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A random packing structure of equal spheres — statistical geometrical analysis of tetrahedral configurations
P o w d e r Technology, 2 0 ( 1 9 7 8 ) ~) Elsevier Sequoia
233 - 242 S.A., Lausanne -- Printed in the Netherlands
233
A Random Packing Structure of Equal Spheres -- Statistical Geometrical Analysis of Tetrahedral Configurations
KEISHI GOTOH*,
W . S. J O D R E Y
a n d E. M . T O R Y
D e p a r t m e n t o f ~fathematics and C o m p u t e r Science, Jtfount Allison University, Sackville, N e w Brunswick (Canada EOA 3C0) ( R e c e i v e d O c t o b e r 8, ! 9 7 7 ; i n r e v i s e d form February 23, 1978)
SUMMARY
A random packing structure constructed by a computer simulation of the slow settling o f r i g i d s p h e r e s i n t o a r a n d o m l y p a c k e d b e d is investigated, where the bulk-mean volume f r a c t i o n o f t h e p a r t i c l e s is 0 . 5 8 2 . E a c h s p h e r e settled on three others forms a tetrahedron, w h o s e d e t a i l e d g e o m e t r i c a l c o n f i g u r a t i o n is successfully examined by a statistical geometrical analysis.
settling of equal spheres into a randomly p a c k e d b e d [ 2 ] w h o s e p a c k i n g d e n s i t y is 0.582. Tory's simulation assumes no bouncing, no bumping of precariously stable spheres to more stable positions, and no consolidation by spreading spheres apart to accommodate i n c o m i n g s p h e r e s . E a c h s p h e r e is i n t r o d u c e d o n l y a f t e r t h e p r e v i o u s o n e is p e r m a n e n t l y in place [2].
CONFIGURATION OF THE TETRAHEDRON FORMED BY FOUR SPHERE CENTERS
INTRODUCTION Studies of random packing can be divided into two categories: computer simulations, and analyses of intrinsic structure. Simulations can be subdivided into two types. The f i r s t , w h i c h is c h m ~ c t e r i z e d b y i t s k i n e t i c nature, includes sedimentation of spheres into a r a n d o m l y p a c k e d b e d [ 1 , 2 ] as w e l l as m o lecular dynamics and Monte Carlo methods in the field of thermodynamics. The second consists of simulations of the growth of clusters [3, 4]. Studies of the structure of random packing [5 - 13] have been concerned mainly with the distributions of coordination number, voidage, and packing density. The geometrical configuration of individual particles in an actual random packing has been reported [X0], but experiments lack the prec i s i o n t o distinguish s t r i c t c o n t a c t f r o m v e r y close approach [8]. In this paper, we study the detailed geometrical structure of a random packing constructed by a computer simulation of the slow
*Present address: Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan.
A r m q d o m p a c k i n g c a n b e c h a r a c t e r i z e d as a collection of tetrahedra [5]. In the computer simulation of the slow settling of equal spheres into a randomly packed bed, the comp u t a t i o n a l g o r i t h m is b a s e d o n t h e f a c t t h a t e a c h s p h e r e is s u p p o r t e d b y t h r e e o t h e r s [ 2 ] . We examine the configurations of these tetrahedra in a random packing whose packing d e n s i t y o f 0 . 5 8 2 i d e n t i f i e s i t as a r a n d o m loose packing. We choose the origin at the center of the sphere which settled on three others (see Fig. la). The centers of the three supporting spheres determine a plane which cuts the spherical envelope on which the centers of t h e s u p p o r t i n g s p h e r e s m u s t lie. T h e p o s i t i o n of the circular cross-section (which corresponds to the line AHB in Fig. la) can be e x p r e s s e d as a f u n c t i o n o f t h e s h o r t e s t d i s tancer from the origin and the angle 0 of t h e i n c l i n a t i o n as s h o w n i n F i g . ! a . F o l l o w ing the usual practice of using capital letters for random variables and lower case for their v a l u e s , w e d e s i g n a t e R a n d O as r a n d o m variables and study their joint probability d e n s i t y f u n c t i o n f ( r , 0 ). T a b l e 1 s h o w s t h e results of the computer experiment and Table
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F ; g . l ( a ) . T h e c o o r d i n a t e s y s t e m ira u a ; L s o r t h e s p h e r e d i a m e t e r . ( b ) O r i e n t a t i o z ~ o f t h r e e s p h e r e c e n t e r s 1, 2 , 3 on the p erlmeter of circular ero~ection ( v i e w f r o m C-*H d i r e c t i o n i n F i g . l a ) . z d e n o t e s t h e c r o s s p o i n t w i t h
z-axis. 1P = 1Q = 1.0 diameters. TABLE2
Averagepropertiesofthetetrahedra
F, 0_, deg ~1, deg ~2, deg ~3, deg
F_~xpt.
Calcd.
0.537 23.9 26.3 103.1 125.5
0.5438 22.96 26.79 102.24 128.88
m e n t a l p r o p e r t i e s listed in T a b l e 2 h a s a p a c k ing density of 0.689. This does not coincide with the average packing density of the tetrah e d r a b e c a u s e e a c h g e o m e t r i c a l p r o p e r t y has been averaged irrespective of the others. The c o m p u t e r - g e n e r a t e d t e t r a h e d r a vary so widely t h a t n o t o n e h a s p r o p e r t i e s c l o s e t o t h o s e in T a b l e 2. DISCUSSION
2 gives a v e r a g e values. T h e j o i n t d i s t r i b u t i o n of r and 0 expresses not only the microscopic s t r u c t u r e o f t h e p a c k i n g , b u t also t h e r o u g h ness o f t h e s u r f a c e o f t h e p a c k e d b e d : a b e d w i t h l a r g e r r a n d s m a l l e r 0 is m o r e d e n s e a n d has a f l a t t e r s u r f a c e . Next we consider the orientation of the sphere centers on the perimeter of the circular c r o s s - s e c t i o n . T a k i n g t h e l o w e s t p o i n t L as t h e d a t u m p o i n t , w e d e f i n e 01 as t h e a n g l e between L and the center of the lowest sphere. T h e n ¢2 is t h e a n g l e b e t w e e n t h e c e n t e r s o f t h e l o w e s t a n d n e x t l o w e s t , a n d ~3 b e t w e e n l o w e s t a n d h i g h e s t (see Fig. l b ) . F i g u r e s 2(a) and 2(b) show experimental estimates of frequency distributions of ¢1 for various r a n d 0 . F i g u r e 3 is a t y p i c a l e s t i m a t e d f r e q u e n c y distribution of ~2- Since the range of ~a d e p e n d s n o t o n l y o n r a n d 0 b u t also o n ¢1 a n d ¢2, o u r data are far t o o sparse to provide a n e m p i r i c a l relationsbA.-p. A v e r a g e v a l u e s f o r t h e s e angles, o b t a i n e d f r o m t h e c o m p u t e r e x p e r i m e n t , a r e l i s t e d i a T a b l e 2. T h e v a l u e s o f R , O , ",,~nd t h e s e t h x e e a n g l e s determine the configtr.~tion of the tetrahedron which we disct~ theoretically below. The tetrahedron with the average experi-
We first c o n s i d e r t h e d o m a i n s o f R a n d O. O b v i o u s l y r I> 0 a n d 0 1> 0. T h r e e s p h e r e c e n t e r s lie o n t h e p e r i m e t e r o f t h e c i r c u l a r c r o s s - s e c t i o n w h o s e r a d i u s is v ~ - - r 2 ; t h e r e f o r e r ~ x/-2/3 in u n i t s o f t h e s p h e r e d i a m e t e r . In order to support the central sphere placed a t t h e o r i g i n , this c i r c u l a r c r o s s - s e c t i o n m u s t c r o s s t h e z-axis. O t h e r w i s e , t h e t e t r a h e d r o n w o u l d b e u n s t a b l e (see Fig. l a ) . H e n c e , t h e d o m a i n s are
O~r~x/-2/3,
040
~cos-lr
(1)
in e x c e l l e n t a g r e e m e n t w i t h t h e e x p e r i m e n t a l r e s u l t s o f T a b l e 1. I n d e v e l o p i n g t h i s m o d e l , we assume t h a t r and 0 are d e t e r m i n e d b e f o r e the positions of the centers of the three spheres. T o facilitate explanations, however, w e will first d i s c u s s t h e o r i e n t a t i o n s o f t h e s u p p o r t i n g spheres.
ORIENTATIONS OF THREE SUPPORTING SPHERES In order to consider t h e orientations o f the three supporting spheres, Tory's simulation
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must be explained in detail. A falling sphere t o u c h e s a t-trst s p h e r e a n d r o l l s d o w n o n t h i s until it touches a second. Then the f~lling sphere rolls again downward on these two spheres until it touches a third to form a stable tetrahedz-al configux-ation [2]. Henceforth we consider the sphere with the lowest z - c o o r d i n a t e a s t h e ~trst s p h e r e , t h e n e x t lowest as the second, and the highest as the
distribu-
third. The orientation of the sphere hit ftrst is c o n s i d e r e d t o b e a r b i t r a r y , s o t h e d i s t r i b u t i o n o f ~3 is t a k e n t o b e u n i f o r m . S i n c e t h i s r a n d o m l o o s e p a c k i n g is c h a r a c t e r i z e d b y a rough surface [2], an incoming sphere rolls d o w n a c o n s i d e r a b l e d i s t a n c e a f t e r t h e fLrst hit. This can be accommodated by placing t h e c e n t e r s o f s p h e r e s o n e a n d t w o as l o w as possible on the perimeter of the cross-
237 0 15
assumed to be proportional to the vertical distance from its highest position (the point M in Fig. lb). This yields the conditional probability density function (p.d.f.):
g ( ¢ ~ Jr, O ) = "~
~ r ~ - - r 2 c o s 01 - - r t a n 0
0 I0
\
z
/1L
o uJ
(,j~__r
(3)
z c o s 01 - - r t a n 0 ) d 0 1
O
in which the denominator is t h e n o r m a l i z a tion factor. The conditional expectation of ,b 1 is
\ 005
\
E ( ( b I Jr, 0 ) =
01g(01
[r, O) d 0 1 =
o 00
v
50
c o s O l r + 0 1 L s i n 01L - - 0 . 5 0 ~ r COS 01L - - 1
150
tO0 C)2
s i n OSL - - 0 1 - c o s 01¥
{DEG }
Fig. 3. Frequency distribution of ~2- o Experimental data falling in the ranges r = 0.5 - 0.6 (--), 0 = 20 - 30 (deg) and 01 = 25 - 30 (deg}; , theoretical from eqn. (6).
section, retaining the st~lbility condition for the supported sphere. This implies that the two lower spheres should be treated differently from the highest. Tory's packing is a n i s o t r o p i c n o t o n l y i n i t s h o r i z o n t a l a n d vertical characteristics [2, 14], but also as regards up and down. Each sphere is supported by exactly three spheres and supports from zero to six others [2]. An inversion of the packing would make more than onequarter of the spheres unsupported [2] and is t h u s p r o h i b i t e d . T h e s c a t t e r i n F i g s . 2 a n d 3 a n d t h e s p a r s e d a t a f o r 03 m a k e i t n e c e s s a r y to compare theoretical and experimental (i.e. c o m p u t e r - g e n e r a t e d ) average properties of tetrahedm. A s s u m i n g t h a t t h e l o w e s t s p h e r e is t h e f i r s t to appear on the perimeter of the circular c r o s s - s e c t i o n , w e i-wst c o n s i d e r t h e p r o b a b i l i t y d i s t r i b u t i o n o f 4)1- F i g u r e s l ( b ) a n d l ( a ) s h o w t h a t t h e d o m a i n o f & l is 0 ~ 01 ~< ~ I L
01L
=
L
M H L = . C O S --1 ~
a n d t h e o v e r a l l m e a n is COS - 1 r
¢1 = f
drf
O
0
E(,~l[r. O)f(r.
0) dO
(5)
where f ( r , 0 ) is the j o i n t p.d.f, given b y eqn. (18). Figures 2(a) and 2(b) compare theoretical c u r v e s c a l c u l a t e d f r o m e q n . ( 3 ) w i t h results from the computer experiment. Since i t is i m p o s s i b l e t o o b t a i n d a t a f o r e x a c t l y t h e same values of r and 0, there are only 100 200 experimental points for each distribution. Although the data scat'oer widely, we conclude that eqn. (3) adequately e~presses the p r o b a b i J i t y d i s t r i b u t i o n o f • 1Assuming that the second lowest sphere is t h e s e c o n d t o a p p e a r , w e p o s t u l a t e t h a t the probability that its center falls between 02 and 02 + d02 is proportional to the vertical distance from its highest possible position, the point R in Fig. l(b). This yields the conditional p.d.f.: h(0R
[01,
I", 0 ) =
COS ( 0 2 2 L [ COS ( 0 2
r tan O
(4)
- - 0 1 ) - - COS ( 0 2 L --01)
--COS
(02L
-- 01) --01)]
dOz
(2)
(6) for fixed r and 0. Since we are choosing the position of the lowest sphere, the probability t h a t i t s c e n t e r l i e s b e t w e e n ~1 a n d ~1 + d 0 1 i s
where the denominator is the normalization factor. The conditional expectation of cD2 is
238
E(~2[0~.
r. o ) =
j2L 02s
c ) 2 h ( ~ 2 [01, r, O) dq) 2
= [COS b - - c o s a + Ozr. s i n b - - 0 2 , s i n a - 1
2
-- o~cos
(ozL - - 0 ~ ) c o s
b] [sin b -- sin a --
(7)
b] - z
where
a - - '#z~ - - O z ,
b -- ¢ 2 L ~ 0 1
~ z , -- m m x ( L 1 H P , L 1 / 4 1 " )
(8)
6 2 z --= m i n ( L 1 H R , 2 ~ - - 2 L I H Q )
already noted, the position of sphere 2 must be lower than that of the point R. Equation (8) expresses both of these restrictions and gives ~he u p p e r l i m i t , ~2L, f o r t h e i n t e g r a t i o n _ Fi£ure 3 compares the theoretical curve calculated from eqn. (6) with a typical result from the computer experiment. Although the data are sparse (about 130 points) and scattered, we conclude that eqn. (6) adequately expresses the probab;lity distribution of (b 2- T h e o v e r a l l m e a n is C:OS__i r
ffz=f
drf
O
f(r. 0) dO X
O
and Fig. l(b) shows that
/ ~ E(,~2Iol. r, o )g(O~[r, 0 )
L 1H1 ~ = 20 z
L X H P = L II--IQ =- A
for distance 1R > 1
(9)
dos
(10)
o
where t h e j o i n t p.d.f, f(r. 0 ) is given b y eqn. ( 1 8 ) . where A--=cos -1
[1
1
2(1--r
2)
]
Finally, we consider the orientation of the third (and highest) sphere. As ment i o n e d e a r l i e r i n t h i s s e c t i o n , t h i s s p h e r e is considered to be uniformly distributed_ The lower and upper limits are, respectively,
and
B - - r t a n O/ ~ f - l - - r z
Ca~ = / - 1 H Q = c o s - 1 C)ZL
Further details are given in AppendLx (A). The upper and lower limits of the random variable ~2 are obtained as follows. Since we placed the lowest sphere at the point i in Fig. l(b}, the centers of the other two spheres must lie on opposite sides of the line 1ZR. Othe~vise, the central sphere cannot be stable on these three spheres. The point P in Fig. l(b) deno[es the position where the distance between the points 1 and P equals the sphere diameter. Thus, the second sphere cannot lie between p o i n t s 1 a n d P . A l s o , t h e s e c o n d s p h e r e is higher than the first, so the former cannot l i e b e t w e e n 1_ a n d 1 " . E q u a t i o n (8} e x presses both of these restrictions and gives the lower limit, 02s, for the integration_ To determine the upper limit, we must take into account the third sphere. In Fig. l(b}, Q is t h e p o i n t w h i c h is o n e d i a m e t e r f r o m the point 1. To leave room for the third sphere, the angle between sphere 2 and the point 1 must be at least twice L 1HQ. As
= 2u -- (02
+
2 ( 1 ~ r 2)
I- X H Q ) = 2re
--
02
Hence, the conditional expectation 1
(11) --
c)3s
is 02
E(,%[~1, .o2, r, 0)-----~(O3L + 0 ~ ) =~--~(12) T h e o v e r a l l m e a n is ~f2/3
Cos
:f
d, f
0
0
x
~ --
-z
r
flL
r(r, O ) d O
g( llr, O)d01 0
h ( O a l O l , r, 0 ) d 0 2
(13)
o2s Splitting this multiple integral into two pieces, we use the definition of a p.d.f, for the part containing u, and eqns. (7) and (10) for that containing ~2]2, to obtain
~2 ~s
=
=
2
(14)
239
In summary, we can determLue the rositions of the three supporting spheres on the perimeter of the circular cross-section only by knowing r and 0. First, the lowest sphere is placed according to a probability t h a t is p r o p o r t i o n a l t o t h e v e r t i c a l d i s t m u c c from its highest position. Next, a second s p h e r e is p l a c e d i n s i m i l a r f a s h i o n . T h e p . d . f , f o r t h e t h i r d s p h e r e is u n i f o r m o v e r the available region._To compute the overall m e a n s , Oz, 0 2 , a n d ¢ 3 , w e n e e d t o k n o w t h e distribution of R and O. JOINT PROBABILITY DISTRIBUTION OF R AND R AND O The position of the circular cross-section can be characterized by its central point. We consider an infinitesimal area r d0 dr and revolve it about the z-axis, obtaining a ring with volume d V = 2 ~ r 2 s i n 0 d r dO
(15)
If we cut a spherical volume with radius equal to 1.0 diameters at random, then whenever the center of the resulting circular crosssection falls in the volume expressed by eqn. ( 1 5 ) , t h e dish,a n t e f r o m t h e o r i g i n t o t h e c e n ter of the cross-section falls between r and r ÷ dr and the angle of inclination falls between 0 and 0 + dO. Accordingly, the probability of f i n d i n g R a n d E) a t t h i s p o s i t i o n is p r o p o r t i o n al to dV. The perimeter of the circular cross-section through the centers of the supporting spheres is 2 ~ J ' ( 1 - - r 2 ), w h i c h r e a c h e s i t s m i n i m u m value of 2~/~/3 when r = x/T2]3). For r > ~/(2/3) there is no case of three sphere centers lying on the perimeter. Alternatively, even if r < ~/(2/3), there are some cases where the central sphere cannot be supported by three others. We must take into account the probability that a circular cross-section has three sphere centers which make the central sphere stable. Thus, we focus attention on the range o f p o s i t i o n s a v a i l a b l e f o r t h e s e c o n d s p h e r e in the presence of the other two supporting s p h e r e s . T h e m a x i m u m a n d m i n i m u m o f ¢b 2 are given by eqn. (8). The probability that 2 m a k e s the" c e n t r a l s p h e r e s t a b l e is a s s u m e d to be proportional to the available arc length p r o j e c t e d o n t h e x - y p l a n e [ 1 0 ] , i.e.
P~--E[V r~-r~
( ~ 2 L - - ~ # 2 , ) COS0 [ r, 0 ]
=~
r~ - r 2
cOs 0
(~2r-
--¢2,)g(Ozlr, O)dCz
0
(16) w h e r e CZL a n d O2s a r e g i v e n b y e q n . ( 8 ) , a n d Pc i s n o t n o r m a l i z e d a t t h i s s t a g e . O f c o u r s e , a space has been taken into account for the third sphere. We have taken the view that a circuIar cross-section is determined first, and then the positions of the three supporting spheres. Equation (16) must be calculated without any knowledge of the positions of the three supporting spheres. We have theref o r e u s e d t h e c o n d i t i o n a l p . d . f , f o r ,P 1 t o average pc over all values of • z - Further details are given in AppendLx (B). Finally, consider a factor attributed to the w a y i n w h i c h t h e T o r y p a c k i n g [ 2 ] is f o r m e d . We consider that a falling sphere C touches a f i r s t s p h e r e a n d t h e n a s e c o n d , cf. A a n d B i n Fig. la. The incoming sphere rolls always downward on these until it touches a third o n e . N o f o r c e is t a k e n i n t o a c c o u n t i n o r d e r to make the packing dense. Hence the transverse movement of the rolling sphere into the available empty space contributes to this purpose. This sphere rolls down toward the central point H. In its rest position, therefore, small distances between H and the z-axis are more likely than large. With 0 = cos-Zr, the d i s t a n c e r s i n O b e t w e e n H a n d t h e z - a x i s is a maximum, and the probability of forming a s t a b l e t e t r a h e d r o n is z e r o . A c c o r d i n g l y , w e consider eqn. (17) as a factor attributed to the method of packing. P s ~- r~/r~ - - r 2 - - r
sin 0
(17)
From eqns. (15) - (17), we finally obtain t h e j o i n t p . d . f , o f R a n d E) w h o s e d o m a i n is prescribed by (1).
r 2 p c p s sin 0 ~os-'~
f(r, O) = ~
f
r dr f
0
0
(18)
pop. sin 0 dO
where the denominator expresses the normalization factor of the p.d.f. Equation (18) is a function of r and 0 only. The averages of R and O are cOS-~ z r
-- f
dr f
0
0
and
0) d0
(19)
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05
-
t o c o m p u t e t h e m u l t i p l e i n t e g r a l s in e q n s . (18) - (20), (5), and (10) to o b t a i n t h e norm a l i z a t i o n f a c t o r , r , O, ~ z , a n d ~-2 r e s p e c t i v e ly_ T h e v a l u e o f ¢3 is o b t a i n e d f r o m ( 1 4 ) . Table 2 shows the agreement between calculated a n d e x p e r i m e n t a l values. T h e probabili t y d e n s i t y f u n c t i o n f(r, O) also a g r e e s w i t h experiments. Figures 4(a) a n d 4(b) d e p i c t t w o t y p i c a l cases.
\"
/
J
• t-}
• /!
/ / -
\
7
CONCLUSION
~%1
I'I, i': ,
\
\
'-,\
t
e ~ t~EG I
(b)
Fig. 4(a). Joint p:ohab~-lity distribution o f r and 0 for 0 = 20 - 25 (d-'g). LkR = 0.05; ~0 = 5 (deg). x , o for 8 0 0 0 a n d 9000 s p h e r e packings respectively. Solid c u r v e is theoretic-,-I for 0 = 22.5 (deg). (b) Joint p r o b ability distributio.~ o f r a n d 0 for r = 0.5 - 0 . 5 5 (--). ~ R = 0.05; ~-%0 = 5 (deg). x , e for S 0 0 0 a n d 9 0 0 0 s p h e r e packings respectively. Solid cffrve is theoretical for r = 0 . 5 2 5 (--)cos-- z r
f 0
dr f
or(r,o)dO
0
T o o b t a i n f(r, 0 ), w e f i r s t c a l c u l a t e P c f r o m e q n . ( 1 6 ) u s i n g ~2T- a n d ~2_, f r o m (8) a n d g ( ~ z Jr, 0 ) f r o m (3). T h e s p e c i f i c a t i o n o f ~2~ a n d 02~ i n t e r m s o f ¢ x , r, a n d 0 is g i v e n in A p p e n d i x ( A ) , w h i l e t h e c a l c u l a t i o n o f p~ is s h o w n in A p p e n d i x (B). U s i n g t h i s r e s u l t a n d e q n . ( 1 7 ) f o r p~, w e u s e t h e R o m b e r g m e t h o d
We h a v e i n v e s t i g a t e d a r a n d o m p a c k i n g structure constructed by a computer simulation of the slow settling of equal spheres into a randomly packed bed. In particular, we have examined the detailed configuration of the tetrahedron formed when one sphere settles o n three others. T h e centers o f the t h r e e s u p p o r t i n g s p h e r e s lie o n a s p h e r i c a l envelope about the central sphere and determine a circular cross-section of that envelope. T h e c e n t e r o f t h e c e n t r a l s p h e r e is t a k e n as t h e origin a n d t h e p o s i t i o n o f t h e cross-sect i o n is g i v e n b y t h e d i s t a n c e r o f its c e n t e r a n d its a n g l e o f i n c l i n a t i o n 0. T h e j o i n t p . d . f . f(r, 0 ), d e f i n e d b y e q n . ( 1 8 ) , is o b t a i n e d f r o m geometrical conditions (15) and (16) and the packing criterion (17). After the cross-section is f i x e d , p o s i t i o n s o f t h e t h r e e s u p p o r t i n g s p h e r e s a r e d e t e r m i n e d o n its p e r i m e t e r . F i r s t , t h e l o w e s t o f t h e t h r e e is p o s i t i o n e d a c c o r d i n g t o a p r o b a b i l i t y w h i c h is p r o p o r t i o n a l t o t h e v e r t i c a l d i s t a n c e f r o m its h i g h e s t p o s s i b l e p o i n t . N e x t , a s e c o n d s p h e r e is s i m i l a r l y p l a c e d . T h e d i s t r i b u t i o n o f t h e t h i r d s p h e r e is
241 uniform o v e r t h e available region. Based u p o n
these ideas, the detailed geometrical properties calculated for the model tetrahedron agree with those obtained from computer experiments. The model should be tested further to see if it yields the same anisotropy as T o r y ' s p a c k i n g [ 2 , 1 4 ] . This paper has dealt with only one random p a c k i n g , i.e_ a r a n d o m p a c k i n g c o n s t r u c t e d by a computer simulation of the slow settling of equal spheres into a randomly packed bed [2]. The bulk-mean volume fraction of the particles-is 0.582, which can be considered to be a random loose packing. A similar statistical geometrical analysis should be tested for other packings. The synthesis of the assembly of tetrahedra to form a random packing should also be considered.
Subscript 1, 2, 3 t h r e e s p h e r e c e n t e r s in o r d e r o f l o w e s t z coordinate L upper limit s lower limit Superscrip t average value APPENDIX (A) Equation
(8)
Czs = L 1 H P f o r ¢ l
<
=L 1HI* for¢l
C
> C
¢2r =2~--2LIHQ
forr>~/2/2, and¢x
= / IHR A C I ~ I O W L E D GE~,%IE N T S W e a c k n o w l e d g e the financial support of the National Research Council of Canada.
for all cases other than the
above, where C ~ sin - x
1 2~/1 -- r z
LIST O F SYMBOLS
E=
E(¢tlr, O)
conditionM expectation of
E(~a[¢~, r, 0)
conditional expectation of &2 j o i n t p r o b a b i l i t y d e n s i t y distribution function of r and 0, eqn. (18) conditional probability d e n s i t y f u n c t i o n o f Cz conditional probability d e n s i t y f u n c t i o n o f ¢2 defined by eqn. (16) defined by eqn. (17) distance from the origin to the center of circular crosss e c t i o n , in u n i t s o f t h e s p h e r e d ; ~ r n e t e r ( s e e Fig_ l a ) r a n d o m v a r i a b l e , i.e. e v e n t of r a n g l e s d e f i n e d in Fig. l ( b ) r a n d o m v a r i a b l e s , i.e. e v e n t s
f(r. O)
g(o~ l", o) h(¢21¢1, r. o) Pc Ps r
R ~I,2.3
~I.2.3
of
8 0
~1.2.3
a n g l e o f i n c l i n a t i o n o f circ u l a r c r o s s - s e c t i o n ( s e e Fig. l a ) r a n d o m v a r i a b l e , i.e. e v e n t o f 8
O >E
tan-l[X/~--rZ
1 l 2 ( 1 Z_r2 )
D = sin-l[~-~t--sin
A -- c o s - 1
[1
A +~B 2 --cos2A}]
1 ] 2 ( 1 - - r 2)
B = r tan Olx/~--r (B) Equation
1t]
2
(16)
Pc = I - - H II
1 fir. ~
--
Ob
~2s( c ° s ~1 - - B ) d P l
2r = A f o r c o s - 1 ~ - - ~ < 0 < cos-Xr. Le. ¢~,_ < C = [AsinC--ABC+2cos~xL
--2cosC+
2 ~ l L Sin ¢ I L - - 2 C s i n C - - B ( ~ L 2r f o r 0 < c o s - 1 ~--ff, i.e. ~ I L > C
-- C2)]/Q
242 where Q -= s i n ¢i"- - - B O L L /IL
1 I ---------
¢z,- ( c o s ¢ i
d¢1
-- B)
Q'o = [ 2 ( = -- A ) ( s i n D - - B D )
F(D) ]/Q
+ F ( ¢ I L ) --
for r > x/2/2
= F(¢x,)/Q
for all cases
and
0 > E
other
than
the
above,
where
F ( x ) -- _f l_ 1 H R
d¢i
- (cos 0 1 - - B )
o
a n d L 1/!R is given b y eqn. (9). Integrating b y parts w h e n
necessary, we
obtain 3
= cos
F(x)
2(sin
x
-- 1 + 2x
-- --Bx
(.1--c~cosx).
9_2 1 -- B 2
z
+
2
(cosx--B x -- Bx) tan -I ~ s~m-~
+l--B21n
"(
sin x
)
--
1 --~, x
y dy
w h ~ r e ~ ---- 2 J ~ [ ( 1 + B 2 ) REFERENCES 1 % V . M . V i s s c h e r a n d ~I. B o l s t e r l i , R a n d o m p a c k i n g of equal and unequal spheres in two and three dimensions, Nature (London), 2 3 9 (197°--) 5 0 4 - 507_
2 E. M. Tory, B. H. Church, M. K. Tam and M. Ratner, Simulated random packing of equal spheres, Can. J. Chem. Eng., 5I (1973) 484 - 493. 3 C . H_ B a n n e t t , S e r i a l l y d e p o s i t e d a m o r p h o u s a g gregates of hard spheres, J. Appl. Phys., 43 (1972) 2727 - 2734. 4 D.J. Adams and J. Matheson, Computation of d e n s e r a n d o m p a c k i n g s c,f h a r d s p h e r e s , J . C h e m . Phys., 56 (1972) 1989 - 1994. 5 J.D. Bernal, The structure of liquids, Proc. R. Soc., A280 (1964) 299 - 322. 6 D . P . H a u g h e y a n d G . S. G . B e v e r i d g e , S t r u c t u r a l properties of packed beds -- A review, Can. J. Chem." E n g . , 4 7 (1969) 1 3 0 - 1 4 0 . 7 (3. D . S c o t t a n d D . M . K i l g o u r , T h e d e n s i t y o f random close packing of spheres, Br. J. Appl. Phys. S e r . 2, 2 ( 1 9 6 9 ) 8 6 3 - 8 6 6 . 8 J . D . B e r n a l , I. A . C h e r r y , J . L . F i n n e y a n d K . 1%. Knight, An optical machine for measuring sphere coordinates in random packings, J. Phys. E, 3 (1970) 388 - 390. 9 T . W . S . P a n g , U . M . F r a n k l i n a n d %V. A . M i l l e r , On the siinulation of liquid structure by random close packing, Mater. Sci. Eng., 12 (1973) 167 172. 10 K. Gotoh and J. L. Finney, Statistical geometrical approach to random packing density of equal spheres, Nature (London), 252 (1974) 202 - 205. 11 H. Iwata and T. Homma, Distribution of coordination numbers in random packing of homogeneous spheres, Powder Technol., 10 (1974) 79 - 83. 12 K. K. Pillai, Voidage variation at the wall of a packed bed of spheres, Chem. Eng_ Sei., 32 (1977) 59 - 61. 13 J. L. Finney, Modelling the structures of amorphous metals and alloys, Nature (London), 266 (1977) 309 - 314. 1 4 E . M . T o r y , N . A . C o c h r a n e a n d S . R . %Vacldell, Anisotropy in simulated random packing of equal spheres, Nature (London), 220 (1968) 1023 - 1024.
233 - 242 S.A., Lausanne -- Printed in the Netherlands
233
A Random Packing Structure of Equal Spheres -- Statistical Geometrical Analysis of Tetrahedral Configurations
KEISHI GOTOH*,
W . S. J O D R E Y
a n d E. M . T O R Y
D e p a r t m e n t o f ~fathematics and C o m p u t e r Science, Jtfount Allison University, Sackville, N e w Brunswick (Canada EOA 3C0) ( R e c e i v e d O c t o b e r 8, ! 9 7 7 ; i n r e v i s e d form February 23, 1978)
SUMMARY
A random packing structure constructed by a computer simulation of the slow settling o f r i g i d s p h e r e s i n t o a r a n d o m l y p a c k e d b e d is investigated, where the bulk-mean volume f r a c t i o n o f t h e p a r t i c l e s is 0 . 5 8 2 . E a c h s p h e r e settled on three others forms a tetrahedron, w h o s e d e t a i l e d g e o m e t r i c a l c o n f i g u r a t i o n is successfully examined by a statistical geometrical analysis.
settling of equal spheres into a randomly p a c k e d b e d [ 2 ] w h o s e p a c k i n g d e n s i t y is 0.582. Tory's simulation assumes no bouncing, no bumping of precariously stable spheres to more stable positions, and no consolidation by spreading spheres apart to accommodate i n c o m i n g s p h e r e s . E a c h s p h e r e is i n t r o d u c e d o n l y a f t e r t h e p r e v i o u s o n e is p e r m a n e n t l y in place [2].
CONFIGURATION OF THE TETRAHEDRON FORMED BY FOUR SPHERE CENTERS
INTRODUCTION Studies of random packing can be divided into two categories: computer simulations, and analyses of intrinsic structure. Simulations can be subdivided into two types. The f i r s t , w h i c h is c h m ~ c t e r i z e d b y i t s k i n e t i c nature, includes sedimentation of spheres into a r a n d o m l y p a c k e d b e d [ 1 , 2 ] as w e l l as m o lecular dynamics and Monte Carlo methods in the field of thermodynamics. The second consists of simulations of the growth of clusters [3, 4]. Studies of the structure of random packing [5 - 13] have been concerned mainly with the distributions of coordination number, voidage, and packing density. The geometrical configuration of individual particles in an actual random packing has been reported [X0], but experiments lack the prec i s i o n t o distinguish s t r i c t c o n t a c t f r o m v e r y close approach [8]. In this paper, we study the detailed geometrical structure of a random packing constructed by a computer simulation of the slow
*Present address: Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan.
A r m q d o m p a c k i n g c a n b e c h a r a c t e r i z e d as a collection of tetrahedra [5]. In the computer simulation of the slow settling of equal spheres into a randomly packed bed, the comp u t a t i o n a l g o r i t h m is b a s e d o n t h e f a c t t h a t e a c h s p h e r e is s u p p o r t e d b y t h r e e o t h e r s [ 2 ] . We examine the configurations of these tetrahedra in a random packing whose packing d e n s i t y o f 0 . 5 8 2 i d e n t i f i e s i t as a r a n d o m loose packing. We choose the origin at the center of the sphere which settled on three others (see Fig. la). The centers of the three supporting spheres determine a plane which cuts the spherical envelope on which the centers of t h e s u p p o r t i n g s p h e r e s m u s t lie. T h e p o s i t i o n of the circular cross-section (which corresponds to the line AHB in Fig. la) can be e x p r e s s e d as a f u n c t i o n o f t h e s h o r t e s t d i s tancer from the origin and the angle 0 of t h e i n c l i n a t i o n as s h o w n i n F i g . ! a . F o l l o w ing the usual practice of using capital letters for random variables and lower case for their v a l u e s , w e d e s i g n a t e R a n d O as r a n d o m variables and study their joint probability d e n s i t y f u n c t i o n f ( r , 0 ). T a b l e 1 s h o w s t h e results of the computer experiment and Table
234
0
0
0
0
L_~ c o 0 ,-~ Ol co ol
oa .~.
~
0
G'%~ o O O.l c~
,---, c o 0 0 0
oa
0
0
0
0
0
0
0
0
0
0
~3 ,-4
..~
0
0
0
0
0
0
0
0
oa ~.-
0 0 0 ,-~ ,-4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
,--4 ,--4 ,.-4 ,.-4 ,-.4 ,.-~
°NNg~NNN°NN~
"~°°°°°
"-4
S
0
N
~
N
N
N
0
~
~N
.
,
.
l
l
.
.
o
l
~
J
l
m
l
,
,
,
235
S
1
.
3
\ [ " %.
1t
p
(a)
(b)
N
i
F ; g . l ( a ) . T h e c o o r d i n a t e s y s t e m ira u a ; L s o r t h e s p h e r e d i a m e t e r . ( b ) O r i e n t a t i o z ~ o f t h r e e s p h e r e c e n t e r s 1, 2 , 3 on the p erlmeter of circular ero~ection ( v i e w f r o m C-*H d i r e c t i o n i n F i g . l a ) . z d e n o t e s t h e c r o s s p o i n t w i t h
z-axis. 1P = 1Q = 1.0 diameters. TABLE2
Averagepropertiesofthetetrahedra
F, 0_, deg ~1, deg ~2, deg ~3, deg
F_~xpt.
Calcd.
0.537 23.9 26.3 103.1 125.5
0.5438 22.96 26.79 102.24 128.88
m e n t a l p r o p e r t i e s listed in T a b l e 2 h a s a p a c k ing density of 0.689. This does not coincide with the average packing density of the tetrah e d r a b e c a u s e e a c h g e o m e t r i c a l p r o p e r t y has been averaged irrespective of the others. The c o m p u t e r - g e n e r a t e d t e t r a h e d r a vary so widely t h a t n o t o n e h a s p r o p e r t i e s c l o s e t o t h o s e in T a b l e 2. DISCUSSION
2 gives a v e r a g e values. T h e j o i n t d i s t r i b u t i o n of r and 0 expresses not only the microscopic s t r u c t u r e o f t h e p a c k i n g , b u t also t h e r o u g h ness o f t h e s u r f a c e o f t h e p a c k e d b e d : a b e d w i t h l a r g e r r a n d s m a l l e r 0 is m o r e d e n s e a n d has a f l a t t e r s u r f a c e . Next we consider the orientation of the sphere centers on the perimeter of the circular c r o s s - s e c t i o n . T a k i n g t h e l o w e s t p o i n t L as t h e d a t u m p o i n t , w e d e f i n e 01 as t h e a n g l e between L and the center of the lowest sphere. T h e n ¢2 is t h e a n g l e b e t w e e n t h e c e n t e r s o f t h e l o w e s t a n d n e x t l o w e s t , a n d ~3 b e t w e e n l o w e s t a n d h i g h e s t (see Fig. l b ) . F i g u r e s 2(a) and 2(b) show experimental estimates of frequency distributions of ¢1 for various r a n d 0 . F i g u r e 3 is a t y p i c a l e s t i m a t e d f r e q u e n c y distribution of ~2- Since the range of ~a d e p e n d s n o t o n l y o n r a n d 0 b u t also o n ¢1 a n d ¢2, o u r data are far t o o sparse to provide a n e m p i r i c a l relationsbA.-p. A v e r a g e v a l u e s f o r t h e s e angles, o b t a i n e d f r o m t h e c o m p u t e r e x p e r i m e n t , a r e l i s t e d i a T a b l e 2. T h e v a l u e s o f R , O , ",,~nd t h e s e t h x e e a n g l e s determine the configtr.~tion of the tetrahedron which we disct~ theoretically below. The tetrahedron with the average experi-
We first c o n s i d e r t h e d o m a i n s o f R a n d O. O b v i o u s l y r I> 0 a n d 0 1> 0. T h r e e s p h e r e c e n t e r s lie o n t h e p e r i m e t e r o f t h e c i r c u l a r c r o s s - s e c t i o n w h o s e r a d i u s is v ~ - - r 2 ; t h e r e f o r e r ~ x/-2/3 in u n i t s o f t h e s p h e r e d i a m e t e r . In order to support the central sphere placed a t t h e o r i g i n , this c i r c u l a r c r o s s - s e c t i o n m u s t c r o s s t h e z-axis. O t h e r w i s e , t h e t e t r a h e d r o n w o u l d b e u n s t a b l e (see Fig. l a ) . H e n c e , t h e d o m a i n s are
O~r~x/-2/3,
040
~cos-lr
(1)
in e x c e l l e n t a g r e e m e n t w i t h t h e e x p e r i m e n t a l r e s u l t s o f T a b l e 1. I n d e v e l o p i n g t h i s m o d e l , we assume t h a t r and 0 are d e t e r m i n e d b e f o r e the positions of the centers of the three spheres. T o facilitate explanations, however, w e will first d i s c u s s t h e o r i e n t a t i o n s o f t h e s u p p o r t i n g spheres.
ORIENTATIONS OF THREE SUPPORTING SPHERES In order to consider t h e orientations o f the three supporting spheres, Tory's simulation
236 015 = 2 0 - 2 . ~ [O--G J
W rc-~ O5-O55 A. 07075
=
0 IO
z
•
"k
LL
005
-\
•
o-ot 0
20
, ~' 0
6'0 {D--G
(a)
"
80
)
OI5
I
r =~_.5-O55 :-~ • a
~ t.DEG ] '~,
-
'5
--[ 2C - 25 , -:'.C- -=~ o
°
°°o
r
eo
T
,o
s'0 0.
\"
\~
i 80
[BEG.}
F i g . 2 ( a ) . F r e q u e n c y d ~ s t N b u t i o n o f ~ l - S o l i d c u r v e s are theoretical for 0 = 2 2 . 5 ( d e g ) . ( b ) F r e q u e n c y tion o f ~ 1 - S o l i d c u r v e s a r e t h e o r e t i c a l f o r r = 0 . 5 2 5 ( - ) .
must be explained in detail. A falling sphere t o u c h e s a t-trst s p h e r e a n d r o l l s d o w n o n t h i s until it touches a second. Then the f~lling sphere rolls again downward on these two spheres until it touches a third to form a stable tetrahedz-al configux-ation [2]. Henceforth we consider the sphere with the lowest z - c o o r d i n a t e a s t h e ~trst s p h e r e , t h e n e x t lowest as the second, and the highest as the
distribu-
third. The orientation of the sphere hit ftrst is c o n s i d e r e d t o b e a r b i t r a r y , s o t h e d i s t r i b u t i o n o f ~3 is t a k e n t o b e u n i f o r m . S i n c e t h i s r a n d o m l o o s e p a c k i n g is c h a r a c t e r i z e d b y a rough surface [2], an incoming sphere rolls d o w n a c o n s i d e r a b l e d i s t a n c e a f t e r t h e fLrst hit. This can be accommodated by placing t h e c e n t e r s o f s p h e r e s o n e a n d t w o as l o w as possible on the perimeter of the cross-
237 0 15
assumed to be proportional to the vertical distance from its highest position (the point M in Fig. lb). This yields the conditional probability density function (p.d.f.):
g ( ¢ ~ Jr, O ) = "~
~ r ~ - - r 2 c o s 01 - - r t a n 0
0 I0
\
z
/1L
o uJ
(,j~__r
(3)
z c o s 01 - - r t a n 0 ) d 0 1
O
in which the denominator is t h e n o r m a l i z a tion factor. The conditional expectation of ,b 1 is
\ 005
\
E ( ( b I Jr, 0 ) =
01g(01
[r, O) d 0 1 =
o 00
v
50
c o s O l r + 0 1 L s i n 01L - - 0 . 5 0 ~ r COS 01L - - 1
150
tO0 C)2
s i n OSL - - 0 1 - c o s 01¥
{DEG }
Fig. 3. Frequency distribution of ~2- o Experimental data falling in the ranges r = 0.5 - 0.6 (--), 0 = 20 - 30 (deg) and 01 = 25 - 30 (deg}; , theoretical from eqn. (6).
section, retaining the st~lbility condition for the supported sphere. This implies that the two lower spheres should be treated differently from the highest. Tory's packing is a n i s o t r o p i c n o t o n l y i n i t s h o r i z o n t a l a n d vertical characteristics [2, 14], but also as regards up and down. Each sphere is supported by exactly three spheres and supports from zero to six others [2]. An inversion of the packing would make more than onequarter of the spheres unsupported [2] and is t h u s p r o h i b i t e d . T h e s c a t t e r i n F i g s . 2 a n d 3 a n d t h e s p a r s e d a t a f o r 03 m a k e i t n e c e s s a r y to compare theoretical and experimental (i.e. c o m p u t e r - g e n e r a t e d ) average properties of tetrahedm. A s s u m i n g t h a t t h e l o w e s t s p h e r e is t h e f i r s t to appear on the perimeter of the circular c r o s s - s e c t i o n , w e i-wst c o n s i d e r t h e p r o b a b i l i t y d i s t r i b u t i o n o f 4)1- F i g u r e s l ( b ) a n d l ( a ) s h o w t h a t t h e d o m a i n o f & l is 0 ~ 01 ~< ~ I L
01L
=
L
M H L = . C O S --1 ~
a n d t h e o v e r a l l m e a n is COS - 1 r
¢1 = f
drf
O
0
E(,~l[r. O)f(r.
0) dO
(5)
where f ( r , 0 ) is the j o i n t p.d.f, given b y eqn. (18). Figures 2(a) and 2(b) compare theoretical c u r v e s c a l c u l a t e d f r o m e q n . ( 3 ) w i t h results from the computer experiment. Since i t is i m p o s s i b l e t o o b t a i n d a t a f o r e x a c t l y t h e same values of r and 0, there are only 100 200 experimental points for each distribution. Although the data scat'oer widely, we conclude that eqn. (3) adequately e~presses the p r o b a b i J i t y d i s t r i b u t i o n o f • 1Assuming that the second lowest sphere is t h e s e c o n d t o a p p e a r , w e p o s t u l a t e t h a t the probability that its center falls between 02 and 02 + d02 is proportional to the vertical distance from its highest possible position, the point R in Fig. l(b). This yields the conditional p.d.f.: h(0R
[01,
I", 0 ) =
COS ( 0 2 2 L [ COS ( 0 2
r tan O
(4)
- - 0 1 ) - - COS ( 0 2 L --01)
--COS
(02L
-- 01) --01)]
dOz
(2)
(6) for fixed r and 0. Since we are choosing the position of the lowest sphere, the probability t h a t i t s c e n t e r l i e s b e t w e e n ~1 a n d ~1 + d 0 1 i s
where the denominator is the normalization factor. The conditional expectation of cD2 is
238
E(~2[0~.
r. o ) =
j2L 02s
c ) 2 h ( ~ 2 [01, r, O) dq) 2
= [COS b - - c o s a + Ozr. s i n b - - 0 2 , s i n a - 1
2
-- o~cos
(ozL - - 0 ~ ) c o s
b] [sin b -- sin a --
(7)
b] - z
where
a - - '#z~ - - O z ,
b -- ¢ 2 L ~ 0 1
~ z , -- m m x ( L 1 H P , L 1 / 4 1 " )
(8)
6 2 z --= m i n ( L 1 H R , 2 ~ - - 2 L I H Q )
already noted, the position of sphere 2 must be lower than that of the point R. Equation (8) expresses both of these restrictions and gives ~he u p p e r l i m i t , ~2L, f o r t h e i n t e g r a t i o n _ Fi£ure 3 compares the theoretical curve calculated from eqn. (6) with a typical result from the computer experiment. Although the data are sparse (about 130 points) and scattered, we conclude that eqn. (6) adequately expresses the probab;lity distribution of (b 2- T h e o v e r a l l m e a n is C:OS__i r
ffz=f
drf
O
f(r. 0) dO X
O
and Fig. l(b) shows that
/ ~ E(,~2Iol. r, o )g(O~[r, 0 )
L 1H1 ~ = 20 z
L X H P = L II--IQ =- A
for distance 1R > 1
(9)
dos
(10)
o
where t h e j o i n t p.d.f, f(r. 0 ) is given b y eqn. ( 1 8 ) . where A--=cos -1
[1
1
2(1--r
2)
]
Finally, we consider the orientation of the third (and highest) sphere. As ment i o n e d e a r l i e r i n t h i s s e c t i o n , t h i s s p h e r e is considered to be uniformly distributed_ The lower and upper limits are, respectively,
and
B - - r t a n O/ ~ f - l - - r z
Ca~ = / - 1 H Q = c o s - 1 C)ZL
Further details are given in AppendLx (A). The upper and lower limits of the random variable ~2 are obtained as follows. Since we placed the lowest sphere at the point i in Fig. l(b}, the centers of the other two spheres must lie on opposite sides of the line 1ZR. Othe~vise, the central sphere cannot be stable on these three spheres. The point P in Fig. l(b) deno[es the position where the distance between the points 1 and P equals the sphere diameter. Thus, the second sphere cannot lie between p o i n t s 1 a n d P . A l s o , t h e s e c o n d s p h e r e is higher than the first, so the former cannot l i e b e t w e e n 1_ a n d 1 " . E q u a t i o n (8} e x presses both of these restrictions and gives the lower limit, 02s, for the integration_ To determine the upper limit, we must take into account the third sphere. In Fig. l(b}, Q is t h e p o i n t w h i c h is o n e d i a m e t e r f r o m the point 1. To leave room for the third sphere, the angle between sphere 2 and the point 1 must be at least twice L 1HQ. As
= 2u -- (02
+
2 ( 1 ~ r 2)
I- X H Q ) = 2re
--
02
Hence, the conditional expectation 1
(11) --
c)3s
is 02
E(,%[~1, .o2, r, 0)-----~(O3L + 0 ~ ) =~--~(12) T h e o v e r a l l m e a n is ~f2/3
Cos
:f
d, f
0
0
x
~ --
-z
r
flL
r(r, O ) d O
g( llr, O)d01 0
h ( O a l O l , r, 0 ) d 0 2
(13)
o2s Splitting this multiple integral into two pieces, we use the definition of a p.d.f, for the part containing u, and eqns. (7) and (10) for that containing ~2]2, to obtain
~2 ~s
=
=
2
(14)
239
In summary, we can determLue the rositions of the three supporting spheres on the perimeter of the circular cross-section only by knowing r and 0. First, the lowest sphere is placed according to a probability t h a t is p r o p o r t i o n a l t o t h e v e r t i c a l d i s t m u c c from its highest position. Next, a second s p h e r e is p l a c e d i n s i m i l a r f a s h i o n . T h e p . d . f , f o r t h e t h i r d s p h e r e is u n i f o r m o v e r the available region._To compute the overall m e a n s , Oz, 0 2 , a n d ¢ 3 , w e n e e d t o k n o w t h e distribution of R and O. JOINT PROBABILITY DISTRIBUTION OF R AND R AND O The position of the circular cross-section can be characterized by its central point. We consider an infinitesimal area r d0 dr and revolve it about the z-axis, obtaining a ring with volume d V = 2 ~ r 2 s i n 0 d r dO
(15)
If we cut a spherical volume with radius equal to 1.0 diameters at random, then whenever the center of the resulting circular crosssection falls in the volume expressed by eqn. ( 1 5 ) , t h e dish,a n t e f r o m t h e o r i g i n t o t h e c e n ter of the cross-section falls between r and r ÷ dr and the angle of inclination falls between 0 and 0 + dO. Accordingly, the probability of f i n d i n g R a n d E) a t t h i s p o s i t i o n is p r o p o r t i o n al to dV. The perimeter of the circular cross-section through the centers of the supporting spheres is 2 ~ J ' ( 1 - - r 2 ), w h i c h r e a c h e s i t s m i n i m u m value of 2~/~/3 when r = x/T2]3). For r > ~/(2/3) there is no case of three sphere centers lying on the perimeter. Alternatively, even if r < ~/(2/3), there are some cases where the central sphere cannot be supported by three others. We must take into account the probability that a circular cross-section has three sphere centers which make the central sphere stable. Thus, we focus attention on the range o f p o s i t i o n s a v a i l a b l e f o r t h e s e c o n d s p h e r e in the presence of the other two supporting s p h e r e s . T h e m a x i m u m a n d m i n i m u m o f ¢b 2 are given by eqn. (8). The probability that 2 m a k e s the" c e n t r a l s p h e r e s t a b l e is a s s u m e d to be proportional to the available arc length p r o j e c t e d o n t h e x - y p l a n e [ 1 0 ] , i.e.
P~--E[V r~-r~
( ~ 2 L - - ~ # 2 , ) COS0 [ r, 0 ]
=~
r~ - r 2
cOs 0
(~2r-
--¢2,)g(Ozlr, O)dCz
0
(16) w h e r e CZL a n d O2s a r e g i v e n b y e q n . ( 8 ) , a n d Pc i s n o t n o r m a l i z e d a t t h i s s t a g e . O f c o u r s e , a space has been taken into account for the third sphere. We have taken the view that a circuIar cross-section is determined first, and then the positions of the three supporting spheres. Equation (16) must be calculated without any knowledge of the positions of the three supporting spheres. We have theref o r e u s e d t h e c o n d i t i o n a l p . d . f , f o r ,P 1 t o average pc over all values of • z - Further details are given in AppendLx (B). Finally, consider a factor attributed to the w a y i n w h i c h t h e T o r y p a c k i n g [ 2 ] is f o r m e d . We consider that a falling sphere C touches a f i r s t s p h e r e a n d t h e n a s e c o n d , cf. A a n d B i n Fig. la. The incoming sphere rolls always downward on these until it touches a third o n e . N o f o r c e is t a k e n i n t o a c c o u n t i n o r d e r to make the packing dense. Hence the transverse movement of the rolling sphere into the available empty space contributes to this purpose. This sphere rolls down toward the central point H. In its rest position, therefore, small distances between H and the z-axis are more likely than large. With 0 = cos-Zr, the d i s t a n c e r s i n O b e t w e e n H a n d t h e z - a x i s is a maximum, and the probability of forming a s t a b l e t e t r a h e d r o n is z e r o . A c c o r d i n g l y , w e consider eqn. (17) as a factor attributed to the method of packing. P s ~- r~/r~ - - r 2 - - r
sin 0
(17)
From eqns. (15) - (17), we finally obtain t h e j o i n t p . d . f , o f R a n d E) w h o s e d o m a i n is prescribed by (1).
r 2 p c p s sin 0 ~os-'~
f(r, O) = ~
f
r dr f
0
0
(18)
pop. sin 0 dO
where the denominator expresses the normalization factor of the p.d.f. Equation (18) is a function of r and 0 only. The averages of R and O are cOS-~ z r
-- f
dr f
0
0
and
0) d0
(19)
240 i
•
t
[
/
•
"
/
:\
p
-
t
-
I
:
L
/
/
\
/;/ / .
,
\
\.
.j:
,\ (a)
~
09
05
-
t o c o m p u t e t h e m u l t i p l e i n t e g r a l s in e q n s . (18) - (20), (5), and (10) to o b t a i n t h e norm a l i z a t i o n f a c t o r , r , O, ~ z , a n d ~-2 r e s p e c t i v e ly_ T h e v a l u e o f ¢3 is o b t a i n e d f r o m ( 1 4 ) . Table 2 shows the agreement between calculated a n d e x p e r i m e n t a l values. T h e probabili t y d e n s i t y f u n c t i o n f(r, O) also a g r e e s w i t h experiments. Figures 4(a) a n d 4(b) d e p i c t t w o t y p i c a l cases.
\"
/
J
• t-}
• /!
/ / -
\
7
CONCLUSION
~%1
I'I, i': ,
\
\
'-,\
t
e ~ t~EG I
(b)
Fig. 4(a). Joint p:ohab~-lity distribution o f r and 0 for 0 = 20 - 25 (d-'g). LkR = 0.05; ~0 = 5 (deg). x , o for 8 0 0 0 a n d 9000 s p h e r e packings respectively. Solid c u r v e is theoretic-,-I for 0 = 22.5 (deg). (b) Joint p r o b ability distributio.~ o f r a n d 0 for r = 0.5 - 0 . 5 5 (--). ~ R = 0.05; ~-%0 = 5 (deg). x , e for S 0 0 0 a n d 9 0 0 0 s p h e r e packings respectively. Solid cffrve is theoretical for r = 0 . 5 2 5 (--)cos-- z r
f 0
dr f
or(r,o)dO
0
T o o b t a i n f(r, 0 ), w e f i r s t c a l c u l a t e P c f r o m e q n . ( 1 6 ) u s i n g ~2T- a n d ~2_, f r o m (8) a n d g ( ~ z Jr, 0 ) f r o m (3). T h e s p e c i f i c a t i o n o f ~2~ a n d 02~ i n t e r m s o f ¢ x , r, a n d 0 is g i v e n in A p p e n d i x ( A ) , w h i l e t h e c a l c u l a t i o n o f p~ is s h o w n in A p p e n d i x (B). U s i n g t h i s r e s u l t a n d e q n . ( 1 7 ) f o r p~, w e u s e t h e R o m b e r g m e t h o d
We h a v e i n v e s t i g a t e d a r a n d o m p a c k i n g structure constructed by a computer simulation of the slow settling of equal spheres into a randomly packed bed. In particular, we have examined the detailed configuration of the tetrahedron formed when one sphere settles o n three others. T h e centers o f the t h r e e s u p p o r t i n g s p h e r e s lie o n a s p h e r i c a l envelope about the central sphere and determine a circular cross-section of that envelope. T h e c e n t e r o f t h e c e n t r a l s p h e r e is t a k e n as t h e origin a n d t h e p o s i t i o n o f t h e cross-sect i o n is g i v e n b y t h e d i s t a n c e r o f its c e n t e r a n d its a n g l e o f i n c l i n a t i o n 0. T h e j o i n t p . d . f . f(r, 0 ), d e f i n e d b y e q n . ( 1 8 ) , is o b t a i n e d f r o m geometrical conditions (15) and (16) and the packing criterion (17). After the cross-section is f i x e d , p o s i t i o n s o f t h e t h r e e s u p p o r t i n g s p h e r e s a r e d e t e r m i n e d o n its p e r i m e t e r . F i r s t , t h e l o w e s t o f t h e t h r e e is p o s i t i o n e d a c c o r d i n g t o a p r o b a b i l i t y w h i c h is p r o p o r t i o n a l t o t h e v e r t i c a l d i s t a n c e f r o m its h i g h e s t p o s s i b l e p o i n t . N e x t , a s e c o n d s p h e r e is s i m i l a r l y p l a c e d . T h e d i s t r i b u t i o n o f t h e t h i r d s p h e r e is
241 uniform o v e r t h e available region. Based u p o n
these ideas, the detailed geometrical properties calculated for the model tetrahedron agree with those obtained from computer experiments. The model should be tested further to see if it yields the same anisotropy as T o r y ' s p a c k i n g [ 2 , 1 4 ] . This paper has dealt with only one random p a c k i n g , i.e_ a r a n d o m p a c k i n g c o n s t r u c t e d by a computer simulation of the slow settling of equal spheres into a randomly packed bed [2]. The bulk-mean volume fraction of the particles-is 0.582, which can be considered to be a random loose packing. A similar statistical geometrical analysis should be tested for other packings. The synthesis of the assembly of tetrahedra to form a random packing should also be considered.
Subscript 1, 2, 3 t h r e e s p h e r e c e n t e r s in o r d e r o f l o w e s t z coordinate L upper limit s lower limit Superscrip t average value APPENDIX (A) Equation
(8)
Czs = L 1 H P f o r ¢ l
<
=L 1HI* for¢l
C
> C
¢2r =2~--2LIHQ
forr>~/2/2, and¢x
= / IHR A C I ~ I O W L E D GE~,%IE N T S W e a c k n o w l e d g e the financial support of the National Research Council of Canada.
for all cases other than the
above, where C ~ sin - x
1 2~/1 -- r z
LIST O F SYMBOLS
E=
E(¢tlr, O)
conditionM expectation of
E(~a[¢~, r, 0)
conditional expectation of &2 j o i n t p r o b a b i l i t y d e n s i t y distribution function of r and 0, eqn. (18) conditional probability d e n s i t y f u n c t i o n o f Cz conditional probability d e n s i t y f u n c t i o n o f ¢2 defined by eqn. (16) defined by eqn. (17) distance from the origin to the center of circular crosss e c t i o n , in u n i t s o f t h e s p h e r e d ; ~ r n e t e r ( s e e Fig_ l a ) r a n d o m v a r i a b l e , i.e. e v e n t of r a n g l e s d e f i n e d in Fig. l ( b ) r a n d o m v a r i a b l e s , i.e. e v e n t s
f(r. O)
g(o~ l", o) h(¢21¢1, r. o) Pc Ps r
R ~I,2.3
~I.2.3
of
8 0
~1.2.3
a n g l e o f i n c l i n a t i o n o f circ u l a r c r o s s - s e c t i o n ( s e e Fig. l a ) r a n d o m v a r i a b l e , i.e. e v e n t o f 8
O >E
tan-l[X/~--rZ
1 l 2 ( 1 Z_r2 )
D = sin-l[~-~t--sin
A -- c o s - 1
[1
A +~B 2 --cos2A}]
1 ] 2 ( 1 - - r 2)
B = r tan Olx/~--r (B) Equation
1t]
2
(16)
Pc = I - - H II
1 fir. ~
--
Ob
~2s( c ° s ~1 - - B ) d P l
2r = A f o r c o s - 1 ~ - - ~ < 0 < cos-Xr. Le. ¢~,_ < C = [AsinC--ABC+2cos~xL
--2cosC+
2 ~ l L Sin ¢ I L - - 2 C s i n C - - B ( ~ L 2r f o r 0 < c o s - 1 ~--ff, i.e. ~ I L > C
-- C2)]/Q
242 where Q -= s i n ¢i"- - - B O L L /IL
1 I ---------
¢z,- ( c o s ¢ i
d¢1
-- B)
Q'o = [ 2 ( = -- A ) ( s i n D - - B D )
F(D) ]/Q
+ F ( ¢ I L ) --
for r > x/2/2
= F(¢x,)/Q
for all cases
and
0 > E
other
than
the
above,
where
F ( x ) -- _f l_ 1 H R
d¢i
- (cos 0 1 - - B )
o
a n d L 1/!R is given b y eqn. (9). Integrating b y parts w h e n
necessary, we
obtain 3
= cos
F(x)
2(sin
x
-- 1 + 2x
-- --Bx
(.1--c~cosx).
9_2 1 -- B 2
z
+
2
(cosx--B x -- Bx) tan -I ~ s~m-~
+l--B21n
"(
sin x
)
--
1 --~, x
y dy
w h ~ r e ~ ---- 2 J ~ [ ( 1 + B 2 ) REFERENCES 1 % V . M . V i s s c h e r a n d ~I. B o l s t e r l i , R a n d o m p a c k i n g of equal and unequal spheres in two and three dimensions, Nature (London), 2 3 9 (197°--) 5 0 4 - 507_
2 E. M. Tory, B. H. Church, M. K. Tam and M. Ratner, Simulated random packing of equal spheres, Can. J. Chem. Eng., 5I (1973) 484 - 493. 3 C . H_ B a n n e t t , S e r i a l l y d e p o s i t e d a m o r p h o u s a g gregates of hard spheres, J. Appl. Phys., 43 (1972) 2727 - 2734. 4 D.J. Adams and J. Matheson, Computation of d e n s e r a n d o m p a c k i n g s c,f h a r d s p h e r e s , J . C h e m . Phys., 56 (1972) 1989 - 1994. 5 J.D. Bernal, The structure of liquids, Proc. R. Soc., A280 (1964) 299 - 322. 6 D . P . H a u g h e y a n d G . S. G . B e v e r i d g e , S t r u c t u r a l properties of packed beds -- A review, Can. J. Chem." E n g . , 4 7 (1969) 1 3 0 - 1 4 0 . 7 (3. D . S c o t t a n d D . M . K i l g o u r , T h e d e n s i t y o f random close packing of spheres, Br. J. Appl. Phys. S e r . 2, 2 ( 1 9 6 9 ) 8 6 3 - 8 6 6 . 8 J . D . B e r n a l , I. A . C h e r r y , J . L . F i n n e y a n d K . 1%. Knight, An optical machine for measuring sphere coordinates in random packings, J. Phys. E, 3 (1970) 388 - 390. 9 T . W . S . P a n g , U . M . F r a n k l i n a n d %V. A . M i l l e r , On the siinulation of liquid structure by random close packing, Mater. Sci. Eng., 12 (1973) 167 172. 10 K. Gotoh and J. L. Finney, Statistical geometrical approach to random packing density of equal spheres, Nature (London), 252 (1974) 202 - 205. 11 H. Iwata and T. Homma, Distribution of coordination numbers in random packing of homogeneous spheres, Powder Technol., 10 (1974) 79 - 83. 12 K. K. Pillai, Voidage variation at the wall of a packed bed of spheres, Chem. Eng_ Sei., 32 (1977) 59 - 61. 13 J. L. Finney, Modelling the structures of amorphous metals and alloys, Nature (London), 266 (1977) 309 - 314. 1 4 E . M . T o r y , N . A . C o c h r a n e a n d S . R . %Vacldell, Anisotropy in simulated random packing of equal spheres, Nature (London), 220 (1968) 1023 - 1024.