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DENSITY LIMIT FOR RANDOM CLOSE PACKING OF EQUAL
FIGURES IN D-DIMENSIONAL EUCLIDEAN SPACE
A.B.Shubin
Institute of Metallurgy, Urals Branch of RAS, Russian Federation
ABSTRACT
Theoretical approach developed describes the conditions of the homogeneity and
maximum density of the random ensemble of equal figures of arbitrary shape in the
D-dimensional Euclidean space. Theorem proven in this work and computer models
of the random packing of hard spheres in 3D-space allowed us to demonstrate a very
good agreement between well-known experimental maximum of packing density
0.637 and calculated value of 0.638  0.003.
INTRODUCTION
The random ensembles of equal figures (mainly spheres) are the convenient
idealized model of the simple liquids [1,2], some metallic melts and amorphous
media [3]. The statistical geometry of the random packings of spherical particles (1-,
2- or 3-dimensional) was the studying subject in many investigations [e.g. 4, 5].
Recently the attention is focused on the more complicate systems like the packings of
spherical and non-spherical figures (particles) in 2D, 3D and higher-dimensional
space [6, 7]. Nevertheless generally accepted definition of the random close packing
(RCP) wasn’t formulated until present time [8] even for 3D-ensembles of equal
spheres. This challenge continues to capture the attention as well as the finding of
density limit for RCP of hard spheres [9]. The attempts to explain well-known
experimental value of the maximum packing fraction of 0.637 have been performed
by many authors. Woodcock [10] found that this packing density (Buffon constant
2/) can be grounded by thermodynamic approach related to liquid-glass transition.
Different definitions of the random sphere packing (RCP) were proposed
particularly in the papers [11, 12] and by the author in [13]. Later the result of the
work [13] was generalized for the case of the convex particles of any shape in 3Dspace [14].
Here we present more general approach to defining RCP of equal figures of
arbitrary shape in Euclidean space of any dimension. Besides, the condition for the
maximum density of such particle ensembles was rigorously established.
DEFINITIONS
Consider the set of congruent non-overlapped figures of arbitrary shape in the
Euclidean space ED of the dimension D. Let this ensemble consists of N figures and
the number N can be arbitrary large. Let the given ensemble occupies the space
region {V} with the numeric value V, and the volume of one figure is u. We suppose
that {V} is convex set and for any pair of its points this space region contains also the
line segment which connects these points.
The packing density in our ensemble can be defined as
 = Nu/V.
(1)
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Consider the single separately taken figure that is identical to any of the figures of
ensemble. Take the point O in this figure. Let this point will be the origin for the
coordinate system (CS) which is rigidly bound with this figure. We can define D
orthogonal coordinate axes for this CS and then assume that the orientation of the
axes and the location of O-point is the same for all the figures. So, any CS origin is
located equally for any figure in the ensemble. Further, the coordinate axes for any
CS are oriented identically relatively each figure. Obviously, the O-points and axes
can be displaced together with the the corresponding figure.
Let us define also the main fixed coordinate system for all the packing of particles
(figures in common case). Now the location and orientation of each i-th figure can be
given as the Oi -point coordinates and the set of the values determining the orientation
of the local axes (Xi, Yi, …).
Let the ensemble in the space region {V} includes N figures and every figure has
its own CS described above. The location and orientation of all the figures of
ensemble are determined in the common coordinate system.
Then let’s perform the analogue of the Voronoi tessellation in our D-dimensional
ensemble. Now all the {V} space is divided to the non-overlapped Voronoi regions
(e.g. polygons in 2D and polyhedra in 3D). We assume that our D-dimensional
particles ensemble is homogeneous if one can not distinguish statistical and
geometric properties of the Voronoi regions (VR) taken from any part of ensemble.
Consider two homogeneous ensembles of the figures. If the statistical and
geometrical properties of VR in these ensembles are equal let us call them
“geometrically equivalent”.
Consider two homogeneous ensembles with equal packing density. Let the first
consists of N figures and the second includes (N+k) figures (k  2) and the N/k ratio
can be specified arbitrarily large. Let {V1} be the space region of the first ensemble
having the numeric value of V1 and {V2} be the space region of the second ensemble
having the numeric value of V2. Let {V2} completely includes {V1} inside.
We can now describe the following procedure. Let us place in {V1} (which
consist of N equal figures) additionally k the same figures. The location change of
“old” N figures is possible. We can place these k figures sequentially and separately
(one by one), in randomly taken locations in the space region {V1} also with random
orientation of each figure. Then propose that the O(N+i) coordinates for any i-th placed
additional figure will be always constant and fixed in the common coordinate system.
Establish the analogous requirement for the i-th figure orientation. Since k<<N, the
probability of the mutual contact for the figures from this additional k group is
negligible (and can be given as arbitrarily small). The requirement of the constancy of
the coordinates and the orientation is related only to the figures from the k group.
Definition 1. Call the first and the second ensembles random ensembles if there
exist an algorithm that allows us to generate the second ensemble (of N+k figures)
from the first ensemble (of N figures) as a result of the procedure described above,
and both the ensembles are geometrically equivalent.
The good example of non-random ensemble is any crystal 3D-packing of hard
spheres. Really, consider it as the “first ensemble”. The geometrically equivalent
“second ensemble” can not be constructed by placing the additional spheres with
random and fixed coordinates. The long-range order will be inevitably broken.
Well-known “random close packing” (RCP) of hard spheres is, evidently, the
typical case of the random ensemble of figures (particles).
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Consider two separately taken figures (figure 1 and figure 2) identical to the
figures of the “first ensemble”. Let the arbitrary point B have the constant coordinates
in the CS related to the figure 1. Investigate all the allowable (without overlapping)
relative positions of the figures 1 and 2 tracking the coordinates of B point in the CS
connected with the figure 2.
One can find the region with the defined boundaries and connected with the
second figure which is inaccessible for the point B. The existence of such a region is
caused by the requirement of figure 1- figure 2 non-overlapping. The volume (h) of
the region mentioned is determined only by the coordinates of the point B in the CS
connected with the figure 1. Obviously the value of h is more or equal to zero (h  0).
Consider, further, the arbitrarily taken point A belonging to the space {V1}. Let us
find the coordinates of this point in each of N local coordinate systems connected
with the figures of the ensemble. Then determine the values h1, h2, … hN for the point
A considering sequentially each i-th figure as the “figure 1” as described above.
Definition 2. Call the most of all values h1, h2, … hN excluded volume (W) for
the given point A.
It is evident that the excluded volume of the points is one of the statistical and
geometric properties of the ensemble.
Divide {V1} space to infinitesimal volume parts (IVP) dV. Assign the value of
excluded volume (W) to each IVP. Consider all the IVP dV{V1} and find W
distribution for them. The probability density for this distribution (W) meets the
following conditions
Wmax
 W(W)dW = <W>,
Wmin
(2)
Wmax
 (W)dW = 1,
Wmin
(3)
Here Wmin, Wmax are, correspondently, minimum and maximum values of the
excluded volume for IVP dV{V1}; <W> is the average excluded volume for all
dV{V1}. So, the arbitrarily taken IVP have the excluded volume value in the
interval [W; W+dW] with the probability (W)dW.
Assign to every IVP the number (i) of the figure which determines its excluded
volume. If W = 0 let us assign to the IVP the number of the figure with nearest
surface.
Definition 3. The set of IVP dV{V1} and having the same number i is h-region
of the i-th figure.
DENSITY LIMIT THEOREM
Let us perform random placement of k additional figures with fixed coordinates
and orientations in the space of {V1}. Keep the numbering of “old” figures from 1 to
N. Assign the numbers (N+1) - (N+k) to the “new” figures. We assumed above that
the first ensemble of N figures is random. Consequently, according to the definition 1,
there exists an algorithm that allows us to generate the second ensemble
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geometrically equivalent to the first ensemble by such a placement. Assume that we
generated just such a second ensemble. Then all the statistical and geometric
characteristics for the first and the second ensemble are equal.
As we specified above, the second ensemble occupies the space {V2} with the
volume value V2. By definition, the packing density in the first and second ensemble
is the same. Consequently,
V2 = (k+N)u/.
(4)
Since
V1 = Nu/,
we have
V2 – V1 = ku/.
(5)
Let us take randomly some h-region in the first or in the second ensemble. Since
both these ensembles are geometrically equivalent we can not find the geometric
characteristics that give us the opportunity to distinguish from what the ensemble the
h-region was taken. The average h-region volume is u/ in both the ensembles. All
the statistical and geometric properties (and, so, (W) distributions) are also equal in
the initial (first) and final (second) ensemble.
Let us call the aggregate of h-regions with the number from 1 to N “old volume”
(Vold ) both for the first and the second ensemble. Let us call the aggregate of hregions for the additional k figures (with the number from (N+1) to (N+k)) “new
volume” (Vnew).
Consider all possible configurations of N figures in {V1} space that are
geometrically equivalent to the first ensemble. Take arbitrary value of the excluded
volume (interval [W; W+dW]) and then determine the region in {V1} where the IVP
with W[W; W+dW] can be found. Call this space as “accessible region” for the IVP
with the excluded volume W. Evidently, for each W we can find corresponding
accessible region.
According to the definition of random ensemble, the additional figure with
arbitrary orientation can be placed in any random location without change of the
statistical and geometrical properties in the new ensemble generated. Consequently, if
the ensemble is random, the accessible region for the IVP with the excluded volume
W is all the {V1} space (if we neglect boundary effects).
At the same time, each of newly placed in {V1} fixed figures decreases the
accessible region for any IVP from “old volume” minimally on the amount of W for
this IVP. Consequently, the accessible region volume for any IVP dV{Vold} within
the boundaries of {V1} space can not be more than (V1 – kW) after k group
placement.
We remember our assumption that all statistical and geometric properties of “old
volume” are indistinguishable before and after this figures addition. Therefore the
“density” of IVP with the excluded volume [W; W+dW] in “old volume” is
unchangeable and equal to (W)dW from the equations (2)-(3). This fact allows us to
find total volume of IVP with the excluded volume [W; W+dW] which will be
displaced from {V1} space boundaries in the second (final) ensemble. This total
volume will be equal to kW(W)dW.
The integral value of “old” IVP displaced from {V1} can be found as follows:
Wmax
k  W(W)dW = k<W>.
(6)
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Wmin
Since for a given condition the difference (V2 – V1) is equal to ku/ and {V1} is
fully contained in {V2}, it is obvious that u/  <W>. The result obtained allows us to
formulate the following theorem.
Theorem. The value of volume occupied by the random figures packing can not
be less than N<W>.
Consequence 1. The density  of the random packing can not be more than
u/<W>.
Consequence 2. If the k group mentioned above is present in the ensemble and
the packing density is more than u/<W> then this packing can not be homogeneous.
Consequence 3. If the k group mentioned above is present in the ensemble and
this packing undergoes the compaction then after the density  reaching u/<W> value
one could observe statistical and geometrical properties (e.g. different coordination
number etc.) which give the opportunity to differ the k group from the other figures.
NUMERIC RESULTS AND DISCUSSION
The packing density corresponding to the ensemble inhomogeneity appearing can
be different for various packing algorithms. But the algorithm that allows us to
generate the homogeneous random ensemble does not exist if  > u/<W> and k
group is present in the packing. To find the numeric values of the density limit
according to the theorem proven one can use the computer models of the random
packing of equal figures (particles) of different shape. The <W>() dependence
obtained by the investigation of the geometrical properties of the model packing
intersects the (u/) curve in the point which corresponds to the maximum density of
the random packing (max).
In particular, computer models of the random packings of hard spheres in 3D
were realized by author in the paper [13]. The main numerical result was a very good
agreement between well-known experimental maximum of packing density max 
0.637 [2] and calculated value [13, 14] max = 0.638  0.003. The methods of the
average excluded volume calculation and then max value finding were described for
the particles of arbitrary shape in [13, 14]. 2D packing of hard discs shows the density
limit at max = 0.682 (in this case max is the fraction of the surface occupied by discs)
[15].
The numerical results obtained are the absolute limits for the packing density in
the random ensembles of spheres. Real max values can not exceed these limits (but
can only be lower or equal to them).
We can note that the different definitions of the random packing can lead us to
different values of the maximum packing density. E.g. authors of [16] observed the
order-disorder transition in hard disc system at the density 0.82  0.03. Nevertheless
if the group of fixed discs was randomly placed in the ensemble one could observe
the differences in the statistical and geometric properties of this group and other discs
already at   0.682 [15].
ACKNOWLEDGEMENTS
This work was supported by the Presidium of Urals Branch of RAS (Project 12-P3-1032).
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