Particle Size Distribution II - American Chemical Society

Chapter 1

Turbidimetric Techniques

Downloaded by UNIV OF ALABAMA TUSCALOOSA on September 3, 2013 | http://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch001

Capability To Provide the Full Particle Size Distribution

Theodora Kourti, John F. MacGregor, and Archie E. Hamielec McMaster Institute for Polymer Production Technology, Department of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada

A detailed theoretical investigation on the capability of turbidimetric methods to provide an estimate of the particle size distribution (PSD) in suspensions of non-absorbing particles is presented. It is shown that turbidimetric methods are not expected to provide the fullPSDin certain cases. The type of information that can be extracted from a turbidimetric method is strongly related to the (m, α) values of the system under consideration, (m is the ratio of the refractive index of the suspended particles to that of the medium; α is proportional to the ratio of the particle diameter to the wavelength of the light in the medium.) Therefore, results and conclusions cannot be extrapolated from one system to another without knowledge of how these parameters affect the results. For small values of m (m i - = S V ^ l

+ ^

(3)

F>.n-P>)

where S ( D , o ) i i s the r e s i d u a l s u m of squares c o r r e s p o n d i n g to the e s t i m a t e d ( D , o), p is the n u m b e r of p a r a m e t e r s t h a t were e s t i m a t e d (p = 2), n i s the n u m b e r of observations (n = 3) a n d F ( v i , V 2 ) gives the upper co% of a n F d i s t r i b u t i o n w i t h v i , V2 degrees of freedom. g

m

n

u

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

g

1. K O U R T I E T A L .

7


0.35


J 5 % \ 9 0 % \95%

K % \ \ VVn\ \

0.30

i

\

\ O 0.25

\ \\\\ \ \\\\\ \ x

\ Yv\\

\

UJ Q

\

\\\\ \ \

Q or

q 0.20

\

\ \ \V \ \ \ \ \ \ \

0.7pm). A s a r e s u l t , u n d e r r a n d o m e r r o r , the e s t i m a t e s for the p o l y s t y r e n e cases a r e v e r y close to the t r u e s o l u t i o n for D v a l u e s as s m a l l as 0.4 urn. T h e m a x i m u m % d e v i a t i o n s f r o m the t r u e s o l u t i o n observed i n the e s t i m a t e d D , D N a n d D v a l u e s , w h e n a 3% e r r o r was i n t r o d u c e d i n the specific t u r b i d i t y m e a s u r e m e n t s , a r e s u m m a r i z e d i n T a b l e II. N o t i c e t h a t for the cases where the specific t u r b i d i t y curves s t a y close together a r o u n d t h e i r i n t e r s e c t i o n v e r y l a r g e d e v i a t i o n s f r o m the t r u e s o l u t i o n were observed i n the e s t i m a t e d D a n d D N ; however, the e s t i m a t e of the D i s a l w a y s v e r y close to the t r u e v a l u e . It i s c l e a r f r o m the above t h a t for the w a v e l e n g t h s used, for p o l y ( v i n y l acetate) latexes ( m = l . l ) a successful estimate of the P S D c a n be obtained o n l y for suspensions w i t h l a r g e p a r t i c l e s , w h i l e for polystyrene (where the v a l u e of m i s l a r g e r ) the r e g i o n where a successful estimate of the P S D c a n be obtained c a n be extended to s m a l l e r sizes. g

g

g

g

g

w

g

w

T A B L E II. M a x i m u m % E r r o r i n E s t i m a t e d P a r a m e t e r s ( W i t h Respect to T h e T r u e Solution) F o r E x p e r i m e n t a l E r r o r 3% Polystyrene Latexes Cases (point i n F i g . 4)

P o l y ( v i n y l acetate) L a t e x e s C a s e s (point i n F i g . 3) parameter

Dg D D N

w

B

D

F

H

I

K

B

C

D

E

F

62.0 57.0 2.0

37.0 32.0 3.0

13.6 11.3 3.6

45.2 44.5 0.7

46.5 39.0 4.6

7.6 6.1 7.4

27.0 24.0 0.9

15.0 13.0 2.1

10.0 7.7 4.3

4.8 4.0 5.1

8.6 8.0 3.6




0.0


0.2 MEAN

0.4

9

0.6

OF LOG-NORMAL,

0.8 Dg

1.0

(/im)

F i g u r e 3. E s t i m a t e s of the p a r t i c l e size d i s t r i b u t i o n w h e n a r a n d o m e r r o r (up to ± 3%) was added to the specific t u r b i d i t i e s . P o i y ( v i n y l acetate) l a t e x .

0.0

0.2

0.4

0.6

M E A N O F L O G - N O R M A L , Dg

0.8

1.0

(/xm)

F i g u r e 4. E s t i m a t e s of the p a r t i c l e size d i s t r i b u t i o n w h e n a r a n d o m e r r o r (up to ± 3%) was added to the specific t u r b i d i t i e s . P o l y s t y r e n e latex.


10

PARTICLE SIZE DISTRIBUTION II


The Diameter Exponent and The Apparent Diameter It is c l e a r t h a t the specific t u r b i d i t y b e h a v i o u r depends: i) o n the p a r t i c l e sizes covered by the P S D of the suspension a n d ii) o n the v a l u e of m of the suspension. T o e x p l a i n t h i s b e h a v i o u r we w i l l m a k e use of two concepts: the d i a m e t e r exponent a n d the a p p a r e n t diameter. F r o m specific t u r b i d i t y m e a s u r e m e n t s one c a n estimate a s i n g l e d i a m e t e r for a suspension, a n apparent diameter. F o r a monodisperse suspension the a p p a r e n t d i a m e t e r , is the t r u e d i a m e t e r o f the p a r t i c l e s . F o r a p o l y d i s p e r s e s u s p e n s i o n the a p p a r e n t d i a m e t e r , o b t a i n e d by t r e a t i n g the s y s t e m as a monodisperse one, i s sometimes n u m e r i c a l l y close to a m e a n i n g f u l a v e r a g e o f 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 ( w e i g h t a v e r a g e , v o l u m e to surface a v e r a g e , etc.). T h e a p p a r e n t d i a m e t e r e s t i m a t e d for a polydisperse suspension f r o m a specific t u r b i d i t y m e a s u r e m e n t (see A p p e n d i x A ) c a n be defined as (see E q u a t i o n 2):

T h e r e l a t i o n between the a p p a r e n t d i a m e t e r a n d a n average d i a m e t e r of the p a r t i c l e size d i s t r i b u t i o n has been discussed by M e e h a n a n d B e a t t i e (18) a n d i t i s b r i e f l y o u t l i n e d below. F o r a monodisperse s u s p e n s i o n , at a g i v e n w a v e l e n g t h , the dependence of the s c a t t e r i n g coefficient o n the d i a m e t e r c a n be a p p r o x i m a t e d by:

K

=

s c a t

k(I X— ,n

=

/

f

k'(DA

(5)

m

and

R

=K s c a t

// n D

n

\ n„ Dn \

n

J

n

— )

VX

JL

t

p

4

2

=

kD

y

(6)

w h e r e y = z + 2; k', k are p r o p o r t i o n a l i t y constants. Rgcat i s the s c a t t e r i n g cross section (17). F o r constant D, K ^ t = ko Xo"*, a n d therefore z has been t e r m e d " t h e w a v e l e n g t h exponent" (19). T h e t u r b i d i t y of a monodisperse suspension w i t h N p a r t i c l e s per cm3 i s g i v e n as: x=NkD

y

( 7 )

I n a polydisperse s y s t e m a t a g i v e n w a v e l e n g t h :

x =

Y n.R. = Y c •f— i i scat i

i

n . k. D * i i l

1

and


(8)



6 4-

1

11

(9)

1


where nj i s the n u m b e r f r a c t i o n of p a r t i c l e s w i t h d i a m e t e r D j . W h e n the sizes of the p a r t i c l e s i n a polydisperse s y s t e m a r e s u c h , t h a t k a n d y do not change s i g n i f i c a n t l y w i t h the d i a m e t e r (i.e., k i - k2 - k i - k a n d y i - y2 — y i ~ y ) , t h e n :

(10)

C o m p a r i n g E q u a t i o n s 4 a n d 10, we conclude t h a t the a p p a r e n t d i a m e t e r obtained by t r e a t i n g the s y s t e m as monodisperse, w o u l d correspond to the (y,3) average d i a m e t e r of the p a r t i c l e size d i s t r i b u t i o n . T h e exponent ( y - 3 ) h a d e a r l i e r been t e r m e d "the r a d i u s exponent" by H e l l e r a n d P a n g o n i s (15), a n d as they pointed out, i t has the v a l u e of 3 (i.e. y = 6) i n the R a y l e i g h r e g i m e , a n d zero (y = 3) at the specific t u r b i d i t y m a x i m a a n d m i n i m a , regardless of the m v a l u e s . Specific t u r b i d i t y m e a s u r e m e n t s r e s u l t i n t u r b i d i t y average d i a m e t e r s ( D = D63) w h e n y = 6, a n d w e i g h t a v e r a g e d i a m e t e r s ( D = D43) w h e n y = 4. F o r v e r y l a r g e p a r t i c l e s , w h e r e the s c a t t e r i n g coefficient is constant ( K = 2 ) , z = 0 a n d y = 2, specific t u r b i d i t i e s a r e a l w a y s p r o p o r t i o n a l to (I/D32). W i t h broad d i s t r i b u t i o n s of sizes, where the v a l u e of y changes throughout the d i s t r i b u t i o n i t is impossible to a s s i g n a n y m e a n i n g f u l average to the a p p a r e n t d i a m e t e r . T h i s is e s p e c i a l l y t r u e for h i g h e r m v a l u e s due to i r r e g u l a r (oscillating) v a r i a t i o n s of Kgcat w i t h a at h i g h m values ( F i g u r e 5). A t constant w a v e l e n g t h , f r o m E q u a t i o n s 5 a n d 6 we c a n w r i t e : x

w

s c a t

€n(K

) = z € n a + €nk' scat c (ID

or €n(R

) = y d € n a + €nk scat * c

T h e v a l u e of y c a n be obtained for a n y a at a n y m v a l u e f r o m the slope of f n (Rgcat) ^ » a n d the v a l u e of z f r o m the slope of tn ( K ^ t ) vs. €n a ; (y = z + 2). S e v e r a l a u t h o r s (18.19) have t a b u l a t e d v a l u e s of y for s e v e r a l p a i r s of m a n d a values. V a l u e s of y c a l c u l a t e d for a wide range of a v a l u e s for m = 1.05, 1.0, 1.15 a n d 1.2 are p l o t t e d i n F i g u r e 6. T h e s c a t t e r i n g coefficient K ( D / A , n p / n ) was c a l c u l a t e d d i r e c t l y f r o m the M i e theory a n d the slope z of InCKgcat) vs tna at a g i v e n £na was c a l c u l a t e d u s i n g a c e n t r a l d i f f e r e n c e d e r i v a t i v e f o r m u l a (16). N o t i c e i n F i g u r e 6 t h a t for a l l m values the y vs a b e h a v i o u r i s a l m o s t i d e n t i c a l for a < 1.6. H e n c e , i n t h i s r e g i m e a p p a r e n t d i a m e t e r s for systems w i t h different m v a l u e s , b u t the same d i s t r i b u t i o n s w i l l be i d e n t i c a l . F o r a < 0 . 5 , y approaches the l i m i t i n g v a l u e of 6, thereby y i e l d i n g t u r b i d i t y average d i a m e t e r s . F o r 0 . 5 < a < 1.6 the y v a l u e f a l l s f r o m 6 to ~ 4 . 4 . Specific t u r b i d i t i e s for p a r t i c l e size d i s t r i b u t i o n s w i t h a v a l u e s i n t h i s range are expected to give a n a p p a r e n t d i a m e t e r n u m e r i c a l l y between the t u r b i d i t y a n d the w e i g h t average d i a m e t e r . T h e n the y v a l u e f a l l s a b r u p t l y , for a v a l u e s up to a - 1.6 e x h i b i t s a n i n d e n t a t i o n (corresponding to the f i r s t i n f l e c t i o n point of the K t vs. a curve) a n d t h e n i t decreases m o n o t o n i c a l l y for m = 1.05 w h i l e i t s t a r t s o s c i l l a t i n g for m = 1.2, f o l l o w i n g the corresponding vs. a o s c i l l a t o r y b e h a v i o u r . F o r s m a l l v a l u e s of m (m = 1.05, 1.1, 1.15), the v a l u e of y is a p p r o x i m a t e l y 4.0 (4.2 ^ y ^ 3.8) for a v e r y wide range of a v a l u e s v s

m

m

s c a


n

a


alpha

(ttD/

A

M

)

F i g u r e 5. T h e s c a t t e r i n g coefficient as a function of a (a = n D / X ) , for m = 1.1 a n d m = 1.2. m

0.0

2.0

alpha

4.0

6.0

8.0

10.0

(=rrD/A ) M

F i g u r e 6. T h e exponents y a n d z as a function of a , for m = 1.05,1.1,1.15 and 1.2.


1.

KOURTI ET AL.

13


(for m = 1.05, 3.0 ^ a ^ 14; for m = 1.15, 3 < a < 6). H e n c e , for s m a l l v a l u e s of m a n d for p a r t i c l e size d i s t r i b u t i o n s w i t h a v a l u e s i n the above r e g i o n , a p p a r e n t d i a m e t e r s o b t a i n e d f r o m specific t u r b i d i t y r e a d i n g s w o u l d be n u m e r i c a l l y very close to the weight average d i a m e t e r . F o r m = 1.2, the a p p a r e n t d i a m e t e r c a n be assigned to the w e i g h t a v e r a g e o n l y i n a r e l a t i v e l y n a r r o w a r e g i m e (3.0 ^4.3). E x p l a n a t i o n for the Specific T u r b i d i t y B e h a v i o u r A constant specific t u r b i d i t y c u r v e i s a c u r v e of constant apparent d i a m e t e r . O n a D - o p l a n e , at g i v e n ( A , m ) , a l l the points o n a constant specific t u r b i d i t y c u r v e correspond to l o g - n o r m a l d i s t r i b u t i o n s w i t h the same a p p a r e n t d i a m e t e r . F o r m = 1.1, the apparent d i a m e t e r is expected to be n u m e r i c a l l y v e r y close to the w e i g h t a v e r a g e d i a m e t e r o f t h e s u s p e n s i o n (y - " 4 . 0 ) , for a n y type 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 ( b i m o d a l , u n i m o d a l ) p r o v i d e d t h a t the p a r t i c l e s i n the suspension correspond to a v a l u e s between 3.0 a n d 8.0 ( F i g u r e 6). F o r p a r t i c l e s suspended i n w a t e r the p a r t i c l e d i a m e t e r s c o r r e s p o n d i n g to these a v a l u e s are:


g

m

X -

for

0

0.28jim^Ds

400 n m

0.76um

0.42 p m «5 D«* 1.14 p m

X = 600 n m 0

( F o r suspensions w i t h l o g - n o r m a l p a r t i c l e size d i s t r i b u t i o n s the a r e g i m e for w h i c h the a p p a r e n t d i a m e t e r , obtained f r o m specific t u r b i d i t y m e a s u r e m e n t s , are n u m e r i c a l l y close to the w e i g h t average of the P S D c a n be extended f u r t h e r (16). It is c l e a r f r o m the above t h a t for suspensions w i t h m = 1.1 a n d p a r t i c l e s w i t h d i a m e t e r s u p to 0.8 u m (and for a n y type of d i s t r i b u t i o n ) , specific t u r b i d i t i e s a t both 400 a n d 600 n m r e s u l t i n apparent d i a m e t e r s n u m e r i c a l l y close to the weight average of the s u s p e n s i o n , a n d therefore n u m e r i c a l l y close to each other. Therefore, constant specific t u r b i d i t y c u r v e s a t these two w a v e l e n g t h s , plotted a g a i n s t the p a r a m e t e r s t h a t define the P S D , s t a y close to each other a n d pass t h r o u g h those points t h a t define constant w e i g h t average d i a m e t e r s . F o r l o g - n o r m a l d i s t r i b u t i o n s , on a D - o p l a n e , these c u r v e s p a s s t h r o u g h points for w h i c h D = D exp(3.5o ) = constant = D where D is the a p p a r e n t d i a m e t e r c o r r e s p o n d i n g to the s p e c i f i c t u r b i d i t y a s s u m i n g the s u s p e n s i o n i s m o n o disperse. T h i s i s the case observed p r e v i o u s l y for the poly ( v i n y l acetate) d i s t r i b u t i o n s c o r r e s p o n d i n g to points B , H , D a n d I ( F i g u r e 1). g

w

g

2

a p

a p

F o r systems w i t h l a r g e r p a r t i c l e s ( D > 0 . 7 u m , o > 0 . 2 ) where the y v a l u e v a r i e s s i g n i f i c a n t l y w i t h i n the d i s t r i b u t i o n , specific t u r b i d i t y m e a s u r e m e n t s r e s u l t i n different a p p a r e n t d i a m e t e r s at the two w a v e l e n g t h s ; therefore, the constant specific t u r b i d i t y curves o n a D - o p l a n e intersect a t a w i d e r angle a n d more c l e a r l y define the p a r t i c l e size d i s t r i b u t i o n (point K , F i g u r e 1). F o r poly (sty rene), where y r e m a i n s close to 4.0 for o n l y a v e r y n a r r o w range of a v a l u e s ( m = 1.2, F i g u r e 6), a constant weight average t r e n d is expected o n l y for n a r r o w e r d i s t r i b u t i o n s of p a r t i c l e s w i t h a v a l u e s i n the above range. T h i s i s the case for t h e d i s t r i b u t i o n s c o r r e s p o n d i n g to points B a n d C ( F i g u r e 4). D i s t r i b u t i o n s of v e r y s m a l l p a r t i c l e s ( a < 1 . 6 ) g i v e a l m o s t the s a m e a p p a r e n t d i a m e t e r for m = 1.1 or m = 1.2, a n d hence the i d e n t i c a l b e h a v i o u r of specific t u r b i d i t y for poly ( v i n y l acetate) a n d poly (sty rene) a t 600 n m (cases A a n d B ) . F r o m the above d i s c u s s i o n , i t i s c l e a r t h a t the difference i n the specific t u r b i d i t y b e h a v i o u r o n a D - o p l a n e , observed between polystyrene a n d polyvinyl acetate) suspensions, i s due to the different v a l u e s of m of these systems w h i c h m e a n s a different w a v e l e n g t h exponent b e h a v i o u r . F o r s m a l l v a l u e s of m (m = 1.05, 1.10), the w a v e l e n g t h exponent y changes v e r y s l o w l y w i t h a (for a > 3.0) a n d r e m a i n s v e r y close to 4 for a w i d e a r e g i m e . Therefore, polydisperse systems w i t h s m a l l v a l u e s of m a n d p a r t i c l e s c o v e r i n g the above a r e g i m e s a r e expected to e x h i b i t a specific t u r b i d i t y b e h a v i o u r o n a D - o p l a n e , g

g

g

g


14

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

s i m i l a r to t h a t of p o l y ( v i n y l acetate) latexes. Specific t u r b i d i t y c u r v e s , at different w a v e l e n g t h s , are expected to stay close together a r o u n d the point of t r u e s o l u t i o n d e f i n i n g constant weight average d i a m e t e r trends. In t h i s case the e s t i m a t i o n of the f u l l P S D is impossible.


The Weight Average Diameter from The Apparent Diameter: Experimental Verification

P a r t i c l e size d i s t r i b u t i o n s were s y n t h e s i z e d by m i x i n g monodisperse p o l y ( v i n y l acetate) latexes. These d i s t r i b u t i o n s were c o v e r i n g p a r t i c l e size from 60 to 420 n m (16). T h e i r apparent diameters c a l c u l a t e d from specific t u r b i d i t y measurements are shown i n T a b l e III, together w i t h t h e i r t r u e w e i g h t average d i a m e t e r . ( M o r e t h a n one v a l u e for D of a s a m p l e i n d i c a t e s repeated measurements.) It c a n be seen t h a t for a l l the cases the a p p a r e n t d i a m e t e r s a t 440 n m a n d 500 n m a r e r e m a r k a b l y close to the t r u e w e i g h t average d i a m e t e r . T h i s e x p e r i m e n t a l o b s e r v a t i o n v e r i f i e s o u r c o n c l u s i o n s f r o m t h e t h e o r e t i c a l a n a l y s i s . It i s obvious t h a t w i t h the a p p a r e n t d i a m e t e r s so close to each other, the p a r a m e t e r s of the P S D c a n not be e s t i m a t e d since the sample appears monodisperse w i t h i n e x p e r i m e n t a l e r r o r ( w i t h a d i a m e t e r e q u a l to the w e i g h t average diameter). a p

The Turbidity Ratio Method

T h e d i f f i c u l t i e s t h a t a r e encountered i n the e s t i m a t i o n of the P S D u s i n g specific t u r b i d i t y m e a s u r e m e n t s have been discussed so far. A n o t h e r t u r b i d i m e t r i c method t h a t has been u t i l i z e d by researchers for the d e t e r m i n a t i o n of p a r t i c l e size i n polydisperse systems is the t u r b i d i t y r a t i o . In t h i s approach the p a r t i c l e size d i s t r i b u t i o n i s r e l a t e d to the r a t i o of two t u r b i d i t y m e a s u r e m e n t s at two w a v e l e n g t h s , one of w h i c h is chosen as basis:

where subscripts A o i , Aob denote t h a t the q u a n t i t i e s x, n , n , A are e v a l u a t e d at the corresponding wavelengths. W a l l a c h a n d H e l l e r (2) r e p o r t e d a s u c c e s s f u l a p p l i c a t i o n o f t h e m e t h o d i n e s t i m a t i n g the P S D i n p o l y s t y r e n e suspensions w i t h large p a r t i c l e s ( 0 . 6 5 < D < 1.3 pm). W h e n M a x i m et a l (3) however, a p p l i e d the method for p o l y ( v i n y l acetate) latexes w i t h 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 r a n g e , i t was stated t h a t the t u r b i d i t y r a t i o " l e a d s to m u l t i v a l u e d solutions a n d unless p r i o r estimates of the a n s w e r are a v a i l a b l e f r o m some other technique, t h e r e a r e no c r i t e r i a for c h o o s i n g b e t w e e n a l t e r n a t i v e s o l u t i o n s " . S i m i l a r observations w i t h M a x i m et a l (3) a r e reported by H a s e l e r (14) for s m a l l v a l u e s of m ( m = 1.15). In the a n a l y s i s t h a t f o l l o w s , i t w i l l be s h o w n t h a t t h i s t e c h n i q u e i s e x t r e m e l y s e n s i t i v e to e x p e r i m e n t a l e r r o r for s u s p e n s i o n s w i t h s m a l l v a l u e s of m (m < 1.15) a n d p a r t i c l e sizes i n the s u b m i c r o m e t e r r a n g e , a n d therefore s h o u l d not be used for the d e t e r m i n a t i o n of the P S D i n s u c h systems. p

m

m



15


T A B L E III. T h e A p p a r e n t D i a m e t e r for m = 1.095 for P o l y ( v i n y l acetate) Suspensions w i t h K n o w n D i s t r i b u t i o n s distribution


DV1

D

w

(true)

D

a p

D

at 440 n m

_

136

a p

at 500 n m

134 134

136 137 136

DV2

163

159 158

160 159

DV3

194.3

195 196.5 196

196 197 193

DV4

227

226 227 229

232 230 232

BDV5

253.2

255 254

263 262

DV6

340.2

338 336 341

342 344 340

DV7

309.3

300 303

302 306

A m i n i m u m of three t u r b i d i t y m e a s u r e m e n t s at three w a v e l e n g t h s , one of w h i c h is chosen as b a s i s , a r e r e q u i r e d i n p r i n c i p l e for the d e t e r m i n a t i o n of a t w o - p a r a m e t e r p a r t i c l e size d i s t r i b u t i o n . A w a v e l e n g t h t h a t v e r y frequently (2^3) has been chosen as basis is X b 546 n m , a n d i t is u s u a l l y recommended t h a t the other two w a v e l e n g t h s are w i d e l y separated. W e t h e r e f o r e chose A i = 350 n m , A02 700 n m , a n d c a l c u l a t e d t u r b i d i t y r a t i o s (1350/1546 a n d X700/X546) f ° s e v e r a l l o g - n o r m a l d i s t r i b u t i o n s , for m = 1.1. W h e n constant t u r b i d i t y r a t i o c u r v e s were plotted for these two w a v e l e n g t h s o n a D - o p l a n e , they looked as i f they were coincident for v e r y l o n g ( D , o) regions a n d there seemed no w a y to define the point of t h e i r t r u e i n t e r s e c t i o n . I n other words, a l a r g e n u m b e r of d i s t r i b u t i o n s s i g n i f i c a n t l y different f r o m each other have, even u n d e r i d e a l conditions (i.e., no e x p e r i m e n t a l e r r o r ) , t u r b i d i t y r a t i o s t h a t are almost e q u a l to each other at more t h a n one w a v e l e n g t h . T h i s i s i l l u s t r a t e d i n T a b l e I V w h e r e t u r b i d i t y r a t i o s w e r e c a l c u l a t e d at two w a v e l e n g t h s for some ( D , o) p a i r s , located a l o n g these a l m o s t coincident c u r v e s ; t h e i r v a l u e s were c o m p a r e d w i t h those of t h e ( 0 . 5 0 0 , 0 . 1 5 ) p a i r for t h e t w o w a v e l e n g t h s a n d the % differences a r e g i v e n . N o t i c e t h a t these differences are a l l below 0.5%. T h e s e n s i t i v i t y of the method to e x p e r i m e n t a l e r r o r is obvious; e r r o r s as s m a l l as 0.5% o n the t u r b i d i t y r a t i o m e a s u r e m e n t s w i l l r e s u l t i n e s t i m a t e d d i s t r i b u t i o n s v e r y m u c h different f r o m the t r u e ones. N o t i c e also t h a t the weight average d i a m e t e r s of these d i s t r i b u t i o n s are not n e c e s s a r i l y v e r y close to the t r u e one; i n other words, n e i t h e r the correct d i s t r i b u t i o n nor a correct w e i g h t average c a n be obtained. 0

=

=

0

r

g

g

g


16


T A B L E I V . T u r b i d i t y R a t i o s for Some P o l y ( v i n y l acetate) Suspensions W i t h L o g - n o r m a l P a r t i c l e Size D i s t r i b u t i o n s ; B a s i s : Xob = 546 n m

a


(pm)

X02 = 700 n m

^01 = 350 n m

D (pm) w

ratio

%di£f.

ratio

%difT.

0.500

0.15

0.541

2.5017

0.465

0.20

0.535

2.5012

0.02

0.5755

0.13

0.425

0.25

0.529

2.4989

0.11

0.5753

0.18

0.380

0.30

0.520

2.4973

0.17

0.5753

0.18

0.330

0.35

0.507

2.5005

0.05

0.5749

0.24

0.285

0.40

0.499

2.4934

0.33

0.5758

0.08

0.5763

T o e x p l a i n the t u r b i d i t y r a t i o b e h a v i o u r , we w i l l m a k e a g a i n use of the d i a m e t e r exponent. Suppose t h a t i n a suspension w i t h a v a l u e of m = 1.1 a n d a r e l a t i v e l y broad 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 p a r t i c l e d i a m e t e r s a r e b e t w e e n D i = 0.3 p m a n d D = 0.8 p m w i t h n u m b e r fractions n i , n 2 , . . . , nq. F r o m E q u a t i o n s 8 a n d 12 the t u r b i d i t y r a t i o c a n be w r i t t e n : q

x _ k' n, D f + . . . . 350 _ 1 1 1 2

*546 ^

+ k' n D ' q q q 3

k, n, D * + . . . . + k n D 1 1 1 q q q 6

6

3 , 9

x„„ 700 ^ 0 , 0

_

k" n , D f 1 1 1

+....

4

+ k" n D q q q 4

k, n, D * * . . . . + k n D ' 1 1 1 q q q 6

3

1

9

where k i ' , k i " , k j denote t h a t the k v a l u e s are different at different w a v e l e n g t h s . T h e exponents y have been c a l c u l a t e d for the a v a l u e s c o r r e s p o n d i n g to each w a v e l e n g t h , for water medium. F r o m these r a t i o s , i t c a n be seen t h a t , even for a broad d i s t r i b u t i o n a n d t u r b i d i t y m e a s u r e m e n t s at two w i d e l y separated w a v e l e n g t h s , the d i a m e t e r exponents between the n u m e r a t o r s a n d the d e n o m i n a t o r do not differ s i g n i f i c a n t l y . Therefore, the t u r b i d i t y r a t i o cannot be v e r y s e n s i t i v e to the d i s t r i b u t i o n c h a r a c t e r i s t i c s ; different d i s t r i b u t i o n s w i t h t h e i r m a i n populations o n the same r e g i m e s , are expected to give the same t u r b i d i t y r a t i o s . T h e difference between the exponents w i l l become even s m a l l e r , for w a v e l e n g t h s closer to 546 n m t h a n the ones used above, a n d for n a r r o w e r p a r t i c l e size d i s t r i b u t i o n s . H e n c e for m = 1.1, w h e r e t h e d i a m e t e r e x p o n e n t c h a n g e s v e r y s l o w l y w i t h a , t h e t u r b i d i t y r a t i o does not change s i g n i f i c a n t l y for different d i s t r i b u t i o n s i n the s u b m i c r o m e t e r range. (For m = 1.05, these p r o b l e m s a r e expected to occur i n w i d e r p a r t i c l e size regimes.) Therefore, s m a l l e x p e r i m e n t a l e r r o r s m a y r e s u l t i n estimates s i g n i f i c a n t l y different f r o m the t r u 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 . I f t h e w a v e l e n g t h s a t w h i c h t h e t u r b i d i t y m e a s u r e m e n t s a r e t a k e n a r e not w i d e l y separated f r o m the w a v e l e n g t h used as b a s i s , t h e n the t u r b i d i t y r a t i o w i l l be p r a c t i c a l l y independent of the p a r t i c l e d i a m e t e r . T h e t u r b i d i t y r a t i o i s expected to be a stronger f u n c t i o n of the p a r t i c l e size d i s t r i b u t i o n i n (m, a) regions where the d i a m e t e r exponent changes s i g n i f i c a n t l y w i t h a. F o r




17


m = 1.2 a n d l a r g e r a v a l u e s the t u r b i d i t y r a t i o technique c a n be used successfully for the d e t e r m i n a t i o n of the f u l l P S D . T h e above a n a l y s i s e x p l a i n s w h y the reports of W a l l a c h a n d H e l l e r (2) a n d M a x i m et a l (3), o n the c a p a b i l i t y of the method to d e t e r m i n e the P S D , for systems w i t h different m v a l u e s a n d different p a r t i c l e sizes c o n t r a d i c t each other. T h e specific t u r b i d i t y has a n advantage over the t u r b i d i t y r a t i o m e t h o d , as a r e s u l t of the different properties m e a s u r e d by the two methods. T h e t u r b i d i t y r a t i o u t i l i z e s o n l y t u r b i d i t y m e a s u r e m e n t s , w h i l e i n the specific t u r b i d i t y , the p a r t i c l e concent r a t i o n i s also m e a s u r e d . T h e p a r t i c l e c o n c e n t r a t i o n i s a l w a y s p r o p o r t i o n a l to the t h i r d m o m e n t of the P S D . I n the cases where y does not change s i g n i f i c a n t l y w i t h a , specific t u r b i d i t y i s p r o p o r t i o n a l to D 3 ( ) A l t h o u g h the P S D d e t e r m i n a t i o n i s not possible i n these cases, a n a p p a r e n t d i a m e t e r , c o r r e s p o n d i n g to the D 3 average of the d i s t r i b u t i o n , c a n a l w a y s be obtained. O n the c o n t r a r y , the t u r b i d i t y r a t i o w i l l be a l m o s t independent of the d i a m e t e r i n these cases a n d no r e l i a b l e e s t i m a t e of the p a r t i c l e size c a n be obtained, (the t u r b i d i t y r a t i o i s p r o p o r t i o n a l to the ( y i - y 2 ) t h power of D y b u t w h e n ( y i —y2) t h e n y i - Y 2 - 0 , andD°=1.0) y

y _ 3

y

y i

9

2

Concluding Remarks T u r b i d i m e t r i c methods cannot be expected to p r o v i d e i n f o r m a t i o n o n the f u l l P S D i n m a n y s i t u a t i o n s . T h e e s t i m a t i o n of the t r u e P S D (even a s s u m i n g t h a t i t i s k n o w n to be of a l o g - n o r m a l form) i s e x t r e m e l y d i f f i c u l t even u n d e r v e r y s m a l l e x p e r i m e n t a l e r r o r , i n (m, a) regions where the v a l u e of the w a v e l e n g t h exponent changes v e r y s l o w l y w i t h the v a l u e of a. I n these regions the p a r a m e t e r s that define the P S D a r e h i g h l y c o r r e l a t e d a n d s m a l l e x p e r i m e n t a l e r r o r s r e s u l t i n l a r g e e r r o r s i n the p a r a m e t e r e s t i m a t i o n . I n these r e g i m e s , i m p r o v i n g d e c o n v o l u t i o n a l g o r i t h m s f o r E q u a t i o n 1, d o e s n o t a l t e r t h e f u n d a m e n t a l r e g r e s s i o n p r o b l e m caused by the h i g h c o r r e l a t i o n a m o n g the p a r a m e t e r s . T h e r e f o r e , t h e t y p e o f i n f o r m a t i o n a b o u t the P S D t h a t c a n be e x t r a c t e d f r o m a n y t u r b i d i m e t r i c method a n d i t s s e n s i t i v i t y to e x p e r i m e n t a l e r r o r are s t r o n g l y r e l a t e d to the m a n d a v a l u e s of the suspension. U n f o r t u n a t e l y t h i s c o n s i d e r a t i o n has not often been t a k e n i n t o account a n d one of the controversies (3.4) i n the l i t e r a t u r e r e s u l t e d w h e n w o r k e r s t r i e d to e x t r a p o l a t e observations a n d conclusions, correct for t h e i r s y s t e m a n d the t u r b i d i m e t r i c method they used, to systems w i t h different m a n d a v a l u e s a n d for a different method. (Discussions on the i n c o n s i s t e n c i e s a n d c o n f l i c t s r e p o r t e d i n t h e l i t e r a t u r e a n d t h e i r r e s o l u t i o n c a n be found i n K o u r t i (16.17).) T h i s study however i n d i c a t e d t h a t specific t u r b i d i t y i s a v e r y r e l i a b l e m e t h o d (and for n o n - a b s o r b i n g p a r t i c l e s , more r e l i a b l e t h a n the t u r b i d i t y ratio). It c a n a l w a y s p r o v i d e i n f o r m a t i o n on the p a r t i c l e size of a polydisperse suspension; t h a t i s , a correct average d i a m e t e r c a n be o b t a i n e d even w h e n the e s t i m a t i o n of the f u l l P S D i s not possible. M o r e s p e c i f i c a l l y , for suspensions of n o n - a b s o r b i n g p a r t i c l e s : i) the t u r b i d i t y average d i a m e t e r a n d the volume-surface average d i a m e t e r (D32) c a n be e s t i m a t e d for v e r y s m a l l a n d v e r y l a r g e p a r t i c l e s , r e s p e c t i v e l y , for a n y v a l u e of m . i i ) for suspensions w i t h v a l u e s of m < 1.15 a n d d i s t r i b u t i o n s c o v e r i n g a values s m a l l e r t h a n a p p r o x i m a t e l y 8.0, t h e w e i g h t average d i a m e t e r ( D ) c a n be c o r r e c t l y e s t i m a t e d ( n u m e r i c a l l y D i s e q u a l to t h e a p p a r e n t d i a m e t e r o b t a i n e d by t r e a t i n g the s y s t e m as monodisperse). F o r the o t h e r ( m , a) r e g i m e s , the e s t i m a t i o n of the f u l l P S D of n o n - a b s o r b i n g p a r t i c l e s s h o u l d be possible. F o r these cases i t has been s h o w n (16.17) t h a t the a s s u m p t i o n of a l o g - n o r m a l P S D w i l l r e s u l t i n a c o r r e c t e s t i m a t e ( w i t h i n 1%) o f t h e w e i g h t a v e r a g e d i a m e t e r ; for c o n t i n u o u s d i s t r i b u t i o n s (i.e., not w i d e l y separated bimodals) i t w i l l also provide the correct l o c a t i o n of the m a i n body o f the d i s t r i b u t i o n o n weight basis. w

w

O u r choice to use l o g - n o r m a l d i s t r i b u t i o n s for t h i s a n a l y s i s , a n d the choice of the w a v e l e n g t h s (400-600 nm) for the specific t u r b i d i t i e s , do not affect the m a i n c o n c l u s i o n of t h i s i n v e s t i g a t i o n for suspensions of n o n - a b s o r b i n g p a r t i c l e s (i.e., t h a t i n f o r m a t i o n o n the f u l l P S D of a suspension is impossible whenever specific t u r b i d i t i e s at d i f f e r e n t w a v e l e n g t h s r e s u l t i n the same apparent diameters). W e covered so m a n y d i s t r i b u t i o n s



18


w i t h d i f f e r e n t sizes (i.e., a w i d e r a n g e o f a v a l u e s ) , t h a t t h e a c t u a l v a l u e s o f t h e w a v e l e n g t h s do not put a n y r e s t r i c t i o n . W e chose to w o r k w i t h p o l y s t y r e n e (m=*1.2) a n d p o l y ( v i n y l acetate) l a t e x e s ( m = l . l ) a t w a v e l e n g t h s t h a t these latexes do not absorb because a l a r g e n u m b e r o f studies i n the l i t e r a t u r e a r e for n o n - a b s o r b i n g p a r t i c l e s a n d s y s t e m s w i t h m v a l u e s between 1.1 a n d 1.2 (17). S y s t e m s w i t h 1 . 0 < m < 1 . 1 5 a r e e x p e c t e d to b e h a v e l i k e p o l y ( v i n y l acetate), a n d systems w i t h m > 1.2 l i k e the polystyrene latexes. W i t h the above a n a l y s i s i t was d e m o n s t r a t e d for the f i r s t t i m e , t h a t t h e o r e t i c a l l y , t u r b i d i m e t r i c techniques a r e not expected to provide i n f o r m a t i o n o n the f u l l P S D o f the suspension i n c e r t a i n cases a n d these cases have been i d e n t i f i e d . These a r e t h e ( m , a) r e g i m e s where the v a r i o u s t u r b i d i t y functions a r e more s e n s i t i v e to e x p e r i m e n t a l e r r o r s , a n d they are d e t e r m i n e d b y the b e h a v i o u r o f the w a v e l e n g t h exponent; i n these r e g i m e s , the w a v e l e n g t h exponent i s not a s t r o n g f u n c t i o n o f the p a r t i c l e size. T h i s c o n c l u s i o n i s g e n e r a l , a n d not r e s t r i c t e d o n l y to n o n - a b s o r b i n g p a r t i c l e s . T h e conclusions f r o m t h e above a n a l y s i s corroborate a n d e x p l a i n e x p e r i m e n t a l r e s u l t s reported i n the l i t e r a t u r e (17) a n d our e x p e r i m e n t a l observations.

Literature Cited 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Barth, H.G., and S-T. Sun, AnalyticalChemistry,57 (1985) 151R. Wallach, M.L., and W. Heller, J. Phys.Chem.,68 (1964) 924. Maxim, L.D., Klein, A., Meyer, M.E., and C.H. Kuist, J. Polym. Sci. PartC,27 (1969) 195. Zollars, R.L., J. Colloid Interface Sci., 74 (1980) 163. Kourti, T., Penlidis, A., Hamielec, A.E., and J.F. MacGregor, ACS PMSE Div. Preprints, 53 (1985) 147. Kourti, T., Penlidis, A., MacGregor, J.F., and A.E. Hamielec, In Provder, Th. (Ed.), Particle Size Distribution - Assessment and Characterization, ACS Symposium Series, 332, Washington, D.C., 1987. Gossen, P.D., Kourti, T., Penlidis, A., MacGregor, J.F., and A.E. Hamielec, PRP-6 Automation Preprints. 6th International IFAC/IFIP/IMEKO Conference on Instrumentation and Automation in the Paper, Rubber, Plastics and Polymerization Industries, Akron, Ohio, U.S.A., 1986. Gossen, P.D., On-Line Particle Size Measurement and Mathematical Modelling of Continuous Poly(vinyl acetate) EmulsionPolymerization,Master's Thesis. Dept. Chem. Eng., McMaster University, Hamilton, Ontario, Canada, 1988. Maron, S.H., Pierce, P.E., and Ulevitch, I.N., J. Coll. Sci., 18 (1963) 470. Van de Hulst, H.C., Light Scattering by SmallParticles,J. Wiley and Sons, Inc., New York, 1957. Kerker, M., The Scattering of Light and Other Electro-magnetic Radiation. Academic Press, New York, 1969. Yang, K.C., and R. Hogg, AnalyticalChemistry,51 (1979) 758. Kourti, T.,MacGregor, J.F.,and A.E. Hamielec, J. Colloid Inter. Sci., 120 (1987) 292. Haseler, S.C., Part.Charact., 2 (1985) 14. Heller, W., and W.J. Pangonis, J. Chem.Phys.,26 (1957) 498. Kourti, T., Polymer Latexes: Production byHomogeneous Nucleation and Methods for Particle Size Determination,Ph.D. Thesis, McMaster University, Hamilton, Ont., Canada, 1989. Kourti, T. and MacGregor, J.F., "Particle Size Determination Using Turbidimetry: Capabilities, Limitations and Evaluation for On-Line Applications", This volume. Meehan, E. J. and Beattie, W.H.. J. Phys.Chem.,64 (1960) 1006. Heller, W., Bhatnagar, H.L., Nakagaki, M., J. Chem. Phys.,36 (1962) 1163.




19


Appendix A D u e to the o s c i l l a t o r y c h a r a c t e r of the specific t u r b i d i t y vs d i a m e t e r c u r v e , one specific t u r b i d i t y v a l u e a t g i v e n w a v e l e n g t h corresponds to more t h a n one d i a m e t e r . If there is not a p r i o r i i n f o r m a t i o n on the p a r t i c l e size range i n the suspension, specific t u b i d i t y m e a s u r e m e n t s at more t h a n one w a v e l e n g t h a r e needed to u n i q u e l y define the correct d i a m e t e r of a monodisperse s y s t e m (or, the a p p a r e n t d i a m e t e r of a polydisperse system). A t each w a v e l e n g t h a specific t u r b i d i t y m e a s u r e m e n t defines a set of d i a m e t e r v a l u e s . F o r a monodisperse suspension there is a v a l u e c o m m o n to a l l sets a n d t h i s i s the t r u e d i a m e t e r of the suspension. F o r a p o l y d i s p e r s e s u s p e n s i o n t h e sets of d i a m e t e r s o b t a i n e d a t d i f f e r e n t w a v e l e n g t h s , do not i n g e n e r a l have a c o m m o n v a l u e . H o w e v e r , i n some cases t h e polydisperse suspensions e x h i b i t a b e h a v i o u r somewhat s i m i l a r to monodisperse ones; these a r e the cases where the a p p a r e n t d i a m e t e r c a n be assigned to a m e a n i n g f u l average of the P S D of the suspension. I n these cases, each one of the sets of d i a m e t e r s obtained a t s e v e r a l w a v e l e n g t h s c o n t a i n s a d i a m e t e r w i t h a v a l u e t h a t (although not constant, i.e., not c o m m o n to a l l sets), appears to change s l o w l y w i t h the w a v e l e n g t h . T h a t d i a m e t e r corresponds to a n average of the P S D a n d its v a l u e at each w a v e l e n g t h gives the correct a p p a r e n t d i a m e t e r of the suspension for t h a t w a v e l e n g t h . T h i s b e h a v i o u r is observed for suspensions of s u b m i c r o n p a r t i c l e s whenever the a p p a r e n t d i a m e t e r s a r e close to D or D of the P S D , or for suspensions of v e r y l a r g e p a r t i c l e s (where K ^ t 2.0) w h e n the apparent d i a m e t e r i s close to D32. I n t h i s w o r k , i n the d i s c u s s i o n of the a p p a r e n t d i a m e t e r a n d i t s r e l a t i o n to averages of the P S D , D represents the correct v a l u e of the apparent d i a m e t e r at g i v e n w a v e l e n g t h , selected i n the m a n n e r e x p l a i n e d above. F i n a l l y , for p r a c t i c a l purposes, i f i t is k n o w n t h a t the p a r t i c l e s i n the suspension have d i a m e t e r s i n the s u b m i c r o n or n e a r m i c r o n range (i.e., D < 2-3 pm), one specific t u r b i d i t y a t one m e a s u r e m e n t c a n be used to define the p a r t i c l e size of a monodisperse o r , the a p p a r e n t d i a m e t e r of a polydisperse suspension, because there is a monotonic r e l a t i o n between the specific t u r b i d i t y a n d the p a r t i c l e d i a m e t e r i n t h a t r e g i m e (i.e., f r o m the set of a l l v a l u e s a v a i l a b l e for t h a t m e a s u r e m e n t we choose t h e one c o r r e s p o n d i n g to a d i a m e t e r D , s u c h t h a t 0 ^ D £ 2-3 pm). x

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R E C E I V E D M a y 14,1991


Particle Size Distribution II - American Chemical Society

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