Measuring Particle Size Distribution of Latex Particles Using Dynamic

Chapter 7

Measuring Particle Size Distribution of Latex Particles Using Dynamic Light Scattering 1

Ruth S. Stock and W. Harmon Ray

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on March 3, 2016 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch007

Chemical Engineering Department, University of Wisconsin, Madison, WI 53706

In a light-scattering experiment, particles that have diameters on the order of the wavelength of incident light and a large refractive index relative to that of the medium, e.g. polymer latex in water, scatter light according to Mie theory. The intensity of the scattered light varies by an order of magnitude in an oscillatory fashion with respect to particle size. It was possible to incorporate the mathematical description of Mie theory into Provencher's constrained regularization method to find the distribution by weight. Alternatively, a simple linear least squares algorithm is used to find the particle size distribution by intensity; this is then transformed to a distribution by weight using Mie theory. Measurements are made at multiple scattering angles and the particle size distributions are compared. A composite distribution formed by averaging the weight distributions found at various angles is considered. These methods are demonstrated on multimodal and broad distributions. Dynamiclightscatteringmaybeusedtodeterminethe particle size distribution or molecular weight distribution in a wide variety of applications. See réf. (1) for discussion of this literature. There are special problems that occur when the particle diameter is large relative to the wavelength of incident light. This is of interest since many latex systems have particles with diameters of 300 to 1000 nm and a large ratio of particle to fluid refractive index (1.2 for polystyrene latex). The most common wavelengths for lasers used in light scattering are on the order of 500 nm. Light entering a particle at two different points will experience different path lengths. The refractive index difference between the particle and the medium will also cause a phase shift in the scattered light leaving the particle (see Figure 1). Due to these effects, the intensity of light scattered by suspended spherical particles depends on the particle diameter relative to the wavelength of incident light, the scattering angle and the refractive index ratio. Plots of intensity as a function of angle for a polystyrene/water system with 514.5 nm incident laser wavelength are shown in Figure 2. 'Current address: General Motors Research Laboratories, Warren, MI 48090-9055 0097-6156/87/0332-0105$06.00/0 © 1987 American Chemical Society In Particle Size Distribution; Provder, Theodore; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


106

PARTICLE SIZE DISTRIBUTION

Mie

Scattering

l-g

ΐβ~* -{^ ι — ι — ι — ι — ι — ι — ι 0 5Θ0 4

ι

ι—ι—ι—ι—ι—ι—ι—ι—ι—ι 1000 1500

Diameter

ι I 2000

(nm)

F i g u r e 2. P r e d i c t i o n s o f i n t e n s i t y p e r u n i t p a r t i c l e v o l u m e a s a f u n c t i o n of p a r t i c l e diameter f o r three scattering angles given i n c i d e n t w a v e l e n g t h 5 1 4 . 5 nm a n d r e f r a c t i v e i n d e x r a t i o p o l y s t y r e n e / w a t e r o f 1.2.

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


7.

107

Dynamic Light Scattering

STOCK AND RAY

I n o r d e r t o a c c o u n t f o r t h e s e e f f e c t s , one may u s e M i e t h e o r y t o o b t a i n t h e v o l u m e o r mass ( i n t h e c a s e o f a h o m o g e n e o u s d e n s i t y system) d i s t r i b u t i o n of l a t e x p a r t i c l e s i n s t e a d of the i n t e n s i t y d i s t r i b u t i o n i n i t i a l l y o b t a i n e d from q u a s i e l a s t i c l i g h t s c a t t e r i n g data, B e d w e l l e t . a l . ( 2 ) h a v e o b t a i n e d t h e mass d i s t r i b u t i o n o f l a r g e p a r t i c l e s by i n c o r p o r a t i n g M i e t h e o r y i n t o t h e h i s t o g r a m method. H e r e we p r o p o s e two m e t h o d s o f d a t a a n a l y s i s . First, P r o v e n c h e r ' s method o f c o n s t r a i n e d r e g u l a r i z a t i o n has been m o d i f i e d to i n c l u d e the Mie s c a t t e r i n g f a c t o r i n the k e r n e l of the i n t e g r a l t o be i n v e r t e d . T h i s method i s s i m i l a r t o t h a t d e v e l o p e d i n d e p e n d e n t l y by S. B o t t ( 3 ) . S e c o n d we c o n s i d e r u s i n g a s i m p l e n o n n e g a t i v e l y c o n s t r a i n e d l i n e a r l e a s t s q u a r e s f i t and c o m b i n i n g i n f o r m a t i o n o b t a i n e d from data taken 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 . Data

Analysis

B o t h P r o v e n c h e r ' s method and t h e n o n - n e g a t i v e l e a s t s q u a r e s (NNLS) method f i r s t o b t a i n the b e s t f i t i n the l e a s t squares sense. In f a c t P r o v e n c h e r ' s p r o g r a m u s e s t h e NNLS p r o g r a m a s p a r t o f t h e a n a l y s i s procedure. P r o v e n c h e r ' s m e t h o d d e p a r t s f r o m t h e NNLS s o l u t i o n by r e g u l a r i z i n g so t h a t t h e s m o o t h e s t 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 o n s i s t e n t w i t h the data i s obtained. T h i s r e g u l a r i z a t i o n i s a c c o m p l i s h e d by i n c l u d i n g a t e r m i n t h e o b j e c t i v e f u n c t i o n w h i c h i s r e l a t e d to the second d e r i v a t i v e of 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 . As t h e p a r t i c l e s i z e d i s t r i b u t i o n becomes s m o o t h e r t h i s t e r m becomes s m a l l e r r e s u l t i n g i n t h e minimum v a l u e o f t h e o b j e c t i v e f u n c t i o n . I n b o t h m e t h o d s t h e l e a s t s q u a r e s p a r a m e t e r s c o n s i s t e d o f 40 h i s t o g r a m s t e p h e i g h t s s p r e a d o v e r a two d e c a d e p a r t i c l e s i z e r a n g e . T h e maximum n u m b e r o f p a r a m e t e r s a l l o w e d i n P r o v e n c h e r ' s p r o g r a m i s 50. The i n t e n s i t y d i s t r i b u t i o n o f p a r t i c l e the e x p e r i m e n t a l l y observed e l e c t r i c f i e l d g ^ ^ ( T ) as f o l l o w s a

g

(

1

)

s i z e , G(a), i s r e l a t e d to autocorrelation function,

max 2

(t) - }

G(a)exp(-k Tk T/3irna)da

(1)

B

a

min

H e r e G ( a ) i s t h e i n t e n s i t y s c a t t e r e d by t h e p a r t i c l e s w i t h d i a m e t e r s between a and a+da, k i s Boltzmann's c o n s t a n t , Τ the temperature, k t h e s c a t t e r i n g v e c t o r ( d e p e n d e n t on s c a t t e r i n g a n g l e and w a v e l e n g t h ) , τ the c o r r e l a t i o n time and η the v i s c o s i t y . To f i n d t h e m a s s f r a c t i o n d i s t r i b u t i o n , F ( a ) , we u s e R

a

g

(D( ) T

max

« j

F( ) a

a

3

2

(i ( )/a )exp(-k Tk T^na)da 1

a

B

(2)

min

w h e r e i^(a) i s the s c a t t e r e d dimensionless p a r t i c l e size

intensity

per p a r t i c l e

(6)

and

α - πβη/λ


α i s the

(3)



108

H e r e 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 medium a n d λ i s t h e w a v e l e n g t h o f i n c i d e n t l i g h t i n a vacuum. We m o d i f i e d P r o v e n c h e r ' ^ p r o g r a m t o c a l l a subroutine which would supply values of ( i ^ ( a ) / a ) f o r the k e r n e l of the i n t e g r a l . The i n i t i a l s o l u t i o n i s t h a t w i t h l i t t l e o r no r e g u l a r i z a t i o n . A chosen s o l u t i o n where the i n c r e a s e i n the o b j e c t i v e f u n c t i o n o v e r the I n i t i a l s o l u t i o n c o u l d about 50% of the t i m e be due t o e x p e r i m e n t a l n o i s e a n d a b o u t 5 0 % o f t h e t i m e be due t o o v e r s m o o t h i n g , i s s e l e c t e d by a s t a t i s t i c a l c r i t e r i o n ( 4 , 5 ) . A s e c o n d a n a l y s i s p r o c e d u r e I n v o l v e s u s i n g t h e r e s u l t s f r o m many s c a t t e r i n g a n g l e s t o e x t r a c t t h e mass f r a c t i o n 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 f r o m t h e d a t a . S i n c e i t w o u l d be h a r d t o f i n d a n o p t i m u m s c a t t e r i n g a n g l e ( t h a t w h i c h has a r e l a t i v e l y h i g h i n t e n s i t y f o r the p a r t i c l e s i z e ( s ) of i n t e r e s t ) w i t h o u t a l r e a d y knowing 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 , one p o s s i b i l i t y i s t o a v e r a g e i n f o r m a t i o n f r o m s e v e r a l s c a t t e r i n g a n g l e s . We u s e t h e n o n - n e g a t i v e l y c o n s t r a i n e d l e a s t s q u a r e s method ( 6 ) , w i t h o u t f u r t h e r r e g u l a r i z a t i o n , t o o b t a i n 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 . T h e n , t h e M i e f a c t o r s ( 7 ) a r e com puted f o r each p a r t i c l e s i z e . These s c a t t e r i n g f a c t o r s are used to c o n v e r t t h e i n t e n s i t y d i s t r i b u t i o n a t e a c h a n g l e t o a mass d i s t r i b u t i o n of p a r t i c l e s i z e s . F i n a l l y , t h e 10 m a s s d i s t r i b u t i o n s ( o n e f o r each angle) are averaged t o g e t h e r to form a composite d i s t r i b u t i o n . O t h e r i n v e s t i g a t o r s (8,9,10) have used a s i m i l a r method i n w h i c h i n t e n s i t y d i s t r i b u t i o n s f r o m NNLS a r e c o n v e r t e d t o m a s s d i s t r i b u t i o n s u s i n g Mie f a c t o r s ; however, they o b t a i n d a t a f o r l o n g e r time periods at a single scattering angle. Experimental F o u r s e p a r a t e l a t e x s a m p l e s w e r e a n a l y s e d . A b i m o d a l m i x t u r e was c o m p o s e d by m i x i n g e q u a l p a r t s o f s o l u t i o n s w i t h 0 . 0 0 3 % s o l i d s o f Dow l a t e x m o n o d i s p e r s e s t a n d a r d s w i t h n o m i n a l d i a m e t e r s 1 0 9 - a n d 497-nm h a v i n g s t a n d a r d d e v i a t i o n s o f 2.7 a n d 5.9 nm r e s p e c t i v e l y . T h e m i x t u r e was s o n i c a t e d t o e l i m i n a t e a g g r e g a t e s . A p o l y s t y r e n e l a t e x w i t h a b r o a d d i s t r i b u t i o n was o b t a i n e d f r o m K o d a k . This d i s t r i b u t i o n h a d b e e n c h a r a c t e r i z e d p r e v i o u s l y by e l e c t r o n m i c r o s c o p y , u l t r a c e n t r i f u g e and C o u l t e r c o u n t e r . A m o n o d i s p e r s e , s u r f a c t a n t f r e e , s u l f a t e d , p o l y s t y r e n e s t a n d a r d a n d a m i x t u r e o f 10 such monodisperse s t a n d a r d s were purchased from I n t e r f a c i a l Dynamics Corporation. A p o l y v i n y l c h l o r i d e l a t e x w i t h a broad d i s t r i b u t i o n was d o n a t e d by B. F. G o o d r i c h . T h i s s a m p l e h a d b e e n c h a r a c t e r i z e d by J o y c e L o e b l d i s c c e n t r i f u g e . T h e s e s a m p l e s w e r e d i l u t e d t o 0.01% s o l i d s and s o n i c a t e d t o e l i m i n a t e a g g r e g a t e s . T h e c o r r e l a t i o n f u n c t i o n was m e a s u r e d u s i n g a M a l v e r n K 7 0 2 5 correlator. The w a v e l e n g t h o f t h e i n c i d e n t l i g h t ( a r g o n i o n l a s e r , L e x e l M o d e l 9 5 ) was 514.5 nm. The t e m p e r a t u r e o f t h e w a t e r b a t h s u r r o u n d i n g t h e s a m p l e c e l l was 3 0 7 . 6 K. T h e same e q u i p m e n t was used i n a p r e v i o u s p u b l i c a t i o n ( 1 ) . However, the d a t a c o l l e c t i o n p r o g r a m was m o d i f i e d t o d i s c r i m i n a t e a g a i n s t d u s t ( 1 1 ) . I n t h i s FORTRAN p r o g r a m t o t a l p h o t o n c o u n t f r o m s e v e r a l s h o r t ( 5 o r 10 second) runs were used t o e s t a b l i s h p h o t o n c o u n t a v e r a g e and s t a n dard deviation. A f t e r e a c h s h o r t e x p e r i m e n t t h e d a t a was a c c e p t e d o n l y i f i t s p h o t o n c o u n t was l e s s t h a n t h e mean p h o t o n c o u n t p l u s three standard deviations. E a c h s h o r t e x p e r i m e n t was n o r m a l i z e d t o i t s own b a s e l i n e . A s e r i e s o f 60 f i v e s e c o n d e x p e r i m e n t s w e r e



7.

STOCK AND RAY


109

a v e r a g e d t o y i e l d one a u t o c o r r e l a t i o n f u n c t i o n . Three c o r r e l a t i o n sample times were used a t each a n g l e . The maximum a n d minimum w e r e c o m p u t e d u s i n g R a c z e k ' s f o r m u l a ( 1 2 ) f o r t h e 4 9 7 - a n d 109-nm p a r ticles respectively. F o r the Kodak d i s t r i b u t i o n the e x p e c t e d upper and l o w e r bounds o f the d i s t r i b u t i o n were used. F o r t h e b r o a d d i s t r i b u t i o n s a n a l y s e d by t h e NNLS m e t h o d , d a t a was t a k e n a t 10 s c a t t e r i n g a n g l e s : 20°, 30°, 40°, 50°, 60°, 70°, 8 0 ° , 9 0 ° , 100° a n d 110°. A t each a n g l e t h r e e sample times were used. These sample t i m e s were computed u s i n g R a c z e k ' s f o r m u l a w i t h d i f f e r e n t v a l u e s f o r the bandwidth 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 : v » 2 , 4 a n d 8. F o r e a c h s e t o f c o n d i t i o n s a s e r i e s o f 30 t e n s e c o n d e x p e r i m e n t s were averaged to y i e l d a s i n g l e a u t o c o r r e l a t i o n f u n c tion. Results We now d e m o n s t r a t e t h e s e m e t h o d s o n e x p e r i m e n t a l d a t a . F i r s t we c o n s i d e r P r o v e n c h e r ' s method as m o d i f i e d u s i n g the examples o f a b i m o d a l d i s t r i b u t i o n and b r o a d d i s t r i b u t i o n . T h e NNLS m e t h o d i s d e m o n s t r a t e d o n a m o n o d i s p e r s e d i s t r i b u t i o n , a m i x t u r e o f 10 monod i s p e r s e s t a n d a r d s and a b r o a d d i s t r i b u t i o n . F o r P r o v e n c h e r s method as m o d i f i e d , the s t a t i s t i c s f o r the i n t e n s i t y a n d mass f r a c t i o n d i s t r i b u t i o n s f o u n d f o r t h e b i m o d a l m i x t u r e a r e s h o w n i n T a b l e I . We s e l e c t t h e s a m p l e t i m e s w h i c h g i v e t h e i n t e n s i t y d i s t r i b u t i o n s h a v i n g t h e p e a k a r e a r a t i o o f t h e 109 nm p e a k t o t h e 497 nm p e a k c l o s e s t t o u n i t y . T h e f i n a l mass d i s t r i b u t i o n s show o v e r a l l means w h i c h a r e q u i t e a c c u r a t e , s e e T a b l e I . F r o m F i g u r e 3 one o b s e r v e s t h a t t h e r e l a t i v e p e a k a r e a s a n d p e a k l o c a t i o n s a r e q u i t e a c c u r a t e . F i g u r e 3 a l s o s h o w s t h a t t h e f i n a l mass f r a c t i o n d i s t r i b u t i o n i s t h e same a t t h r e e d i f f e r e n t s c a t t e r i n g a n g l e s a l t h o u g h the i n t e n s i t y d i s t r i b u t i o n f o r a s c a t t e r i n g a n g l e of 30° was v e r y d i f f e r e n t . T a b l e I I shows d a t a o b t a i n e d f r o m a b r o a d p o l y s t y r e n e l a t e x s a m p l e d o n a t e d by K o d a k . The m a s s a v e r a g e a n d s t a n d a r d d e v i a t i o n c o m p a r e w e l l w i t h t h e s t a t i s t i c s f r o m t h e mass d i s t r i b u t i o n f o u n d by e l e c t r o n m i c r o s c o p y . The s o l u t i o n s f o r s c a t t e r i n g a n g l e s o f 30° a n d 90° r e p r e s e n t t h e m a j o r f e a t u r e s o f t h e d i s t r i b u t i o n f a i r l y w e l l b u t they a r e too smooth. Note that the d i s t r i b u t i o n f o u n d a t 60° h a s t h e h i g h e s t d u s t l e v e l o f a l l t h e s e solutions; i t i s l i k e l y t h a t t h e low mean i s due t o o v e r c o m p e n s a t i o n o f the d u s t term. Thus P r o v e n c h e r ' s method as m o d i f i e d y i e l d s s a t i s f y i n g r e s u l t s f o r b o t h the b i m o d a l and b r o a d d i s t r i b u t i o n e x a m p l e s e x c e p t when t h e d u s t t e r m i s l a r g e . 1


110



Table I .

Angle Actual

b

Initial 0

b

Initial Chosen

Sample Time

a

b

Initial

Results

Relative Standard

Average D i a m e t e r ( nm) Mass Intensity

(microsec)

Chosen

Chosen

Bimodal D i s t r i b u t i o n

Intensity

Deviation Mass 0.64

303

30

140

478

281

0.23

0.74

30

140

472

281

0.15

0.66

60

8

337

243

0.80

0.89

60

8

313

328

0.68

0.62

90

9

347

353

0.56

0.56

90

9

346

348

0.55

0.57

* The c h o s e n s o l u t i o n h a s t h e o p t i m a l amount o f r e g u l a r i z a t i o n . The i n i t i a l s o l u t i o n has l i t t l e o r no r e g u l a r i z a t i o n .

T a b l e I I . Kodak D i s t r i b u t i o n

Relative Standard

Average D i a m e t e r ( nm) Angle Actual

Sample Time

Intensity

(microsec)

Results

Mass

Deviation

Intensity

Mass

Dust (%)

0.30

1050

90

14

1000

1044

0.21

0.24

11

Chosen*

60

98

906

892

0.08

0.09

15

0

30

28

1112

1075

0.13

0.16

0

Chosen

Chosen

0

The c h o s e n s o l u t i o n

has the o p t i m a l

amount o f

regularization.



7.

STOCK A N D RAY

111


700 Diameter

1ΘΘ

F i g u r e 3. C o n s t r a i n e d d i s t r i b u t i o n f o r three

ι

I

(nm)

I I I I

200 300 400 500 Diameter (nm)

700

r e g u l a r i z a t i o n solutions f o r the bimodal s c a t t e r i n g a n g l e s shown.

( a ) No c o r r e c t i o n w i t h M i e f a c t o r . (b) I n c l u d i n g c o r r e c t i o n w i t h M i e f a c t o r .



112


F o r t h e second method, c o m p o s i t e d i s t r i b u t i o n r e s u l t s from t h e n o n - n e g a t i v e l y c o n s t r a i n e d l e a s t s q u a r e s method u s i n g s e v e r a l a n g l e s a r e shown i n T a b l e I I I . The c o m p o s i t e mass d i s t r i b u t i o n s l i g h t l y u n d e r e s t i m a t e s t h e mean o f t h e b r o a d d i s t r i b u t i o n s s h o w n . However, t h e e s t i m a t e o f t h e mean n e v e r d i s p l a y s a g r e a t e r e r r o r t h a n 5%. T h e s t a n d a r d d e v i a t i o n o f t h e c o m p o s i t e mass d i s t r i b u t i o n o v e r e s t i m a t e s t h a t o f a l l d i s t r i b u t i o n s . I n T a b l e I I I a n d i n F i g u r e 4 we c o n s i d e r t h e r e s u l t s f r o m t h e B. F . G o o d r i c h s a m p l e . B o t h F i g u r e s 4 a and 4b a r e t h e a v e r a g e o f 10 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 e a c h f r o m a n a u t o c o r r e l a t i o n f u n c t i o n c o l l e c t e d f o r 15 m i n u t e s . I n Figure 4a each a u t o c o r r e l a t i o n f u n c t i o n was f r o m s c a t t e r i n g a t a d i f f e r e n t s c a t t e r i n g a n g l e w h i l e i n F i g u r e 4b a l l t h e a u t o c o r r e l a t i o n f u n c t i o n s w e r e f r o m s c a t t e r i n g a t 90°. N o t e t h a t somewhat b e t t e r r e s u l t s a r e o b t a i n e d by c o m b i n i n g d a t a f r o m s c a t t e r i n g a t s e v e r a l a n g l e s .

Table I I I .

M o d i f i e d NNLS M e t h o d R e s u l t s ; C o m p o s i t e D i s t r i b u t i o n s from A l l Angles

Average diameter(nm) Intensity Mass A c t u a l

Relative Standard Deviation Actual Intensity Mass

412-nm PS S t a n d a r d

451

428

412

0.181

0.077

0.02

Mixture of Standards

564

612

648

0.66

0.54

0.40

BFG

PVC1 C o m p o s i t e

618

641

658

0.44

0.50

0.41

BFG

P V C 1 90

571

617

658

0.46

0.39

0.41

only

Conclusion By i n c o r p o r a t i n g t h e o r e t i c a l p r e d i c t i o n s o f t h e i n t e n s i t y a s a f u n c t i o n o f p a r t i c l e s i z e f o reach s c a t t e r i n g angle given the i n c i d e n t w a v e l e n g t h and approximate r e f r a c t i v e i n d e x r a t i o i n t o t h e data anal y s i s ( v i a c o n s t r a i n e d r e g u l a r i z a t i o n and NNLS), i t i s p o s s i b l e t o o b t a i n t h e mass f r a c t i o n 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 f r o m t h e p h o ton a u t o c o r r e l a t i o n function of Mie s c a t t e r e r s without separate measurements o f i n t e n s i t y s c a t t e r i n g a s a f u n c t i o n o f a n g l e and without extrapolation to zero angle. The M i e s c a t t e r i n g f a c t o r was i n c l u d e d i n t h e k e r n e l o f t h e i n t e g r a l a n a l y s e d by c o n s t r a i n e d regularization. F o r t h e NNLS m e t h o d , t h e i n t e n s i t y d i s t r i b u t i o n d e t e r m i n e d by a n a l y s i s o f t h e a u t o c o r r e l a t i o n f u n c t i o n was c o n v e r t e d t o a mass d i s t r i b u t i o n by m u l t i p l y i n g e a c h h i s t o g r a m s t e p a r e a by t h e appropriate s c a t t e r i n g f a c t o r (average over the p a r t i c l e s i z e repres e n t e d by t h e s t e p ) . We h a v e s h o w n t h a t t h e c o m p o s i t e mass d i s t r i b u t i o n a v e r a g e d o v e r s e v e r a l s c a t t e r i n g a n g l e s i s s o m e t i m e s more a c c u r a t e t h a n t h e c o m p o s i t e m a s s d i s t r i b u t i o n a v e r a g e d o v e r t h e same t o t a l time from a s i n g l e s c a t t e r i n g angle. The m o d i f i c a t i o n o f t h e NNLS m e t h o d i n v o l v e s s i m p l e r a n d f a s t e r c o m p u t a t i o n t h a n t h e m o d i f i c a t i o n o f t h e c o n s t r a i n e d r e g u l a r i z a t i o n method.


113


STOCK AND RAY


7.

b

0

900

1Θ00

1900

2000

Diameter (nm) F i g u r e 4. NNLS s o l u t i o n s f o r t h e B. F. G o o d r i c h s a m p l e . Both s o l u t i o n s i n c l u d i n g the M i e f a c t o r (mass) and w i t h o u t c o r r e c t i o n ( i n t e n s i t y ) a r e shown. (a) Average o f r e s u l t s from c o r r e l a t i o n f u n c t i o n s taken a t t e n scattering angles, (b) Average o f r e s u l t s

from

ten correlation

functions

t a k e n a t 90?



114


Literature Cited 1. R. S. Stock and W. H. Ray, J. Polym. Sci. Polym. Phys. Ed., 23, 1393, (1985). 2. Β. Bedwell, Erd. Gulari and D. Melnik in "Measurement of Suspended Particles by Quasielastic Light Scattering", Β. E. Dahneke Ed., Wiley, New York, p. 237, 1983. 3. S. Bott, in this volume. 4. S.W. Provencher, J. Hendrix, L. DeMaeyer and N. Paulussen, J. Chem. Phys., 69, 4273 (1978). 5. S. W. Provencher, Comput. Phys. Commun., 27, 213 and 219, (1982). 6. E. F. Grabowski and I. D. Morrison in "Measurement of Suspended Particles by Quasielastic Light Scattering", Β. E. Dahneke Ed., Wiley, New York, p. 199, 1983. 7. H. H. Denman, W. Heller and W. J. Pangonis, "Angular Scattering Functions for Spheres", Wayne State University Press, Detroit, 1966. 8. C. A. Herb, I. D. Morrison and E. F. Grabowski, in this volume. 9. C. A. Herb, I. D. Morrison and E. F. Grabowski, in "Magnetic Resonance and Scattering in Surfactant Systems", L. Magid, ed., Plenum Press, to be published. 10. I. D. Morrison, E. F. Grabowski and C. A. Herb, Langmuir, 1, 496, (1985). 11. Erd. Gulari, Es. Gulari, S. Tsunashima and B. Chu, Polymer, 20, 347, (1979). 12. J. Raczek, Eur. Polym. J., 18, 863 (1982). RECEIVED August 7, 1986


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