Extremely Wide Dynamic Range, High-Resolution Particle Sizing by

Chapter 7

Extremely Wide Dynamic Range, High-Resolution Particle Sizing by Light Scattering

Downloaded by CORNELL UNIV on August 2, 2012 | http://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch007

S. E. Bott and W. H. Hart Scientific Instruments Division, Coulter Electronics, Inc., 29 Cottage Street, Amherst, MA 01002

Light diffraction is one of the most commonly employed methods of measuring sizes of particles in the range of 0.1μm to around 1000μm (1,2). The popularity of the method stems from its ability to make quick, facile, precise measurements as well as its adaptability to measuring samples presented in various forms, for example, stirred or pumped suspensions of particles in any clear liquid, aerosols, dry powders in an air stream, particles adhering to a glass slide, etc (3,4). Diffraction measurements are based on sensing the angular patterns of light scattered by particles of different sizes when they are exposed to a collimated beam of light. Because the patterns of light scattered by particles of different sizes are highly characteristic of the particle size, a mathematical analysis of the pattern of light scattered from a sample of particles can extract an accurate, reproducible measure of the size distribution. The aforementioned flexibility of the method in measuring particles presented in a variety of ways, combined with the remoteness of the detection method (i.e. the light source and detectors are located far from the particles and therefore do not perturb the system under measurement) gives diffraction the potential for making measurements of very broadly dispersed distributions of particles. This article presents work done by the authors in extending the potentially large dynamic range of diffraction measurements as well as extending the size range over which the light scattering methods can accurately resolve particle size distributions. W h e n a b e a m o f l i g h t i s projected onto a p a r t i c l e , i n the l i m i t o f p a r t i c l e s i z e m u c h greater than the w a v e l e n g t h o f the l i g h t , the interactions o f the p a r t i c l e w i t h l i g h t c a n be b r o k e n d o w n i n t o three p h e n o m e n a : r e f l e c t i o n , r e f r a c t i o n a n d d i f f r a c t i o n . T h e f i r s t t w o p h e n o m e n a are e x p e r i e n c e d d a i l y ; the last i s a b e n d i n g o f l i g h t a r o u n d a p a r t i c l e as the p a r t i c l e interrupts o r b l o c k s part o f the p l a n a r w a v e f r o n t s o f d i e l i g h t i n c i d e n t o n the p a r t i c l e . It s h o u l d be n o t e d that i n the general case, e n c o m p a s s i n g p a r t i c l e s o f a r b i t r a r y s i z e a n d l i g h t o f a n y w a v e l e n g t h , the interaction o f tight w i t h p a r t i c l e s , c o l l e c t i v e l y termed scattering, d o e s not separate i n t o the three d i s t i n c t p h e n o m e n a m e n t i o n e d a b o v e . T h e extent o f the b e n d i n g o f the l i g h t a r o u n d a p a r t i c l e depends o n the r a t i o o f 0097-6156/91/0472-0106$06.00/0 © 1991 American Chemical Society

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


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the w a v e l e n g t h o f the l i g h t to the p a r t i c l e size. I n general, l a r g e r p a r t i c l e s d i f f r a c t the l i g h t i n t o s m a l l e r scattering angles (i.e. i n t o the n e a r l y f o r w a r d d i r e c t i o n ) w h i l e the d i f f r a c t e d l i g h t from s m a l l e r particles i s sent i n t o larger scattering angles, w h e r e the scattering angle i s d e f i n e d as the angle between the d i r e c t i o n o f p r o p a g a t i o n o f the i n c i d e n t l i g h t a n d the d i r e c t i o n from w h i c h the d i f f r a c t e d l i g h t i n o b s e r v e d . T o q u a n t i f y the d i f f r a c t e d l i g h t , a n o p t i c a l s y s t e m s u c h as that s h o w n i n F i g u r e 1, i s u s e d . T h e F o u r i e r lens i n this s y s t e m , l o c a t e d at its f o c a l l e n g t h from the p l a n e o f the detector array, o r F o u r i e r p l a n e , serves to f o c u s a l l the u n d i f f r a c t e d tight i n t o a s i n g l e p o i n t i n the center o f the array o f s e n s i n g detectors. T h i s u n d i f f r a c t e d l i g h t i s g e n e r a l l y u s e d o n l y to m o n i t o r the b e a m strength. A l l l i g h t that i s d i f f r a c t e d from a n y o f d i e p a r t i c l e s i n the b e a m i n t o a g i v e n scattering a n g l e , i s f o c u s e d b y the F o u r i e r l e n s i n t o a t h i n a n n u l u s , centered a r o u n d the p o i n t w h e r e the u n d i f f r a c t e d l i g h t i s f o c u s e d . L i g h t d i f f r a c t e d i n t o s m a l l angles p r o d u c e s a n n u l i o f s m a l l r a d i u s ; that d i f f r a c t e d i n t o l a r g e r angles results i n a n n u l i o f l a r g e r r a d i i . A detection s y s t e m c o m p r i s i n g a n array o f a n n u l a r detectors measures the pattern o f l i g h t d i f f r a c t e d from the particles i n the b e a m . F i g u r e 2 s h o w s the d i f f r a c t e d l i g h t p a t t e r n , as a f u n c t i o n o f scattering angle f o r p a r t i c l e s o f 5 5 0 a n d 2 7 5 | i m . I f particles o f m o r e than o n e s i z e are i n the b e a m , the c o m p o s i t e d i f f r a c t i o n pattern w i l l be the l i n e a r s u p e r p o s i t i o n o f the patterns c o r r e s p o n d i n g to the i n d i v i d u a l p a r t i c l e s . I n the l i m i t o f particles m u c h larger than the w a v e l e n g t h o f l i g h t a n d s m a l l scattering angles, the pattern o f d i f f r a c t e d l i g h t i s d e s c r i b e d b y the Fraunhofer Diffraction a p p r o x i m a t i o n (5). I n the m o r e g e n e r a l case, i n c l u d i n g h i g h e r scattering angles a n d arbitrary ratio o f l i g h t w a v e l e n g t h to p a r t i c l e s i z e , v a r i o u s other a p p r o x i m a t i o n s c a n be u s e d to c a l c u l a t e the scattered l i g h t pattern. F o r the s p e c i f i c case o f i s o t r o p i c s p h e r i c a l particles o f arbitrary s i z e , an e x a c t g e n e r a l e x p r e s s i o n f o r the scattered l i g h t pattern i s g i v e n b y the Mie theory ( £ ) . T h e general r e l a t i o n s h i p between the l i g h t pattern m e a s u r e d at n discrete detectors a n d the d i s t r i b u t i o n o f the particles i n m selected s i z e c l a s s i f i c a t i o n s i s g i v e n b y :

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w h e r e f^ i s the l i g h t f l u x m e a s u r e d at the i t h detector, aj i s the a m o u n t o f l i g h t f l u x d u e to p a r t i c l e s o f size c l a s s i f i c a t i o n j , d i f f r a c t e d i n t o the i t h detector a n d the x j , j = l , m are the d e s i r e d v o l u m e s o f particles o f e a c h s i z e c l a s s i f i c a t i o n . I n this e q u a t i o n , the{ a| }are c a l c u l a t e d u s i n g the F r a u n h o f e r d i f f r a c t i o n a p p r o x i m a t i o n , the M i e theory o r s o m e other e x p r e s s i o n r e l a t i n g p a r t i c l e size to d i f f r a c t i o n pattern. E q u a t i o n s o f the f o r m o f (1) h a v e standard m e a n s o f s o l u t i o n ( 7 , f i ) . I n a d i f f r a c t i o n m e a s u r e m e n t , the c h o i c e o f angles c o v e r e d b y the n detectors g e n e r a l l y determines the n u m b e r a n d range o f the m size c l a s s i f i c a t i o n s c o v e r e d b y the i n s t r u m e n t . A s a general r u l e o f t h u m b , the d y n a m i c range o f scattering angles c o v e r e d b y a n i n s t r u m e n t r o u g h l y equals the d y n a m i c r a n g e o f sizes w h i c h the i n s t r u m e n t i s c a p a b l e o f m e a s u r i n g . ( T h e d y n a m i c range o f a s a m p l e i s d e f i n e d as the r a t i o o f the largest to the s m a l l e s t particle c o n t a i n e d i n s i g n i f i c a n t percentage i n a s a m p l e . D y n a m i c range i n scattering angle refers to the ratio o f the largest to the s m a l l e s t scattering angle o v e r w h i c h l i g h t c a n be sensed i n a s i n g l e m e a s u r e m e n t . ) T h u s a n i n s t r u m e n t w h i c h measures angles from 0 . 0 3 ° to 3 ° , a d y n a m i c r a n g e o f 1 0 0 : 1 , w o u l d be e x p e c t e d to m e a s u r e p a r t i c l e s i z e from a r o u n d 8|im to a r o u n d 8 0 0 ^ m ,



PARTICLE SIZE DISTRIBUTION II

Light Source

Particle Field

Fourier Lens

Detector Array

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Angle (degrees) F i g u r e 2 . Scattered l i g h t f l u x (intensity/area) pattern f o r 2 7 5 | i m a n d 5 5 0 | i m particles.


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also a d y n a m i c range o f 1 0 0 : 1 . ( T h e center o f the s i z e range c o v e r e d b y a m e a s u r e m e n t o f a g i v e n a n g u l a r r a n g e w o u l d d e p e n d o n the w a v e l e n g t h o f l i g h t u s e d ; h o w e v e r , the d y n a m i c range o f the s i z e m e a s u r e m e n t w o u l d a l w a y s be about the same as the d y n a m i c r a n g e o f angles o v e r w h i c h the m e a s u r e m e n t w a s m a d e . ) T h i s r u l e o f t h u m b b r e a k s d o w n f o r s m a l l p a r t i c l e s as w i l l be seen later. F r o m the r u l e o f t h u m b g i v e n a b o v e , it i s apparent that to increase the d y n a m i c range o f a s i z i n g m e a s u r e m e n t , the a n g u l a r r a n g e o v e r w h i c h the scattered l i g h t i s m e a s u r e d m u s t be i n c r e a s e d . H e n c e , h i s t o r i c a l l y , the first c o m m e r c i a l l i g h t d i f f r a c t i o n i n s t r u m e n t s c o v e r e d a d y n a m i c range o f a r o u n d 1 0 0 : 1 . S e v e r a l years later c o m m e r c i a l i n s t r u m e n t s c o v e r i n g a d y n a m i c range o f several h u n d r e d to o n e b e c a m e a v a i l a b l e . T h e i n s t r u m e n t u s e d b y the authors c o v e r s a d y n a m i c r a n g e e x c e e d i n g s e v e r a l t h o u s a n d to one.


Broad

Size Distributions

M a n y i m p o r t a n t i m p o r t a n t classes o f p a r t i c l e s are d i s p e r s e d i n e x t r e m e l y b r o a d d i s t r i b u t i o n s , d i s t r i b u t i o n s w h o s e d y n a m i c range m a y e x c e e d 1 0 0 0 : 1 . A l t h o u g h d i f f r a c t i o n offers the p o t e n t i a l o f m e a s u r i n g o v e r s u c h a n e x t r e m e l y b r o a d d y n a m i c r a n g e , there are s o m e p h y s i c a l p r o b l e m s w h i c h m a k e this w i d e d y n a m i c r a n g e d i f f i c u l t , i n p r a c t i c e , to a c h i e v e . F i g u r e 3 i s a l o g - l o g p l o t o f the d i f f r a c t i o n pattern e x p e c t e d f o r e q u a l v o l u m e s o f a l a r g e ( 7 0 0 ( i m ) a n d s m a l l (l|xm) p a r t i c l e s . I n o r d e r to a c c u r a t e l y measure these p a r t i c l e s , d i f f r a c t i o n patterns s i m i l a r to these m u s t m e a s u r a b l e b y the instrument T h e d i f f i c u l t i e s w i t h m e a s u r i n g samples w h i c h c o n t a i n p a r t i c l e s o f sizes as disparate as those s h o w n i n F i g u r e 3 are t w o f o l d . T h e f i r s t part l i e s i n the t r e m e n d o u s range o f l i g h t f l u x e s w h i c h m u s t be sensed. T h e detectors m u s t be a b l e to sense l i g h t l e v e l s w i t h h i g h p r e c i s i o n o v e r m a n y orders o f m a g n i t u d e to a c c u r a t e l y characterize t w o d i f f r a c t i o n patterns l i k e those s h o w n . T h i s d i f f i c u l t y i s f r e q u e n t l y a d d r e s s e d b y u s i n g a detector a r r a y w h i c h has detectors o f i n c r e a s i n g s i z e s f o r the l a r g e r a n g l e s . I n t h i s w a y , the h i g h l i g h t f l u x l e v e l s ( f l u x i s l i g h t p o w e r p e r u n i t area), w h i c h o c c u r p r i m a r i l y at s m a l l scattering angles, are detected b y v e r y s m a l l detectors; c o n v e r s e l y , the l o w e r f l u x l e v e l s at h i g h e r scattering angles are detected b y larger detectors. T h e s i g n a l l e v e l s are thus b a l a n c e d b y t a i l o r i n g the detector sizes to the l i g h t l e v e l they are l i k e l y to sense. F i g u r e 4 A s h o w s the a m o u n t o f l i g h t f l u x f o r a m i x t u r e o f 2 7 5 a n d 5|im p a r t i c l e s , r e c o r d e d b y a series o f detectors i n c r e a s i n g i n area g e o m e t r i c a l l y w i t h scattering angle. N o t e that i n this g r a p h the ordinate i s p l o t t e d o n a l i n e a r rather t h a n a l o g scale. F i g u r e 4 A c a n be contrasted w i t h F i g u r e 2, to illustrate the effect o f e m p l o y i n g detectors o f i n c r e a s i n g sizes to b a l a n c e the l i g h t l e v e l s r e c o r d e d b y the detectors. T h e i m p r o v e m e n t m a d e p o s s i b l e b y the g e o m e t r i c a l l y s c a l e d detector s i z e s , is apparent i n the a d d i t i o n a l o s c i l l a t i o n s w h i c h b e c o m e v i s i b l e f o r the 275|Ltm p a r t i c l e s i n this w e i g h t e d f l u x pattern. A s F i g u r e 4 A also s h o w s , the c o m p o s i t e d i f f r a c t i o n pattern r e s u l t i n g from a m i x t u r e o f p a r t i c l e s o f different sizes i s the s u m o f the d i f f r a c t i o n patterns f o r the i n d i v i d u a l p a r t i c l e s , w e i g h t e d b y the a m o u n t o f p a r t i c l e s o f e a c h s i z e . T h e s e c o n d a n d m o r e persistant d i f f i c u l t y i n m e a s u r i n g scattering patterns o v e r an e x t r e m e l y w i d e d y n a m i c range i s i n the e n o r m o u s difference i n the shape o f the d i f f r a c t i o n patterns o f the s m a l l v e r s u s large p a r t i c l e s . It is this characteristic d i f f e r e n c e i n the shape o f the patterns f o r different s i z e d p a r t i c l e s w h i c h p e r m i t s the m e a s u r e m e n t to d i s t i n g u i s h b e t w e e n p a r t i c l e s o f different sizes. H o w e v e r , d i s t i n g u i s h i n g b e t w e e n the d i f f r a c t i o n patterns o f the t w o large particles r e q u i r e s m e a s u r i n g the l i g h t patterns e x t r e m e l y accurately at s m a l l ( < 0 . 1 ° ) scattering angles. C o n v e r s e l y , d i s t i n g u i s h i n g b e t w e e n the t w o s m a l l p a r t i c l e s requires m e a s u r i n g the v e r y s m a l l a m o u n t o f c u r v a t u r e d i f f e r e n c e i n the d i f f r a c t i o n patterns at h i g h scattering angles. F i g u r e 4 B s h o w s the true (ideal) d i f f r a c t i o n pattern w h i c h w o u l d b e e x h i b i t e d b y 164|im p a r t i c l e s . T o extract a l l the i n f o r m a t i o n about the p a r t i c l e size that i s i n t r i n s i c a l l y


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


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BOTT & HART


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112

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c o n t a i n e d i n the d i f f r a c t i o n pattern, a n i n s t r u m e n t w o u l d n e e d to h a v e e n o u g h detectors to characterize the pattern t h r o u g h a l l the o s c i l l a t i o n s e x h i b i t e d b y the pattern. S u p e r i m p o s e d o n the true d i f f r a c t i o n pattern i s the pattern w h i c h w o u l d be m e a s u r e d b y a r e a l i n s t r u m e n t c o n t a i n i n g 3 2 discrete l i g h t detectors s p a c e d g e o m e t r i c a l l y across the a n g u l a r r a n g e o f interest. A s i s apparent from the g r a p h , these 3 2 detectors are s u f f i c i e n t to capture m o s t , but not a l l , o f the character o f the true d i f f r a c t i o n pattern. A d d i t i o n a l detectors, l o c a t e d b e t w e e n those s h o w n w o u l d p e r m i t the true d i f f r a c t i o n pattern to b e m o r e c o m p l e t e l y m e a s u r e d . T o measure the d i f f r a c t i o n pattern accurately at b o t h h i g h a n d v e r y l o w angles requires a l a r g e n u m b e r o f detectors spread o v e r a v e r y w i d e area o n the p l a n e o f the detector. B e c a u s e o f the e x p e n s e o f c o n s t r u c t i n g a s u i t a b l e detector a r r a y a n d because o f the a d d e d d i f f i c u l t y o f the o p t i c a l aberations (blur) w h i c h o c c u r w h e n a l e n s i s u s e d to f o r m a n i m a g e o v e r s u c h a w i d e part o f the F o u r i e r p l a n e , the a n g u l a r range o v e r w h i c h d i f f r a c t i o n patterns are c o n v e n t i o n a l l y m e a s u r e d i n p a r t i c l e s i z i n g measurements i s g e n e r a l l y l i m i t e d to a range o f s e v e r a l h u n d r e d to o n e , i.e. the largest a n g l e m e a s u r e d i s , at m o s t , s e v e r a l h u n d r e d t i m e s the s m a l l e s t a n g l e m e a s u r e d i n a s i n g l e m e a s u r e m e n t A s d i s c u s s e d e a r l i e r , this l i m i t s the d y n a m i c range o f a s i z e m e a s u r e m e n t to a s i m i l a r s e v e r a l h u n d r e d to one. I f a l l the p a r t i c l e s to be m e a s u r e d f a l l w i t h i n the s e v e r a l h u n d r e d to o n e s i z e range w h i c h the l i m i t s o f the a n g u l a r range o f the scattering m e a s u r e m e n t p e r m i t , the l i m i t a t i o n s i n the a n g u l a r range s h o u l d have little effect o n the r e c o v e r e d s i z e d i s t r i b u t i o n . H o w e v e r , f o r w i d e d y n a m i c r a n g e s a m p l e s , the f o r e g o i n g c o n d i t i o n w i l l not a l w a y s be met. I n a d d i t i o n , l i m i t e d d y n a m i c range m a y a l s o a d v e r s e l y affect s a m p l e s f o r w h i c h part o f the s i z e d i s t r i b u t i o n f a l l s outside o f the r a n g e o f the m e a s u r e m e n t e v e n i f the d y n a m i c range o f the d i s t r i b u t i o n d o e s n o t n e c e s s a r i l y e x c e e d the d y n a m i c r a n g e o f the m e a s u r e m e n t T h i s i s w o r r i s o m e , s i n c e m a n y s a m p l e s c o m p r i s e u n k n o w n 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 w h i c h m i g h t c o n t a i n p a r t i c l e s outside the range o f the instruments. It m i g h t be e x p e c t e d that out o f range m a t e r i a l w o u l d be i g n o r e d b y the d i f f r a c t i o n measurement; h o w e v e r , out o f range m a t e r i a l s t i l l contributes scattered l i g h t w i t h i n the measurement range a n d therefore w i l l result i n m e a s u r e m e n t a n o m a l i e s o f the sort d i s c u s s e d b e l o w . O b v i o u s l y a n i m p o r t a n t advantage o f the w i d e s t p o s s i b l e a n g u l a r scattering range i s i m m u n i t y to p r o b l e m s o f t h i s s o r t T o test the effects o f l i m i t e d a n g u l a r range, restricted size range d i f f r a c t i o n measurements on a s i z e d i s t r i b u t i o n w h i c h does not f a l l c o m p l e t e l y w i t h i n the measurement range, the f o l l o w i n g e x p e r i m e n t w a s p e r f o r m e d . T h e d i f f r a c t i o n pattern e x p e c t e d f o r a 1:1 m i x t u r e o f 2 7 5 a n d 5 |im spheres w a s c a l c u l a t e d over a l a r g e a n g u l a r range: 0.03 to 3 5 ° . N o i s e c o r r e s p o n d i n g to 1% o f the l i g h t f l u x s i g n a l strength w a s added to the d i f f r a c t i o n pattern to s i m u l a t e the noise levels generally contained i n s u c h measurements. T h i s c o m p u t e r s i m u l a t e d data w a s then a n a l y z e d o v e r the f u l l a n g u l a r r a n g e , a n d the a m o u n t o f m a t e r i a l i n 7 2 s i z e classes from .8 to 8 0 0 |im w a s r e p o r t e d . T h a t result w a s c o m p a r e d to a s i m i l a r a n a l y s i s c o v e r i n g a m o r e c o n v e n t i o n a l , l i m i t e d a n g u l a r range o f .03 to 3.5° a n a l y z e d o v e r a c o r r e s p o n d i n g l y s m a l l e r range o f s i z e c l a s s i f i c a t i o n s : 8 - 8 0 0 | i m . T h e h i s t o g r a m p l o t t e d w i t h a f i n e l i n e i n F i g u r e 5 i s the s i z e d i s t r i b u t i o n r e c o v e r e d u s i n g the f u l l a n g u l a r range measaurement, the h i s t o g r a m p l o t t e d w i t h a b o l d l i n e i s the result w i t h the l i m i t e d a n g u l a r range m e a s u r e m e n t T h e i m p o r t a n t result is that i n the l i m i t e d angular/size range result, the 5[im p a r t i c l e s , t h o u g h out o f the s i z e range o v e r w h i c h the a n a l y s i s i s p e r f o r m e d , are n o t s i m p l y i g n o r e d b y the a n a l y s i s . Instead, an artifact p e a k at at a r o u n d 150|im i s r e p o r t e d , as a r e s u l t o f the p r e s e n c e o f t h e 5 ^ m p a r t i c l e s . U s i n g the broader a n g u l a r r a n g e , the b i m o d a l size d i s t r i b u t i o n i s accurately r e c o v e r e d . T h e source o f the artifact peak c a n be f o u n d b y r e f e r r i n g again to F i g u r e 4. T h e c o m p o s i t e d i f f r a c t i o n pattern s h o w n is a result o f the s u m o f the i n d i v i d u a l d i f f r a c t i o n patterns f r o m the t w o p o p u l a t i o n s o f particles. A l t h o u g h the larger a n d s m a l l e r p a r t i c l e s diffract l i g h t p r e d o m i n a n t l y i n t o the l o w e r a n d h i g h e r scattering angles, r e s p e c t i v e l y , the


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p o p u l a t i o n o f s m a l l e r particles d o e s diffract a s i g n i f i c a n t a m o u n t o f l i g h t i n t o the .03 - 3 ° a n g u l a r range d o m i n a t e d b y the larger p a r t i c l e s d i f f r a c t i o n pattern. T h i s l i g h t leads to the r e p o r t i n g o f the 150|im a r t i f a c t N o t e that i f the a n a l y s i s o f the data f r o m the l i m i t e d a n g u l a r range m e a s u r e m e n t is e x t e n d e d to i n c l u d e a w i d e r range o f s i z e c l a s s i f i c a t i o n s , other artifacts w i l l be i n t r o d u c e d b y the attempt to size particles w h o s e d o m i n a n t d i f f r a c t i o n p e a k f a l l s out o f the r a n g e o v e r w h i c h the d i f f r a c t i o n i s sensed. ,


Double

Fourier Lens Optical Collection

System

A l t h o u g h to the k n o w l e d g e o f the authors it has not b e e n p r e v i o u s l y d o n e , the r e s t r i c t e d a n g u l a r range u s e d b y m a n y d i f f r a c t i o n instruments c a n be e x t e n d e d s t r a i g h t f o r w a r d l y b y a d d i n g a s e c o n d F o u r i e r lens to capture l i g h t d i f f r a c t e d i n t o h i g h e r scattering angles. S u c h a m e a s u r e m e n t s y s t e m i s s h o w n s c h e m a t i c a l l y i n F i g u r e 6. T h e a d d i t i o n o f the s e c o n d F o u r i e r l e n s h e l p s to p r e v e n t artifacts s u c h as those d e s c r i b e d a b o v e , i n a d d i t i o n to e x t e n d i n g the d y n a m i c range o f the measurement. Loss of Resolution

at S m a l l Size

End

A n e x a m i n a t i o n o f F i g u r e 7 s h o w s that f o r s m a l l p a r t i c l e s , e s p e c i a l l y p a r t i c l e s b e l o w a r o u n d 0.5|im i n d i a m e t e r , the d i f f r a c t i o n patterns are rather s i m i l a r . F o r t h i s r e a s o n , the d i f f r a c t i o n patterns o f s m a l l particles d o not p r o v i d e as m u c h i n f o r m a t i o n about p a r t i c l e s i z e as d o the m o r e oscillatory, i n f o r m a t i o n r i c h , patterns e x h i b i t e d b y l a r g e r particles. T h e d i f f r a c t i o n patterns f o r v e r y s m a l l particles are a l l c h a r a c t e r i z e d b y a n e v e n scattering intensity out to a f a i r l y h i g h a n g l e , f o l l o w e d b y a r e g i o n at h i g h e r angle i n w h i c h the scattered l i g h t intensity f a l l s o f f s l i g h t l y w i t h angle. T h e b a s i c s i m i l a r i t y i n the d i f f r a c t i o n patterns f o r s m a l l particles m e a n s that the r e s o l u t i o n o f d i f f r a c t i o n measurements i n this s i z e r e g i m e i s i n t r i n s i c a l l y l i m i t e d . T h e r e s o l u t i o n o f the measurement c a n be e x t e n d e d b y m e a s u r i n g to as h i g h a scattering angle as p o s s i b l e , i n order to detect s m a l l d i f f e r e n c e s i n the shape o f the pattern i n the r e g i o n i n w h i c h the patterns are f a l l i n g i n a m p l i t u d e . H o w e v e r , because o f the b a s i c s i m i l a r i t y i n the shape o f the patterns, measurements at h i g h e r angles p r o v i d e d i m i n i s h i n g returns. Polarization

Intensity

Differential Scattering

(PIPS)

T h e authors h a v e d e v i s e d an alternate m e t h o d f o r c h a r a c t e r i z i n g p a r t i c l e s i n the s u b m i c r o m e t e r s i z e range. T h i s m e t h o d i s based o n a p a r t i c u l a r feature o f the scattered l i g h t pattern f o r particles s m a l l e r than the w a v e l e n g t h o f the i n c i d e n t l i g h t , m o r e specifically, o n the s e n s i t i v i t y o f scattering b y these s m a l l p a t i c l e s to the p o l a r i z a t i o n o f the i n c i d e n t l i g h t V a r i o u s m e t h o d s h a v e b e e n u s e d p r e v i o u s l y w h i c h e x p l o i t the p o l a r i z a t i o n s e n s i t i v i t y o f scattering to size s m a l l particles ( 2 , 1 0 ) . F i g u r e 8 s h o w s the e x p e r i m e n t a l geometry. T h e i n c i d e n t l i g h t m a y be p o l a r i z e d either p e r p e n d i c u l a r to o r p a r a l l e l to the scattering p l a n e , i.e. the p l a n e d e f i n e d b y the l i g h t b e a m a n d a l i n e f r o m the scattering v o l u m e to the detector. H e u r i s t i c a l l y , the o r i g i n o f the p o l a r i z a t i o n s e n s i t i v i t y o f the scattering o f s m a l l p a r t i c l e s c a n be u n d e r s t o o d i n the f o l l o w i n g w a y . L i g h t is a transverse e l e c t r o m a g n e t i c w a v e , i.e. the electric a n d m a g n e t i c f i e l d s o f the l i g h t o s c i l l a t e i n a d i r e c t i o n p e r p e n d i c u l a r to the d i r e c t i o n o f p r o p a g a t i o n o f the b e a m . I f a s m a l l p a r t i c l e i s l o c a t e d i n a l i g h t b e a m , the o s c i l l a t i n g electric f i e l d i n d u c e s i n the p a r t i c l e a n o s c i l l a t i n g d i p o l e m o m e n t ; the electrons i n the a t o m s c o m p r i s i n g the p a r t i c l e m o v e b a c k a n d f o r t h w i t h respect to the r e l a t i v e l y stationary p a r t i c l e . T h e d i r e c t i o n o f i n d u c e d m o t i o n o f the electrons w i l l be i n the d i r e c t i o n o f o s c i l l a t i o n o f electric f i e l d , a n d therefore p e r p e n d i c u l a r to the d i r e c t i o n o f p r o p a g a t i o n o f the l i g h t b e a m . B e c a u s e o f the transverse nature o f l i g h t , the o s c i l l a t i n g d i p o l e radiates l i g h t i n a l l d i r e c t i o n s e x c e p t i n the d i r e c t i o n o f o s c i l l a t i o n ; there i s , b y d e f i n i t i o n , n o c o m p o n e n t o f transverse


114


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F i g u r e 5. T r u e d i s t r i b u t i o n a n d r e c o v e r e d d i s t r i b u t i o n s h o w i n g artifact p e a k ( 1 5 0 ( i m ) r e s u l t i n g f r o m inadequate d y n a m i c range.

Light Source

Particle Field

Fourier Lenses

Detector Arrays

F i g u r e 6. O p t i c a l train f o r w i d e d y n a m i c range d i f f r a c t i o n m e a s u r e m e n t .


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Angle (degrees) F i g u r e 7. L i g h t f l u x patterns f o r s m a l l p a r t i c l e s .

Observer Source/ Molecule

F i g u r e 8. E x p e r i m e n t a l g e o m e t r y f o r P I D S scattering m e a s u r e m e n t



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o s c i l l a t i o n o f the electrons i n this d i r e c t i o n . R e f e r r i n g a g a i n to F i g u r e 8, the l i g h t i s p o l a r i z e d i n the x - z p l a n e ; the scattering intensity i n the d i r e c t i o n o f the ± z a x i s w i l l be zero. I f the l i g h t w e r e p o l a r i z e d i n the x - y p l a n e , there w o u l d be a p p r e c i a b l e scattered tight i n t e n s i t y i n the z d i r e c t i o n , since the d i r e c t i o n o f o s c i l l a t i o n o f the electrons w o u l d then be p e r p e n d i c u l a r to the z a x i s . T h u s the scattering response o f the s m a l l p a r t i c l e i s h e a v i l y dependent o n the p o l a r i z a t i o n o f the l i g h t T h e electric f i e l d f r o m a n o s c i l l a t i n g d i p o l e i s s h o w n i n F i g u r e 9. T h e f o r e g o i n g e x p l a n a t i o n w a s p r e d i c a t e d o n the p a r t i c l e s b e i n g m u c h s m a l l e r than the w a v e l e n g t h o f the l i g h t . A s the p a r t i c l e s i z e increases (or, e q u i v a l e n t l y , as the l i g h t w a v e l e n g t h decreases), the p a r t i c l e w i l l n o l o n g e r act l i k e a s i m p l e d i p o l e a n d the scattering pattern w i l l b e c o m e m o r e c o m p l e x . F i g u r e 10 i s a p l o t , as a f u n c t i o n o f scattering a n g l e , o f the difference i n scattered l i g h t i n t e n s i t y f o r i n c i d e n t l i g h t p o l a r i z e d p e r p e n d i c u l a r to v s p a r a l l e l to the scattering p l a n e . T h i s d i f f e r e n c e i s t e r m e d a P U D S s i g n a l . F o r the s m a l l e r p a r t i c l e , the scattering pattern i s a r o u g h l y q u a d r a t i c c u r v e centered at 9 0 ° . F o r p a r t i c l e s w h i c h are larger, the pattern shifts s o m e w h a t to the left. F o r s t i l l l a r g e r p a r t i c l e s (or a g a i n , shorter l i g h t w a v e l e n g t h s ) , the pattern w i l l shift further to the left a n d s e c o n d a r y peaks w i l l appear. S i n c e the P I D S s i g n a l i s d e p e n d e n t o n p a r t i c l e s i z e r e l a t i v e to the l i g h t w a v e l e n g t h , b y m e a s u r i n g the P I D S s i g n a l at a v a r i e t y o f l i g h t w a v e l e n g t h s , v a l u a b l e i n f o r m a t i o n about the p a r t i c l e s i z e d i s t r i b u t i o n c a n be o b t a i n e d . T h e p h y s i c s d e s c r i b e d a b o v e c a n be u t i l i z e d to measure s u b m i c r o m e t e r particles i n a m e a s u r e m e n t s y s t e m s u c h as that s h o w n i n F i g u r e 1 1 . A w h i t e l i g h t s o u r c e i s c o l l i m a t e d to f o r m a b e a m o f l i g h t O n e filter o n a c a r o u s e l c o n t a i n i n g s e v e r a l filterp o l a r i z e r s selects l i g h t o f one p a r t i c u l a r w a v e l e n g t h a n d o f p o l a r i z a t i o n either p e r p e n d i c u l a r to o r p a r a l l e l to the scattering plane. T h e s y m m e t r y o f the scattering pattern a r o u n d 9 0 ° scattering angle i s m e a s u r e d f o r that filter-polarizer b y s e v e r a l detectors. T h e filter w h e e l i s rotated to measure, sequentially, the scattered l i g h t pattern r e s u l t i n g f r o m the other filter-polarizers ( c o v e r i n g the t w o p o l a r i z a t i o n s at other selected l i g h t w a v e l e n g t h s ) o n the c a r o u s e l . T h e P I D S pattern f o r e a c h o f s e v e r a l l i g h t w a v e l e n g t h s c a n i n this w a y be m e a s u r e d a n d r e c o r d e d b y the i n s t r u m e n t F i g u r e 12 s h o w the P I D S patterns, f o r three l i g h t w a v e l e n g t h s , e x p e c t e d f o r p a r t i c l e s o f 0 . 1 - 0 . 4 ^ m . T h e s e P I D S patterns are i n m a r k e d contrast to the d i f f r a c t i o n patterns f o r p a r t i c l e s o f s i m i l a r s i z e s , s h o w n i n F i g u r e 7. W i t h o u t d e l v i n g i n t o the m a t h e m a t i c s , i t i s c l e a r that the P I D S patterns are m u c h m o r e characteristic o f p a r t i c l e s i z e i n this range, than are the d i f f r a c t i o n patterns. T h e s i m i l a r i t y b e t w e e n the d i f f r a c t i o n pattens for s m a l l p a r t i c l e s restricts the r e s o l u t i o n a n d a c c u r a c y w i t h w h i c h the p a r t i c l e s c a n be s i z e d . C o n v e r s e l y , the e x t r e m e l y characteristic P I D S patterns f o r v a r i o u s s m a l l p a r t i c l e s m a k e s it easy to i d e n t i f y the size o f p a r t i c l e s b y o b s e r v i n g the P I D S s i g n a l s . A l t h o u g h i t i s b e y o n d the s c o p e o f this a r t i c l e , a m a t h e m a t i c a l a n a l y s i s m u s t be u s e d to r e d u c e the P I D S patterns at s e v e r a l w a v e l e n g t h s to a s i z e d i s t r i b u t i o n . S u c h a n a n a l y s i s m u s t also c o m b i n e c o n v e n t i o n a l d i f f r a c t i o n data w i t h the a d d i t i o n a l i n f o r m a t i o n present i n the P I D S s i g n a l s . F i g u r e s 13 a n d 14 s h o w s i z e d i s t r i b u t i o n s r e c o v e r e d f o r t w o s m a l l (0.137 a n d 0 . 3 5 9 n m ) p o l y s t y r e n e latex spheres ( P S L ) u s i n g d i f f r a c t i o n a l o n e , F i g u r e 13, a n d d i f f r a c t i o n i n c o n j u n c t i o n w i t h P I D S , F i g u r e 14. R e s u l t s w e r e o b t a i n e d u s i n g the C o u l t e r m o d e l L S I 3 0 p a r t i c l e analyzer. T h e d i f f r a c t i o n results s h o w that the m e t h o d i s f a i r l y i n s e n s i t i v e to p a r t i c l e s i z e i n t h i s s i z e r e g i m e . T h e results f o r p a r t i c l e s o f b o t h sizes are r e p o r t e d as a s i n g l e b r o a d p e a k c e n t e r e d at a r o u n d 0 . 2 5 ^ m . U s i n g P I D S i n a d d i t i o n to the d i f f r a c t i o n , the t w o P S L s are r e p o r t e d as the correct, n a r r o w p e a k s , w i t h m e a n sizes q u i t e c l o s e to the n o m i n a l sizes.


BOTT&HART



7.

Figure 10. PIDS signal for particles much smaller (0.07(im) than light wavelength (0.45|im) and particles comparable to light wavelength (0.2|im). In Particle Size Distribution II; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.

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Conclusion T h e d y n a m i c range o f a d i f f r a c t i o n s i z i n g measurement c a n be e x t e n d e d b y e x t e n d i n g the a n g u l a r range o v e r w h i c h the d i f f r a c t e d l i g h t i s m e a s u r e d . A n n e w o p t i c a l s c h e m e e m p l o y i n g t w o sets o f F o u r i e r c o l l e c t i o n o p t i c s , o f d i f f e r e n t o p t i c a l p o w e r s , h a s b e e n presented. T h e n e w s c h e m e extends the a n g u l a r range o v e r w h i c h the d i f f r a c t i o n pattern c a n be a c c u r a t e l y c h a r a c t e r i z e d i n a s i n g l e measurement. T h e use o f the s e c o n d o p t i c a l train has been s h o w n to h e l p a l l e v i a t e c e r t a i n artifacts that o c c u r w h e n b r o a d s i z e d i s t r i b u t i o n s are m e a s u r e d w i t h c o n v e n t i o n a l s i n g l e c o l l e c t i o n train o p t i c s . A n e w p o l a r i z a t i o n sensitive l i g h t scattering s c h e m e , P I D S , has b e e n d e s c r i b e d f o r m e a s u r e m e n t s o f s u b m i c r o m e t e r p a r t i c l e s . T h i s s c h e m e extends the u s e f u l s i z i n g r a n g e f o r d i f f r a c t i o n i n s t r u m e n t s d o w n to 0.1 |im. U s i n g g r a p h i c a l c o m p a r i s o n s o f scattered l i g h t patterns, reasons f o r the h i g h e r s i z i n g r e s o l u t i o n o f P I D S v e r s u s s i m p l e d i f f r a c t i o n h a v e been o u t l i n e d . I n a d d i t i o n , e x p e r i m e n t a l d a t a u s i n g the P I D S m e a s u r e m e n t o n s u b m i c r o m e t e r p o l y s t y r e n e latex spheres demonstrates the c o n s i d e r a b l y better r e s o l u t i o n a n d a c c u r a c y o f P I D S v e r s u s c o n v e n t i o n a l d i f f r a c t i o n f o r p a r t i c l e s less than 0 . 4 | i m . U s i n g a c o m b i n a t i o n o f the d o u b l e F o u r i e r l e n s c o l l e c t i o n s y s t e m a n d the n e w p o l a r i z a t i o n sensitive scattering m e t h o d f o r s u b m i c r o m e t e r p a r t i c l e s , a s i n g l e the entire s i z e range f r o m O . l ^ m to a r o u n d l O O O j i m c a n be m e a s u r e d i n a s i n g l e m e a s u r e m e n t


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