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Derivation of pacticle size distributions from measurements on plane sections through particle beds. Effect of regularity of particle packing
P o w d e r Technology, 11 (1975) 1--10 © Elsevier Sequoia S.A., Lausanne - - Printed in The Netherlands
Derivation of particle Size Distributions from Measttrements on Plane Sections through Particle Beds. Effect of Regularity of Particle Pacbing
R.I~L ECKHOFF and G. ENSTAD Powder Research Laboratory. Chr. Michelsen Institute, Dept. o f A p p l i e d Physics, N - 5 0 0 0 Bergen (Norway)
(Received January 8, 1974)
SUMMARY W h e n t h e size d i s t r i b u t i o n o f p a r t i c l e s e m bedded in an opaque, continuous solid phase is r e q u i r e d , t h e g e n e r a l a p p r o a c h is t o d e d u c e t h e d i s t r i b u t i o n f r o m t h e size d i s t r i b u t i o n o f particle cross-sections in a plane cut through the particle bed. When the particles are approximately spherical, this deduction can be performed by tasking the assumption that the distances from the plane of cut to the particle centers are rectangularly distributed. The validity of this aa~umption does not, however, appear to have been investigated in previously published work, and in the present contribution the assumption has been considered more
closely. The di~ibution of the distances between t h e s p h e r e c e n t e r s a~nd a r a n d o m p l a n e h a s been investigated both theoretically and experi m e n t a l l y , s t a r t i n g w i t h regll|Ar p a c k i n g s o f monosized spheres, and extending the treatment to monosized spheres packed at random a n d t-ma~ly t o p a c k i n g s o f p o l y s i z e d s p h e r e s . The distribution of partic4e center to cutring plane distances will approach a rectangul a r d i ~ i b u t i o n w i t h i n c r e a s i n g s a m p l e size eve~. f o r bt~ictly r e g u l a r p a c k i n g s . H o w e v e r , f o r f m i ~ s a m p l e s , t h e p a c k i n g r e g u l a r i t y m a y si~nifica~atly affect the extent to which such a di~ibut i o n is r e a l i z e d . 1. INTRODUCTION T h e p r o b l e m o f d e t e r m l n l n g t h e s i z e distribution of particles embedded in a contintto~ts solid phase from which they cannot easily be
l i b e r a t e d arises in a w i d e r a n g e o f f i e l d s o f r e search, product development and product control. In medicine and biology the particles will b e cells in t i s s u e , in c o n c r e t e a n d p o l y e s t e r t h e y will b e gas b u b b l e s a n d i n r o c k m i n e r a l grains. T h e s o l u t i o n w h i c h is g e n e r a l l y a d o p t e d w h e n o n e is f a c e d w i t h t h i s p r o b l e m is t o d e duce the size distribution of the three-r]imensional bed of particles from the size distribution of particle cross-sections produced by a random plane through the specimen, or from the distribution of chord intercepts produced by a random straight line through the specimen. T h e d e d u c t i o n is p e r f o r m e d b y m e a n s o f a transformation theory involving two basic a s s u m p t i o n s . F i r s t l y i t is a s s u m e d t h a t t h e particles are spheres and secondly that the distances from the sphere centers to a random plane through the particle bed are rectangularly distributed. T h e p r e s e n t c o n t r i b u t i o n is c o n c e r n e d w i t h the last assumption, the validity of which does n o t a p p e a r t o h a v e b e e n i n v e s t i g a t e d in p r e viously published work. 2. REVIEW OF THE L I T E R A T U R E A l t h o u g h t h e r e v i e w t o b e g i v e n in +..he f o l l o w i n g is m a i n l y i n t e n d e d t o s h o w t h e s t a t e o f t h e a r t as f a r as t h e a c t u a l p r o b l e m is c o n cerned, it also illustrates how communication b e t w e e n d i f - r e r e n t b r a n c h e s o f s c i e n c e in which an identical powder problem has been treated has up to recently been lacking to a sLr;lring e x t e n t . T h e r e v i e w t h e r e f o r e a l s o demonstrates the justification of powder tech-
2 n o l o g y as a s p e c i f i c f i e l d o f r e s e a r c h a n d c o m munication. T h e f i r s t c o n t r i b u t i o n t o b e c o n s i d e r e d is t h e o n e b y Delesse [ 1 ] f r o m 1 8 4 8 , i n w h i c h the problem of determining the content of a s p e c i f i c m i n e r a l c o m p o n e n t in r o c k s p e c i m e n s was c o n s i d e r e d . S t a r t i n g w i t h a p l a n e , p o l i s h e d s u r f a c e o f a r e a F o f t h e a c t u a l m a t e r i a l . Delesse was a b l e t o s h o w t h e o r e t i c a l l y t h a t
f~vv_ F
V
w h e r e f is t h e a r e a w i t h i n F o c c u p i e d b y t h e desired mineral component, V the volume of a r e p r e s e n t a t i v e p i e c e o f r o c k a n d v t h e volu m e o f the desired mineral within t h a t piece. Delesse's m e t h o d o f d e t e r m i n i n g v o l u m e r a t i o s was l a t e r m o d i f i e d b y R o s i w a l [ 2 ] , w h o was a b l e t o s h o w t h a t L
V
w h e r e L is t h e l e n g t h o f a s t r a i g h t line t h r o u g h the specimen and l the sum of the intercepts a l o n g L c r o s s i n g areas o c c u p i e d b y t h e d e s i r e d mineral. T h e first steps t o w a r d s a t r a n s f o r m a t i o n t h e o r y f o r c a l c u l a t i n g size d i s t r i b u t i o n s w e r e c a r r i e d o u t b y H a g e r m a n n [ 3 ] a n d Wicksell [ 4 ] in 1 9 2 4 . H a g e r m a n n c o n s i d e r e d t h e g e n e ral p r o b l e m o f r a n d o m l y d i s t r i b u t e d m o n o sized s p h e r e s , w h i l s t Wicksell w a s c o n c e r n e d w i t h t h e s p e c i f i c p r o b l e m o f c o m p u t i n g size d i s t r i b u t i o n s o f s t a r clusters. A y e a r l a t e r Wicksell [ 5 ] c a r r i e d his w o r k f u r t h e r , p r e s e n ting a general t h e o r y for the calculation o f t h e size d i s t r i b u t i o n s o f e m b e d d e d p a r t i c l e s o n t h e basis o f t h e size d i s t r i b u t i o n o f p a r t i c l e c r o s s - s e c t i o n s in a r a n d o m p l a n e t h r o u g h t h e particle bed. Starting with a contribution by Tarnmann and Crone [ 6 ] , Scheil [7] developed a t h e o r y f o r c a l c u l a t i n g size d i s t r i b u t i o n s o n t h e basis o f size d i s t r i b u t i o n s o f p a r t i c l e c u t s i n : a p l a n e s e c t i o n , w h i c h w a s p u b l i s h e d in 1 9 3 1 . S c h e i l ' s m e t h o d was similar t o t h a t d e v e l o p e d b y Wicksell [ 5 ] , b u t a p p a r e n t l y S c h e ~ d i d n o t k n o w o f Wicksell's work. A n i m p r o v e d version o f 9eheil's t h e o r y was published b y Scheil and W u r s t [ 8 ] in 1 9 3 6 , a n d f u r t h e r i m p r o v e m e n t s b y S c h e i l a l o n e in 1 9 3 5 [ 9 ] a n d 1 9 3 7 [ 1 0 ] , . b u t t h e r e s u l t i n g m e t h o d still s u f f e r e d f r o m c o n s i d e r a b l e d i s a d v a n t a g e s as f a r as p r a c t i c a l application was concerned.
In 1934 Schwartz [11] had published an i m p r o v e d version o f Schefl's first m e t h o d , a n d in 1 9 5 8 S a l t y k o v [ 1 2 ] p r e s e n t e d a c o n siderably more efficient method starting with Schwartz's work. The important new contrib u t i o n in S a l t y k o v ' s m e t h o d w a s a g e n e r a l l y applicable table of coefficients, which reduced the required amount of computation for performing the desired transformation quite considerably. In the beginning of the nineteen-sixties Oel [ 1 3 ] p u b l i s h e d a m e t h o d w h i c h is basically identical with Scheil's first method from 1 9 3 1 . A c c o r d i n g t o t ~ e r e f e r e n c e list, h o w e v e r , Oel w a s n e i t h e r a w a r e o f W i c k s e l l ' s c o n t r i b u tion n o r o f Scheil's. A s l a t e as 1 9 6 7 M e i s n e r [ 1 4 ] p r e s e n t e d a n elegant derivation, based on Wicksell's and Scheil's work, of a method very similar to that p r o d u c e d b y S a l t y k o v t e n y e a r s earlier. Evidently Meisner neither knew the work of Schwartz nor that of Saltykov. The same year, 1967, Rose [15] published a m e t h o d d e v e l o p e d o n t h e basis o f t h e c o n t r i b u t i o n s b y H a g e r m a n n , Wieksell, I Q m m b e i n and Pettijohn. Again a m e t h o d very similar t o that of Saltykov, involving tables of generally applicable coefficients, was the result. Acc o r d i n g t o t h e r e f e r e n c e list, t h e c o n t r i b u t i o n s by Schwartz, Saltykov and Meisner were not known to Rose. D u r i n g t h e y e a r s several i n d e p e n d e n t w o r k e r s have published derivations of exact solutions t o t h e b a s i c i n t e g r a l e q u a t i o n s i n v o l v e d in t h e actual problem. The contributions by Whi~er and W a t s o n [ 1 6 J , D r a p a l a n d H o r a l e k [ 1 7 ] and Kendall and Moran [18] should be ment i o n e d . I t is r e g r e t t a b l e , h o w e v e r , t h a t t h e e x a c t s o l u t i o n is n o t a p p l i c a b l e in p r a c t i c e . Several a u t h o r s h a v e p u b l i s h e d l i t e r a t u r e surveys covering the actual problem, and the very comprehensive survey by Exner [19] from 1972 ought to be mentioned here. A l t h o u g h t h e m e t h o d s p r e s e n t e d b y t h e various workers referred t o so far differ s o m e w h a t i n t h e details, t h e b a s i c f e a t u r e s are i d e n t i c a l . All t h e c o n t r i b u t i o n s , as f a r as c a l c u l a t i o n o f size d i s t r i b u t i o n s is c o n c e r n e d , d e a l w i t h s p h e r i c a l p a r t i c l e s . F u r t h e r m o r e , a n d t h i s is t h e m a i n c o n c e r n o f t h i s p a p e r , all t h e c o n t r i b u t i o n s i n v o l v e t h e a s s u m p t i o n t h a t t h e distances of particle centers from a random plane o f intersection are rectanguIarly distributed. So far the survey has been Hmited to methods
o f c a l c , , l s t i n g t r u e size d i s t r i b u t i o n s o n t h e basis o f size d i s t r i b u t i o n s o f p a r t i c l e c r o s s s e c t i o n s in a r a n d o m p l a n e t h r o u g h a b e d o f particles. In o r d e r t o m a k e t h e p i c t u r e c o m plete, it o u g h t t o be m e n t i o n e d t h a t several w o r k e r s have described m e t h o d s f o r calculat i n g t h e t r u e size d i s t r i b u t i o n f r o m t h e distribution of chord intercepts produced by a s t r a i g h t line t h r o u g h t h e p a r t i c l e b e d . T h e c o n t r i b u t i o n s b y P o w e r s [ 2 0 ] , Willis [ 2 1 ] , L o r d a n d Willis [ 2 2 ] , S p e c t o r [ 2 3 ] , F u l l m a n [ 2 4 ] , Miinzer and SchneiderhDhn [25], Horikwa [26], Cahn and Fullman [27] and Bockstiegel [ 2 8 ] , all o f w h i c h h a v e t h e classical c o n t r i b u t i o n o f R o s i w a l [ 2 ] as t h e s t a r t i n g p o i n t , s h o u l d be m e n t i o n e d s p e c i f i c a l l y . C o n t r a r y t o t h e m e t h o d b a s e d o n t h e diameters of particIe cross-sections, the chord approach for spherical particles has a very s i m ~ l e , a n a l y t i c a l s o l u t i o n , w h i c h is r e a d i l y applicable in practice. T h e s~lution does, however, rest on the assumption that the p a r t i c l e c e n t e r s a r e r a n d o m l y d i s t r i b u t e d in s p a c e . I n f a c t t h i s a s s u m p t i o n is i d e n t i c a l w i t h t h e o n e s t a t e d in t h e h e a d i n g o f t h i s p a p e r , a n d h e n c e t h e c o n c l u s i o n s f r o m t h e p r e s e n t inv e s t i g a t i o n will b e d i r e c t l y a p p l i c a b l e also t o the chord methods. T h e a c c o u n t g i v e n in t h e f o l l o w i n g is a condensed version of a comprehensive report [ 2 9 ] in w h i c h t h e t h e o r e t i c a l a n d e x p e r i m e n t a l i n v e s t i g a t i o n s a r e d e s c r i b e d in d e t a i l .
3. THEORY
t o t h e t o t a l f r e q u e n c y N , w h i l s t H is t h e m e a n o f H i f o r all classes. I n case o f d i s t r i b u t i o n s which differ considerably from the rectangular, a ~ will b e large, a n d f o r d i s t r i b u t i o n s w h i c h are n e a r l y r e c t a n g u l a r , t h i s p a r a m e t e r ~d_ll a t t a i n s m a l l values. F o r p e r f e c t r e c t a n g u l a r d i s t r i b u t i o n s , it will b e z e r o . F o r a r a n d o m sample extracted from a rectangular distribut i o n , o n e will h a v e a~ -* 0
(2)
N-+oo
T h i s in t u r n m e a n s t h a t i f i t c a n b e s h o w n t h a t e q n . (2) h o l d s , t h e dista-:bution in q u e s t i o n must be rectangular. It must be pointed out, however, that o~ for a r a n d o m s a m p l e f r o m a given p o p u l a t i o n will vary statistically a l t h o u g h K and N are k e p t c o n s t a n t . F o r t h i s r e a s o n 0 2 as s u c h is n o t suitable for characterizing the extent to which a given d i s t r i b u t i o n d i f f e r s f r o m t h e r e c t a n g u lar. I t w a s t h e r e f o r e d e c i d e d t o a d o p t t h e p r o b a b i l i t y PK (N) f o r a 2 b e i n g s m a l l e r t h a n a c h o s e n p o s i t i v e c o n s t a n t g , as t h e d e s i r e d criterion. P r o v i d e d it c a n b e s h o w n that PK(IV)
-~ I , N
-+ ~
n o matter h o w small is the value of g, the distribution m u s t b e rectangular. A s will b e c o m e apparent in the following~ it w a s in the theoretical approach decided to consider the case of a continuous distribution, since this facilitated the theoretical work. It is, h o w e v e r , p o s s i b l e t o s h o w [ 2 9 ] t h a t
3.1 Criterion o f degree o f rectangular distribu tion
a~ <
Since the object of the present work was to investigate the extent to which the distances f r o m s p h e r e c e n t e r s t o a r a n d o m p l a n e o f int e r s e c t i o n w e r e r e c t a n g u l a r l y distributed, a convenient quantitative criterion for characterizing the extent to which the distribution was rectangular had to be found. As a starting point, it was decided to consider the parameter
w h e r e o~ is t h e v a r i a n c e o f t h e c o n t i n u o u s frequency distribution, and
K
a~
=
X(H, -- ~)z =~Ar~,--~
(3)
A-
a~
1 A = ~ - (total width of distribution)
(4)
(5)
will a l w a y s h o l d . H e n c e , i f (3) c a n b e s h o w n t o h o l d f o r t h e c o n t i n u o u s case f o r ~-q~ smaller t h a n a n y p o s i t i v e v a l u e o f g , i t also h o l d s for the discrete distribution with frequency variance a~.
(I)
i
which denotes the variance of a frequency d i s t r i b u t i o n h a v i n g K e q u a l l y w i d e classes: H i is t h e r a t i o o f t h e f r e q u e n c y N i i n t h e i t h class
3. 2 I~egular, cubical p a c k i n g s o f m o n o s i z e d spheres One would intuitively anticipate that the assumption to be investigated would be faced w i t h its g r e a t e s t c h a l l e n g e w h e n a p p l i e d t o
4 Since the constant d does not influence t h e d i s t r i b u t i o n o f distmmnes o f s p h e r e c e n t e r s from the plane, the considerationscan be l i m i t e d t o t h e special case w h e r e d ecp!Alg z e r o , i.e. t o a p l a n e t h r o u g h t h e o r i g i n , i n t h e c a s e of which eqn. (9)simplifies to
It, re.n}
,
h z , , , ffi a l + b m
, Fig. I . ~ regular, cubical packing of monosized spheres in a Ca~'tcsian coordinate system. p e r f e c t l y r e g u l a r packings o f s p h e r e s . H e n c e it was d e c i d e d t o c o n s i d e r a s i m p l e c u b i c a l p a c k i n g o f m o n o s i z e d spheres, w h i c h is ill u s t r a t e d in Fig. 1. I f o n e s p h e r e d i a m e t e r is t a k e n as t h e u n i t a l o n g t h e c o o r d i n a t e axis i n Fig. 1, t h e r a d i u s v e c t o r f r o m t h e origin t o a s p h e r e c e n t e r w i t h c o o r d i n a t e s l, m a n d n will b e t h e ~,,,,, "
=
,;÷mr
(6)
+ nk
w h e r e i, j a n d ~ a r e t h e u n i t v e c t o : s o f t h e c o o r d i n a t e s y s t e m , a n d l, m a n d n a r e integers. T h e g e n e r a l e q u a t i o n f o r a p l a n e is (7)
ax + by + cz + d = O
and the unit vector ; = .T+
bf+
(8)
will b e p e r p e n d i c u l a r t o t h e p l a n e , w h i l s t d will b e t h e d i s t a n c e f r o m t h e o r i g i n t o t h e plane. T h e d i s t a n c e f r o m t h e s p h e r e c e n t e r ~dth c o o r d i n a t e r , l, m a n d n t o t h e p l a n e will be ~lrrtn
~
~'[rl~ n
*
7 + d = c: + bm + cn + d
as can be deduced
i /~ \
(9)
f r o m Fig. 2.
"~ . ~ . . ; " l a n e
(10)
+ cn
The position of a plane of intersection with r e s p e c t t o t h e s y s t e m i l i n s t r a t e d i n F i g . I will then be fully determined by the direction of th~ u n i t v e c t o r ~, L ~ b y t h e c o n s t a n t s a, b a n 0 c. I f n o w b'~a d e n o t e s t h e t o t a l s o l i d a n g l e within which ~ c a n n o t be located if t h e con° ditio~ c¢} < g
(11)
h a s t o b e satisfied, t h e n Px = 1
~2t 4~
(12)
expresses t h e probability t h a t a r a n d o m plane through the cubical packing of monosized s p h e r e s will s a t i s f y c o n d i t i o n ( 1 1 ) . I t is, h o w ever, n e c e s s a r y t o f i n d a n e x p r e s s i o n f o r £~a before eqn. (12) can be o f a n y use. One characteristic feature of the process of intersecting a regular packing o f spheres w i t h a r a n d o m p l a n e is t h a t t h e r e m ~ c e r t a i n s p e c i f i c d i r e c t i o n s o f ~ f o r w h i c h i t is o n l y possible t o obtain a limited n u m b e r o f sphere c e n t e r t o p l a n e distances. C l m ~ o n s of particles having these particulax distances will t~len o c c u r p e r i o d i c a l l y a l o n g t h e p l a n e o f i n t e r s e c t i o n . S u c h d i r e c t i o n s o f ~ will i n the following be denoted periodic directions o r p e ~ o d i c i t i e s . T h e z e w 0 1 b e ~a~ i n ~ t e numb e r o f e v e n l y d i s t r i b u t e d p~-z~odie d i r e c t i o n s . A n y d i r e c t i o n of. ~ wi~ t h e r e f o r e b e l o c a t e d i n t h e v i c i n i t y o f s o m e p e r i o d i c "dizection. I t is p o s s i b l e t o s h 0 w [ 2 9 ] t h a t f o r a n y p e r i o d i c d i r e c t i o n o f P, t h e p a r a m e t e r s a. b and ein eqn. (10) muat equal ,,
=
±L
b --
±IW a n d
e
=
(13)
where L, M a n d N are intege; n u m b e r s with no common factor, and where Ar = ~/L 2 + ~
Fig. 2. A regular, eubic~d packing of mono•zed spheres cut by a randomly oriented plane.
+/V ~
(14)
is t w i c e t h e t o t a l n ~ of ~;f~erent i~'~de center t 0 p l a n e distances for t h a t c u m b i n a t i o n o f L , M a n d N , i . e . w h i c h o c c u r o n b o t h aides of the plane for that particularpefi~.
In the following AL will be denoted the pe~od. I t is f u r t h e r p o s s i b I e t o s h o w [ 2 9 ] t h a t i n t h e c a s e A L ;~ A ~ m , w h e r e
Az.
Iz
(15)
~V~g
m.,
~ o ~ , i G o n (11) ~ always be ~ , ~gst f o r A L < A r m this c o n d i t i o n will in most n o t be fnlfmed. A p e r i o d i c i t y f o r w h i c h
Y
c o n d i t i o n ( 1 1 ) is n o t s a t i s f i e d w i l l b e su~oun.d e d b y a m,~rl s o l i d a n g l e A&'2t(A r ) W i t h ~ w h i c h ( 1 1 ) is n o t sati~-f~,d e i t h e r . I n or¢!~~v t o f i n d thi~ sOHd a n g l e ~ b ' ~ ( A L ) , an ~ f¢¢ o~ in the vicinity of the peciodlcity should be found. This, however, would be extremely difficult. For that reason the ~dis~bufion of paxticle center to p]mle distances was appro-~im~ by a continuous dist~fl~ion. The approximation holds providodA .~Az, whereA i s t h e t o t a l a r e a o f t h e p l a n e o f i n t e r s e c t i o n c o n s i d e r e d , w h i c h in turn will be equivalent to the total number of i n t a c s e c t ~ spheres. F o r t h e c o n t i n u o u s distrFDution, t h e v a r i a n c e o ~ ( A 0 ) w a s f o u n d [ 2 9 ]
F'¢g. 3 . H l ~ t r a t l o n o f t h e s o l i d ~mg]e ~ b ' ~ ( A L ) r o u n d i n g a p e r i o d i c d i r e c t i o n ( L , M. N ) .
toeq~
the symbol [] denoting "the integer part of". The pattern of intersection between the [
I+1
where Im
=
~
F g ( g ) = 2 ( 1 + K g ) ~ f=o
(2r + z) (19)
and
,.
(V:÷
(20)
s o l i d angle Ag'~'g(A L ) a n d a spherical surface
--1)
(AL~/, ° - - x")
sup
(16)
where Z~0 is the angle between the peciodic d i r e c t i o n a n d P. • is t h e i n t e g e r p a r t o f t h e
with the center in the origin will be a set of c o n c e n t r i c r i n g s as i l l u s t x a t e d i n F i g . 3. I n o r d e r t o e s t i m a t e the t o t a l s o l i d angle
w i t h i n which condition ( 1 1 ) is n o t satisfied, eqn. (18) was summed for the periodicities wif;h A L "~ Al.m.
ALm
E q u a t i o n (4) sires T h e summation is tO be performed f o r all
~(~0) > ~- - 2K~
c o m b i n a t i o n s o f L, M a n d N h a v i n g n o c o m m o n f a c t o r and f o r A L <; ALto-
H e n c e , t h e solid m~gle Ab--Z'~(AL) w i t h i n which
I n order t o perform this summation, the f o l l o w i n g approximate m e t h o d was adopted.
w ~ l be greater than the solid angle ~ Z z ( A ~ ) w i t h i n which condition (ZZ) is n o t s a t i r e d . B y substituting ~ ( , e ~ ) ~,, eqn. ( 16) b y 2 ~ g , a q u a d r a ~ e q ~ o n r e s ~ , yielding tWO v a I u e s o f ~B(/}, f r o m w h i c h t h e s o l i d
angle Z ~ ( A ~ )
can be d e d u c e d . I t c a n be
~ o w n [29] t~st
Consider a spherical shell with trait thickness, w i t h t h e c e n t e r in origin a n d w i t h r a d i u s RL e ~ u a I t o t h e i n t e g e r p s x t o f A L ~ ! . T h i s shell wl]] c o n t a l n 4 7 r R ~ p a r t l c l e c e n t ~ z s . I f n o w t h e t ~ - t i o n po o f t h e s e p a r t i c l e c e n t e r s h a v i n g c o o r d i n a t e s (LJHrJV,) w i t h n o c o m m o n f a c t o r is c o n s i d e r e d c o n s t a n t , t h e n u m b e r o f p a r t i c l e c e n t a r s w i t h i n t h e s p h e r i c a l s h e ] / w h i c h re-
p r e s e n t p e ~ o d i c i t i e s v n ~ b e 41¢R~pv. T h e mag-
"%~,;(AL) ffi A I A
(18)
nitude of the periods AL within a shell are now considered constant and equal to RL - Equation
(21) then takes the approximate form
g
~
0.6
w h i c h in c o m b i n a t i o n w i t h e q n . ( 1 8 ) gives
5 o.~ 0.2
47rGK (g) A
(22)
GK (g) = FK ( g ) A n m P~,
(23)
0.0 ~'0 60 IODIAO~I
where
(24) w i g a p p l y . T h i s means t h a t
150
100 200 300
150 300 4SO
~
~ 1
0.2 0.0
GK (g) N
2 0 60 100 140
1.0
15
0.6 0.4
0.0
Since the diameter of the monosized spheres h a s b e e n t a k e n as t h e u n i t o f l e n g t h in t h e treatment above, the numerical value of A will f o r t h e r e g u l a r c u b i c a l p a c k i n g c o n s i d e r e d be approximately equal to the number N of particle cross-sections within A. For this reason, and using eqn. (22), one has GK (g)
o.g
10 0.4
S
0.2
PK = 1 - - ~ - ~ > 1 - - 4--~
(25)
G K ( g ) g i v e n in e q n . ( 2 3 ) is f u l l y d e t e r m i n e d w h e n K a n d g a r e given. H e n c e f o r a n y f i n i t e p o s i t i v e v a l u e s o f K a n d g, e q n . ( 2 5 ) m e a n s that PK -~ 1 N->~
300/.SO
IOD2OD3DO
1.0
The error involved by adopting the approxim a t e s u m m a t i o n will d e c r e a s e w i t h i n c r e a s i n g value of ALto. M a i n l y b e c a u s e o f e q n . (4), b u t a l s o b e c a u s e condition (11) may be satisfied for some comb i n a t i o n s Of A L a n d K e v e n w h e n A L < A L t o , the relation
PK > 1
0.005
4~R2pvA~2"g(Rr.)
RL=I
~2~ ~
0.01
~I K 7.0 P.~.
ALm
~2"~
0.05
(26)
w h i c h in t u r n m e a n s t h a t t h e d i s t r i b u t i o n o f particle center to plane distances must be rectangular even when intersecting regular cubical packings. T h e s o l i d l i n e in Fig. 4 s h o w s (1 - - G K ( g ) / N ) as a f u n c t i o n o f N f o r v a r i o u s v a l u e s o f K a n d g . Although the analysis presented above was performed for regular cubical packings, it can relatively easily be shown [29] that the general f o r m o f e q n . ( 2 5 ) will a p p l y t o all regular packings, each particular packing geometry having a specific form of GK(g).
. . . . . 20 60 100 140
S
100 200 300
150 3O9 450
F i g . 4 . PK(N) s h o w n g r a p h i c a l l y f o r d i f f e r e n t c o m b i n a tions of K and g. The theoretical values given by eqn. (25);--X-- results from computer simulation of cubical packings cut by random planes; o experiment with random numbers, simul~/ting perfect random packings.
Hence, the general and important conclusion can be drawn that cutting regular packings by r a n d o m p l a n e s will a l w a y s y i e l d r e c t a n g u l a r d i s t r i b u t i o n s o f p a r t i c l e c e n t e r t o p l a n e distances, provided the sampled number N of p a r t i c l e c r o s s - s e c t i o n s is s u f f i c i e n t l y large. 3. 3 E x t e n s i o n o f c o n c l u s i o n t o irregular p a c k ings a n d p o l y s i z e d s p h e r e s i z e d i s t r i b u t i o n s As pointed out in Section 3.2, one would expect that the assumption to be tested would be faced with its greatest challenge in the case of perfect regular packing geometries. It was possible to show that even for a perfect regular packing of monosized spheres, the distribution of distances from particle centers to the plane of intersection would approach the rectangular form with increasing ~mple size. H e n c e , i t a p p e a r s s e n s i b l e t o e x p e c t t h a t this would be so for any practical packing geometry. In fact, one would even anticipate t h a t t h e s a m p l e size r e q u i r e d f o r o b t a i n i n g a rect~nguler distribution with a specified degree
o f a c c u r a c y w o u l d d e c r e a s e w i t h i n c r e a s i n g irregularity of the packing geometry. It appears to be quite easy to extend the conclusion to polysized sphere size distributions, since such distributions may be cons i d e r e d as a m i x t u r e o f m o n o s i z e d f r a c t i o n s , for which the conclusion holds independently.
4. EXPERIMENTAL
4.1 C o m p u t e r s i m u l a t i o n o f p l a n e c u t s t hr o ugh cubical p a c k i n g s o f m o n o s i z e d spheres -
In order to test the validity of the theory developed in Section 3.2 for regular packings, a simulation programme for a regular cubical packing intersected by a randomly oriented plane was made. The center to plane distances were calculated by eqn. (10), while the frequency variance was determined by eqn. (1). For each orientat-ion o f t h e c u t t i n g p l a n e t h e c a l c u l a t i o n s w e r e performed for successively increasing areas of the planes of cut. This was done for 100 different orientations chosen at random. The number of orientations satisfying condition (11) for a certain area A of the plane of cut, d i v i d e d b y 1 0 0 w i l l t h e n g i v e t h e v a l u e o f Pg for the area A, containing N ~ A particle crosssections.
The intersection was carried out for g equal t o 0 . 0 5 , 0 . 0 1 a n d 0 . 0 0 5 , a n d f o r K e q u a l t o 5, 10 and 15. The results are presented by the d o t t e d l i n e s i n F i g . 4. A s c a n b e s e e n f r o m F i g . 4, t h e a g r e e m e n t between the results from the simulation experiments and those derived theoretically ( e q n . ( 2 5 ) ) is r e a s o n a b l e . A s w o u l d b e e x p e c t e d from a consideration of the approximations involved in the theoretical treatment, the agreement becomes better with increasing values ofN, K andg.
4.8 Experiment with random numbers H a v i n g n o w e s t a b l i s h e d PK ( N ) f o r r e g u l a r , c u b i c a l p a c k i n g s , i t is o f c o n s i d e r a b l e i n t e r e s t to compare the results obtained with those which would have been produced by a perfect random packing, defined as a packing geometry for which the particle center to cutting plane d i s t a n c e s a r e r e c t a n g u l a r l y d i s t r i b u t e d . C l e a r l y , d u e t o p u r e s t a t i s t i c a l s c a t t e r , PK ( N ) for such a packing would not be unity for finite values for N, but increase towardS unity
w i t h i n c r e a s i n g AT, f o l l o w i n g a c u r v e o f t h e same type as found for the cubical pacldng. It was decided to simulate the perfect random packing by a simple random number experiment performed by means of a computer. N random numbers in the range 0 - 9 were extracted from a random number table and grouped in K equally wide classes. The variance of frequency was calculated. This exercise was performed 100 times, using a new set of N random numbers each time. By means of the collection of the 100 variances so obtained, PK ( N ) w a s c a l c u l a t e d as t h e f r a c t i o n o f t h e 100 samples for which condition (11) was satisfied. This was done for a range of N-values, for K = 5 and 10, g = 0.05, 0.01 and 0.005. The results are represented by the circles in F i g . 4. F o r p r a c t i c a l r e a s o n s K = 1 5 w a s n o t included in the experiment, and for similar reasons the calculations for g = 0.005 are t e r m i n a t e d b e f o r e PK ( N ) is p r a c t i c a l l y e q u a l t o 1. T h e t r e n d s a r e n e v e r t h e l e s s Very d i s t i n c t . For low N-values the random geometry gives lower PK(N) values than does the regular system, whilst for higher N-values the random g e o m e t r y g i v e s t h e g r e a t e s t PK ( N ) v a l u e s , PE (N) approaching unity fairly quickly. The reason for this difference between the two p a c k i n g g e o m e t r i e s is t h e p e r i o d i c n a t u r e o f the regular packing. This periodic nature causes the center to plane distances to be rather evenly distributed already for low N-values, whilst for higher N-values the periodic n a t u r e l e a d s t o o v e r - r e p r e s e n t a t i o n o f c e r tain size groups of flistance at the expense of others.
4. 3 _Practical e x p e r i m e n t wi t h c u t t i n g a solid block contaiTHng irregularly p a c k e d m o n o s i z e d spheres A solid specimen containing monosized spheres was made by molding a bed of suitable spheres in a continuous polyester phase. The spheres had diameter of 5.90 ram, with a standard deviation of 0.07. The specimen was cut into a number of slices. E a c h slice c o n t a i n e d a g r e a t n u m b e r o f intersected spheres, each leaving a circular cross-section in the plane of cut. Knowing the d i a m e t e r Df t h e s p h e r e s a n d b y m e a s u r i n g t h e dismeters of the cross-sections, it was possible to calculate the distribution of particle center to plane distances.
8
0.8
numbers / ~ /
0.6 0.5
I/._I ~ hysica! experiment
o,~
_! /
03
~
K~s
g=Q05
/
0.2 S
0.1 0.0
L
,
t
l0
I
1
~
1
I
I
I
I
]
1
,
N
--
20 30 40 50 60 70Fig. 5. PK(N) obtained from physical experiment
with cutting a fixed bed of irregularly packed monosized spheres ( ), a n d e x p e r i m e n t w i t h r a n d o m numbers (X--X).
The tests were carried out for K = 5 and g = 0 . 0 5 . 1 0 5 s m a l l a r e a s o n t h e slices, e a c h c o n t a i n i n g N p a r t i c l e c r o s s - s e c t i o n s , w e r e investigated f o r different values of N. T h e number of areas satisfying condition (!1) was d i v i d e d b y 1 0 5 , giving PK(N). T h e r e s u l t s o f t h i s series o f t e s t s a r e r e p r e s e n t e d b y t h e s o l i d line in Fig. 5. T h e r e s u l t s from the corresponding random number test are g i v e n b y t h e d o t t e d line. A s c a n b e s e e n f r o m these t w o curves, the physical experim e n t gave r e s u l t s v e r y s i m i l a r t o t h o s e f r o m the experiment with random numbers. The differences are n o t greater t h a n the scatter t h a t should be e x p e c t e d f r o m statistical reasons. H e n c e t h e i m p o r t a n t c o n c l u s i o n m a y be d r a w n t h a t a real " ' i r r e g u l a r " p a c k i n g o f s p h e r e s , as it would occur in practical life,does in fact, as far as the present problem is concerned, behave as a perfect r a n d o m packing.
CONCLUSIONS
(1) S t a r t i n g w i t h a r e g u l a r , c u b i c a l p a c k i n g o f m o n o s i z e d spheres, a t h e o r y has been dev e l o p e d , y i e l d i n g t h e f u n c t i o n ( l - - GK(g)/2V), which constitutes an approximate estimate of t h e p r o b a b i l i t y PK ( N ) o f t h e v a r i a n c e o f t h e d i s t r i b u t i o n in q u e s t i o n b e i n g less t h a n a spec i f i e d Ievel, g. K is t h e n u m b e r o f classes o f t h e d i s t r i b u t i o n a n d N t h e n u m b e r o f p a r t i c l e intersections considered. The approy'mations i n v o l v e d e n s u r e t h a t (1 GK(g)/]NT) will al-
-
w a y s b e less t h a n PK (N), e x c e p t f o r v e r y l o w N - v a l u e s . C l e a r l y (1 - - G K ( g ) / N ) will a p p r o a c h u n i t y w i t h increasing N n o m a t t e r h o w small is t h e s e l e c t e d v a l u e o f g. H e n c e , i t m u s t b e concluded that even for a perfect regular packing geometry, provided the intersecting p l a n e is o r i e n t e d a t r a n d o m , p a r t i c l e c e n t e r t o p l a n e d i s t a n c e s will b e l o n g t o a r e c t a n g u l a r l y distributed universe. (2) T h e c o n c l u s i o n d e r i v e d f r o m t h e o r e t i c a l analysis was tested by means of an experiment in w h i c h a r e g u l a r c u b i c a l p a c k i n g o f m o n o sized spheres cut b y randomly oriented planes was simulated on a computer. T h e simulated experiment confirmed the theoretical conclusions. (3) In order to c o m p a r e the results from cutting cubical packings with those which w o u l d have been obtained with a perfect r a n d o m packing, the beha .viour of a perfect r a n d o m packing was simulated b y a r a n d o m n u m b e r experiment. T h e plots of P K (At)produced in this w a y revealed that the r a n d o m packings in fact produced lower P K (AT) values than did the cubical packing in the range of l o w N values. T h e rev~on for this is that the particle center to plane distances, in the case of a regular packing, will be distributed in a systematic w a y over the whole range of the distances in question, the considerable scatter to be expected f r o m a p u r e l y x a n d o m selection at l o w N-values thus being avoided. For higher N-values the situation was brought in agreem e n t with w h a t w o u l d intuitively be anticipated, the P K ( N ) values for the r a n d o m packing being systematically higher than those for the cubical packing. (4) A real, physical experiment, in which plane cuts were m a d e through a bed of irregularly packed monosized spheres, e m b e d d e d in polyester, again produced P ~ (IV) values in general agreement with those already found. Considerable effort was concentrated on a v o i d i n g a n y r e g u l a r i t y in t h e p a c k i n g g e o m e t r y . C l e a r l y , h o w e v e r , i t w a s i m p o s s i b l e t o define the structure more precisely than by the word "'irregularly". C o m p a r i n g the PK(N) values f o r this p a c k i n g with t h e c o r r e s p o n d i n g values from the random packing test showed that the d i f f e r e n c e s w e r e w i t h i n t h e l i m i t s t o b e exp e c t e d f r o m statisticals c a t t e r . T h i s m e a n s t h a t t h e real " ' i r r e g u l a r " p a c k i n g d o e s in f a c t b e h a v e as a r a n d o m p a c k i n g . (5) O n t h e b a s i s o f t h e p r e v i o u s c o n c l u s i o n s ,
9
a n d since packings o f p o l y s i z e d s p h e r e s m a y be c o n s i d e r e d as s u p e r i m p o s e d p a c k i n g s o f m o n o s i z e d f r a c t i o n s , i t is c o n c l u d e d t h a t t h e h y p o t h e s i s t o b e t e s t e d d o e s in f a c t h o l d w i t h i n t h e llrnlts r e q u i r e d , f o r a n y b e d o f spherical particles, p r o v i d e d t h e n u m b e r N o f p a r t i c l e cross-sections i n c l u d e d in t h e analysis is s u f f i c i e n t l y large.
AL
ALm A~g ( A L )
ACKNOWLEDGEMENTS
T h e a u t h o r s wish t o e xpr es s t h e i r g r a t i t u d e t o t h e R o y a l N o r w e ~ n C o u n c i l f o r Sci ent i fi c and Industrial Research for the f i na nc i a l supp o r t w h i c h r e n d e r e d this i n v e s t i g a t i o n possible. T h a n k s are also d u e t o d i r e c t o r Dr. J.A. A n d e r sen of t h e Chr. Michelsen I n s t i t u t e for his stimulating interest.
A
AO
o2(Ae)
LIST OF SYMBOLS
K
Ni N Hf
H g
Px (N), A
(L m, n) rim n
r, Tand ~ = ai + b]
n u m b e r o f e q u a l l y w i de classes the number of sampled particle cross-sections b e l o n g i n g t o size class i t h e t o t a l n u m b e r o f s a m p l e d particle cross-sections the relative frequency of class i m e a n o f H t f o r all classes v a r i a n c e o f t h e f r e q u e n c y distribution c h o s e n l i m i t f o r o~ t h e p r o b a b i l i t y o f o~ < g class w i d t h , A = 1 / 2 K v ar ian c e o f t h e c o n t i n u o u s frequency distribution coordinates for a sphere center r ad ius v e c t o r f o r a s p h e r e c e n t e r u n i t v e c t o r s in t h e c o o r d i n a t e system unit vector normal to the plane
+c~ d
hlmn
(& M,N)
distance from origin to the plane distance f r o m the sphere center in (/, m, n) to the plane total solid angle within which cannot be orientated if o~ ~ g coordinates for a sphere center denoting a periodicity. The numbers h a v e n o c o m m o n f a c t o r . (AT also denotes the n-tuber of crosss e c t i o n s o n a p l a n e c u t , as previously mentioned)
~g(AL)
t h e di st ance f r o m origin t o t h e s p h e r e c e n t e r (L, M, AT). T h e integer part of Ar-equals twice the n u m b e r o f d i f f e r e n t c e n t e r t o pla n e distances o c c u r r i n g f o r t h e peri o d i c i t y , H e n c e AL is d e n o t e d t h e period the m a x i m u m period for which G~ > g is possible the solid angle surrounding the periodicity (L, M, N ) within which o~ > g t h e t o t a l area o f t h e p l a n e c u t , w h i c h f o r a cubical p a c k i n g geom e t r y will a p p r o x i m a t e l y equal t h e t o t a l n u m b e r N o f cross-sections angle between F and the periodicity variance of frequency for the continuous distribution of the center to plane distances w h e n } lies in the vicinity of the periodicity the solid angle within which o~(AO) ~ 2 K g i n t h e vicinity o f the periodicity total solid angle w i t h i n w h i c h
G~ ~ 2 K g
GK (g)
FK P,,
p a r a m e t e r o c c u r r i n g in t h e exp r e s s i o n f o r t h e t h e o r e t i c a l l y derived l o w e r l i m i t f o r P~c (N), given in eqn. (25). G K ( g ) is given in (23) given in eqn. (19) the fraction of sphere centers w i t h i n a spherical shell o f u n i t t h i c k n e s s having c o o r d i n a t e s (L, M, N ) w i t h n o c o m m o n f a c t o r
REFERENCES I M. Delesse, Pour d~terminer la composition des roches, Procede Mecan., (1862) 379. 2 A. Rosiwal, Ueber geometrische Gesteinsanalysen, Verhandlungen, 5, 6 (1898) 1 4 ~ 3 T.H. Hagermann, A m e t h o d for assessment of particle size and its distribution in finely grained sedimentary rocks, Geol. Foren. Stockholm Forh., 46 (1925) 325 (in Swedish). 4 S.D. Wicksell, A study o f the p~operties of globular distributions, Arkiv Mat. Astron. Fysik., 18 ( 1 9 2 4 ) 1. 5 S.I).WickselI, T h e corpuscle problem: A mathematical study of a biometrie problem, Biometriea, 1 7 ( 1 9 2 5 ) 84. 6 G. T a m m a n a n d W. C r o n e , Z. A n o r g . A l l g e m . Chem., 187 (1930) 289.
10 7 E. S c h e i l , Die B e r e c h n u n g d e r A n z a h l u n d G r ~ s s e n v e r t e i l u n g kugelir~rmiger K r i s t a l l e i n u n d u r c h s i c h tigen KDrpern mit Hilfe der d u r c h e i n e n e b e n e n S e h n i t t e r h a l t e n e n S c h n i t t k r e i s e , Z. A n o r g . A l i g e m . C h e m . , 201 ( 1 9 3 1 ) 259. 8 E. S c h e i l a n d H. W u r s t , S t a t i s t i s c h e Gef~dgeunters u c h u n g e n II, Z. M e t a l l k . , 28 ( 1 9 3 6 ) 3 2 0 . 9 E. S c h e i l , S t a t i s t i s c h e G e f d g e u n t a r s u c h u n g e n I, Z. M e t a l l k . , 27 ( 1 9 3 5 ) 1 9 9 . 1 0 E. S c h e i l a n d A . L a n g e w e i s e , A r c h l y . E i s e n h i i t t e n w e s e n , 11 ( 1 9 3 7 ) 93. 11 H . A . S c h w a r t z , T h e m e t a l l o g r a p h i c d e t e r m i n a t i o n o f t h e size d i s t r i b u t i o n o f t e m p e r c a r b o n n o d u l e s , Metals Alloys, 5 (1934) 139. 1 2 S . A . S a l t y k o v , S t e r e o m e t r i c m e t a l l o g r a p h y , Met a l i u r g i z d a t , M o s c o w , 2 n d e d n . , 1 9 5 8 , p. 4 4 6 . 13 H . J . Oel, K r i s t a l l w a c h s t u ~ b e i m S i n t e r n y o n Hochtemparaturoxiden, Rapport yon dem Max P l a n k I n s t i t u t f'dr S i l i k a t f o r s c h u n g , 87 W i i r z b u r g , Deutschland. 1 4 J . t~Ieisner, E s t i m a t i o n o f t h e d i s t r i b u t i o n o f diam e t e r s o f spherical particles f r o m a given g r o u p e d distribution of diameters of observed circles formed by a plane section, Statist. Neerlandica, 1 ( 1 9 6 7 ) 11. o 15 H . E . R o s e , T h e d e t e r m i n a t i o n o f t h e g r a i n - s i z e distribuLion o f a spherical granular material emb e d d e d in a m a t r i x , S e d i m e n t o l o g y , 1 0 ( 1 9 6 8 ) 293. 1 6 E.T. W h i t t a k e r a n d G.N. W a t s o n , M o d e r n A n a l y s e r , C a m b r i d g e U n i v . Press, C a m b r i d g e , 4 t h e d n . , 1 9 2 7 , p. 229. 17 S. D r a p a l a n d V . H o r a l e k , S o m e r e l a t i o n s b e t w e e n parameters of structure of metallographic specimen surface and in t h e space o f m etal s p e c i m e n s , A c t a Tech., 6 (1959) 474. 18 M.G. K e n d a l l a n d P . A . P . M o r a n , G e o m e t r i c a l P r o b a b i l i t y , G r i f f i n , L o n d o n , 1 9 3 6 , p. 86.
1 9 H . E . E x n e r , A n a l y s i s o f grain- a n d p a r t i c l e - s i z e d i s t r i b u t i o n s in metallic materials, Int. Met. Rev., 17 ( 1 9 7 2 ) 25. 2 0 T . C . P o w e r s , T h e air r e q u i r e m e n t s o f f r o s t resistant concrete, Proceedings Highway Research Board, 29 ( 1 9 4 9 ) 189. 21 T . F . Willis, D i s c u s s i o n o f a p a p e r b y T . C . P o w e r s : T h e air r e q u i r e m e n t s o f f r o s t r e s i s t a n t c o n c r e t e , P r o c e e d i n g s Highway R e s e a r c h Board, 29 (1949) 203. 22 G.W. L o r d a n d T . F . Willis, C a l c u l a t i o n o f air b u b b l e size d i s t r i b u t i o n f r o m r e s u l t s o f a R o s i w a l t r a v e r s e o f a e r a t e d c o n c r e t e , A S T M Bull. 1 0 ( 1 9 5 1 ) 56. 23 A . G . S p e k t o r , A n a l y s i s o f d i s t r i b u t i o n o f s p h e r i c a l particles in n o n - t r a n s p a r e n t structures, Zavodsk. L a b . , 16 (2) ( 1 9 5 9 ) 1 7 3 . 24 R . L . F u l l m a n , M e a s u r e m e n t o f p a r t i c l e s sizes i n opaque bodies, J. Metals, 5 (1953) 447. 25 H. M i i n z e r a n d P. S e h n e i d e r h b h n , D a s S e h n e n s c h n i t t v e r f a h r e n , Heidelbergar Beitr. Mineral. Petrog., 3 (1953) 456. 26 E. H o r i k w a , O n t h e n e w m e t h o d o f r e p r e s e n t a t i o n o f t h e m i x t u r e o f several g r a i n sizes, T e t s u t o H a g a n e , 40 (1954) 991. 27 J.W. C a h n a n d R . L . F u l l m a n , O n t h e u s e o f l i n e a l a n a l y s i s f o r o b t a i n i n g p a r t i c l e size d i s t r i b u t i o n f u n c t i o n s in o p a q u e samples, J. Metals, 5 (1956) 610. 28 G. B o c k s t i e g e l , E i n e e i n f a c h e F o r m e l z u r B e r e c h nung rKumlicher Gr~ssenverteilung aus durch L i n e a r a n a l y s e e r h a l t e n e n D a t e n , Z. M e t a l l k . , 57
(1966) 647. 29 G. E n s t a d , A c r i t i c a l a n a l y s i s o f t h e a s s u m p t i o n that the distances from the sphere centers to a r a n d o m plane t h r o u g h a b e d o f s p h e r e s are rectangularly distributed, CMI R e p t . 7 2 0 0 3 - 1 / G E ( 1 9 7 2 ) (in N o r w e g i a n ) .
Derivation of particle Size Distributions from Measttrements on Plane Sections through Particle Beds. Effect of Regularity of Particle Pacbing
R.I~L ECKHOFF and G. ENSTAD Powder Research Laboratory. Chr. Michelsen Institute, Dept. o f A p p l i e d Physics, N - 5 0 0 0 Bergen (Norway)
(Received January 8, 1974)
SUMMARY W h e n t h e size d i s t r i b u t i o n o f p a r t i c l e s e m bedded in an opaque, continuous solid phase is r e q u i r e d , t h e g e n e r a l a p p r o a c h is t o d e d u c e t h e d i s t r i b u t i o n f r o m t h e size d i s t r i b u t i o n o f particle cross-sections in a plane cut through the particle bed. When the particles are approximately spherical, this deduction can be performed by tasking the assumption that the distances from the plane of cut to the particle centers are rectangularly distributed. The validity of this aa~umption does not, however, appear to have been investigated in previously published work, and in the present contribution the assumption has been considered more
closely. The di~ibution of the distances between t h e s p h e r e c e n t e r s a~nd a r a n d o m p l a n e h a s been investigated both theoretically and experi m e n t a l l y , s t a r t i n g w i t h regll|Ar p a c k i n g s o f monosized spheres, and extending the treatment to monosized spheres packed at random a n d t-ma~ly t o p a c k i n g s o f p o l y s i z e d s p h e r e s . The distribution of partic4e center to cutring plane distances will approach a rectangul a r d i ~ i b u t i o n w i t h i n c r e a s i n g s a m p l e size eve~. f o r bt~ictly r e g u l a r p a c k i n g s . H o w e v e r , f o r f m i ~ s a m p l e s , t h e p a c k i n g r e g u l a r i t y m a y si~nifica~atly affect the extent to which such a di~ibut i o n is r e a l i z e d . 1. INTRODUCTION T h e p r o b l e m o f d e t e r m l n l n g t h e s i z e distribution of particles embedded in a contintto~ts solid phase from which they cannot easily be
l i b e r a t e d arises in a w i d e r a n g e o f f i e l d s o f r e search, product development and product control. In medicine and biology the particles will b e cells in t i s s u e , in c o n c r e t e a n d p o l y e s t e r t h e y will b e gas b u b b l e s a n d i n r o c k m i n e r a l grains. T h e s o l u t i o n w h i c h is g e n e r a l l y a d o p t e d w h e n o n e is f a c e d w i t h t h i s p r o b l e m is t o d e duce the size distribution of the three-r]imensional bed of particles from the size distribution of particle cross-sections produced by a random plane through the specimen, or from the distribution of chord intercepts produced by a random straight line through the specimen. T h e d e d u c t i o n is p e r f o r m e d b y m e a n s o f a transformation theory involving two basic a s s u m p t i o n s . F i r s t l y i t is a s s u m e d t h a t t h e particles are spheres and secondly that the distances from the sphere centers to a random plane through the particle bed are rectangularly distributed. T h e p r e s e n t c o n t r i b u t i o n is c o n c e r n e d w i t h the last assumption, the validity of which does n o t a p p e a r t o h a v e b e e n i n v e s t i g a t e d in p r e viously published work. 2. REVIEW OF THE L I T E R A T U R E A l t h o u g h t h e r e v i e w t o b e g i v e n in +..he f o l l o w i n g is m a i n l y i n t e n d e d t o s h o w t h e s t a t e o f t h e a r t as f a r as t h e a c t u a l p r o b l e m is c o n cerned, it also illustrates how communication b e t w e e n d i f - r e r e n t b r a n c h e s o f s c i e n c e in which an identical powder problem has been treated has up to recently been lacking to a sLr;lring e x t e n t . T h e r e v i e w t h e r e f o r e a l s o demonstrates the justification of powder tech-
2 n o l o g y as a s p e c i f i c f i e l d o f r e s e a r c h a n d c o m munication. T h e f i r s t c o n t r i b u t i o n t o b e c o n s i d e r e d is t h e o n e b y Delesse [ 1 ] f r o m 1 8 4 8 , i n w h i c h the problem of determining the content of a s p e c i f i c m i n e r a l c o m p o n e n t in r o c k s p e c i m e n s was c o n s i d e r e d . S t a r t i n g w i t h a p l a n e , p o l i s h e d s u r f a c e o f a r e a F o f t h e a c t u a l m a t e r i a l . Delesse was a b l e t o s h o w t h e o r e t i c a l l y t h a t
f~vv_ F
V
w h e r e f is t h e a r e a w i t h i n F o c c u p i e d b y t h e desired mineral component, V the volume of a r e p r e s e n t a t i v e p i e c e o f r o c k a n d v t h e volu m e o f the desired mineral within t h a t piece. Delesse's m e t h o d o f d e t e r m i n i n g v o l u m e r a t i o s was l a t e r m o d i f i e d b y R o s i w a l [ 2 ] , w h o was a b l e t o s h o w t h a t L
V
w h e r e L is t h e l e n g t h o f a s t r a i g h t line t h r o u g h the specimen and l the sum of the intercepts a l o n g L c r o s s i n g areas o c c u p i e d b y t h e d e s i r e d mineral. T h e first steps t o w a r d s a t r a n s f o r m a t i o n t h e o r y f o r c a l c u l a t i n g size d i s t r i b u t i o n s w e r e c a r r i e d o u t b y H a g e r m a n n [ 3 ] a n d Wicksell [ 4 ] in 1 9 2 4 . H a g e r m a n n c o n s i d e r e d t h e g e n e ral p r o b l e m o f r a n d o m l y d i s t r i b u t e d m o n o sized s p h e r e s , w h i l s t Wicksell w a s c o n c e r n e d w i t h t h e s p e c i f i c p r o b l e m o f c o m p u t i n g size d i s t r i b u t i o n s o f s t a r clusters. A y e a r l a t e r Wicksell [ 5 ] c a r r i e d his w o r k f u r t h e r , p r e s e n ting a general t h e o r y for the calculation o f t h e size d i s t r i b u t i o n s o f e m b e d d e d p a r t i c l e s o n t h e basis o f t h e size d i s t r i b u t i o n o f p a r t i c l e c r o s s - s e c t i o n s in a r a n d o m p l a n e t h r o u g h t h e particle bed. Starting with a contribution by Tarnmann and Crone [ 6 ] , Scheil [7] developed a t h e o r y f o r c a l c u l a t i n g size d i s t r i b u t i o n s o n t h e basis o f size d i s t r i b u t i o n s o f p a r t i c l e c u t s i n : a p l a n e s e c t i o n , w h i c h w a s p u b l i s h e d in 1 9 3 1 . S c h e i l ' s m e t h o d was similar t o t h a t d e v e l o p e d b y Wicksell [ 5 ] , b u t a p p a r e n t l y S c h e ~ d i d n o t k n o w o f Wicksell's work. A n i m p r o v e d version o f 9eheil's t h e o r y was published b y Scheil and W u r s t [ 8 ] in 1 9 3 6 , a n d f u r t h e r i m p r o v e m e n t s b y S c h e i l a l o n e in 1 9 3 5 [ 9 ] a n d 1 9 3 7 [ 1 0 ] , . b u t t h e r e s u l t i n g m e t h o d still s u f f e r e d f r o m c o n s i d e r a b l e d i s a d v a n t a g e s as f a r as p r a c t i c a l application was concerned.
In 1934 Schwartz [11] had published an i m p r o v e d version o f Schefl's first m e t h o d , a n d in 1 9 5 8 S a l t y k o v [ 1 2 ] p r e s e n t e d a c o n siderably more efficient method starting with Schwartz's work. The important new contrib u t i o n in S a l t y k o v ' s m e t h o d w a s a g e n e r a l l y applicable table of coefficients, which reduced the required amount of computation for performing the desired transformation quite considerably. In the beginning of the nineteen-sixties Oel [ 1 3 ] p u b l i s h e d a m e t h o d w h i c h is basically identical with Scheil's first method from 1 9 3 1 . A c c o r d i n g t o t ~ e r e f e r e n c e list, h o w e v e r , Oel w a s n e i t h e r a w a r e o f W i c k s e l l ' s c o n t r i b u tion n o r o f Scheil's. A s l a t e as 1 9 6 7 M e i s n e r [ 1 4 ] p r e s e n t e d a n elegant derivation, based on Wicksell's and Scheil's work, of a method very similar to that p r o d u c e d b y S a l t y k o v t e n y e a r s earlier. Evidently Meisner neither knew the work of Schwartz nor that of Saltykov. The same year, 1967, Rose [15] published a m e t h o d d e v e l o p e d o n t h e basis o f t h e c o n t r i b u t i o n s b y H a g e r m a n n , Wieksell, I Q m m b e i n and Pettijohn. Again a m e t h o d very similar t o that of Saltykov, involving tables of generally applicable coefficients, was the result. Acc o r d i n g t o t h e r e f e r e n c e list, t h e c o n t r i b u t i o n s by Schwartz, Saltykov and Meisner were not known to Rose. D u r i n g t h e y e a r s several i n d e p e n d e n t w o r k e r s have published derivations of exact solutions t o t h e b a s i c i n t e g r a l e q u a t i o n s i n v o l v e d in t h e actual problem. The contributions by Whi~er and W a t s o n [ 1 6 J , D r a p a l a n d H o r a l e k [ 1 7 ] and Kendall and Moran [18] should be ment i o n e d . I t is r e g r e t t a b l e , h o w e v e r , t h a t t h e e x a c t s o l u t i o n is n o t a p p l i c a b l e in p r a c t i c e . Several a u t h o r s h a v e p u b l i s h e d l i t e r a t u r e surveys covering the actual problem, and the very comprehensive survey by Exner [19] from 1972 ought to be mentioned here. A l t h o u g h t h e m e t h o d s p r e s e n t e d b y t h e various workers referred t o so far differ s o m e w h a t i n t h e details, t h e b a s i c f e a t u r e s are i d e n t i c a l . All t h e c o n t r i b u t i o n s , as f a r as c a l c u l a t i o n o f size d i s t r i b u t i o n s is c o n c e r n e d , d e a l w i t h s p h e r i c a l p a r t i c l e s . F u r t h e r m o r e , a n d t h i s is t h e m a i n c o n c e r n o f t h i s p a p e r , all t h e c o n t r i b u t i o n s i n v o l v e t h e a s s u m p t i o n t h a t t h e distances of particle centers from a random plane o f intersection are rectanguIarly distributed. So far the survey has been Hmited to methods
o f c a l c , , l s t i n g t r u e size d i s t r i b u t i o n s o n t h e basis o f size d i s t r i b u t i o n s o f p a r t i c l e c r o s s s e c t i o n s in a r a n d o m p l a n e t h r o u g h a b e d o f particles. In o r d e r t o m a k e t h e p i c t u r e c o m plete, it o u g h t t o be m e n t i o n e d t h a t several w o r k e r s have described m e t h o d s f o r calculat i n g t h e t r u e size d i s t r i b u t i o n f r o m t h e distribution of chord intercepts produced by a s t r a i g h t line t h r o u g h t h e p a r t i c l e b e d . T h e c o n t r i b u t i o n s b y P o w e r s [ 2 0 ] , Willis [ 2 1 ] , L o r d a n d Willis [ 2 2 ] , S p e c t o r [ 2 3 ] , F u l l m a n [ 2 4 ] , Miinzer and SchneiderhDhn [25], Horikwa [26], Cahn and Fullman [27] and Bockstiegel [ 2 8 ] , all o f w h i c h h a v e t h e classical c o n t r i b u t i o n o f R o s i w a l [ 2 ] as t h e s t a r t i n g p o i n t , s h o u l d be m e n t i o n e d s p e c i f i c a l l y . C o n t r a r y t o t h e m e t h o d b a s e d o n t h e diameters of particIe cross-sections, the chord approach for spherical particles has a very s i m ~ l e , a n a l y t i c a l s o l u t i o n , w h i c h is r e a d i l y applicable in practice. T h e s~lution does, however, rest on the assumption that the p a r t i c l e c e n t e r s a r e r a n d o m l y d i s t r i b u t e d in s p a c e . I n f a c t t h i s a s s u m p t i o n is i d e n t i c a l w i t h t h e o n e s t a t e d in t h e h e a d i n g o f t h i s p a p e r , a n d h e n c e t h e c o n c l u s i o n s f r o m t h e p r e s e n t inv e s t i g a t i o n will b e d i r e c t l y a p p l i c a b l e also t o the chord methods. T h e a c c o u n t g i v e n in t h e f o l l o w i n g is a condensed version of a comprehensive report [ 2 9 ] in w h i c h t h e t h e o r e t i c a l a n d e x p e r i m e n t a l i n v e s t i g a t i o n s a r e d e s c r i b e d in d e t a i l .
3. THEORY
t o t h e t o t a l f r e q u e n c y N , w h i l s t H is t h e m e a n o f H i f o r all classes. I n case o f d i s t r i b u t i o n s which differ considerably from the rectangular, a ~ will b e large, a n d f o r d i s t r i b u t i o n s w h i c h are n e a r l y r e c t a n g u l a r , t h i s p a r a m e t e r ~d_ll a t t a i n s m a l l values. F o r p e r f e c t r e c t a n g u l a r d i s t r i b u t i o n s , it will b e z e r o . F o r a r a n d o m sample extracted from a rectangular distribut i o n , o n e will h a v e a~ -* 0
(2)
N-+oo
T h i s in t u r n m e a n s t h a t i f i t c a n b e s h o w n t h a t e q n . (2) h o l d s , t h e dista-:bution in q u e s t i o n must be rectangular. It must be pointed out, however, that o~ for a r a n d o m s a m p l e f r o m a given p o p u l a t i o n will vary statistically a l t h o u g h K and N are k e p t c o n s t a n t . F o r t h i s r e a s o n 0 2 as s u c h is n o t suitable for characterizing the extent to which a given d i s t r i b u t i o n d i f f e r s f r o m t h e r e c t a n g u lar. I t w a s t h e r e f o r e d e c i d e d t o a d o p t t h e p r o b a b i l i t y PK (N) f o r a 2 b e i n g s m a l l e r t h a n a c h o s e n p o s i t i v e c o n s t a n t g , as t h e d e s i r e d criterion. P r o v i d e d it c a n b e s h o w n that PK(IV)
-~ I , N
-+ ~
n o matter h o w small is the value of g, the distribution m u s t b e rectangular. A s will b e c o m e apparent in the following~ it w a s in the theoretical approach decided to consider the case of a continuous distribution, since this facilitated the theoretical work. It is, h o w e v e r , p o s s i b l e t o s h o w [ 2 9 ] t h a t
3.1 Criterion o f degree o f rectangular distribu tion
a~ <
Since the object of the present work was to investigate the extent to which the distances f r o m s p h e r e c e n t e r s t o a r a n d o m p l a n e o f int e r s e c t i o n w e r e r e c t a n g u l a r l y distributed, a convenient quantitative criterion for characterizing the extent to which the distribution was rectangular had to be found. As a starting point, it was decided to consider the parameter
w h e r e o~ is t h e v a r i a n c e o f t h e c o n t i n u o u s frequency distribution, and
K
a~
=
X(H, -- ~)z =~Ar~,--~
(3)
A-
a~
1 A = ~ - (total width of distribution)
(4)
(5)
will a l w a y s h o l d . H e n c e , i f (3) c a n b e s h o w n t o h o l d f o r t h e c o n t i n u o u s case f o r ~-q~ smaller t h a n a n y p o s i t i v e v a l u e o f g , i t also h o l d s for the discrete distribution with frequency variance a~.
(I)
i
which denotes the variance of a frequency d i s t r i b u t i o n h a v i n g K e q u a l l y w i d e classes: H i is t h e r a t i o o f t h e f r e q u e n c y N i i n t h e i t h class
3. 2 I~egular, cubical p a c k i n g s o f m o n o s i z e d spheres One would intuitively anticipate that the assumption to be investigated would be faced w i t h its g r e a t e s t c h a l l e n g e w h e n a p p l i e d t o
4 Since the constant d does not influence t h e d i s t r i b u t i o n o f distmmnes o f s p h e r e c e n t e r s from the plane, the considerationscan be l i m i t e d t o t h e special case w h e r e d ecp!Alg z e r o , i.e. t o a p l a n e t h r o u g h t h e o r i g i n , i n t h e c a s e of which eqn. (9)simplifies to
It, re.n}
,
h z , , , ffi a l + b m
, Fig. I . ~ regular, cubical packing of monosized spheres in a Ca~'tcsian coordinate system. p e r f e c t l y r e g u l a r packings o f s p h e r e s . H e n c e it was d e c i d e d t o c o n s i d e r a s i m p l e c u b i c a l p a c k i n g o f m o n o s i z e d spheres, w h i c h is ill u s t r a t e d in Fig. 1. I f o n e s p h e r e d i a m e t e r is t a k e n as t h e u n i t a l o n g t h e c o o r d i n a t e axis i n Fig. 1, t h e r a d i u s v e c t o r f r o m t h e origin t o a s p h e r e c e n t e r w i t h c o o r d i n a t e s l, m a n d n will b e t h e ~,,,,, "
=
,;÷mr
(6)
+ nk
w h e r e i, j a n d ~ a r e t h e u n i t v e c t o : s o f t h e c o o r d i n a t e s y s t e m , a n d l, m a n d n a r e integers. T h e g e n e r a l e q u a t i o n f o r a p l a n e is (7)
ax + by + cz + d = O
and the unit vector ; = .T+
bf+
(8)
will b e p e r p e n d i c u l a r t o t h e p l a n e , w h i l s t d will b e t h e d i s t a n c e f r o m t h e o r i g i n t o t h e plane. T h e d i s t a n c e f r o m t h e s p h e r e c e n t e r ~dth c o o r d i n a t e r , l, m a n d n t o t h e p l a n e will be ~lrrtn
~
~'[rl~ n
*
7 + d = c: + bm + cn + d
as can be deduced
i /~ \
(9)
f r o m Fig. 2.
"~ . ~ . . ; " l a n e
(10)
+ cn
The position of a plane of intersection with r e s p e c t t o t h e s y s t e m i l i n s t r a t e d i n F i g . I will then be fully determined by the direction of th~ u n i t v e c t o r ~, L ~ b y t h e c o n s t a n t s a, b a n 0 c. I f n o w b'~a d e n o t e s t h e t o t a l s o l i d a n g l e within which ~ c a n n o t be located if t h e con° ditio~ c¢} < g
(11)
h a s t o b e satisfied, t h e n Px = 1
~2t 4~
(12)
expresses t h e probability t h a t a r a n d o m plane through the cubical packing of monosized s p h e r e s will s a t i s f y c o n d i t i o n ( 1 1 ) . I t is, h o w ever, n e c e s s a r y t o f i n d a n e x p r e s s i o n f o r £~a before eqn. (12) can be o f a n y use. One characteristic feature of the process of intersecting a regular packing o f spheres w i t h a r a n d o m p l a n e is t h a t t h e r e m ~ c e r t a i n s p e c i f i c d i r e c t i o n s o f ~ f o r w h i c h i t is o n l y possible t o obtain a limited n u m b e r o f sphere c e n t e r t o p l a n e distances. C l m ~ o n s of particles having these particulax distances will t~len o c c u r p e r i o d i c a l l y a l o n g t h e p l a n e o f i n t e r s e c t i o n . S u c h d i r e c t i o n s o f ~ will i n the following be denoted periodic directions o r p e ~ o d i c i t i e s . T h e z e w 0 1 b e ~a~ i n ~ t e numb e r o f e v e n l y d i s t r i b u t e d p~-z~odie d i r e c t i o n s . A n y d i r e c t i o n of. ~ wi~ t h e r e f o r e b e l o c a t e d i n t h e v i c i n i t y o f s o m e p e r i o d i c "dizection. I t is p o s s i b l e t o s h 0 w [ 2 9 ] t h a t f o r a n y p e r i o d i c d i r e c t i o n o f P, t h e p a r a m e t e r s a. b and ein eqn. (10) muat equal ,,
=
±L
b --
±IW a n d
e
=
(13)
where L, M a n d N are intege; n u m b e r s with no common factor, and where Ar = ~/L 2 + ~
Fig. 2. A regular, eubic~d packing of mono•zed spheres cut by a randomly oriented plane.
+/V ~
(14)
is t w i c e t h e t o t a l n ~ of ~;f~erent i~'~de center t 0 p l a n e distances for t h a t c u m b i n a t i o n o f L , M a n d N , i . e . w h i c h o c c u r o n b o t h aides of the plane for that particularpefi~.
In the following AL will be denoted the pe~od. I t is f u r t h e r p o s s i b I e t o s h o w [ 2 9 ] t h a t i n t h e c a s e A L ;~ A ~ m , w h e r e
Az.
Iz
(15)
~V~g
m.,
~ o ~ , i G o n (11) ~ always be ~ , ~gst f o r A L < A r m this c o n d i t i o n will in most n o t be fnlfmed. A p e r i o d i c i t y f o r w h i c h
Y
c o n d i t i o n ( 1 1 ) is n o t s a t i s f i e d w i l l b e su~oun.d e d b y a m,~rl s o l i d a n g l e A&'2t(A r ) W i t h ~ w h i c h ( 1 1 ) is n o t sati~-f~,d e i t h e r . I n or¢!~~v t o f i n d thi~ sOHd a n g l e ~ b ' ~ ( A L ) , an ~ f¢¢ o~ in the vicinity of the peciodlcity should be found. This, however, would be extremely difficult. For that reason the ~dis~bufion of paxticle center to p]mle distances was appro-~im~ by a continuous dist~fl~ion. The approximation holds providodA .~Az, whereA i s t h e t o t a l a r e a o f t h e p l a n e o f i n t e r s e c t i o n c o n s i d e r e d , w h i c h in turn will be equivalent to the total number of i n t a c s e c t ~ spheres. F o r t h e c o n t i n u o u s distrFDution, t h e v a r i a n c e o ~ ( A 0 ) w a s f o u n d [ 2 9 ]
F'¢g. 3 . H l ~ t r a t l o n o f t h e s o l i d ~mg]e ~ b ' ~ ( A L ) r o u n d i n g a p e r i o d i c d i r e c t i o n ( L , M. N ) .
toeq~
the symbol [] denoting "the integer part of". The pattern of intersection between the [
I+1
where Im
=
~
F g ( g ) = 2 ( 1 + K g ) ~ f=o
(2r + z) (19)
and
,.
(V:÷
(20)
s o l i d angle Ag'~'g(A L ) a n d a spherical surface
--1)
(AL~/, ° - - x")
sup
(16)
where Z~0 is the angle between the peciodic d i r e c t i o n a n d P. • is t h e i n t e g e r p a r t o f t h e
with the center in the origin will be a set of c o n c e n t r i c r i n g s as i l l u s t x a t e d i n F i g . 3. I n o r d e r t o e s t i m a t e the t o t a l s o l i d angle
w i t h i n which condition ( 1 1 ) is n o t satisfied, eqn. (18) was summed for the periodicities wif;h A L "~ Al.m.
ALm
E q u a t i o n (4) sires T h e summation is tO be performed f o r all
~(~0) > ~- - 2K~
c o m b i n a t i o n s o f L, M a n d N h a v i n g n o c o m m o n f a c t o r and f o r A L <; ALto-
H e n c e , t h e solid m~gle Ab--Z'~(AL) w i t h i n which
I n order t o perform this summation, the f o l l o w i n g approximate m e t h o d was adopted.
w ~ l be greater than the solid angle ~ Z z ( A ~ ) w i t h i n which condition (ZZ) is n o t s a t i r e d . B y substituting ~ ( , e ~ ) ~,, eqn. ( 16) b y 2 ~ g , a q u a d r a ~ e q ~ o n r e s ~ , yielding tWO v a I u e s o f ~B(/}, f r o m w h i c h t h e s o l i d
angle Z ~ ( A ~ )
can be d e d u c e d . I t c a n be
~ o w n [29] t~st
Consider a spherical shell with trait thickness, w i t h t h e c e n t e r in origin a n d w i t h r a d i u s RL e ~ u a I t o t h e i n t e g e r p s x t o f A L ~ ! . T h i s shell wl]] c o n t a l n 4 7 r R ~ p a r t l c l e c e n t ~ z s . I f n o w t h e t ~ - t i o n po o f t h e s e p a r t i c l e c e n t e r s h a v i n g c o o r d i n a t e s (LJHrJV,) w i t h n o c o m m o n f a c t o r is c o n s i d e r e d c o n s t a n t , t h e n u m b e r o f p a r t i c l e c e n t a r s w i t h i n t h e s p h e r i c a l s h e ] / w h i c h re-
p r e s e n t p e ~ o d i c i t i e s v n ~ b e 41¢R~pv. T h e mag-
"%~,;(AL) ffi A I A
(18)
nitude of the periods AL within a shell are now considered constant and equal to RL - Equation
(21) then takes the approximate form
g
~
0.6
w h i c h in c o m b i n a t i o n w i t h e q n . ( 1 8 ) gives
5 o.~ 0.2
47rGK (g) A
(22)
GK (g) = FK ( g ) A n m P~,
(23)
0.0 ~'0 60 IODIAO~I
where
(24) w i g a p p l y . T h i s means t h a t
150
100 200 300
150 300 4SO
~
~ 1
0.2 0.0
GK (g) N
2 0 60 100 140
1.0
15
0.6 0.4
0.0
Since the diameter of the monosized spheres h a s b e e n t a k e n as t h e u n i t o f l e n g t h in t h e treatment above, the numerical value of A will f o r t h e r e g u l a r c u b i c a l p a c k i n g c o n s i d e r e d be approximately equal to the number N of particle cross-sections within A. For this reason, and using eqn. (22), one has GK (g)
o.g
10 0.4
S
0.2
PK = 1 - - ~ - ~ > 1 - - 4--~
(25)
G K ( g ) g i v e n in e q n . ( 2 3 ) is f u l l y d e t e r m i n e d w h e n K a n d g a r e given. H e n c e f o r a n y f i n i t e p o s i t i v e v a l u e s o f K a n d g, e q n . ( 2 5 ) m e a n s that PK -~ 1 N->~
300/.SO
IOD2OD3DO
1.0
The error involved by adopting the approxim a t e s u m m a t i o n will d e c r e a s e w i t h i n c r e a s i n g value of ALto. M a i n l y b e c a u s e o f e q n . (4), b u t a l s o b e c a u s e condition (11) may be satisfied for some comb i n a t i o n s Of A L a n d K e v e n w h e n A L < A L t o , the relation
PK > 1
0.005
4~R2pvA~2"g(Rr.)
RL=I
~2~ ~
0.01
~I K 7.0 P.~.
ALm
~2"~
0.05
(26)
w h i c h in t u r n m e a n s t h a t t h e d i s t r i b u t i o n o f particle center to plane distances must be rectangular even when intersecting regular cubical packings. T h e s o l i d l i n e in Fig. 4 s h o w s (1 - - G K ( g ) / N ) as a f u n c t i o n o f N f o r v a r i o u s v a l u e s o f K a n d g . Although the analysis presented above was performed for regular cubical packings, it can relatively easily be shown [29] that the general f o r m o f e q n . ( 2 5 ) will a p p l y t o all regular packings, each particular packing geometry having a specific form of GK(g).
. . . . . 20 60 100 140
S
100 200 300
150 3O9 450
F i g . 4 . PK(N) s h o w n g r a p h i c a l l y f o r d i f f e r e n t c o m b i n a tions of K and g. The theoretical values given by eqn. (25);--X-- results from computer simulation of cubical packings cut by random planes; o experiment with random numbers, simul~/ting perfect random packings.
Hence, the general and important conclusion can be drawn that cutting regular packings by r a n d o m p l a n e s will a l w a y s y i e l d r e c t a n g u l a r d i s t r i b u t i o n s o f p a r t i c l e c e n t e r t o p l a n e distances, provided the sampled number N of p a r t i c l e c r o s s - s e c t i o n s is s u f f i c i e n t l y large. 3. 3 E x t e n s i o n o f c o n c l u s i o n t o irregular p a c k ings a n d p o l y s i z e d s p h e r e s i z e d i s t r i b u t i o n s As pointed out in Section 3.2, one would expect that the assumption to be tested would be faced with its greatest challenge in the case of perfect regular packing geometries. It was possible to show that even for a perfect regular packing of monosized spheres, the distribution of distances from particle centers to the plane of intersection would approach the rectangular form with increasing ~mple size. H e n c e , i t a p p e a r s s e n s i b l e t o e x p e c t t h a t this would be so for any practical packing geometry. In fact, one would even anticipate t h a t t h e s a m p l e size r e q u i r e d f o r o b t a i n i n g a rect~nguler distribution with a specified degree
o f a c c u r a c y w o u l d d e c r e a s e w i t h i n c r e a s i n g irregularity of the packing geometry. It appears to be quite easy to extend the conclusion to polysized sphere size distributions, since such distributions may be cons i d e r e d as a m i x t u r e o f m o n o s i z e d f r a c t i o n s , for which the conclusion holds independently.
4. EXPERIMENTAL
4.1 C o m p u t e r s i m u l a t i o n o f p l a n e c u t s t hr o ugh cubical p a c k i n g s o f m o n o s i z e d spheres -
In order to test the validity of the theory developed in Section 3.2 for regular packings, a simulation programme for a regular cubical packing intersected by a randomly oriented plane was made. The center to plane distances were calculated by eqn. (10), while the frequency variance was determined by eqn. (1). For each orientat-ion o f t h e c u t t i n g p l a n e t h e c a l c u l a t i o n s w e r e performed for successively increasing areas of the planes of cut. This was done for 100 different orientations chosen at random. The number of orientations satisfying condition (11) for a certain area A of the plane of cut, d i v i d e d b y 1 0 0 w i l l t h e n g i v e t h e v a l u e o f Pg for the area A, containing N ~ A particle crosssections.
The intersection was carried out for g equal t o 0 . 0 5 , 0 . 0 1 a n d 0 . 0 0 5 , a n d f o r K e q u a l t o 5, 10 and 15. The results are presented by the d o t t e d l i n e s i n F i g . 4. A s c a n b e s e e n f r o m F i g . 4, t h e a g r e e m e n t between the results from the simulation experiments and those derived theoretically ( e q n . ( 2 5 ) ) is r e a s o n a b l e . A s w o u l d b e e x p e c t e d from a consideration of the approximations involved in the theoretical treatment, the agreement becomes better with increasing values ofN, K andg.
4.8 Experiment with random numbers H a v i n g n o w e s t a b l i s h e d PK ( N ) f o r r e g u l a r , c u b i c a l p a c k i n g s , i t is o f c o n s i d e r a b l e i n t e r e s t to compare the results obtained with those which would have been produced by a perfect random packing, defined as a packing geometry for which the particle center to cutting plane d i s t a n c e s a r e r e c t a n g u l a r l y d i s t r i b u t e d . C l e a r l y , d u e t o p u r e s t a t i s t i c a l s c a t t e r , PK ( N ) for such a packing would not be unity for finite values for N, but increase towardS unity
w i t h i n c r e a s i n g AT, f o l l o w i n g a c u r v e o f t h e same type as found for the cubical pacldng. It was decided to simulate the perfect random packing by a simple random number experiment performed by means of a computer. N random numbers in the range 0 - 9 were extracted from a random number table and grouped in K equally wide classes. The variance of frequency was calculated. This exercise was performed 100 times, using a new set of N random numbers each time. By means of the collection of the 100 variances so obtained, PK ( N ) w a s c a l c u l a t e d as t h e f r a c t i o n o f t h e 100 samples for which condition (11) was satisfied. This was done for a range of N-values, for K = 5 and 10, g = 0.05, 0.01 and 0.005. The results are represented by the circles in F i g . 4. F o r p r a c t i c a l r e a s o n s K = 1 5 w a s n o t included in the experiment, and for similar reasons the calculations for g = 0.005 are t e r m i n a t e d b e f o r e PK ( N ) is p r a c t i c a l l y e q u a l t o 1. T h e t r e n d s a r e n e v e r t h e l e s s Very d i s t i n c t . For low N-values the random geometry gives lower PK(N) values than does the regular system, whilst for higher N-values the random g e o m e t r y g i v e s t h e g r e a t e s t PK ( N ) v a l u e s , PE (N) approaching unity fairly quickly. The reason for this difference between the two p a c k i n g g e o m e t r i e s is t h e p e r i o d i c n a t u r e o f the regular packing. This periodic nature causes the center to plane distances to be rather evenly distributed already for low N-values, whilst for higher N-values the periodic n a t u r e l e a d s t o o v e r - r e p r e s e n t a t i o n o f c e r tain size groups of flistance at the expense of others.
4. 3 _Practical e x p e r i m e n t wi t h c u t t i n g a solid block contaiTHng irregularly p a c k e d m o n o s i z e d spheres A solid specimen containing monosized spheres was made by molding a bed of suitable spheres in a continuous polyester phase. The spheres had diameter of 5.90 ram, with a standard deviation of 0.07. The specimen was cut into a number of slices. E a c h slice c o n t a i n e d a g r e a t n u m b e r o f intersected spheres, each leaving a circular cross-section in the plane of cut. Knowing the d i a m e t e r Df t h e s p h e r e s a n d b y m e a s u r i n g t h e dismeters of the cross-sections, it was possible to calculate the distribution of particle center to plane distances.
8
0.8
numbers / ~ /
0.6 0.5
I/._I ~ hysica! experiment
o,~
_! /
03
~
K~s
g=Q05
/
0.2 S
0.1 0.0
L
,
t
l0
I
1
~
1
I
I
I
I
]
1
,
N
--
20 30 40 50 60 70Fig. 5. PK(N) obtained from physical experiment
with cutting a fixed bed of irregularly packed monosized spheres ( ), a n d e x p e r i m e n t w i t h r a n d o m numbers (X--X).
The tests were carried out for K = 5 and g = 0 . 0 5 . 1 0 5 s m a l l a r e a s o n t h e slices, e a c h c o n t a i n i n g N p a r t i c l e c r o s s - s e c t i o n s , w e r e investigated f o r different values of N. T h e number of areas satisfying condition (!1) was d i v i d e d b y 1 0 5 , giving PK(N). T h e r e s u l t s o f t h i s series o f t e s t s a r e r e p r e s e n t e d b y t h e s o l i d line in Fig. 5. T h e r e s u l t s from the corresponding random number test are g i v e n b y t h e d o t t e d line. A s c a n b e s e e n f r o m these t w o curves, the physical experim e n t gave r e s u l t s v e r y s i m i l a r t o t h o s e f r o m the experiment with random numbers. The differences are n o t greater t h a n the scatter t h a t should be e x p e c t e d f r o m statistical reasons. H e n c e t h e i m p o r t a n t c o n c l u s i o n m a y be d r a w n t h a t a real " ' i r r e g u l a r " p a c k i n g o f s p h e r e s , as it would occur in practical life,does in fact, as far as the present problem is concerned, behave as a perfect r a n d o m packing.
CONCLUSIONS
(1) S t a r t i n g w i t h a r e g u l a r , c u b i c a l p a c k i n g o f m o n o s i z e d spheres, a t h e o r y has been dev e l o p e d , y i e l d i n g t h e f u n c t i o n ( l - - GK(g)/2V), which constitutes an approximate estimate of t h e p r o b a b i l i t y PK ( N ) o f t h e v a r i a n c e o f t h e d i s t r i b u t i o n in q u e s t i o n b e i n g less t h a n a spec i f i e d Ievel, g. K is t h e n u m b e r o f classes o f t h e d i s t r i b u t i o n a n d N t h e n u m b e r o f p a r t i c l e intersections considered. The approy'mations i n v o l v e d e n s u r e t h a t (1 GK(g)/]NT) will al-
-
w a y s b e less t h a n PK (N), e x c e p t f o r v e r y l o w N - v a l u e s . C l e a r l y (1 - - G K ( g ) / N ) will a p p r o a c h u n i t y w i t h increasing N n o m a t t e r h o w small is t h e s e l e c t e d v a l u e o f g. H e n c e , i t m u s t b e concluded that even for a perfect regular packing geometry, provided the intersecting p l a n e is o r i e n t e d a t r a n d o m , p a r t i c l e c e n t e r t o p l a n e d i s t a n c e s will b e l o n g t o a r e c t a n g u l a r l y distributed universe. (2) T h e c o n c l u s i o n d e r i v e d f r o m t h e o r e t i c a l analysis was tested by means of an experiment in w h i c h a r e g u l a r c u b i c a l p a c k i n g o f m o n o sized spheres cut b y randomly oriented planes was simulated on a computer. T h e simulated experiment confirmed the theoretical conclusions. (3) In order to c o m p a r e the results from cutting cubical packings with those which w o u l d have been obtained with a perfect r a n d o m packing, the beha .viour of a perfect r a n d o m packing was simulated b y a r a n d o m n u m b e r experiment. T h e plots of P K (At)produced in this w a y revealed that the r a n d o m packings in fact produced lower P K (AT) values than did the cubical packing in the range of l o w N values. T h e rev~on for this is that the particle center to plane distances, in the case of a regular packing, will be distributed in a systematic w a y over the whole range of the distances in question, the considerable scatter to be expected f r o m a p u r e l y x a n d o m selection at l o w N-values thus being avoided. For higher N-values the situation was brought in agreem e n t with w h a t w o u l d intuitively be anticipated, the P K ( N ) values for the r a n d o m packing being systematically higher than those for the cubical packing. (4) A real, physical experiment, in which plane cuts were m a d e through a bed of irregularly packed monosized spheres, e m b e d d e d in polyester, again produced P ~ (IV) values in general agreement with those already found. Considerable effort was concentrated on a v o i d i n g a n y r e g u l a r i t y in t h e p a c k i n g g e o m e t r y . C l e a r l y , h o w e v e r , i t w a s i m p o s s i b l e t o define the structure more precisely than by the word "'irregularly". C o m p a r i n g the PK(N) values f o r this p a c k i n g with t h e c o r r e s p o n d i n g values from the random packing test showed that the d i f f e r e n c e s w e r e w i t h i n t h e l i m i t s t o b e exp e c t e d f r o m statisticals c a t t e r . T h i s m e a n s t h a t t h e real " ' i r r e g u l a r " p a c k i n g d o e s in f a c t b e h a v e as a r a n d o m p a c k i n g . (5) O n t h e b a s i s o f t h e p r e v i o u s c o n c l u s i o n s ,
9
a n d since packings o f p o l y s i z e d s p h e r e s m a y be c o n s i d e r e d as s u p e r i m p o s e d p a c k i n g s o f m o n o s i z e d f r a c t i o n s , i t is c o n c l u d e d t h a t t h e h y p o t h e s i s t o b e t e s t e d d o e s in f a c t h o l d w i t h i n t h e llrnlts r e q u i r e d , f o r a n y b e d o f spherical particles, p r o v i d e d t h e n u m b e r N o f p a r t i c l e cross-sections i n c l u d e d in t h e analysis is s u f f i c i e n t l y large.
AL
ALm A~g ( A L )
ACKNOWLEDGEMENTS
T h e a u t h o r s wish t o e xpr es s t h e i r g r a t i t u d e t o t h e R o y a l N o r w e ~ n C o u n c i l f o r Sci ent i fi c and Industrial Research for the f i na nc i a l supp o r t w h i c h r e n d e r e d this i n v e s t i g a t i o n possible. T h a n k s are also d u e t o d i r e c t o r Dr. J.A. A n d e r sen of t h e Chr. Michelsen I n s t i t u t e for his stimulating interest.
A
AO
o2(Ae)
LIST OF SYMBOLS
K
Ni N Hf
H g
Px (N), A
(L m, n) rim n
r, Tand ~ = ai + b]
n u m b e r o f e q u a l l y w i de classes the number of sampled particle cross-sections b e l o n g i n g t o size class i t h e t o t a l n u m b e r o f s a m p l e d particle cross-sections the relative frequency of class i m e a n o f H t f o r all classes v a r i a n c e o f t h e f r e q u e n c y distribution c h o s e n l i m i t f o r o~ t h e p r o b a b i l i t y o f o~ < g class w i d t h , A = 1 / 2 K v ar ian c e o f t h e c o n t i n u o u s frequency distribution coordinates for a sphere center r ad ius v e c t o r f o r a s p h e r e c e n t e r u n i t v e c t o r s in t h e c o o r d i n a t e system unit vector normal to the plane
+c~ d
hlmn
(& M,N)
distance from origin to the plane distance f r o m the sphere center in (/, m, n) to the plane total solid angle within which cannot be orientated if o~ ~ g coordinates for a sphere center denoting a periodicity. The numbers h a v e n o c o m m o n f a c t o r . (AT also denotes the n-tuber of crosss e c t i o n s o n a p l a n e c u t , as previously mentioned)
~g(AL)
t h e di st ance f r o m origin t o t h e s p h e r e c e n t e r (L, M, AT). T h e integer part of Ar-equals twice the n u m b e r o f d i f f e r e n t c e n t e r t o pla n e distances o c c u r r i n g f o r t h e peri o d i c i t y , H e n c e AL is d e n o t e d t h e period the m a x i m u m period for which G~ > g is possible the solid angle surrounding the periodicity (L, M, N ) within which o~ > g t h e t o t a l area o f t h e p l a n e c u t , w h i c h f o r a cubical p a c k i n g geom e t r y will a p p r o x i m a t e l y equal t h e t o t a l n u m b e r N o f cross-sections angle between F and the periodicity variance of frequency for the continuous distribution of the center to plane distances w h e n } lies in the vicinity of the periodicity the solid angle within which o~(AO) ~ 2 K g i n t h e vicinity o f the periodicity total solid angle w i t h i n w h i c h
G~ ~ 2 K g
GK (g)
FK P,,
p a r a m e t e r o c c u r r i n g in t h e exp r e s s i o n f o r t h e t h e o r e t i c a l l y derived l o w e r l i m i t f o r P~c (N), given in eqn. (25). G K ( g ) is given in (23) given in eqn. (19) the fraction of sphere centers w i t h i n a spherical shell o f u n i t t h i c k n e s s having c o o r d i n a t e s (L, M, N ) w i t h n o c o m m o n f a c t o r
REFERENCES I M. Delesse, Pour d~terminer la composition des roches, Procede Mecan., (1862) 379. 2 A. Rosiwal, Ueber geometrische Gesteinsanalysen, Verhandlungen, 5, 6 (1898) 1 4 ~ 3 T.H. Hagermann, A m e t h o d for assessment of particle size and its distribution in finely grained sedimentary rocks, Geol. Foren. Stockholm Forh., 46 (1925) 325 (in Swedish). 4 S.D. Wicksell, A study o f the p~operties of globular distributions, Arkiv Mat. Astron. Fysik., 18 ( 1 9 2 4 ) 1. 5 S.I).WickselI, T h e corpuscle problem: A mathematical study of a biometrie problem, Biometriea, 1 7 ( 1 9 2 5 ) 84. 6 G. T a m m a n a n d W. C r o n e , Z. A n o r g . A l l g e m . Chem., 187 (1930) 289.
10 7 E. S c h e i l , Die B e r e c h n u n g d e r A n z a h l u n d G r ~ s s e n v e r t e i l u n g kugelir~rmiger K r i s t a l l e i n u n d u r c h s i c h tigen KDrpern mit Hilfe der d u r c h e i n e n e b e n e n S e h n i t t e r h a l t e n e n S c h n i t t k r e i s e , Z. A n o r g . A l i g e m . C h e m . , 201 ( 1 9 3 1 ) 259. 8 E. S c h e i l a n d H. W u r s t , S t a t i s t i s c h e Gef~dgeunters u c h u n g e n II, Z. M e t a l l k . , 28 ( 1 9 3 6 ) 3 2 0 . 9 E. S c h e i l , S t a t i s t i s c h e G e f d g e u n t a r s u c h u n g e n I, Z. M e t a l l k . , 27 ( 1 9 3 5 ) 1 9 9 . 1 0 E. S c h e i l a n d A . L a n g e w e i s e , A r c h l y . E i s e n h i i t t e n w e s e n , 11 ( 1 9 3 7 ) 93. 11 H . A . S c h w a r t z , T h e m e t a l l o g r a p h i c d e t e r m i n a t i o n o f t h e size d i s t r i b u t i o n o f t e m p e r c a r b o n n o d u l e s , Metals Alloys, 5 (1934) 139. 1 2 S . A . S a l t y k o v , S t e r e o m e t r i c m e t a l l o g r a p h y , Met a l i u r g i z d a t , M o s c o w , 2 n d e d n . , 1 9 5 8 , p. 4 4 6 . 13 H . J . Oel, K r i s t a l l w a c h s t u ~ b e i m S i n t e r n y o n Hochtemparaturoxiden, Rapport yon dem Max P l a n k I n s t i t u t f'dr S i l i k a t f o r s c h u n g , 87 W i i r z b u r g , Deutschland. 1 4 J . t~Ieisner, E s t i m a t i o n o f t h e d i s t r i b u t i o n o f diam e t e r s o f spherical particles f r o m a given g r o u p e d distribution of diameters of observed circles formed by a plane section, Statist. Neerlandica, 1 ( 1 9 6 7 ) 11. o 15 H . E . R o s e , T h e d e t e r m i n a t i o n o f t h e g r a i n - s i z e distribuLion o f a spherical granular material emb e d d e d in a m a t r i x , S e d i m e n t o l o g y , 1 0 ( 1 9 6 8 ) 293. 1 6 E.T. W h i t t a k e r a n d G.N. W a t s o n , M o d e r n A n a l y s e r , C a m b r i d g e U n i v . Press, C a m b r i d g e , 4 t h e d n . , 1 9 2 7 , p. 229. 17 S. D r a p a l a n d V . H o r a l e k , S o m e r e l a t i o n s b e t w e e n parameters of structure of metallographic specimen surface and in t h e space o f m etal s p e c i m e n s , A c t a Tech., 6 (1959) 474. 18 M.G. K e n d a l l a n d P . A . P . M o r a n , G e o m e t r i c a l P r o b a b i l i t y , G r i f f i n , L o n d o n , 1 9 3 6 , p. 86.
1 9 H . E . E x n e r , A n a l y s i s o f grain- a n d p a r t i c l e - s i z e d i s t r i b u t i o n s in metallic materials, Int. Met. Rev., 17 ( 1 9 7 2 ) 25. 2 0 T . C . P o w e r s , T h e air r e q u i r e m e n t s o f f r o s t resistant concrete, Proceedings Highway Research Board, 29 ( 1 9 4 9 ) 189. 21 T . F . Willis, D i s c u s s i o n o f a p a p e r b y T . C . P o w e r s : T h e air r e q u i r e m e n t s o f f r o s t r e s i s t a n t c o n c r e t e , P r o c e e d i n g s Highway R e s e a r c h Board, 29 (1949) 203. 22 G.W. L o r d a n d T . F . Willis, C a l c u l a t i o n o f air b u b b l e size d i s t r i b u t i o n f r o m r e s u l t s o f a R o s i w a l t r a v e r s e o f a e r a t e d c o n c r e t e , A S T M Bull. 1 0 ( 1 9 5 1 ) 56. 23 A . G . S p e k t o r , A n a l y s i s o f d i s t r i b u t i o n o f s p h e r i c a l particles in n o n - t r a n s p a r e n t structures, Zavodsk. L a b . , 16 (2) ( 1 9 5 9 ) 1 7 3 . 24 R . L . F u l l m a n , M e a s u r e m e n t o f p a r t i c l e s sizes i n opaque bodies, J. Metals, 5 (1953) 447. 25 H. M i i n z e r a n d P. S e h n e i d e r h b h n , D a s S e h n e n s c h n i t t v e r f a h r e n , Heidelbergar Beitr. Mineral. Petrog., 3 (1953) 456. 26 E. H o r i k w a , O n t h e n e w m e t h o d o f r e p r e s e n t a t i o n o f t h e m i x t u r e o f several g r a i n sizes, T e t s u t o H a g a n e , 40 (1954) 991. 27 J.W. C a h n a n d R . L . F u l l m a n , O n t h e u s e o f l i n e a l a n a l y s i s f o r o b t a i n i n g p a r t i c l e size d i s t r i b u t i o n f u n c t i o n s in o p a q u e samples, J. Metals, 5 (1956) 610. 28 G. B o c k s t i e g e l , E i n e e i n f a c h e F o r m e l z u r B e r e c h nung rKumlicher Gr~ssenverteilung aus durch L i n e a r a n a l y s e e r h a l t e n e n D a t e n , Z. M e t a l l k . , 57
(1966) 647. 29 G. E n s t a d , A c r i t i c a l a n a l y s i s o f t h e a s s u m p t i o n that the distances from the sphere centers to a r a n d o m plane t h r o u g h a b e d o f s p h e r e s are rectangularly distributed, CMI R e p t . 7 2 0 0 3 - 1 / G E ( 1 9 7 2 ) (in N o r w e g i a n ) .