Particle Size Distribution - American Chemical Society

Chapter 8 Photon Correlation Spectroscopy, Transient Electric Birefringence, and Characterization of Particle Size

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Distributions in Colloidal Suspensions Renliang Xu, James R. Ford, and Benjamin Chu Department of Chemistry, State University of New York at Stony Brook, Long Island, NY 11794-3400 By using a combination of static and dynamic laser light scattering (LLS) and transient electric birefringence (TEB) we have been able to determine structural characteristics and size distributions of polydisperse disk-shaped particles (bentonite) in suspensions. In the limit of low concentration and scattering angle we obtained the weight-average molecular weight M w , the z -average radius of gyration 1/2 and the second virial coefficient A 2 from static light scattering measurements; at higher scattering angles we were able to estimate an average particle thickness. Photon correlation function measurements of both the polarized and the depolarized components of scattered light give us the average diffusion coefficients DT (translational) and DR (rotational) which can in turn be converted to average particle dimensions. Detailed analysis of characteristic linewidth distributions yield particle size distributions consistent with direct observations using electron microscopy. The TEB experiment provides us with the average optical polarizability difference Δα°, the ratio of permanent dipolar moment to electric polarizability difference, and the average rotational diffusion coefficient DR (TEB). Profile analysis of the decay curve yields a distribution of particle sizes consistent with the results from LLS.

Laser light scattering (LLS), which takes advantage of both static (intensity) and dynamic (intensity correlation/linewidth) measurements, has been used successfully in determining the molecular weight distribution (MWD) of polymers in solution. The essential steps are: 1. to use static light scattering to measure the weight-average molecular weight Mw which calibrates the MWD, 2. to use dynamic light scattering to measure the intensity-intensity 0097-6156/87/0332-0115$06.00/0 © 1987 American Chemical Society Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

116

PARTICLE SIZE DISTRIBUTION

t i m e c o r r e l a t i o n . f u n c t i o n G^ ^(T), and 3· t o p e r f o r m a L a p l a c e i n v e r s i o n of the '(τ) i n o r d e r t o o b t a i n a n a p p r o x i m a t i o n t o the c h a r a c t e r i s t i c l i n e w i d t h d i s t r i b u t i o n f u n c t i o n G(r) w h i c h i s r e l a t e d t o t h e MWD by m e a n s o f a n e m p i r i c a l s c a l i n g r e l a t i o n . I n c o l l o i d a l s u s p e n s i o n s of a n i s o t r o p i c p a r t i c l e s , the static structure factor plays a prominent role i n p a r t i c l e s i z e analysis. We h a v e u s e d t r a n s i e n t e l e c t r i c b i r e f r i n g e n c e ( T E B ) a n d electron m i c r o s c o p y , i n a d d i t i o n to l a s e r l i g h t s c a t t e r i n g , to c o r r e l a t e our a n a l y s i s of 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 of bentonite suspensions. The c o m p l e m e n t a r y n a t u r e o f TEB and p h o t o n c o r r e l a t i o n s p e c t r o s c o p y (PCS) i n p a r t i c l e s i z e a n a l y s i s w i l l be discussed. 2

Theory TEB. Transient e l e c t r i c b i r e f r i n g e n c e (TEB) w i t h s i n g l e , r e v e r s e d o r s i n u s o i d a l e l e c t r i c p u l s e s has become a u s e f u l t o o l f o r studying the structure of l a r g e anisotropic particles in solution (or s u s p e n s i o n ) s i n c e i t was f i r s t p r o p o s e d b y O ' K o n s k i a n d Z i m m ( ] ) . By a p p l y i n g l i n e a r l y p o l a r i z e d l i g h t t o a s y s t e m o f a n i s o t r o p i c p a r t i c l e s a l i g n e d b y a n e x t e r n a l e l e c t r i c f i e l d and b y o b s e r v i n g t h e depolarized i n t e n s i t y I of the t r a n s m i t t e d l i g h t as the system r e l a x e s a f t e r r e m o v a l o f t h e e x t e r n a l f i e l d , we c a n o b s e r v e the change i n the r e f r a c t i v e index difference of two orthogonal d i r e c t i o n s w i t h r e s p e c t t o t i m e and t h e r e b y d e t e r m i n e t h e r o t a t i o n a l diffusion c o e f f i c i e n t and obtain information on the optical p o l a r i z a b i l i t y d i f f e r e n c e and t h e r a t i o o f p e r m a n e n t d i p o l e moment to e l e c t r i c p o l a r i z a b i l i t y d i f f e r e n c e . The r e l a t i o n s h i p between the depolarized time-dependent t r a n s m i t t e d i n t e n s i t y l ( t ) and t h e r e f r a c t i v e i n d e x d i f f e r e n c e i n t w o o r t h o g o n a l d i r e c t i o n s χ and y, i n w h i c h one i s i n t h e d i r e c t i o n o f t h e a p p l i e d e l e c t r i c f i e l d and b o t h a r e p e r p e n d i c u l a r to the d i r e c t i o n of p r o p a g a t i o n of the i n c i d e n t l i g h t , i s g i v e n b y (2)

I(t)-I

b

= k l

p

s i n

2

( a

+

^ )

When t h e e x t e r n a l f i e l d particle fractions, An Q i

E

6

i ss u f f i c i e n t l y strong to f u l l y a l i g n a l l reaches i t s saturation value Δη ^ 3

2πφ. A

û

s i

=

(

δ

8 - ι

}

(

7

)

3

where i s t h e v o l u m e f r a c t i o n o f t h e i t h p a r t i c l e f r a c t i o n and g^ and g-j a r e t h e m a j o r a n d t h e m i n o r p a r t i c l e a x i s o p t i c a l a n i s o t r o p y f a c t o r s given by ο α. g.

= τ-2— ο

(8)

w i t h a*? b e i n g t h e o p t i c a l p o l a r i z a b i l i t y i n t h e j t h d i r e c t i o n ; ε , the p e r m i t t i v i t y i nvacuo; v, t h e volume o f a s i n g l e p a r t i c l e ; and n, 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 u t i o n . 0

Static Light Scattering. For a solution of polydisperse nonabsorbing particles, the Rayleigh r a t i o R (9) i s the excess absolute scattered i n t e n s i t y o f the s o l u t i o n over that o f the pure v v

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

8.

Particle Size Distributions in Colloidal Suspensions

X U ET AL.

119

s o l v e n t a t s c a t t e r i n g a n g l e ft, w h e r e t h e s u b s c r i p t v v i n d i c a t e s v e r t i c a l l y p o l a r i z e d i n c i d e n t and s c a t t e r e d l i g h t , and h a s t h e f o r m R (Θ) vv ^ —

= ζ ν

±

[ Ά

±

Η ^

±

) - 2 λ

2

ο

±

( κ

±

Η κ ι *

±

) )

2

(9)

]

where and P(KL^) a r e t h e c o n c e n t r a t i o n and t h e p a r t i c l e s c a t t e r i n g f a c t o r o f s p e c i e s i h a v i n g m o l e c u l a r w e i g h t M^, Κ i s t h e m a g n i t u d e o f t h e momentum t r a n s f e r v e c t o r g i v e n b y Κ = ( 4 f T A ) s i n ( 6 /2), L^ i s t h e e q u i v a l e n t d i a m e t e r o f t h e i t h p a r t i c l e f r a c t i o n , k i s t h e second v i r i a l c o e f f i c i e n t , and t h e o p t i c a l constant Η i s g i v e n by 2

2

2

4π η Οη/8θ Η

2

= A

ο

with N and (3n/3c) being Avogadro's number and t h e r e f r a c t i v e i n d e x i n c r e m e n t , r e s p e c t i v e l y . F o r s u f f i c i e n t l y s m a l l v a l u e s o f Θ, e q u a t i o n (9) c a n be w r i t t e n a s ( 4 ) A

2

_ = R

(θ)

w

i M Ρ (KL)

+ +

2

w

OA r = — 2 M A

C

w

Π+ —S—-—3

K

2

+

h 2A M C l 2 w Z A

L J

(10) u

u

;

where M i s the weight-average molecular weight; C, the t o t a l concentration of particles; P(KL), the z-average p a r t i c l e scattering f a c t o r ; and < R | ( C ) > , t h e mean s q u a r e " r a d i u s o f g y r a t i o n " a t f i n i t e c o n c e n t r a t i o n C , w h i c h c a n be c o r r e c t e d f o r c o n c e n t r a t i o n by W

z

z

-

[1 + 2 A M C ]

2

Z

2

(11)

W

2

with ^ being the radius of gyration. For disk-like p a r t i c l e s , s u c h a s t h e b e n t o n i t e p a r t i c l e s , e q s . (9), ( l O ) a n d ( l l ) are only approximate because of o p t i c a l anisotropy. In fact, p o l a r i z e d l i g h t s c a t t e r i n g (vv-component) y i e l d s only apparent v a l u e s f o rt h e m o l e c u l a r w e i g h t , t h e r a d i u s o f g y r a t i o n and t h e second v i r i a l c o e f f i c i e n t . T h e c o r r e c t v a l u e s c a n b e c o m p u t e d i f we know t h e d e p o l a r i z a t i o n r a t i o and t h e p a r t i c l e shape. For bentonite p a r t i c l e s , t h e d e p o l a r i z a t i o n r a t i o i s v e r y s m a l l (~0.02). Thus, t h e c o r r e c t i o n f a c t o r i s o f t h e o r d e r o f a f e w p e r c e n t w h i c h we s h a l l ignore i n the present study. z

D y n a m i c L i g h t S c a t t e r i n g . The m e a s u r e d s i n g l e c l i p p e d p h o t o e l e c t r o n count a u t o c o r r e l a t i o n f u n c t i o n f o r the s e l f - b e a t i n g experiment has the form (5) 2

G< )(T,K)

= N [l s

k

x

+

2

b| g< >(τ , ) | ]

(12)

K

w h e r e τ i s t h e d e l a y t i m e ; k, t h e c l i p p i n g l e v e l ; N , t h e t o t a l n u m b e r o f s a m p l e s ; < n > a n d t h e m e a n c l i p p e d a n d u n d i p p e d Q

k


120


counts, respectively; b i s a s p a t i a l coherence f a c t o r g e n e r a l l y tak.en a s an u n k n o w n p a r a m e t e r i n t h e f i t t i n g p r o c e d u r e ; a n d g** '(τ,Κ) i s the n o r m a l i z e d first order electric field autocorrelation function. E q u a t i o n (12) i s v a l i d f o r G a u s s i a n s i g n a l s (6)· For monodisperse systems the normalized polarized d e p o l a r i z e d t i m e c o r r e l a t i o n f u n c t i o n s can be e x p r e s s e d as ( 7 )

β^(τ,Κ) -

β^(τ,Κ) -

and

P (KL)exp[(-D K -m(m+l)D^T] 2

Σ m=0,even

m

(13)

T

2

exp[(-D K -6D )x] T

(Η)

R

w h e r e Dy and D a r e t h e t r a n s l a t i o n a l and r o t a t i o n a l diffusion c o e f f i c i e n t s and t h e p a r t i c l e s c a t t e r i n g i n t e n s i t y f a c t o r s P ( K L ) can be c a l c u l a t e d t h e o r e t i c a l l y f o r p a r t i c l e s o f d i f f e r e n t s h a p e s ( 7 , 8 ) . E q . ( 1 4 ) i s v a l i d o n l y f o r s m a l l KL. S c a t t e r i n g f a c t o r s f o r d i s k - l i k e p a r t i c l e s h a v e b e e n p u b l i s h e d e l s e w h e r e ( 8 ) and we r e c a p i t u l a t e them i n F i g u r e 2. At s u f f i c i e n t l y s m a l l s c a t t e r i n g a n g l e s we can u s e e q u a t i o n s ( 1 3 ) and ( 1 4 ) t o d e t e r m i n e t h e v a l u e s o f D and D f r o m t h e m e a s u r e d p o l a r i z e d and d e p o l a r i z e d i n t e n s i t y intensity autocorrelation f u n c t i o n s (£^ν(τ , K ) a n d G ^ ( T , K ) . It s h o u l d be noted t h a t the model ( 7 , 8 ) assumed i s o t r o p i c t r a n s l a t i o n a l d i f f u s i v e m o t i o n , w h i l e i n r e a l i t y , we m u s t t a k e i n t o a c c o u n t anisotropy in translational diffusion (9.). However, i n e x t r a p o l a t i o n t o i n f i n i t e d i l u t i o n and z e r o s c a t t e r i n g a n g l e , we c a n retrieve the correct translational diffusion coefficient e x p e r i m e n t a l l y , r e g a r d l e s s o f model. R

m

T

R

For |g

polydisperse-particles, ( l )

(x,K)|

= / 0(Γ,κ)β-Γτ

we can w r i t e

ά

Γ

(15)

where G(r,K) i s the n o r m a l i z e d d i s t r i b u t i o n o f c h a r a c t e r i s t i c l i n e w i d t h Γ a t a g i v e n v a l u e o f K. I n v e r s i o n o f e q u a t i o n ( 1 5 ) i n t h e presence of e x p e r i m e n t a l noise to o b t a i n the c h a r a c t e r i s t i c l i n e w i d t h d i s t r i b u t i o n G(r,K) i s an i l l - c o n d i t i o n e d p r o b l e m w h i c h has r e c e i v e d much a t t e n t i o n i n the r e c e n t l i t e r a t u r e ( 6 , 1 0 , 1 1 ) . F o r the sake o f c l a r i t y we have o m i t t e d t h e p o l a r i z a t i o n s u b s c r i p t s ( w o r vh) i n t h i s s e c t i o n . , . 1

The m e t h o d o f c u m u l a n t s (1 2) e x p a n d s g ' ' ( O i n t e r m s o f t h e moments o f the c h a r a c t e r i s t i c l i n e w i d t h d i s t r i b u t i o n f,\ |

( ΐ ) δ

Ρ (Κ)τ + \

2

μ.(Κ)τ

9

(τ,Κ)|

- A exp[-r(K)t

3

+

s

...]

1

(16)

w h e r e Γ i s t h e a v e r a g e l i n e w i d t h and yi - A m o d i f i e d M a r q u a d t - L e v e n b e r g n o n l i n e a r l e a s t s q u a r e s r o u t i n e was u s e d t o d e t e r m i n e t h e b e s t v a l u e s o f Γ a n d μ ·. A n o t h e r s t r a i g h t f o r w a r d a p p r o a c h i n v o l v e d a p p r o x i m a t i n g G(r , K T a s t h e w e i g h t e d sum o f two c h a r a c t e r i s t i c decay t i m e s Γ^"^ and Γ 2


8.

X U ET AL.

G(r,K)

121

Particle Size Distributions in Colloidal Suspensions -Γι τ τ

1+

we

(17)

9

The same nonlinear f i t t i n g routine could determine the best values for w^ and which might not necessarily have physical meaning but could be used to compute Γ and ν · used a regularized inversion of equation (15) that incorporates a p o s i t i v i t y constraint on G(r,K) i n order to obtain more detailed information on the c h a r a c t e r i s t i c linewidth p r o f i l e , A discussion of this approach may be pursued i n reference (lO). W

e

2

Experimental Methods Sample P r e p a r a t i o n , The b e n t o n i t e powder obtained from F i s h e r S c i e n t i f i c Company was dissolved i n doubly d i s t i l l e d and f i l t e r e d water to make a stock s o l u t i o n with C = 3·65 x 10"^ g/g. T h i s stock solution was further diluted to the concentrations l i s t e d i n Table I. A l l measurements were made within 10 days of sample preparation i n order to avoid possible aging problems. TEB. Our TEB apparatus c o n s i s t s of a 1 5 mW He-Ne l a s e r w i t h X = 6 3 2 . 8 nm., three Glan-Thomson p o l a r i z e r s and a q u a r t e r wave p l a t e arranged as shown i n figure 1. The sample was contained i n a Beckman 1 cm q u a r t z c e l l w i t h two platinum electrodes, h e l d at 25·ΟΟ+Ο.Ο5°0. The t r a n s m i t t e d l i g h t i n t e n s i t y was measured by a photomultiplier tube (lP-28) and the output sent to a Biomation 8100 t r a n s i e n t recorder. The e x t e r n a l e l e c t r i c f i e l d was provided by a Cober 605P high voltage pulse generator. The entire experiment was under the control of a microcomputer. High voltage square pulses from 0.5 to 3·3 KV/cm w i t h pulse widths of around 2 msec were a p p l i e d to the f i v e b e n t o n i t e suspensions. The p u l s e t r a c e s and the s i g n a l f r o m the photomultiplier tube were simultaneously recorded at a resolution of 1 0 2 4 p o i n t s each i n the t r a n s i e n t r e c o r d e r using a sample time increment of 40 usee and d i s p l a y e d on an o s c i l l o s c o p e . A f t e r 10 passes the data were transferred to the microcomputer and saved on floppy disk f o r subsequent analysis. The entire measurement sequence for one sample took a few minutes. Q

Light Scattering. The l i g h t scattering table consists of an argon i o n l a s e r , s u i t a b l e lenses and p i n h o l e s to c o n d i t i o n the i n c i d e n t beam, a sample c e l l i n a brass thermostat (25·00_+ 0.05° C) and a rotatable detector arm on which are mounted o p t i c a l parts to define the s c a t t e r i n g geometry and the p h o t o m u l t i p l i e r tube as shown i n f i g u r e 1. The s i g n a l from the p h o t o m u l t i p l i e r tube i s channeled through a preamplifier/discriminator to either a pulse counter for i n t e n s i t y measurements or a Malvern 7027 s i n g l e - c l i p p i n g correlator for the time c o r r e l a t i o n measurements. Intensity measurements of the f i v e suspensions were performed u s i n g the l a s e r at an i n c i d e n t wavelength A = 5 1 4 . 5 nm over an angular range of 1 5 to 36 degrees. Dynamic measurements were done using i n c i d e n t wavelengths of A = 5 1 4 . 5 and 488.0 nm over angular ranges of 1 5 to 5 4 degrees and of 3 5 to 80 degrees, respectively. Q

Q



122 1. 0

F i g u r e 2. S c a t t e r i n g f a c t o r o f d i s k - l i k e p a r t i c l e s w h e r e Κ = 4n" s i n ( 6 / 2 ) / X , and L, t h e p a r t i c l e d i a m e t e r . P r e p r e s e n t s the remaining higher order terms. h

Table I.

Sample

Concentrations

S a m p l e Number

1

TEB

Concentration

(10"^

2.14

2

1.18

3

0.83

4

2.93

5

6.43

1

LLS

g/g

1.02

2

0.67

3

0.38

4

0.21

5

0.13


water)

8.


X U ET AL.

123

( g j i / j C ) . The r e f r a c t i v e i n d e x i n c r e m e n t (9n/3C) o f t h e b e n t o n i t e s u s p e n s i o n was determined using a B r i c e Phoenix differential r e f T a c t o m e t e r a t w a v e l e n g t h s 436 and 546 nm. By i n t e r p o l a t i o n , we e s t i m a t e d (3n/3C) t o be 0.086 a t 514-5 nm. and 0.082 a t 488 nm, i n r e a s o n a b l e a g r e e m e n t w i t h t h o s e v a l u e s r e p o r t e d by o t h e r a u t h o r s (13,14,15). R e s u l t s and D i s c u s s i o n S t a t i c L i g h t S c a t t e r i n g Measurements. A Zimm p l o t f o r the b e n t o n i t e s u s p e n s i o n s i s shown i n f i g u r e 3. H e r e we h a v e p l o t t e d HC/R (6 ) a g a i n s t C K ' + s i n ( θ / 2 ) w h e r e K* i s an a r b i t r a r y c o n s t a n t , and determined M = 3-57 x 10^ g / m o l e , A = 2.93 x 10"^ m o l e - g / g , and =1.51 x 10" cm from yv

2

w

2

9

2

Z

lim

HC

00

R

_

(θ)

w

lim HC Θ+0 R (θ) w

1 M

=

_1 M

k 2 < R U

|

Î

( 0 ) > Z ]

(18)

w

(19) 2

w

and

We h a v e a l s o b e e n a b l e t o e s t i m a t e t h e a v e r a g e t h i c k n e s s of b e n t o n i t e d i s k s from t h e s l o p e o f t h e Zimm p l o t a t z e r o c o n c e n t r a t i o n and h i g h s c a t t e r i n g a n g l e s . I f we take ο 2 J (KL ) P(KL ) = — [ 1 * ] (21 ) (KL.) ^ i 1

±

2

as t h e p a r t i c l e s c a t t e r i n g f a c t o r f o r a d i s k s h a p e d p a r t i c l e ( 8 ) , w h e r e J-j (KL^) i s t h e f i r s t o r d e r B e s s e l f u n c t i o n . F o r KL > 3·5, e q u a t i o n (21; becomes V(KL )

= 8/(KLi)

±

2

(22)

w i t h an e r r o r o f l e s s than 13$. R ^

(θ) - = H

, ^ 2 CK

C

8

-

1

C

T h e r e f o r e , e q u a t i o n (9) becomes

A

— L. 1

(23)

By assuming t h a t a l l the b e n t o n i t e p a r t i c l e s have the same t h i c k n e s s d and d e n s i t y p, and u s i n g the r e l a t i o n L? . — = (24) 4


124


we o b t a i n an e q u a t i o n from d i s k s can be d e t e r m i n e d :

HC R (θΤ w

8πη

-

=

2

S

7 2 —

i

n

(

λ dpN ο A

2,θ 2>

which

the t h i c k n e s s

d o f the b e n t o n i t e

, -\

λ

0ΐ

(

2

5

)

A

I f we t a k e a v a l u e o f 2.0 g / m l f o r t h e d e n s i t y o f t h e b e n t o n i t e disks, we o b t a i n an e s t i m a t e o f 30 A f o r t h e t h i c k n e s s o f a b e n t o n i t e d i s k f r o m t h e s l o p e o f t h e Zimm p l o t i n f i g u r e 3 a t z e r o c o n c e n t r a t i o n and h i g h s c a t t e r i n g a n g l e s . 0

Dynamic L i g h t S c a t t e r i n g Measurements. At s u f f i c i e n t l y low s c a t t e r i n g a n g l e s where o n l y the f i r s t term o f t h e s t a t i c s t r u c t u r e f a c t o r P Q ( K L ) i n e q u a t i o n ( 1 3 ) i s i m p o r t a n t , we c a n e x p r e s s t h e linewidth obtained from p o l a r i z e d dynamic l i g h t scattering measurements as (17)

r

v y

(L,K)

- D (L)K

2

2

2

(1 + f < R ( c ) ^ K )

T

(26)

where D (L)

= D°(L)

T

(1 + k C )

(27)

d

with D ( L ) being the t r a n s l a t i o n a l diffusion c o e f f i c i e n t of p a r t i c l e s w i t h main d i m e n s i o n L e x t r a p o l a t e d t o i n f i n i t e d i l u t i o n . k i sa system s p e c i f i c second v i r i a l c o e f f i c i e n t which combines h y d r o d y n a m i c and t h e r m o d y n a m i c f a c t o r s , and f i s a d i m e n s i o n l e s s number d e p e n d e n t upon t h e p a r t i c l e s t r u c t u r e , p o l y d i s p e r s i t y and solvent. _ T

d

By u s i n g t h e m e t h o d o f c u m u l a n t s , we d e t e r m i n e d r . With v a l u e s o f , A and M ^ o b t a i n e d from s t a t i c l i g h t s c a t t e r i n g m e a s u r e m e n t s , we o b t a i n e d D a n d f f r o m t h e s l o p e o f _ a p l o t o f r /K v e r s u s Κ a t d i f f e r e n t _ c o n c e n t r a t i o n s . By p l o t t i n g D a g a i n s t c o n c e n t r a t i o n , we e s t i m a t e d D ^ - 9·70 χ 10"^ c m / s e c and k = -5-10 χ 10 " g / g . T h e c o n c e n t r a t i o n d e p e n d e n c e o f t h e f v a l u e s s u g g e s t s t h a t f o r l a r g e p a r t i c l e s , i n t e r n a l m o t i o n s have a m e a s u r a b l e e f f e c t on t r a n s l a t i o n a l motions. T h u s , we e x p e c t that D has a c o n c e n t r a t i o n dependence. We were a l s o a b l e to o b t a i n a c o n t i n u o u s l i n e w i d t h d i s t r i b u t i o n c u r v e G ( r , K ) from t h e r e g u l a r i z a t i o n a p p r o a c h by i n v e r s i o n o f e q u a t i o n (15)· The r e s u l t o b t a i n e d f r o m t h e L a p l a c e inversion procedure i s a s e t of d e l t a f u n c t i o n s G ( K ) which a p p r o x i m a t e t h e continuous G ( T , K ) curve sampled at equal i n t e r v a l s . The s u b s c r i p t w s h a l l h e n c e f o r t h be o m i t t e d i n the i n t e r e s t o f c l a r i t y in notation. I n o r d e r to o b t a i n a s i z e d i s t r i b u t i o n from G ( I ^ K ) we n e e d t o c o n v e r t b o t h t h e o r d i n a t e and t h e a b s c i s s a ( 1^) i n t h e f o l l o w i n g way. From e q u a t i o n s (26) and (27) we o b t a i n y v

z

2

T

V Y

T

2

d

4

R

Y V

R

I F

V V

F

k Τ 2

Γ -

2

K ( l + k C ) ( l + f K ) d

z

Ο


(28)

8.


X U ET AL.

where n

i s the s o l v e n t v i s c o s i t y and

Q

we

have used

125 (5)

the r e l a t i o n

k Τ D

T

-

6^L

Particle Size Distribution - American Chemical Society

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