Simulation of small particle penetration in a random
medium
Paul Meakin, Rémi Jullien
To cite this version:
Paul Meakin, Rémi Jullien. Simulation of small particle penetration in a random medium.
Journal de Physique, 1990, 51 (23), pp.2673-2680. <10.1051/jphys:0199000510230267300>.
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J.
Phys.
France 51
1 er DÉCEMBRE
(1990) 2673-2680
2673
1990,
Classification
Physics Abstracts
61.40 81.90
Simulation of small
Paul Meakin
(1)
particle penetration
and Rémi Jullien
in
a
random medium
(2)
(1) Experimental Station, E. I. du Pont de Nemours and Co., Wilmington, DE 19898,
(2) Physique des Solides, Bât. 510, Université Paris-Sud, Centre d’Orsay, 91405 Orsay,
(Received
26 June 1990,
accepted
in
final form
21
U.S.A.
France
August 1990)
Random packings of identical spheres of unit diameter were built according to a
procedure in which spheres are released one after another along randomly positionned vertical
trajectories and then follow the path of steepest descent on the others until they reach a stable
position under gravity. Once a packing has been built, smaller spheres, of diameter
d 1, are allowed to penetrate into it, also following the path of steepest descent and their
penetration depth 0394Z is studied as a function of the parameter 03B5 = (1 - d)/(1 + d). Previous
results that demonstrate the existence of a treshold 03B5c = 31/2 - 1 (corresponding to the
« apollonian » ratio of diameters 2/31/2 - 1), above which 0394Z is infinite, are confirmed. New
results are presented concerning the behavior of 0394Z when the threshold is approached from
below : the mean value 0394Z&#x3E; does not diverge but saturates to 0394Z&#x3E;c ~ 11 and the histogram
N 0394Z&#x3E; reaches an exponential shape whose large 0394Z tail is well fitted by
N 0394Z&#x3E; a exp (- 0.103 0394Z). It is shown that such behavior is due to a non zero proportion of
equilateral triangles of tangent spheres in the random packing.
Abstract.
2014
The mechanism under which small particles can penetrate into a uniform, but random,
medium is a problem of both experimental and theoretical interest, with many potential
technological applications. While in some cases such mechanism can be closely related to
percolation [1]or segregation [2], more generally it exhibits its own specific features and thus
it is worth studying it via adequate numerical simulations.
Here, we address this problem in a very simplified manner. The random medium is
represented by a random packing of identical hard spheres built according to the « steepestdescent » rules, first introduced by Visscher and Bosterlii [3], and reinvented by us in the
context of off-lattice ballistic deposition models [4]. The penetrating particle is also assumed
to be a hard sphere, but with a smaller diameter, which is allowed to move into the packing
according to the same rules. Some preliminary results have already been presented [5] but,
since the efficiency and accuracy of our computer program have recently been improved, we
are now able to present much more precise results : in particular new conclusions, which
correct the previous ones, are given for the behavior of the penetration depth near the
threshold above which it is infinite.
The packing is built by releasing Nb spheres of unit diameter do
1, one after another, on a
basal plane of size L x L, with periodic boundary conditions at the edge of the square. Once
the packing is built, a small sphere of diameter d «-- 1, is released from above the top of the
=
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0199000510230267300
2674
packing along
a
randomly positioned vertical trajectory
and the
calculated, where Zi and Zf are the vertical coordinates of its
OZ
Zi - Zf is
C, when the sphere first
quantity
center
=
contacts the
packing, and when it finds a final stable position, respectively. If it is able to
reach the basal plane, it is again released from above the packing at random and the quantities
Zi - Z f are summed to calculate I1Z. If àZ becomes larger than a given (large) value
I1Zmax, which is another input parameter, the program stops and it is assumed that
OZ is infinite. In practice, we used random packings that were sufficiently deep that
penetrating spheres with diameters larger than the threshold value very rarely reached the
basal plane. To calculate the histogram of OZ as well as its mean value, Np statistically
independent packings (with the same parameters L and Nb) were been built and, for each
packing, a great number of successive independent trials, Nt, were performed. The small ball
was released from independent random initial positions (after each trial the small ball is
obviously discarded from the packing). The size of the small spheres that penetrate into the
random packing of large spheres is characterized by the parameter e that varies from zero to 1
and is related to d by E
0 when d
+ d) = (1 - d)l(l + d), e
(do do and
c == 1 for infinitesimally small particles, i.e. when d « do.
The procedure used to simulate the trajectory of the center C of the moving particle (MP)
in both the packing construction and particle penetration stages consists of a series of steps
illustrated in figure 1 [4-6]. These steps are :
d)l (do
=
(a)
(1)
(2)
In
Vertical motion : MP falls
contacts the basal
contacts a
case
(1)
sphere
plane,
verically.
=
=
This steps ends when MP either :
or
of center
the motion stops. In
in the
packing.
case (2) step (b) is performed.
CIl
Fig. 1. - Some typical motions of the moving particle (MP). In this figure the motion of MP is described
by following the motion of its center C on a sphere of diameter equal to the sum of its diameter and the
one of the contacting sphere. In (a) MP contacts one sphere after a vertical fall, (point A) rolls on it and
escapes at an equatorial position (point E). In (b) MP, after rolling on a sphere, contacts a second
sphere, (point B) rolls on both spheres, reaches a position (point D) where it looses its contact with one
sphere and roll on the other and then escape at an equatorial position (point E). In (c) MP, after rolling
on two spheres Ci and C2 reaches a position (point F) in which it contacts a third one C3 and has to
choose between the following possibilities : stopping, rolling around ClC3 (direction ni), rolling around
C2C3 (direction nD, rolling on C3 alone.
2675
(b) Rolling
on one
sphere :
C rotates around
CI
in
a
vertical
plane. This steps ends when
MP either :
(1) contacts the basal plane, or
(2) reaches an « equatorial » position where C and CI have equal vertical coordinates
(point E in Fig. la), or
(3) contacts another sphere C2.
In case (1) the motion stops. In case (2) the contact with Ci is broken and MP begins to fall
freely (step a). In case (3) step (c) is executed.
(c) Rolling on two spheres : C rotates around the axis Cl C2. This steps ends when MP
either :
(1) contacts the basal plane, or
(2) reaches a « taking off » position, or
(3) contacts a third sphere C3.
The taking off position corresponds to a situation where MP must abandon its contact with
one of the two spheres to roll on the other. This happens when the circular trajectory of C
becomes tangent to a vertical plane (point D in Fig. 1 b). In case (1) the motions stops. In case
(2) step (b) is carried out, after re-labelling the remaining contacting particle as Ci. In case (3)
a stability test must be carried out before continuing. When C is at a position where three
contacts are realized (point F in Fig. 1 c), the unit vectors niand n2 defining the directions of
motion in the next stage are determined, i.e. rotation around CIC3 or C2C3, respectively.
Then the choice depends on the signs of their vertical projections, n 1 , and n2 Z. If they are both
positives, the position is stable and the motion stops. If only one is negative this indicates the
direction for the future motion. If both are negative, the direction with the largest absolute
projection is chosen. In these two last cases, another test is made to determine if MP would
eventually roll on C3 only (by comparing the relative altitudes of point F and point D of the
future motion). If F is higher than D the simulation proceeds to step (b) (after re-labelling C3
as Ci). If F is lower than D another step (c) is carried out (after re-labelling the two new
contacting particles as Ci and C2)Although quite simple in principle, this three-dimensional computer program is quite
difficult to write, especially since accidental overlaps which might occur after an accumulation
of small errors in the determination of the successive positions for C must be avoided. When
searching for a new contacting particle, great care must be taken in degenerate situations (i.e.
situations in which two, or more, contacting positions are found together within the
uncertainties of the calculation) because the wrong solution may be selected. In practice, the
above procedure must be slightly modified to resolve these degeneracies. In the new version
of our program, all the contacting positions found within a given uncertainty (which is taken
to be 10-9, i.e. between single and double precision round-off errors) are treated as true
contacting positions and our program automatically goes to the step corresponding to this
number of contacts (in practice, this means that sometimes a step of the above described
procedure might be skipped). This becomes efficient for very small spheres that can penetrate
very deeply inside the packing, i.e. when E reaches its threshold value ec above which
AZ becomes infinite. In our previous work [5] we were limitated to Ae = ec - E of order
3 x 10- 2, due to such kind of errors. With the new, revised, version of our program we are
able to reach à e
5 x 10- 5 with great confidence in the results.
=
We have run our program with different system sizes and we have checked that the results
do not depend significantly on size. All the results presented here were obtained with
32 and with Nb equal 4 L or 8 L . Since the density of the packing is 0.58 [4], this
L
corresponds to heights of about 115 and 230 respectively. We have also checked that
AZ becomes infinite as soon as c passes Ec
0.73205, by less than 5 x 10- 5. By « infinite » we
=
=
2676
that
get (dZ) = (dZ) ,
for all values
of (dZ) , up to (dZ) =
320. We recall that this threshold value corresponds to the apollonian ratio [7] of
10 L
diameters d/do
2/3 1/2 _1 at which the moving particles exactly fits the hole between three
tangent particles of the packing whose centers from an equilateral triangle of unit edge.
In figure 2 we report the results for the mean value (dZ), which has been obtained by
averaging over N, 2 000 trials times Np 10 packings. Results for the two different heights
have been superimposed. Results are available down to In (Ac) of order - 10, compared with
- 3.5 in our previous publication [5]. While in the range - 1.5, - 3.5, the curve
log (AZ) versus (dE) can roughly be fitted by a straight line [8], it is evident that this is no
longer true when e approaches closer to its threshold value. When Ae tends to zero, a clear
saturation is observed which cannot be due to finite size effects, since it does not depend on
the height of the deposit. From these data the saturation value can be estimated to be
(dZ)c 11.0 ± 0.5.
mean
we
=
=
=
=
=
Fig. 2. - Log-log plot of the mean penetration depth (AZ) as a function of de = ec - s, where e is
related to the particle diameter d by e
(1 - d) / ( 1 + d ) and , 31/2 1 = 0.73205... This plot
results
simulations with the parameters
32, Np 10, Nt = 8 000. 101 points with
Nb 4L 3 = 131 072 and 121 points with Nb 8L 3= 262 144 have been superimposed.
=
=
from
=
=
L
=
=
the histogram of penetration depths for an E value very close to the
i.e.
for AE
5 x 10-4. In this calculation we have considered the
0.7320,
we
and
have
taken
largest height
Nt 8 000 and Np 10. Results from 10 simulations were
combined. When the curve of figure 3 is fitted by a straight line the following exponential
distribution of the penetration depths is obtained :
In
figure 3
threshold £
we
=
show
on
=
=
=
2677
Fig. 3. - Plot of the logarithm of the penetration depth histogram N (AZ) as a function of
AZ for 0.732, i.e. Ae 5 x 10-4. This plot results from simulations with the parameters
L
32, Nb 262 144, Np 10, Nt 8 000.
=
=
=
=
=
=
This is very close to exp ( - AZI (AZ&#x3E; c).
In figure 4 we show the results for the mean lateral displacemcnt (AZ.), where
OL is defined by OL2 - (X; - X f)2 + (Yi - yf)2 , as a function of (I1Z). This figure was
obtained from an average of OL 2 over 8 000 trials with L
32 and Nb
4L . In this
calculation, we have taken care of periodic boundary conditions i.e. Xi - X f is increased
=
Fig.
4.
-
=
Log-log plot of the mean lateral displacement (AL ) as a function of the mean penetration
32 and Nb
This plot results from an average over Nt
8 000 trials with L
131 072.
depth AZ&#x3E;.
=
=
=
2678
(resp. decreased) by L each time C
corrected in the
same
crosses
way. The results
can
the edge X = L
be fitted by :
(resp.
X
=
0)
and
Yi - Yf is
with
Here the error bars are statistical errors coming from the least square fit and the reasonable
result x
0.5, which is consistent with a random horizontal motion, cannot be excluded. The
quite large value found for the coefficient A means that there are strong directional
correlations between successive steps : it results a large « effective mean free path » for the
motion projected on an horizontal plane. If the exponent x is larger than 1/2 this might be
because such correlations could persist at long distances.
The results of figures 2 and 3 can be interpreted as follows. The length of the penetration
depth is essentially determined by the number of constrictions in the packing that the moving
particle is able to cross. A characteristic void diameter d(A, B, C ) =1 B - C 1 /sin (BAC) - 1
is associated with three neighboring spheres with centers at A, B, C. Particles with diameters
larger than d(A, B, C ) are unable to pass between the three spheres at these positions. Let us
define &#x26;z as being the mean vertical distance that the moving particle covers between two
successive triangles and P (d ) the proportion of such triangles with d(A, B, C ) - d. Since
d(A, B, C ) cannot be smaller than dc 2/3 1/2 1, which is realized when ABC form ’an
equilateral triangle of unit edge, P (d) 0 for d - dc. The probability q for a moving particle
of diameter d to penetrate a triangle is q
1
P (d) and consequently, the probability
Q (AZ) of obtaining a penetration depth of AZ, i.e. to penetrate an average number of
k
AZI&#x26;Z successive triangles, is given by :
=
=
=
=
-
=
if size correlations between successive triangles are neglected. The histogram N (AZ), which
is the derivative of Q(AZ) with respect to AZ has essentially the same exponential form.
Assuming P(d) « 1, the large-AZ behavior of N(AZ) has the form :
Then comparing with the results of figure 3, which show that N(AZ) keeps an exponential
form very near to the threshold, we conclude that P (d ) does not tend to zero when d tends to
dc from above, but instead tends to a nonzero value Pc, i.e. P (d ) must exhibit a jump at
dc which corresponds to a nonzero proportion of equilateral triangles of unit edge among all
the Voronoï triangles. From this analysis, we estimate that
The behavior
calculated by :
of (AZ)
Thus the saturation of
P ( d ) does not tend to
figure 2, we find that
can
along
the
same
lines. (dZ)
can
be
when d tend to d,, is also directly linked with the fact that
[9]. Assuming again Pc « 1 and comparing with the result of
(dZ)
zero
also be understood
2679
ten percent, is consistent with the above estimate. The small discrepancy can be
attributed to the assumption Pc « 1 and to the effect of correlations that have been neglected
in our estimate.
It is interesting to test this conclusion directly, by quantitatively analyzing the structure of
the random packing. However, the determination of nearest neighbors as well as « Voronoï »
cells [10] in a random packing of hard spheres can be a very computer time consuming
process, especially for the large packings that are used here. We have used instead a
procedure which takes advantage of the algorithm already written to build the packing. Each
time a particle is added to the packing we record all the triangles of the type C1C2C3 (see
Fig. 1) it encounters on its ways before it stops and we determine the corresponding void
diameters d(Cb C2, C3). We then calculate the quantity II ( d ), which is the proportion of such
triangles wich d(CI, C2, C3) -- d. The curve shown in figure 5 results from an average over
Np 10 packings, with L 32 and Nb 8L 3. In this figure the jump expected from the
above analysis can be seen. The only problem is that too many triangles are counted with this
procedure, since some of the triangles are obviously too large to be considered as triangles
made with nearest neighbors. Therefore we expect that II (d ) is not equal but only
proportional to (and smaller than) P ( d ) near d dc. Cônsequently the jump Hc 0.008 seen
in figure 5 can only give a lower bound for Pc. Anyway, comparing with Pcl&#x26;z = 0.1, it
follows that Sz should be of order a few tenths of the large particle diameter, which seems to
be very reasonable.
which, within
=
=
=
=
=
5. - The dependence of
smaller than d, on d.
Fig.
II(d),
the
proportion
of
triangles
in the
packing
whose void diameter is
In conclusion, we have shown that the mean penetration depth of a small spherical particle
into a random packing of equal spheres does not diverge, but saturates, as the Apollonian
ratio of diameters is approached. This behavior is directly linked with the presence of a non
zero proportion of equilateral triangles of tangent spheres in the packing. As a natural
extension of this work, we intend to study the same penetration problem in a random packing
made with polydisperse spheres. In the case of a binary mixture of spheres with two different
diameters, the saturation value of the penetration depth should depend on the concentration.
In the case of a continuous distribution of diameters with a lower cut-off, we expect to recover
a diverging penetration depth, and the characteristics of the divergence should be related to
the behavior of the diameter distribution near the cut-off. It could also be interesting to study
2680
the penetration of small spheres in « compact » random packings, i.e. packings whose density
is of order 0.62-0.64. In that case most of the triangles made with neighboring spheres are
equilateral with length 1 [11]. The step in P (d) at de should be larger and consequently the
saturated penetration depth should be smaller.
Acknowledgements.
Calculations done in France
to
acknowledge
very
were performed at CIRCE, Orsay.
interesting discussions with J. F. Sadoc.
One of
us
(R. J.) would like
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
BRIDGEWATER J., J. Powder Technol. 15 (1976) 215.
WILLIAMS J. C., J. Powder Technol. 15 (1976) 245.
VISSCHER W. T. and BOSTERLII M., Nature 239 (1972) 504.
JULLIEN R. and MEAKIN P., Europhys. Lett. 4 (1987) 1385.
JULLIEN R. and MEAKIN P., Europhys. Lett. 6 (1988) 629.
JULLIEN R. and MEAKIN P., Nature 344 (1990) 425.
DODDS J. A., J. Colloid Interface Sci. 77 (1980) 317 ;
WILLIAMS D. E. G., Philos. Mag. B 50 (1984) 363 ;
OMNES R., J. Phys. France 46 (1985) 139.
When comparing with the results of reference (5) there is another difference arising from the
definition of Zi, which was then taken to be Zi
Zmax + 1, where Zmax is the highest vertical
coordinate for the spheres of the packing. It results an absolute difference of about 1.5 in
0394Z : the actual 0394Z is smaller and consequently the apparent slope in the range
1.5, - 3.5 is larger.
This corrects the reasoning of reference (5) where the influence of the jump (or a delta peak in the
=
-
[9]
derivative)
[10]
[11]
was
ignored.
MOSSERI R., Geometry of disordered systems, in the
school.
BERNAL J. D., Proc. Roy. Soc. A 280 (1964) 299 ;
GOTO K. and FINNEY J. L., Nature 252 (1974) 202.
proceedings
of the
Beg Grohu
summer