Submicrometer Particle Sizing by Photon Correlation Spectroscopy

Chapter 5

Submicrometer Particle Sizing by Photon Correlation Spectroscopy: Use of Multiple-Angle Detection S. E. Bott

Downloaded by FUDAN UNIV on February 19, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch005

Coulter Electronics, Inc., 29 Cottage Street, Amherst, MA 01002

Photon correlation spectroscopy (PCS) has become a method of choice for sizing particles in the 3-3000nm range. Advances in the analysis of PCS data permit extraction of the particle size distribution as well as the mean diameter. An important problem in PCS involves converting intensity averaged distributions, which are measured by PCS but have little direct meaning, to physically meaningful weight (volume) averaged distributions. For spherical particles the intensity to weight conversion at each scattering angle is given by the 'Mie' equations. However, the 'Mie' conversion is often an oscillatory function in which particles of certain sizes contribute virtually no scattered light. It is shown here that weight distributions from single angle PCS measurements can have huge errors. Complementary information obtained by measurement at an additional angle allow a good measure of the weight averaged size distribution. As new techniques to produce material are developed and older processes are refined, there has been a general trend toward the use of component materials of smaller and smaller size. Using smaller size components often results in bulk materials of greater strength or uniformity or possessing other advantageous qualities. Some examples of this trend can be found in coatings, ceramics and latices 1 2 · Concomitant with the trend has been a requirement for measuring size distributions of smaller particles in order to control or to characterize the industrial process. For particles smaller than around .5 to 1 micron, measuring methods based on single particle detection generally break down because the signals (e.g. scattered light, electrical conductivity through a pore, etc.) obtainable from individual particles below this size are simply too small to detect. A few years ago, a commercial instrument, based on a new principle, photon correlation spectroscopy (PCS)3 , became available for measurement of mean particle sizes in the submicron range. PCS (also known as dynamic light scattering (DLS) or quasi-elastic 0097-6156/87/0332-0074$06.00/0 © 1987 American Chemical Society Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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Submkrometer Particle Sizing

75


l i g h t s c a t t e r i n g (QELS)) o f f e r s s e v e r a l o u t s t a n d i n g advantages o v e r o t h e r m e a s u r e m e n t m e t h o d s : 1> t h e m e a s u r e m e n t i s a b s o l u t e , i . e . i t r e q u i r e s no c a l i b r a t i o n s t a n d a r d s , 2> t h e m e a s u r e m e n t i s n o n i n v a s i v e - t h e sample c a n be r e c o v e r e d unchanged a f t e r t h e m e a s u r e m e n t a n d 3> t h e m e a s u r e m e n t i s q u i c k , u s u a l l y r e q u i r i n g only a few minutes. PCS c i r c u m v e n t s t h e p r o b l e m o f i n a d e q u a t e s i g n a l l e v e l f o r i n d i v i d u a l p a r t i c l e s by simultaneously d e t e c t i n g o f t h e o r d e r o f t h o u s a n d s t o many m i l l i o n s o f particles. I n t h e l a s t f e w y e a r s , PCS h a s b e c o m e a w i d e l y u s e d p a r t i c l e s i z i n g t o o l f o r mean p a r t i c l e s i z e s i n t h e s u b m i c r o n range. The p o s i t i v e a t t r i b u t e s o f p a r t i c l e s i z i n g b y PCS a r e o p p o s e d b y two n e g a t i v e o n e s : t h e m e t h o d , b e c a u s e i t i s b a s e d o n l i g h t s c a t t e r i n g , measures angle dependent ( s c a t t e r e d l i g h t ) i n t e n s i t y w e i g h t e d r a t h e r t h a n t h e more u s e f u l volume w e i g h t e d p a r t i c l e d i s t r i b u t i o n s ; and, the p a r t i c l e s i z i n g has r e l a t i v e l y low resolution. The a n g l e dependent i n t e n s i t y w e i g h t e d a v e r a g e s must u s u a l l y be c o n v e r t e d t o volume a v e r a g e s f o r i n t e r p r e t a t i o n and c o m p a r i s o n w i t h o t h e r p a r t i c l e s i z i n g m e t h o d s . The r e l a t i v e l y l o w r e s o l u t i o n o f PCS c o m b i n e d w i t h t h e s e v e r i t y o f the c o n v e r s i o n o f i n t e n s i t y averages t o weight averages and t h e heavy dependence o f i n t e n s i t y averaged d i s t r i b u t i o n s on t h e s c a t t e r i n g a n g l e a t w h i c h t h e m e a s u r e m e n t i s p e r f o r m e d , means t h a t s i n g l e PCS m e a s u r e m e n t s made a t o n e s c a t t e r i n g a n g l e w i l l often give misleading r e s u l t s . This a r t i c l e presents data which show t h a t b y m a k i n g m u l t i p l e m e a s u r e m e n t s a t s e v e r a l s c a t t e r i n g a n g l e s , much m o r e r e l i a b l e r e s u l t s c a n b e o b t a i n e d . PHOTON CORRELATION

SPECTROSCOPY

F i q u r e 1 s h o w s t h e g e o m e t r y o f a PCS m e a s u r e m e n t . L a s e r l i g h t i s i n c i d e n t on a sample o f p a r t i c l e s suspended o r d i s s o l v e d i n a transparent liquid. The p a r t i c l e s , u n d e r g o i n g B r o w n i a n m o t i o n , d i f f u s e thoughout t h e s o l u t i o n , A t any g i v e n i n s t a n t , t h e l i g h t i n t e n s i t y sensed a t t h e detector i s p r o p o r t i o n a l t o the square of the e l e c t r i c f i e l d a t t h e d e t e c t o r . The phase o f t h e l i g h t s c a t t e r e d from each p a r t i c l e and measured a t t h e f i x e d d e t e c t o r , w i l l d e p e n d o n t h e p o s i t i o n o f t h e p a r t i c l e i n t h e beam a n d o n the d i s t a n c e between t h e p a r t i c l e and t h e d e t e c t o r . Thus t h e composite l i g h t s c a t t e r e d from t h e group o f p a r t i c l e s i nt h e s c a t t e r i n g volume w i l l form an i n t e r f e r e n c e p a t t e r n a t t h e detector. F i g u r e 2 shows a n i n t e r f e r e n c e p a t t e r n r e s u l t i n g f r o m the l i g h t s c a t t e r i n g f r o m two p a r t i c l e s . The two s e t s o f c o n c e n t r i c c i r c l e s r e p r e s e n t t h e wave f r o n t s o f t h e s c a t t e r e d light. The d a r k ' r a y s ' f o r m e d when two s e t s o f c o n c e n t r i c c i r c l e s a r e superimposed, as i n t h e f i g u r e , a r e r e g i o n s where t h e l i g h t s c a t t e r e d f r o m t h e two p a r t i c l e s c o n s t r u c t i v e l y interferes. I f t h e d e t e c t o r i s l o c a t e d i n one o f t h e s e d a r k r a y s , a h i g h l i g h t i n t e n s i t y w i l l be s e n s e d . As t h e p a r t i c l e s r a n d o m l y d i f f u s e t h r o u g h t h e s o l u t i o n , t h e l o c a t i o n o f the dark rays o f constructive interference w i l l change; t h e r e f o r e the i n t e n s i t y sensed by t h e f i x e d d e t e c t o r w i l l v a r y , i . e . t h e i n t e r f e r e n c e p a t t e r n produced by t h e s c a t t e r i n g p a r t i c l e s w i l l be m o d u l a t e d b y t h e p a r t i c l e m o t i o n s . The i n t e n s i t y f l u c t u a t i o n s a t t h e d e t e c t o r , t h o u g h r a n d o m , w i l l be m o r e r a p i d f o r s m a l l , r a p i d l y m o v i n g p a r t i c l e s t h a n f o r

Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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PARTICLE SIZE DISTRIBUTION

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F i g u r e 1. G e o m e t r y o f a P C S m e a s u r e m e n t . The s c a t t e r i n g a n g l e , can v a r y between 0 d e g r e e s a n d 180 d e g r e e s .

F i g u r e 2. A n i n t e r f e r e n c e f r o m two p a r t i c l e s .

pattern

r e s u l t i n g from t h e s c a t t e r i n g


Θ,

5.

BOTT

Submicrometer Particle Sizing

l a r g e r more s l o w l y d i f f u s i n g p a r t i c l e s . The ' n o i s e ' s i g n a l p r o d u c e d b y t h e l i g h t s c a t t e r e d f r o m d i f f u s i n g p a r t i c l e s c a n be c h a r a c t e r i z e d by i t s a u t o c o r r e l a t i o n function, g ( t ) , defined by

!D

g(t) = < I(T) Kt+T) >


where I (T) i s t h e i n t e n s i t y a t t h e d e t e c t o r a t t i m e Τ , and where t h e a n g u l a r b r a c k e t s r e p r e s e n t a t i m e a v e r a g e o v e r Τ . The autocorrelation f u n c t i o n measures t h e s i m i l a r i t y , o r c o r r e l a t i o n between t h e c o n f i g u r a t i o n o f p a r t i c l e s c o n t r i b u t i n g t o an i n t e n s i t y I a t Τ t o t h a t a t i m e *t l a t e r . F o r s y s t e m o f Brownian p a r t i c l e s o f u n i f o r m s i z e and shape, t h e autocorrelation f u n c t i o n w i l l be a d e c a y i n g e x p o n e n t i a l :

g(x) = Α θ "

(2)

2 Γ τ

+Β

where Γ i s a decay c o n s t a n t c h a r a c t e r i s t i c o f p a r t i c l e s o f t h a t s i z e and A and Β a r e c o n s t a n t s dependent on t h e sample, e x p e r i m e n t a l geometry and c o u n t i n g e f f i c i e n c y o f t h e o p t i c s and electronics. 1/Γ i s t h e decay time o f t h e Brownian motion; r o u g h l y speaking, i ti s t h e time r e q u i r e d f o r any p a r t i c u l a r c o n f i g u r a t i o n o f p a r t i c l e p o s i t i o n s w i t h i n t h e s c a t t e r i n g volume to ' r e l a x ' . A f t e r s e v e r a l decay t i m e s , t h e p a r t i c l e s w i l l have d i f f u s e d s u c h t h a t t h e i r new p o s i t i o n s w i l l b e statistically u n c o r r e l a t e d w i t h t h e i r former p o s i t i o n s . The d e c a y c o n s t a n t i s r e l a t e d t o t h e d i f f u s i o n c o n s t a n t o f t h e p a r t i c l e s and t o t h e geometry o f t h e experiment through t h e equation,

T = q D ; q = 4ir η sin (θ/2) / λ 2

(3)

where η i s t h e r e f r a c t i v e i n d e x o f t h e s o l v e n t , θ i s t h e scattering angle, D i s the d i f f u s i o n constant o f the p a r t i c l e s , λ i s t h e w a v e l e n g t h o f t h e l a s e r i n vacuum and q i s t h e m a g n i t u d e o f t h e so c a l l e d ' s c a t t e r i n g v e c t o r ' . In turn, f o r spherical p a r t i c l e s , the d i f f u s i o n constant i s related t othe p a r t i c l e diameter through the Stokes-Einstein equation: ( 4 )

D = kT / 3 ι ί η ο ·

where k i s Boltzmann's c o n s t a n t , Τ i s t h e a b s o l u t e t e m p e r a t u r e , η i s t h e v i s c o s i t y o f t h e s o l u t i o n and d i s t h e p a r t i c l e diameter. U s i n g (3) a n d (4), t h e measured decay constant Γ c a n be d i r e c t l y r e l a t e d t o t h e p a r t i c l e d i a m e t e r . When t h e s a m p l e c o n t a i n s p a r t i c l e s o f d i f f e r e n t s i z e s , t h e autocorrelation f u n c t i o n w i l l b e a sum o f d e c a y i n g e x p o n t e n t i a l s weighted by t h e i n t e n s i t y o f l i g h t s c a t t e r e d from p a r t i c l e s o f each c h a r a c t e r i s t i c s i z e . T h i s c a n be d e s c r i b e d i n g e n e r a l b y the e q u a t i o n :

(5)

g(x) = £ X(s) e "

r ( s )

τ

ds


78

PARTICLE SIZE DISTRIBUTION where the i n d e x , s, l a b e l s the s i z e of the p a r t i c l e s and X ( s ) i s 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 , which g i v e s the r e l a t i v e p r o p o r t i o n of the s c a t t e r i n g from p a r t i c l e s o f s i z e s. The c o n s t a n t s a and b are the lower and upper l i m i t s of p a r t i c l e size. T h i s i n t e g r a l e q u a t i o n must be n u m e r i c a l l y i n v e r t e d t o e x t r a c t the s i z e d i s t r i b u t i o n , X ( s ) , from the measured autocorrelation function. The i n v e r s i o n p r o c e d u r e i s non-trivial. A l t h o u g h v a r i o u s a l g o r i t h m s have been d e v e l o p e d f o r the i n v e r s i o n , by f a r the most w i d e l y used a n a l y s i s program f o r t h i s type of d a t a a n a l y s i s i s a FORTRAN program c a l l e d CONTIN, d e t a i l s o f which are a v a i l a b l e i n the open l i t e r a t u r e ' .


4

5

I t might be imagined t h a t w i t h a r e a s o n a b l e measurement of the a u t o c o r r e l a t i o n f u n c t i o n and an a p p r o p r i a t e i n v e r s i o n a l g o r i t h m , t h a t a v e r y a c c u r a t e s i z e d i s t r i b u t i o n measurement c o u l d be made w i t h PCS. However, i n f a c t , the a u t o c o r r e l a t i o n f u n c t i o n i t s e l f has u s e f u l , but l i m i t e d i n f o r m a t i o n about the d i s t r i b u t i o n o f p a r t i c l e s i z e s . To get an i d e a about the p r e c i s i o n o f PCS measurements on b i m o d a l samples, b i m o d a l m i x t u r e s of p o l y s t y r e n e l a t e x spheres (PSL) were p r e p a r e d t o g i v e e q u a l s c a t t e r i n g i n t e n s i t i e s f r o m each p o p u l a t i o n at a 90 degree s c a t t e r i n g a n g l e . Twenty measurements were made w i t h each m i x t u r e . The c o e f f i c i e n t of v a r i a t i o n ( c . v . standard d e v i a t i o n / m e a n ) o f p e r c e n t a g e of s c a t t e r i n g i n t e n s i t y from each peak o v e r the twenty runs was computed. The r e s u l t s are g i v e n below. 6

β

PSL

#1

(nm)

90 90 170 310

PSL

#2

310 822 1100 1300

(nm)

Run

Time ( s e c ) 120 600 600 120

c.v.

of

#1

12% 5% 6% 16%

c.v.

of

#2

15% 8% 6% 32%

As can be seen, the c.v.s v a r y from sample to sample but are of the o r d e r of 10%. As measurement t i m e s i n c r e a s e , the c.v.β of the measurements w i l l , n a t u r a l l y , d e c r e a s e . INTENSITY, WEIGHT AND

NUMBER AVERAGES

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 , X ( s ) , above i s , u n f o r t u n a t e l y , not i n a form which i s u s e f u l f o r most a p p l i c a t i o n s . This i s because i t i s a s c a t t e r e d i n t e n s i t y w e i g h t e d d i s t r i b u t i o n ( f o r b r e v i t y , ' i n t e n s i t y d i s t r i b u t i o n ' ) r a t h e r than a s i z e d i s t r i b u t i o n based on the volume ( w e i g h t ) or number of particles. The d i f f e r e n c e between d i s t r i b u t i o n s w e i g h t e d i n d i f f e r e n t ways can be most e a s i l y e x p l a i n e d by r e l a t i n g the v a r i o u s d i s t r i b u t i o n s t o a number d i s t r i b u t i o n .


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A n u m b e r d i s t r i b u t i o n g i v e s t h e r e l a t i v e n u m b e r , N, o f p a r t i c l e s o f s i z e s. A volume or weight d i s t r i b u t i o n V(s) i s r e l a t e d to a number d i s t r i b u t i o n by (6)

V(s)

=

N(s)

C (s) v

where C y i s a f a c t o r t o c o n v e r t number t o v o l u m e ; C (s) i s e q u a l t o t h e v o l u m e o f a p a r t i c l e o f s i z e s. S i m i l a r l y , an i n t e n s i t y d i s t r i b u t i o n i s r e l a t e d t o a number d i s t r i b u t i o n by v


(7

)

X(s,8) = N(s) C|(s,8)

C j ( s , Θ ) i s t h e c o n v e r s i o n f a c t o r o f number t o i n t e n s i t y , i . e . t h e amount o f s c a t t e r i n g f r o m a p a r t i c l e o f s i z e s . N o t i c e the e x p l i c i t d e p e n d e n c e o f X and C j on θ · T h i s dependence i s i n c l u d e d to emphasize t h a t the i n t e n s i t y d i s t r i b u t i o n measured by PCS a t one s c a t t e r i n g a n g l e w i l l be d i f f e r e n t t h a n t h a t a t a n o t h e r s c a t t e r i n g a n g l e . W e i g h t and n u m b e r d i s t r i b u t i o n s are, of c o u r s e , independent of s c a t t e r i n g a n g l e . The i m p l i c a t i o n o f e q u a t i o n ( 7 ) a s i t r e l a t e s t o PCS, is that l a r g e r p a r t i c l e s s c a t t e r c o n s i d e r a b l y more l i g h t per p a r t i c l e than smaller p a r t i c l e s . Thus the i n t e n s i t y distributions m e a s u r e d b y PCS h e a v i l y e m p h a s i z e t h e p r e s e n c e o f larger particles. I f the i n t e n s i t y d i s t r i b u t i o n s measured w i t h PCS w e r e e x t r e m e l y a c c u r a t e , t h e f a c t t h a t PCS m e a s u r e s i n t e n s i t y r a t h e r t h a n v o l u m e o r n u m b e r d i s t r i b u t i o n s w o u l d be o f little consequence, p r o v i d e d t h a t the proper c o n v e r s i o n f a c t o r , C j ( s , 8 ) w e r e k n o w . A s w i l l be s e e n i n t h e n e x t s e c t i o n , fairly good c o n v e r s i o n s are i n f a c t , a v a i l a b l e . H o w e v e r , as was p o i n t e d o u t i n t h e p r e v i o u s s e c t i o n , PCS i s a l o w resolution s i z i n g method; t h i s i m p l i e s t h a t the i n t e n s i t y d i s t r i b u t i o n , X ( s , 6 ) , i s m e a s u r e d w i t h l o w p r e c i s i o n and a c c u r a c y . Typically, t h e i n t e n s i t i e s i n t h e two p e a k s i n a b i m o d e l s a m p l e , f o r e x a m p l e , w i l l be m e a s u r e d t o 10% o f t h e t o t a l scattering intensity. P e a k s w h i c h c o m p r i s e l e s s t h a n 10% o f t h e total s c a t t e r i n g i n t e n s i t y , t h e r e f o r e , c a n be a r t i f a c t s . For s m a l l r e a l p e a k s , s m a l l e r r o r s i n the measurement of the scattering i n t e n s i t y from a p o p u l a t i o n of s m a l l p a r t i c l e s w i l l o f t e n l e a d t o huge e r r o r s i n volume d i s t r i b u t i o n s d e r i v e d f r o m the measured i n t e n s i t y d i s t r i b u t i o n because the c o n v e r s i o n i n v o l v e s a h i g h a m p l i f i c a t i o n f a c t o r on t h e s m a l l p a r t i c l e s t o compensate f o r t h e i r low s c a t t e r i n g i n t e n s i t i e s compared t o l a r g e r p a r t i c l e s .

R A Y L E I G H AND

R A Y L E I G H DEBYE REGIMES

When p a r t i c l e s a r e s m a l l c o m p a r e d t o t h e m a g n i t u d e o f s c a t t e r i n g v e c t o r ( q = 4 n n s i n ( e / 2 ) / A ) , C j (β,θ) • s ; 6

the i.e.


the

80

PARTICLE SIZE DISTRIBUTION


c o n v e r s i o n o f i n t e n s i t y t o number d i s t r i b u t i o n s i s a n g l e independent and s c a l e s as t h e square o f t h e volume o f t h e p a r t i c l e . I n t h i s regime, c a l l e d t h e Rayleigh regime, t h e s c a t t e r i n g i n t e n s i t y i s independent o f p a r t i c l e shape. F o r p a r t i c l e s o f a s i z e o f t h e order o f t h e magnitude o f t h e i n v e r s e of t h e s c a t t e r i n g v e c t o r q, C p c a n be found by c o n c e p t u a l l y d i v i d i n g t h e p a r t i c l e i n t o many s m a l l s u b s e g m e n t s , a n d s u m m i n g the s c a t t e r e d l i g h t c o n t r i b u t i o n o f each subsegment, t a k i n g i n t o a c c o u n t t h e r e l a t i v e p o s i t i o n s o f each subsegment a n d t h e d i s t a n c e o f each from t h e d e t e c t o r (andi n c l u d i n g , i f n e c e s s a r y , n o n - i s o t r o p i c p o l a r i z a b i l i t i e s o f t h e subsegments and p o l a r i z a t i o n o f incident and detected l i g h t ) . The r e s u l t o f this process f o r spherical, isotropic p a r t i c l e s i s

C, = {[3/(qr) ][sin qr - qr cos qr]} r 3

2

6

where r = s / 2 . T h e a n g l e d e p e n d e n c e o f C j comes t h r o u g h t h e a n g l e dependence o f q . T h i s i s t h e R a y l e i g h Debye a p p r o x i m a t i o n f o r i s o t r o p i c s p h e r e s . I t i s commonly u s e d b e c a u s e o f i t s simplicity. I n t h i s a p p r o x i m a t i o n , C j c o m p r i s e s two f a c t o r s : the volume squared ( r ) f a c t o r and an a t t e n u t i n g f a c t o r , {3/(qr) ] [ s i nq r - q r cos q r ] } ! , which i s always between 0 and 1, a n d w h i c h q u a n t i f i e s t h e d e g r e e o f i n t r a p a r t i c l e d e s t r u c t i v e i n t e r f e r e n c e i n t h e s c a t t e r e d l i g h t from such a spherical particle. A s w i l l be seen below, t h e range o f a p p l i c a b i l i t y o f t h e R a y l e i g h Debye a p p r o x i m a t i o n i s l i m i t e d . R a y l e i g h Debye a p p r o x i m a t i o n s c a n be f o u n d f o r n o n - s p h e r i c a l p a r t i c l e s , e.g. e l l i p s e s o f r e v o l u t i o n , r i g i d rods and Gaussian coils . 6

3

2

- 6

3

MIE

EQUATION

For i s o t r o p i c s p h e r i c a l p a r t i c l e s o f given r e f r a c t i v e index i na medium o f known r e f r a c t i v e i n d e x , t h e e x a c t f o r m o f C c a n be found by matching t h e i n c i d e n t , i n t e r n a l and s c a t t e r e d e l e c t r o m a g n e t i c waves a t t h e p a r t i c l e s u r f a c e , s u b j e c t t o c e r t a i n boundary c o n d i t i o n s . T h e s o l u t i o n comes i n t h e f o r m of an i n f i n i t e s e r i e s : 7

C,(s,8) = Σ l(2n+1)/(n(n+1))Ka ir (cos β) + b x (cos 8) }(-1) n

n

n

n

where

ir (cos 8) = P n

(1 n

\cos 8) / sin 8

τ (cos 8) = d/d8 (P η

a = n

(1) n

(cos 8))

{ψ (α)ψ (β)-Γηψ (β)ψ (α)}/{< (α)ψ (β)-Γηψ (β)< (α)} /

η

/

η

η

η

/

/

η

η

η

b = {Γηψ (α)ψ (β)-ψ (β)ψ (α)}/{ηΓ ζ (α)ψ (β)-ψ (β)< (α)} /

n

η

η

/

η

η

/

>

η

/

η

η

Submicrometer Particle Sizing by Photon Correlation Spectroscopy

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