the larger particles with smaller particles without disrupting the original packing. Such arrangements
are often called Apollonian packings, [13, 14] after
Apollonius of Perga who studied the problem of recursively 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 are
practical difficulties in constructing such tight recursive
packings. In reality, the best that can be hoped for is to
fill the interstices with the same efficiency as the larger
particles fill space.
C 2002 Kluwer Academic Publishers
0261–8028 In the case of unit spheres, the smallest possible hole
in a randomly packed structure is a triangular pore made
by three spheres in contact, which has a radius of approximately 0.154. This leads to the condition that the
secondary spheres should be at least 6.46 times smaller
than the primary ones if they are to percolate through
the packed assembly and fill all the interstices in it. This
idea was suggested in passing for binary sphere mixtures by Yerazunis and co-workers, [15] and we have
developed it further as follows.
To illustrate our approach, let x be the packing fraction attained by randomly packing particles all of the
same size and shape. Then, the free volume remaining is (1 − x). Suppose now that this free volume may
be packed to the same efficiency by a second population of particles of similar shape. The packing fraction
obtained is then given by Equation 1:
Q = x + (1 − x)x.
(1)
Equation 1 requires not only a significant size difference between the two sets of particles, but also that the
relative volume fractions of the particles take particular values. The volume fractions for the densest total
packing of the two sizes of particle will be:
c1 = x/Q =
x
1
=
x + (1 − x)x
2−x
and c2 = (1−x)x/Q =
(2)
1−x
(1 − x)x
=
. (3)
x + (1 − x)x 2 − x
So, if x = 0.625, the experimental value for spheres
found by McGeary, [5] then Q = 0.625 + 0.235 =
0.859, and the volume fractions of two components as
a proportion of the total packing fraction are c1 = 0.727
and c2 = 0.273. We can now generalize this argument to mixtures with more than two components as
follows.
If we define Q N as the maximum packing fraction
that can be achieved using an N -component mixture of
particles of different sizes then by definition, Q 1 = x.
So, assuming the interstices of each particle packing
are filled with the same efficiency as the packing of
particles of a single size, we have:
Q N +1 = Q N + (1 − Q N )x.
(4)
Transforming this recursion relation into an explicit
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T A B 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
2
3
4
1.000
0.727
0.660
0.638
1.000
0.723
0.670
0.607
–
0.273
0.247
0.239
–
0.277
0.230
0.230
–
–
0.093
0.090
–
–
0.100
0.102
–
–
–
0.034
–
–
–
0.061
0.625
0.859
0.947
0.980
0.625
0.840
0.900
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 Q N , we obtain:
Q N = 1 − (1 − x) N .
(5)
This implies Q N → 1 as N → ∞ independently of x,
provided x ≤ 1, which is the expected asymptotic limit
for a mixture with an infinite number of components.
In order to construct the recursively packed mixtures
with a finite number of components, the relative proportions of each component cn , where c1 is proportion
of the largest size component, can be calculated from
the ratio of the partial packing fraction qn of component
n to the total packing fraction Q N . The total packing
fraction is related to the partial packing fractions of
each component by Equation 6, and the values of cn by
Equation 7.
QN =
N
qn .
(6)
cn = qn /Q N .
(7)
n=1
The values of qn 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 yields
an explicit expression for cn :
cn = qn /Q N =
x(1 − x)n
1 − (1 − x) N
(10)
which can be used to calculate the appropriate proportions of particles with which to construct an maximally
dense randomly packed N -component particle mixture.
The predicted relative proportions for binary, ternary
and quaternary mixtures of spheres, calculated assuming x = 0.625, are compared with the experimental data
of 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 successive
components is sufficiently large so that random packing
is achieved within all interstices. Although the calculations are generally in good agreement with experiment,
there are some discrepancies in the relative amounts
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of the smaller components due to practical limits on particle size. This issue can be made clear using the model
in the following way.
From Equation 5 we can derive the inversion formula:
x = 1 − (1 − Q N )1/N
(11)
So, given Q N we can infer x from the successive orders of experimental data in addition to the first, and use
this to assess the quality of our model. Specifically, we
can test the assumption that x remains constant at each
filling iteration. The results for McGeary’s data [5] are
given in Table II, and it is found that the value of x obtained varies somewhat with N . In general, the values
become smaller as N becomes larger. This is due to the
practical limitations in obtaining monodisperse spheres
with which to construct a packed assembly whose relative sizes are sufficiently different that each successive level of interstices is maximally filled. The size
ratios for the sphere packings of McGeary are given in
Table III.
In general, a size ratio of at least 10 : 1 is needed between each component for the packings to be independent, as assumed in the model. As noted by McGeary,
[5] this is just above the percolation limit dictated by the
triangular pore size. If the size ratio is not sufficiently
large then the packings will be less dense due to finite
size effects at the boundaries of the interstices. So, if
the largest particles are of the order of centimeters, then
the smallest must be of the order of microns for a quaternary mixture. This explains why, in practice, it is rare
to find particle mixes with more than four size grades
T A B L E I I Application of Equation 8 to infer value for x from
McGeary’s [5] maximally dense N -component sphere mixtures
N
QN
x
1
2
3
4
0.625
0.840
0.900
0.951
0.625
0.600
0.536
0.530
T A B L E I I I Size ratios of McGeary’s [5] maximally dense
N -component sphere mixtures
N
Size ratio(s)
1
2
3
4
1
1 : 77
1 : 7 : 77
1 : 7 : 38 : 316
of components, as it is not possible to fill the interstices
more efficiently using additional finer grades due to the
influence of non-geometric forces.
It is worth noting that we have not assumed anything
about the shapes of the particles involved in the recursive filling model. Thus, the arguments presented could
apply equally to similar angular particles which pack
randomly with a different value of x to spheres. However, it would be necessary to define some mean size
parameter for the particles, and it might be expected
that the boundary effects in the interstices would have
a different, possibly more severe, effect on the nested
packing of particles which are not sufficiently different
in size.
We finish by reiterating our main conclusion that it
is possible to predict quantitatively both the maximum
packing fraction and relative proportions of components in a mixture of particles of similar shape under the assumption of random packing by using a
simple recursive procedure, provided that the relative
sizes of each component are such that they pack independently.
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Received 18 March
and accepted 22 April 2002
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