Formation Dynamics Information - ACS Publications - American

Chapter 24

Formation Dynamics

Information

Downloaded by UNIV OF MASSACHUSETTS AMHERST on May 21, 2018 | https://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch024

Can It Be Derived from the Fractal Structure of Fumed Fineparticles? Brian H. Kaye and G. G. Clark Physics Department, Laurentian University, Sudbury, Ontario P3E 2C6, Canada In recent years several publications have described the fractal structure of aerosol systems. In this communication a study of the distribution of boundary fractal dimensions of two populations of two commercially available carbonblack profiles are reported. Both sets of carbonblack profiles manifested structural and textural fractals. It is suggested that the type of distribution function describing the many different fractal boundaries manifest by the profiles indicates that the systems are produced by the interaction of many small causes of the same magnitude. It is also suggested that the difference between the two populations indicates possible differences in the formation dynamics for the two sets of carbonblack profiles. F r a c t a l g e o m e t r y i s t h e study o f the structure o f r u g g e d systems. It h a s been s h o w n b y s e v e r a l w o r k e r s that t h e s t r u c t u r e o f f i n e p a r t i c l e s c r e a t e d by a f u m i n g process results i n agglomerated fineparticles w h i c h have fractal structure (1.2.3). These fineparticle systems include important c o m m e r c i a l p i g m e n t s s u c h as c a r b o n b l a c k a n d t i t a n i u m d i o x i d e ( 4 ) . It has been s h o w n that t h e v a r i o u s aspects o f t h e structure o f a n a g g l o m e r a t e d f i n e p a r t i c l e f o r m e d b y t h e c o l l i s i o n a n d s t i c k i n g t o g e t h e r o f u n i t spheres c a n be d e s c r i b e d b y t h e u s e o f f r a c t a l d i m e n s i o n s . T h e fractal dimension o f a s y s t e m i s an a d d e n d u m to the t o p o l o g i c a l d i m e n s i o n o f a s y s t e m w h i c h d e s c r i b e s t h e space f i l l i n g a b i l i t y o f the c u r v e . T h u s i n F i g u r e 1 the fractal d i m e n s i o n s a n d t o p o l o g i c a l d i m e n s i o n s o f a series o f r u g g e d l i n e s are shown. H i g h r e s o l u t i o n e l e c t r o n m i c r o g r a p h s s h o w that c a r b o n b l a c k a g g l o m e r a t e s are f o r m e d b y spheres o f t h e same s i z e w h i c h c o l l i d e w i t h e a c h o t h e r i n t h e t u r b u l e n t r e g i o n s o f the f l a m e to p r o d u c e a g g l o m e r a t e s o f v a r i o u s shape a n d s i z e . In Figure 2 a digitized carbonblack profile w h i c h h a s been s t u d i e d b y several w o r k e r s i s s h o w n (5.,6_,Z,iL2)c h a r a c t e r i z e t h e f r a c t a l d i m e n s i o n o f the b o u n d a r y o f t h e a g g l o m e r a t e , i t s h o u l d b e n o t e d that s o m e other w o r k e r s w h o h a v e s t u d i e d t h e i n t e r n a l structure o f agglomerates h a v e also used a f r a c t a l d i m e n s i o n r e l a t e d to t h e i n t e r n a l s t r u c t u r e o f the a g g l o m e r a t e s . T h i s fractal dimension w h i c h is T

o

0097-6156/91/0472-0372$06.00/0 © 1991 American Chemical Society

Provder; Particle Size Distribution II ACS Symposium Series; American Chemical Society: Washington, DC, 1991.


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Topological Dimension

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Formation Dynamics Information

Fractal Dimension

1.00

1.00

1.00

1.02

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1.00

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Figure 1: The ruggedness of a line can be described by a fractional addendum to the topological dimension. The combination is known as the fractal dimension of the line. (Reproduced with permission from ref. 2. Copyright 1990 V C H Publishers.)



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P A R T I C L E S I Z E D I S T R I B U T I O N II

Digitized Profile

5 Step Chords

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Structure 8 =I.35

Texture 8 «I.I3

5.0-

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Figure 2: In the equipaced method for deducing the fractal dimension of a boundary, polygons of increasing side length are constructed on a digitized version of the boundary, (a) Various polygons used to estimate the perimeter of the carbonblack profile, (b) Richardson plot of normalized perimeter—resolution parameter data generated by equipaced exploration of the carbonblack profile. A = number of digitized steps along the profile to form a chord; P = perimeter estimate; and 6 = fractal dimension over a given range of inspection resolutions. A and P are normalized with respect to the maximum projected length of the profile. (Reproduced with permission from ref. 2. Copyright 1990 V C H Publishers.)



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d e s c r i b e d as a d e n s i t y o r mass f r a c t a l d i m e n s i o n is d i f f e r e n t f r o m the b o u n d a r y f r a c t a l d i s c u s s e d i n t h i s c o m m u n i c a t i o n (3. 4 ) . W h e n studying the b o u n d a r y f r a c t a l d i m e n s i o n o f an a g g l o m e r a t e s u c h as a c a r b o n b l a c k p r o f i l e , one is s t u d y i n g a t w o d i m e n s i o n a l p r o j e c t i o n o f the structure o f the system. F o r t h i s reason the p r o j e c t e d b o u n d a r y f r a c t a l d i m e n s i o n i s o f t e n less r u g g e d t h a n the f r a c t a l d i m e n s i o n that w o u l d be c h a r a c t e r i z e d b y e m b e d d i n g the c a r b o n b l a c k i n a r e s i n a n d t a k i n g a s e c t i o n t h r o u g h the agglomerate. A l t h o u g h information is lost by s t u d y i n g projected boundary f r a c t a l d i m e n s i o n s i t appears that there is s t i l l a u s e f u l a m o u n t o f information in such fractal boundaries. Obtaining electron micrographs o f c a r b o n b l a c k s i s an e x p e n s i v e p r o c e s s . T e c h n i q u e s f o r l o o k i n g at s e c t i o n s t h r o u g h c a r b o n b l a c k a g g l o m e r a t e s set i n r e s i n w i l l r e q u i r e a d i f f e r e n t set o f b o u n d a r y fractals to d e s c r i b e s u c h structures. A l l o f the f r a c t a l d i m e n s i o n s d i s c u s s e d i n t h i s c o m m u n i c a t i o n are b o u n d a r y f r a c t a l s of projected images. T h i s fact s h o u l d be r e m e m b e r e d w h e n e v a l u a t i n g the information presented i n this c o m m u n i c a t i o n . E s s e n t i a l l y the f r a c t a l d i m e n s i o n o f a b o u n d a r y s u c h as that o f the c a r b o n b l a c k p r o f i l e s h o w n i n F i g u r e 2 is s t u d i e d b y e s t i m a t i n g the b o u n d a r y at a series o f r e s o l u t i o n s . These boundary estimates and an appropriate resolution parameter, i n n o r m a l i z e d f o r m a t , are p l o t t e d o n a l o g - l o g g r a p h w h i c h is k n o w n as a Richardson plot (10). It has been s h o w n that the slope o f the data l i n e s o n s u c h a g r a p h c a n be u s e d to d e d u c e the f r a c t a l d i m e n s i o n o f the b o u n d a r y .

Experimental

Studies

T h e e x p e r i m e n t a l p r o c e d u r e u s e d i n t h i s c o m m u n i c a t i o n to c h a r a c t e r i z e the b o u n d a r i e s o f the c a r b o n b l a c k p r o f i l e s i s the e q u i p a c e d p o l y g o n e x p l o r a t i o n t e c h n i q u e (2). T h e basic c o n c e p t s o f t h i s t e c h n i q u e c a n be a p p r e c i a t e d f r o m F i g u r e 2. T h e d i g i t i z e d o u t l i n e o f the p r o f i l e is stored i n a c o m p u t e r m e m o r y a n d t h e n p o l y g o n s o f d e c r e a s i n g r e s o l u t i o n are c o n s t r u c t e d o n the o r i g i n a l p r o f i l e b y d r a w i n g c o r d s to the b e g i n n i n g a n d e n d o f the distances p a c e d out a r o u n d the p r o f i l e . T h u s i n part (a) o f F i g u r e 2 the b a s i c d i g i t i z a t i o n o f the p r o f i l e and three d i f f e r e n t p o l y g o n s u s e d to e s t i m a t e the p r o f i l e are s h o w n . F o r m a n y c a r b o n b l a c k p r o f i l e s i t has b e e n d i s c o v e r e d e x p e r i m e n t a l l y that the d a t a l i n e s f o r the e x p l o r a t i o n o f the p r o f i l e m a n i f e s t t w o o r m o r e l i n e a r r e l a t i o n s h i p s l e a d i n g to the e s t i m a t i o n o f at least t w o different fractal d i m e n s i o n s as s h o w n i n the F i g u r e 2. It i s b e c o m i n g apparent that the f r a c t a l d i m e n s i o n 8 f o r s y s t e m s s u c h as b o u n d a r y f r a c t a l s o f a e r o s o l f i n e p a r t i c l e s c o n t a i n i n f o r m a t i o n o n the f o r m a t i o n d y n a m i c s and the d e t a i l e d structure o f the p r o f i l e at d i f f e r e n t r e s o l u t i o n s (2. 6). T h u s the data l i n e o f F i g u r e 2 at h i g h r e s o l u t i o n c o r r e s p o n d s to h i g h l e v e l s c r u t i n y o f the texture o f the b o u n d a r y o f the a g g l o m e r a t e s w h e r e a s the data l i n e at coarse r e s o l u t i o n c o m e s f r o m e x p l o r i n g the c o a r s e s t r u c t u r e o f the a g g l o m e r a t e . (the f r a c t a l d i m e n s i o n o f the g r a p h is related to the slope o f the data l i n e by the r e l a t i o n s h i p 6=1+ Iml w h e r e m is the slope o f the l i n e (20) T h e data l i n e o n the R i c h a r d s o n p l o t f o r coarse r e s o l u t i o n is related to the gross s t r u c t u r a l features o f the a g g l o m e r a t e a n d t h i s f r a c t a l d i m e n s i o n i s d e s c r i b e d as the s t r u c t u r a l boundary fractal dimension. Discussion T h e p h y s i c a l s i g n i f i c a n c e o f the s t r u c t u r a l f r a c t a l d i m e n s i o n c a n be a p p r e c i a t e d f r o m the data presented f o r three m o d e l p r o f i l e s i n F i g u r e 3



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Figure 3: The magnitude of the boundary fractal dimension at coarse resolution is related to the gross structure of the profile. (Reproduced with permission from ref. 2. Copyright 1990 V C H Publishers.)



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c r e a t e d b y j o i n i n g g l a s s spheres together to create d i f f e r e n t t y p e s o f agglomerates. A s i m p l e c o m p a c t a g g l o m e r a t e has a s t r u c t u r a l f r a c t a l d i m e n s i o n o f the o r d e r o f 1.17 w h e r e a s m o r e c o m p l e x a g g l o m e r a t e s formed b y the c o l l i s i o n o f s e v e r a l c o m p a c t a g g l o m e r a t e s h a v e h i g h e r b o u n d a r y fractal d i m e n s i o n s o f the order o f 1.37 to 1.42. T h e quantity denoted b y F Q o f F i g u r e 3 is the n o r m a l i z i n g factor used to prepare the data o f the F i g u r e 3. E x p e r i m e n t a l studies s h o w that at h i g h r e s o l u t i o n the p a c k i n g o f the spheres for real carbonblack agglomerates produces a textural fractal o f the o r d e r 0.08 ( 2 j . T h u s i n F i g u r e 4 some h i g h r e s o l u t i o n data f o r the c a r b o n b l a c k o f F i g u r e 2 o b t a i n e d b y a t e c h n i q u e k n o w n as e r o s i o n d i l a t i o n l o g i c is s h o w n ( i ) . T h i s s h o w s a textural fractal o f 1.07 and a structural f r a c t a l o f 1.34 i n d i c a t i n g that the a g g l o m e r a t e has p r o b a b l y b e e n formed b y the c o l l i s i o n o f s e v e r a l s u b - a g g l o m e r a t e s as suggested b y the s k e t c h g i v e n i n F i g u r e 4. (2). In e r o s i o n treatment o f an i m a g e o f an agglomerate the c o m p u t e r c o n t r o l o f the i m a g e a n a l y s i s system strips o f f a l a y e r o f p i x e l s (the i m a g e screen is d i v i d e d b y the c o m p u t e r into a m o s a i c o f s m a l l squares a n d the b a s i c s m a l l square o f s u c h a m o s a i c i s k n o w n t e c h n i c a l l y as a p i x e l ) . A c o m p u t e r c a n repeat t h i s o p e r a t i o n s e v e r a l t i m e s . W h e n s u b j e c t e d to the s t r i p p i n g r o u t i n e the a g g l o m e r a t e s t r u c t u r e c h a n g e s as i n d i c a t e d i n the d i a g r a m s o f F i g u r e 5(a). W e c a n interpret t h i s as that b y stage 9 the structure l o o k s as i f it has 5 s u b s i d i a r y agglomerates. B y the t i m e w e get to 14 p i x e l s t r i p s s o m e c o n t r i b u t o r y a g g l o m e r a t e s h a v e d i s a p p e a r e d o n l y 3 b a s i c c o n t r i b u t i n g structures r e m a i n a n d o n e o f those (the b o t t o m r i g h t h a n d agglomerate) l o o k s as i f it w i l l break d o w n into 2 w i t h f u r t h e r erosion. B y u s i n g p i x e l s t r i p p i n g to estimate the p o s s i b l e n u m b e r o f c o n t r i b u t o r y p r i m a r y a g g l o m e r a t e s to a s e c o n d a r y a g g l o m e r a t e o n e c a n d e d u c e i n f o r m a t i o n o n the f o r m a t i o n d y n a m i c s a n d m a t c h it w i t h the measured fractal dimensions. T h e s e c o n d a p p r o a c h to u n d e r s t a n d i n g the p h y s i c a l s i g n i f i c a n c e o f the c a r b o n b l a c k s t r u c t u r e i s to m o d e l the g r o w t h o f v a r i o u s a g g l o m e r a t e s o n a c o m p u t e r a n d m a t c h the resultant s t r u c t u r e o f the m o d e l s y s t e m to s y s t e m s generated e x p e r i m e n t a l l y ( U J . The modelling p r o c e s s u s e d to s i m u l a t e a g g l o m e r a t e g r o w t h c a n be i l l u s t r a t e d i n t w o d i m e n s i o n s by c o n s i d e r i n g the system as s h o w n i n F i g u r e 6(b). The s y s t e m s o f F i g u r e 6(b) have been g r e a t l y s i m p l i f i e d to e x p l a i n the a c t u a l p r o c e s s rather t h a n to d e p i c t d e t a i l e d e x p e r i m e n t s . A real m o d e l l i n g e x p e r i m e n t w o u l d use a m u c h s m a l l e r p i x e l s i z e w i t h the m o d e l l i n g area c o n t a i n i n g at least 5 0 0 x 5 0 0 p i c t u r e elements. A p i c t u r e element i s t a k e n to be r e p r e s e n t a t i v e o f the s u b - u n i t s j o i n i n g the g r o w i n g a g g l o m e r a t e s . The centre o f the p i x e l space is designated as a n u c l e a t i n g centre f o r the g r o w i n g agglomerate. A b l a c k p i x e l r e p r e s e n t i n g a b a s i c u n i t o f the a g g l o m e r a t e s t r u c t u r e i s then a l l o w e d to a p p r o a c h the n u c l e a t i n g c e n t r e b y a r a n d o m w a l k p r o c e s s u n t i l b y c h a n c e it i m p i n g e s o n the n u c l e a t i n g centre. A s t h i s t i m e a d e c i s i o n is taken as to the p r o b a b i l i t y o f the i m p i n g i n g p i x e l j o i n i n g the g r o w i n g a g g l o m e r a t e . T h u s i n the s i m p l e s t case it i s a s s u m e d that e v e r y s t a g g e r i n g p i x e l a p p r o a c h i n g the n u c l e a t i n g centre has 1 0 0 % c h a n c e o f j o i n i n g the a g g l o m e r a t e i f it i m p i n g e s o r t h o g o n a l l y o n the n u c l e a t i n g centre. A two-dimensional model of this t y p e o f g r o w t h generates the type o f a g g l o m e r a t e s h o w n i n F i g u r e 6(a) w h i c h is k n o w n as W h i l t e n and S a n d e r a g g l o m e r a t e (12). If one n o w changes the s l i c k i n g rules so that for e x a m p l e w h e n the s t a g g e r i n g p i x e l e n c o u n t e r s the g r o w i n g a g g l o m e r a t e there is o n l y a 5 0 % c h a n c e o f j o i n i n g the a g g l o m e r a t e and the s t a g g e r i n g p i x e l m a y m o v e a w a y u n t i l it a g a i n e n c o u n t e r s the g r o w i n g a g g l o m e r a t e s . In F i g u r e 6(b) the a p p e a r a n c e o f the t w o - d i m e n s i o n a l a g g l o m e r a t e s g r o w n w i t h d i f f e r e n t



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3.5 I 0.01

j

i

i

0.02

I i 0.05

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i i i I 0.10

i 0.2

Figure 4: A boundary structural fractal dimension of the order of 1.40 is manifest by agglomerates which appear to have been formed by the collision of several compact agglomerates. (Reproduced with permission from ref. 2. Copyright 1990 V C H Publishers.)

Original

9 Erosions

14 Erosions

Figure 5: Pixel stripping routines called erosion routines can "dissolve" the image of a profile into probable constituent parts. (Reproduced with permission from ref. 2. Provder; Copyright 1990 C H Publishers.) Particle SizeVDistribution II ACS Symposium Series; American Chemical Society: Washington, DC, 1991.


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s t i c k i n g r u l e s are s h o w n . S t u d i e s o n the t h r e e - d i m e n s i o n a l s t r u c t u r e o f a g g l o m e r a t e s m o d e l l e d i n t h i s w a y are b e i n g c a r r i e d out b y s e v e r a l g r o u p s o f scientists ( 1 3 . 1 4 ) . T h e m o d e l l i n g process o f F i g u r e 6(a) o b v i o u s l y matches the g r o w t h o f agglomerates i n t u r b u l e n c e o f a f l a m e . It m a y be that the g r o w t h o f a c o m p l e x agglomerate c a n take p l a c e i n t w o stages w i t h the s t i c k i n g r u l e s c h a n g i n g d u r i n g the f o r m a t i o n p e r i o d . I n F i g u r e 6(c) the g r o w t h o f an a g g l o m e r a t e m o d e l l e d o n a c o m p u t e r w h e n after a c e r t a i n p e r i o d the s t i c k i n g rules c h a n g e s h o w h o w the a g g l o m e r a t e c a n h a v e a s p i n d l y centre w i t h denser f o r m a t i o n s t o w a r d the o u t s i d e . I f one l o o k s at the series o f p r o f i l e s s h o w n i n F i g u r e 6(b) it w o u l d seem to be a reasonable h y p o t h e s i s to suggest that 1 0 0 % s t i c k i n g a g g l o m e r a t e w o u l d be the m o s t d e s i r a b l e c o m m e r c i a l p i g m e n t s i n c e it w o u l d o f f e r the greater s u r f a c e area to the i n t e r a c t i o n o f p h o t o n s i m p i n g i n g u p o n its structure i n a p a i n t f i l m . F u r t h e r m o r e , u n d e r the c o n d i t i o n s o f shear o p e r a t i n g d u r i n g the p h y s i c a l m a n u f a c t u r e o f p a i n t the a g g l o m e r a t e i s l i k e l y to break d o w n to p r o d u c e m o r e i n d e p e n d e n t s c a t t e r i n g centres. T h e p o s s i b i l i t y that t h i s i s a w o r k i n g h y p o t h e s i s a i m e d at o p t i m i z i n g i s u n d e r i n v e s t i g a t i o n ( 1 5 ) . If one e x a m i n e s a set o f p r o f i l e s f r o m a c o m m e r c i a l product it s o o n b e c o m e s apparent that w i t h i n any one set o f f i n e p a r t i c l e s there i s a w h o l e range o f b o u n d a r y f r a c t a l d i m e n s i o n s . T h u s i n F i g u r e 7 the p r o f i l e s o f t w o d i f f e r e n t c o m m e r c i a l l y a v a i l a b l e c a r b o n b l a c k s y s t e m s are s h o w n . These p r o f i l e s w e r e t r a c e d f r o m h i g h m a g n i f i c a t i o n e l e c t r o n m i c r o g r a p h s o f the c a r b o n b l a c k p r o v i d e d b y N . M a c e o f the C a b o t C o r p o r a t i o n (JJL). T h e trade names o f these t w o different c a r b o n b l a c k s m a n u f a c t u r e d and s o l d b y the C a b o t C o r p o r a t i o n are V u l c a n 7 H and S t e r l i n g N S I (16). A p o p u l a t i o n o f p r o f i l e s t a k e n f r o m a series o f h i g h m a g n i f i c a t i o n e l e c t r o n m i c r o g r a p h s are s h o w n i n F i g u r e 8. T h e p r o b l e m addressed i n t h i s c o m m u n i c a t i o n i s an e x p l o r a t i o n o f the p o s s i b i l i t y that the b o u n d a r y f r a c t a l d i m e n s i o n d i s t r i b u t i o n f u n c t i o n o f the t w o sets o f p i g m e n t s are c h a r a c t e r i s t i c o f the t w o p r o d u c t s . E a c h o f the F i g u r e 8 p r o f i l e s o f the t w o sets o f p i g m e n t s w e r e c h a r a c t e r i z e d b y u s i n g the e q u i p a c e d t e c h n i q u e f o r e x p l o r i n g the s t r u c t u r e o f the p r o f i l e . T y p i c a l data plots f o r representative p r o f i l e s f r o m the t w o sets o f f i n e p a r t i c l e s are s h o w n i n F i g u r e 9. T h e s e p r o f i l e s e x h i b i t t w o ranges o f l i n e a r i t y i n t h e i r R i c h a r d s o n ' s p l o t s d e f i n i n g the s t r u c t u r a l a n d t e x t u r a l f r a c t a l d i m e n s i o n o f the p r o f i l e . T h u s at c o a r s e r e s o l u t i o n the s t r u c t u r a l b o u n d a r y f r a c t a l d i m e n s i o n o f the V u l c a n p r o f i l e s h o w n i n F i g u r e 9 i s 1.41. A t h i g h r e s o l u t i o n the t e x t u r a l b o u n d a r y f r a c t a l d i m e n s i o n o f the p r o f i l e is 1.12 a n d this i s d e s c r i b e d as the textural fractal o f the p r o f i l e . The s t r u c t u r a l a n d t e x t u r a l f r a c t a l f o r the r e p r e s e n t a t i v e S t e r l i n g b l a c k p r o f i l e s are s h o w n i n F i g u r e 9(b) are 1.23 and 1.08 r e s p e c t i v e l y . The i n f o r m a t i o n o n the p o p u l a t i o n s t r u c t u r a l a n d t e x t u r a l b o u n d a r y fractals f o r the t w o arrays o f F i g u r e 8 have been plotted i n F i g u r e 10. The s t r u c t u r a l a n d t e x t u r a l d i s t r i b u t i o n data appears to be d e s c r i b a b l e by a G a u s s i a n d i s t r i b u t i o n w i t h the d i s t r i b u t i o n f o r the t w o p i g m e n t s b e i n g clearly different. T h e data presented i n these t w o g r a p h s is a d m i t t e d l y sparse but it s h o u l d be a p p r e c i a t e d that the cost and d i f f i c u l t y o f a c q u i r i n g the n e c e s s a r y e l e c t r o n m i c r o g r a p h s and the cost o f c a r r y i n g out the a n a l y s i s o f each p r o f i l e is an e x p e n s i v e o p e r a t i o n . In fact it was o n l y p o s s i b l e to c a r r y out this e x p e r i m e n t because students were a s k e d to c h a r a c t e r i z e the p r o f i l e s o f F i g u r e 8 as part o f t h e i r u n d e r g r a d u a t e laboratory projects. T h e mean fractal d i m e n s i o n and standard d e v i a t i o n seem to be q u i t e c h a r a c t e r i s t i c o f the t w o different p o p u l a t i o n s . T h e data o f F i g u r e 10 appears to suggest that the v a r i a t i o n i n the f r a c t a l d i m e n s i o n s o f a p o p u l a t i o n o f f i n e p a r t i c l e s is a p o t e n t i a l m e t h o d f o r c h a r a c t e r i z i n g the


380


a)


B

4 5 6 7 8 9 10 11 12 13

i! I 2 3 4 5 6 7 8 9 10II 12 13

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{§J|f 100%

50%

25%

10%

5%

1%

C)

400pixels 800 pixels 1500pixels

1800pixels

2200pixels

1% Sticking

100% Sticking

Figure 6: Randomwalk modeling of cluster growth can be used to create models of agglomerates which may be useful when interpreting the physical significance of observed rugged structures, (a) Basic modeling proceeding and Whitten and Sander agglomerate; (b) agglomerates grown using different sticking probabilities; and (c) agglomerates whose structure manifest changing sticking probabilities.

Stirling NSI

Vulcan 7H

Magnified 250 000 X

Figure 7: Typical profiles micrographs.

of

carbonblacks

traced

from


electron


381


24. K A Y E & C L A R K

Figure 8: Carbonblack profiles taken from high magnification electron micrographs of two commercially available carbonblacks {5} from the Cabot Carbon Corporation, (a) Vulcan 7 H and (b) Sterling NSI. (Reproduced with permission from ref. 16. Copyright Cabot Corporation.)




382



383


24. K A Y E & C L A R K

Figure 10: Distribution data for the variation of structural and textural boundary fractal dimensions of the two populations of carbonblack profiles displayed in Figure 1. (a) Vulcan 7 H and (b) Sterling NSI.



384


differences i n populations o f commercial pigments. T h i s i s a m e t h o d that w i l l b e c o m e m o r e f e a s i b l e as a research a n d q u a l i t y c o n t r o l t o o l as m o r e sophisticated image characterization equipment becomes available for fractal boundary characterization and automated image analysis systems. T h e fact that the t e x t u r a l a n d structural d i s t r i b u t i o n f u n c t i o n s f o r b o t h sets o f p r o f i l e s are p r o b a b l y G a u s s i a n d i s t r i b u t e d w o u l d i n d i c a t e that t h e y h a v e been f o r m e d b y t h e r a n d o m i n t e r a c t i o n o f m a n y causes o f approximately equal strength i.e. there i s n o d o m i n a n t f o r m a t i o n mechanism. T h e fact that the s t r u c t u r a l f r a c t a l d i m e n s i o n o f the V u l c a n 7 H carbonblacks is considerably h i g h e r than the structural fractal d i m e n s i o n o f the S t e r l i n g c a r b o n b l a c k s w o u l d seem t o i n d i c a t e that i n the f o r m a t i o n process agglomerates were c o l l i d i n g with each other f o r a longer period i n the p r o d u c t i o n o f t h e V u l c a n c a r b o n b l a c k t h a n i n the case o f the S t e r l i n g carbonblack. I n other w o r d s the f o r m a t i o n o f the a g g l o m e r a t e s w a s q u e n c h e d m o r e q u i c k l y i n the case o f the S t e r l i n g p r o d u c t t h a n f o r the V u l c a n product. F u r t h e r w o r k i s i n progress to test the h y p o t h e s e s that less rugged p r o f i l e s have been created b y p r e v e n t i n g the a g g l o m e r a t i o n process from continuing ( 1 7 ) .

Literature Cited 1. Mandelbrot, B.B. The Fractal Geometry of Nature; W. Freeman: San Francisco, 1983. 2. Kaye, B.H. A Randomwalk Through Fractal Dimensions; VCH Publishers; Weinheim, Federal Republic of Germany, 1990. 3. Meakin, P.: "Simulations of Aggregation Process," Chapter 3, in the Fractal Approach to Heterogeneous Chemistry, edited by Avnir, D., John Wiley and Sons Limited, 1989. 4. Kaye, B.H. "Characterizing the Structure of Fumed Pigments Using the Concepts of Fractal Geometry", (In press), Particle & Particle Systems Characterization. 5. A.I. Medalia, G.J. Hornik, "Pattern Recognition Problems in the Study of Carbonblack", Pattern Recognition, 4 (1975) 155. 6. A.I. Medalia, "Dynamic Shape Factors of Particles", Powder Technol., 4 (1970-1971) Pgs.. 117-138 7. A.G. Flook, "The Use of Dilation Logic on the Quantimet to Achieve Fractal Dimension Characterization of Texture and Structure of Profiles", Powder Technol., 21 (1978) Pgs. 295-298. 8. H. Schwarz, H.E. Exner, "The Implementation of the Concept of Fractal Dimensions on a Semi-automatic Image Analyzer", Powder Technol., 27 (1980) Pgs. 207-213. 9. B.H. Kaye, "Multi Fractal Description of A Rugged Fineparticle Profile", Part, Charact. 1 (1984), 14-21 10. B.H. Kaye "Direct Characterization of Fineparticles", Wiley & Sons, New York, 1981, See Chapter 10 11. The relationship between the fractal dimension and the formation dynamics of a system has recently been discussed in a lecture entitled "Fractalicious Structures and Probable Events" an address given to the Math Educators of Canada Conference held in Vancouver, May 23-25 by Professor B.H. Kaye, copies of the lecture are available from Laurentian University. 12. T.A Whitten, L.M. Sander, "Diffusion Limited Aggregation; A Kinetic Critical Pnenomenon",Phys. Rev. Lett., 47 (1981) 1400.



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&

CLARK


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13. R. Richter, L.M. Sander, Z. Cheng, "Computer Simulations of Soot Aggregation", J. Colloid Interface Sci., 100 (1984) Pgs. 203-209. 14. P. Meakin, R. Jullien, "The Effects of Random Bond Breaking on Diffusion Limited Cluster-Cluster Aggregation", J. Phys., 46, (1985) Pg. 1543. 15. B.H. Kaye, G.G. Clark, "Evaluating the Physical Significance and Health Hazards of Agglomerate Structure in Combustion Generated Aerosols", in preparation. 16. Cabot Corporation, Concord Road, Billerica, MA, U.S.A., 01821. 17. Experimental work in fractal systems has been developing faster than the appearance of detailed scientific publications which are presented in scientific journals. The information on the formation dynamics embedded in the distribution functions of systems such as carbonblack are explored in a publication in preparation entitled "Discovering the Surprising Patterns of Chaos and Complexity" by B.H. Kaye. RECEIVED

A p r i l 12, 1991


Formation Dynamics Information - ACS Publications - American

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