Chapter 19
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
Application of High-Speed, Integrated, Computerized, Hydrodynamic Chromatography for Monitoring Particle Growth During Latex Polymerization R.
L.
1
2
Van Gilder andM.A.Langhorst
1
Designed Latexes and Resins, The Dow Chemical Company, Midland,MI48640 Analytical Laboratory, The Dow Chemical Company, Midland,MI46840 2
The computerized hydrodynamic chromatograph (HDC) technique has been used successfully to detect agglomeration and new-particle generation as significant deviations from the controlled particle growth during latex polymerizations. It was possible to use this high speed integrated computerized HDC technique to determine when these deviations started and how the growth pattern developed during the polymerization. Transmission electron micrograph data supported the results by the computerized HDC analysis. In the earlier publications (1,2) i t was shown how hydrodynamic chromatography (HDC) could be applied in the study of polymer latexes to determine particle-size. An improved technique for the HDC was developed which utilized higher efficiency and resolving power columns to significantly reduce the analysis time (3). A high speed integrated computer was included in this improvement so that both particle-size and particle-size distribution of latexes could be quantified in the relatively short period of time. This high speed computerized version of the HDC has been used extensively for measurements on the final latex. There is considerable interest in monitoring an emulsion polymerization by following the growth rate of latex particles. It is well known that in addition to a well-controlled particle growth pattern significant deviations can result from particle association or nucleation during the growth stage. A method which would define when such deviations occur during the latex polymerization would be of obvious value and would lead possibly to more efficient optimizations of different latex polymerizations. Information on particle growth during either a seeded polymerization or during the growth stage of an un-seeded polymerization at different degrees of conversion also could enhance the understanding of the kinetics. In earlier work (4,5) the rate of polymerization, for polystyrene latexes primarily, has been related to the latex particle diameter and the total number of particles in the reactor. It would be useful to obtain kinetic data and develop the kinetic relationships for styrène (S)-butadiene (B) latexes. 0097-6156/87/0332-0272$06.00/0 © 1987 American Chemical Society
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
19.
VAN GILDER AND LANGHORST
Hydrodynamic Chromatography
273
In t h i s publication monitoring of d i f f e r e n t p a r t i c l e growth patterns during latex polymerizations using the high speed computerized HDC w i l l be described f o r S/B latexes. K i n e t i c information w i l l not be dealt with i n t h i s paper.
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
Experimental The carboxylated S/B latexes were prepared by an emulsion polymerization process. Samples were taken throughout the runs for analysis on the computerized HDC. The latex p a r t i c l e - s i z e and p a r t i c l e - s i z e d i s t r i b u t i o n was determined using an integrated, high speed chromatograph technique using the delayed marker i n j e c t i o n (3). An eluant composition was chosen f o r HDC measurements so that changes i n the hydrodynamic volume of the carboxylated S/B latexes were i n s i g n i f i cant (2). The reactor samples were diluted with t h i s eluant upon removal from the reactor followed by i n j e c t i o n into the computerized HDC, and elution through the column. The data from the detector was stored i n the computer and mathematically treated using the i n t e grated high speed computer system to y i e l d the p a r t i c l e - s i z e and p a r t i c l e - s i z e d i s t r i b u t i o n data. This entire procedure f o r each reactor latex sample required a r e l a t i v e l y short period of time, approximately f i f t e e n minutes. P r i o r to the measurements of the d i f f e r e n t reactor latex samples the computerized HDC was calibrated f o r p a r t i c l e - s i z e using the standard procedure (3) and also f o r p a r t i c l e - s i z e d i s t r i b u t i o n quantification. For the p a r t i c l e - s i z e d i s t r i b u t i o n c a l i b r a t i o n two d i f f e r e n t p a r t i c l e - s i z e monodisperse carboxylated S/B latexes were polymerized. Various mixtures of these latexes were prepared by blending the large 2100A and the small 700Â latexes i n d i f f e r e n t r a t i o s by weight: 60/40, 70/30, 80/20 and 90/100 respectively. These prepared latex blends were then measured using the HDC technique. Transmission electron microscopy (TEM) analysis (6-8) was used also to characterize the latexes i n t h i s p a r t i c l e growth monitoring study. The electron micrographs were taken at a magnification of 30,000X. Results and Discussion Calibration of the Computerized HDC
for Particle-Size
Distribution.
The p a r t i c l e - s i z e s and p a r t i c l e - s i z e d i s t r i b u t i o n of the two monodisperse carboxylated S/B latexes are shown i n the hydrodynamic chromatograms of Figure 1. The average diameters of the large, D L , and the small, Dg, r e l a t i v e l y monodisperse latexes were 2100A and 720Arespectively. The r e l a t i v e amount of each latex (VL, Vg) i n the d i f f e r e n t blends, as determined by HDC are shown i n Figures 2 and 3. There was excellent correlation between the actual and the measured quant i t i e s of each component i n the d i f f e r e n t binary mixtures. This c a l i b r a t i o n f o r p a r t i c l e - s i z e d i s t r i b u t i o n demonstrated that the computerized HDC could be used to determine the r e l a t i v e amount of each latex i n the various binary mixtures within 1%.
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
274
PARTICLE SIZE DISTRIBUTION
il
Relative Volume Distribution
ι Ί0
I
1 Illlll 100
Diameter
1000
J . L 1 11 M i l l (Angstroms) 10000
100
Diameter
1000
(Angstroms)
Relative Volume Distribution
10
F i g u r e 1. S/B l a t e x p a r t i c l e - s i z e d i s t r i b u t i o n v i a HDC. w i t h p e r m i s s i o n f r o m R e f . 1 1 . C o p y r i g h t 1983 T a p p i J .
10000 Reproduced
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
19.
Relative Volume Distribution
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
Hydrodynamic Chromatography
VAN GILDER A N D LANGHORST
-ι
l
D V D
I
10
Relative Volume Distribution
L s s
= 2100A = 39.7% = 720Â
\»
I I IIΜ 100
275
Diameter
1000
ι ι I MM
(Angstroms)
10000
D = 2100A V = 29.8% Do = 720Â L
s
I 10
I I II I I 100
J L Diameter
AJ
1000
•
(Angstroms)
F i g u r e 2. S/B l a t e x p a r t i c l e - s i z e d i s t r i b u t i o n v i a HDC. w i t h p e r m i s s i o n f r o m R e f . 1 1 . C o p y r i g h t 1983 T a p p i J .
10000
Reproduced
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
PARTICLE SIZE DISTRIBUTION
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
276
Relative Volume Distribution D
I
I I
L
II M
_l L
Diameter
100
10
= 2100A 20.4% 720Â
ι 1000
i d I III I
(Angstroms)
10000
Relative Volume Distribution D = 2100A V = 9.2% Do = 720Â L
s
.4
10
ι
I
I I I
II I 100
I I Diameter
/Nil
/ • \\ 1000
ι
(Angstroms)
F i g u r e 3. S/B l a t e x p a r t i c l e - s i z e d i s t r i b u t i o n v i a HDC. w i t h p e r m i s s i o n f r o m R e f . 11. C o p y r i g h t 1983 T a p p i J .
11
ΜΙ
10000 " Reproduced
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
19.
VAN GILDER AND LANGHORST
Hydrodynamic Chromatography
211
Also i t should be noted that t h i s computerized HDC analysis f o r p a r t i c l e - s i z e measurements of these latexes was within 2% of that measured by TEM. Since the p a r t i c l e - s i z e determinations by the two d i f f e r e n t methods were i n close agreement i t was accepted that the HDC eluant composition was minimizing the p a r t i c l e - s w e l l i n g phenomenon of the carboxylated S/B p a r t i c l e s .
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
Computerized HDC for Monitoring P a r t i c l e Growth Patterns.
the
Latex Emulsion Polymerization
When p a r t i c l e - s i z e and p a r t i c l e - s i z e d i s t r i b u t i o n assessments are required during the latex polymerizations to provide information on the p a r t i c l e growth patterns the size that i s most pertinent i s the i n - s i t u p a r t i c l e - s i z e . As a r e s u l t of t h i s , certain widely used methods of p a r t i c l e - s i z e determination are considered unacceptable. For instance i n electron microscopy the polymer i s removed from i t s aqueous environment, sometimes being chemically modified and always dried before measurement. The latex p a r t i c l e - s i z e s may be quite d i f f e r e n t from that i n t h e i r o r i g i n a l aqueous environment. Light scattering methods u t i l i z e the latex i n i t s natural environment. Light scattering methods u t i l i z e the latex i n i t s natural environ ment. However complications do r e s u l t i n l i g h t scattering i f compositional changes take place i n the polymer phase. Since the eluant i n the HDC can be varied widely i n composition t h i s method appeared to be the most appropriate f o r i n - s i t u measure ments of p a r t i c l e - s i z e . In t h i s work the e f f e c t i v e p a r t i c l e - s i z e was desired. In general when carboxylated S/B latexes are involved the changes i n apparent diameter are due to swelling of the latex p a r t i c l e s by water from the continuous phase. This swelling can r e s u l t when the carboxyl groups, as shown i n Figure 4, become neu t r a l i z e d as the pH of the latex i s raised and water associates to a greater extent with the r e s u l t i n g ionized s i t e s . As mentioned previously the choice of HDC eluant composition becomes important i n order to minimize these swelling e f f e c t s . A l l conventional latex polymerizations can be divided into two stages: a nucleation stage i n which a l l the p a r t i c l e s are formed and growth stage i n which the formed p a r t i c l e s grow to a larger size with no further nucleation. In a seeded emulsion polymerization a fixed number of p a r t i c l e s i n the form of a seed latex are added to the reactor. Monomer and i n i t i a t o r are added then to t h i s seed latex and the p a r t i c l e s grow to a larger s i z e . For both the seeded and un-seeded types the number of p a r t i c l e s should remain constant, during the growth stage without new p a r t i c l e generation or agglomeration of existing p a r t i c l e s . F i r s t , consider a polymerization with a well-controlled growth stage. The aim point i n p a r t i c l e - s i z e f o r t h i s latex polymerization was 2200A. Sampling the reactor latex at an early i n t e r v a l A and analyzing the sample by computerized HDC method showed that the latex was r e l a t i v e l y monodisperse and at the expected average p a r t i cle diameter (Figure 5 ) . This monodispersity 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 as well as the expected p a r t i c l e growth continued through l a t e r intervals Β and C (Figure 5 ) . The f i n a l p a r t i c l e - s i z e of the latex agreed well with the expected diameter. The latex
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
φ
4
Figure
/
W
W
W
4.
\
S0
Styrene-butadiene
^COOH H-C = C Η ή
W
Na*
latex
4
4
ν 2
initiated).
UNREACTED VINYL ACID
COPOLYMERIZED
SURFACE SULFATION
Ο SOLUBLE POLYMER
SURFACTANT
Na* J ^ Η
(carboxylated-persulfate
/ V V V V V ^ V V V ^ \ V N S0 COOH ΛΛΛυ COOH A V W W \ S 0 Na*
COOH COOH
AQUEOUS
PARTICLE
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
19.
VAN GILDER A N D LANGHORST
279
Hydrodynamic Chromatography
Relative Volume Distribution
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
D = 1006Â
ι I I MM 10
100
ι ι I Diameter
rill
j
1000
(Angstroms)
I I I I Mil 10000
Κ
Relative Volume Distribution
D = 1699Â
ι
ι I I M 11
10
100
ι
_L ι l I MM J Diameter
1000
ι I I MM
(Angstroms)
10000
Relative Volume Distribution
D = 2227Â
ι 10
I I I
Ml 100
Figure 5.
» Diameter
S/B l a t e x p a r t i c l e - s i z e
I 1000
/• λ (Angstroms)
distribution
10000
b y HDC.
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
280
PARTICLE SIZE DISTRIBUTION
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
films of thedifferent reactor l a t e x s a m p l i n g s were homogeneous a n d d i d n o t i n d i c a t e anyn o t i c e a b l e signs o f i n s t a b i l i t y o r coagulum f o r m u l a t i o n due t o t h e method o f r e a c t o r s a m p l i n g . As a r e s u l t t h e r e a c t o r sample and the c o r r e s p o n d i n g p a r t i c l e g r o w t h d a t a w h i c h was g e n e r a t e d was i n t e r p r e t e d t o b e v e r y r e p r e s e n t a t i v e . The t r a n s m i s s i o n e l e c t r o n m i c r o g r a p h d a t a , s h o w n i n F i g u r e 6, v e r i f i e d t h a t the p a r t i c l e s were uniform i n size and s p h e r i c a l i n shape t h u s excluding the possibility o f anyn o t i c e a b l e d e s t a b i l i z a t i o n and e s t a b l i s h i n g the f a c t that this l a t e x p o l y m e r i z a t i o n was a w e l l c o n t r o l l e d p a r t i c l e growth case. The s e c o n d p a r t i c l e growth pattern was d e r i v e d from a l a t e x polymerization that was c a r r i e d out using a simple electrolyte addition t othe reactor. Monitoring t h e p o l y m e r i z a t i o n b y t h e c o m p u t e r i z e d HDC i n d i c a t e d that theparticle-size distribution was r e l a t i v e l y m o n o d i s p e r s e d a t an e a r l y i n t e r v a l A ( F i g u r e 7 ) . However, a t a l a t e r i n t e r v a l Β ( F i g u r e 7) 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 became skewed t o w a r d s t h e l a r g e p a r t i c l e - s i z e range. This r e s u l t e d i na multi-component 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 o f t h e f i n a l l a t e x , a s shown a t i n t e r v a l C i n F i g u r e 7. Certain p a r t i c l e p o p u l a t i o n s w e r e l a r g e r t h a n t h a t expected. It w a s s p e c u l a t e d t h a t t h e r e w a s some l o s s o f t h e s t a b i l i z a t i o n charge layer on the particles due t o t h e e l e c t r o l y t e a d d i t i o n (9,10). When t h i s p h e n o m e n o n s t a r t e d t o o c c u r p o s s i b l y t h e e x i s t i n g p a r t i c l e s a s s o c i a t e d w i t h each other t o form d i f f e r e n t size aggregates. E l e c t r o n micrographs o f t h i s l a t e x ( F i g u r e 8) sup p o r t e d t h i s e x p l a n a t i o n s i n c e a c o n s i d e r a b l e number o f t h e l a t e x p a r t i c l e s were no longer s p h e r i c a l and appeared t o consist of d i s t i n c t aggregates. There w a s some i n d i c a t i o n o f new p a r t i c l e generation occurring also. In t h e t h i r d i n s t a n c e t h e HDC m o n i t o r i n g of a s u p p o s e d l y mono disperse l a t e x i n d i c a t e d skewing of thep a r t i c l e - s i z e distribution t h i s time toward the lower p a r t i c l e - s i z e range. HDC a n a l y s i s o f t h e l a t e x a t an e a r l y i n t e r v a l A again i n d i c a t e d a r e l a t i v e l y monodis perse p a r t i c l e - s i z e d i s t r i b u t i o n (Figure 9). At a later interval Β the main particle population had increased i n average diameter s i g n i f i c a n t l y o v e r t h a t o f i n t e r v a l A w i t h a s m a l l (1000A) p a r t i c l e s i z e f r a c t i o n also appearing (Figure 9). As t h i s polymerization c o n t i n u e d , t h e s m a l l and l a r g e p a r t i c l e - s i z e components b o t h increased i ns i z e with t h e volume f r a c t i o n o f t h e s m a l l component also increasing. The t w o p a r t i c l e - s i z e f r a c t i o n s merged i n t o a skewed p a r t i c l e - s i z e d i s t r i b u t i o n , a s shown b y t h e l a t e r i n t e r v a l C c h r o m a t o g r a m i n F i g u r e 9. I t i s p o s s i b l e t h a t t h e u n e x p e c t e d n e w l y g e n e r a t e d p a r t i c l e s r e s u l t e d from t h e p a r t i c u l a r c o n d i t i o n s u s e d i n this polymerization.
It is a l s o p o s s i b l e that the u n e x p e c t e d newly g e n e r a t e d p a r t i c l e s were r e l a t i v e l y unstable and t e n d e d to a s s o c i a t e w i t h t h o s e of t h e m a i n p o p u l a t i o n . The m a i n p a r t i c l e p o p u l a t i o n average d i a m e t e r , a s a r e s u l t , was l a r g e r t h a n t h a t e x p e c t e d a n d t h e d i s t r i b u t i o n d e v i a t e d s i g n i f i c a n t l y f r o m t h e desired m o n o d i s p e r s i t y . The p a r t i c l e g r o w t h d a t a t h a t was g e n e r a t e d by t h e c o m p u t e r i z e d HDC a n a l y s i s c a n be c o m b i n e d w i t h t h e c o n v e r s i o n - p o l y m e r i z a t i o n time d a t a ( F i g u r e 10) t o y i e l d t h e t o t a l number o f p a r t i c l e s N^, i n t h e reactor a tvarious i n t e r v a l s o f sampling (4,5). T h i s i s a n o t h e r way t o r e p r e s e n t t h e d e g r e e o f c o n t r o l i n t h e p a r t i c l e grow-up. In t h i s w o r k t h e t o t a l number o f p a r t i c l e s , N, was p l o t t e d a s a f u n c t i o n o f p o l y m e r i z a t i o n time i n Figure 11. F o r t h e c o n t r o l l e d p a r t i c l e growth
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
VAN GILDER AND LANGHORST
Hydrodynamic Chromatography
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
19.
281
u CO •H
Ο C
3
00
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
282
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
PARTICLE SIZE DISTRIBUTION
Figure
7.
S/B
latex particle-size
distribution
by
HDC.
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
Hydrodynamic Chromatography
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
VAN GILDER A N D LANGHORST
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
PARTICLE SIZE DISTRIBUTION
284
I
Relative Volume Distribution
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
A
D =
ι ι
10
1941Â
. . Diameter
1 I IIII 100
i
III II 1000
J
1 kι 1 1 M i l l (Angstroms) 10000
1
Relative Volume Distribution
D = 2594Â D = 1074Â V =4%
Β
L
s
s
10
ι
11
Mill
l
100
II
J
l I I I LU.
Diameter
1000
Relative Volume Distribution
= 3575Â = 2300Â v ~ 20%
C
s
ι
ι
10
1
M i l l
1
100 F i g u r e 9.
ι
—
I
Diameter
S/B l a t e x p a r t i c l e - s i z e
1000
i
I I
Mill
(Angstroms)
10000
1
ll ι m i
(Angstroms)
distribution
10000
b y HDC.
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
19.
Hydrodynamic Chromatography
VAN GILDER A N D LANGHORST
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
Conversion
Polymerization Time Figure
10,
2 2 0 0 A m o n o d i s p e r s e S/B l a t e x
conversion
profile.
Total No. of Particles, Ν
Nucleation
V
f
/
/
/
Control
/ \
Agglomeration
Aim Pt.
Polymerization Time
Figure
11.
S t a n d a r d S/B l a t e x
polymerization.
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
285
PARTICLE SIZE DISTRIBUTION
286
Downloaded by NORTH CAROLINA STATE UNIV on December 27, 2017 | http://pubs.acs.org Publication Date: February 12, 1987 | doi: 10.1021/bk-1987-0332.ch019
s i t u a t i o n there was good agreement between the aim-point and the calculated t o t a l number of p a r t i c l e s from the experimental data on p a r t i c l e - s i z e and conversion. As could be expected the t o t a l number of p a r t i c l e s , Νγ, decreased f o r the case involving agglomeration and increased when new p a r t i c l e s were generated. I t i s possible that both agglomeration and new particle-generation could occur during the same polymerization, as was indicated i n the second p a r t i c l e growth case described above. As a r e s u l t the t o t a l number of p a r t i cles could change s i g n i f i c a n t l y depending upon which phenomenon was predominating. Conclusions The computerized HDC technique was used to quantify the r e l a t i v e amounts of the large 2100A and small 700A monodisperse latexes i n d i f f e r e n t binary mixtures. The HDC measurements f o r the r e l a t i v e amounts of the two components were within 1% of the actual amounts. In addition to the measurement of the f i n a l p a r t i c l e diameter and p a r t i c l e size d i s t r i b u t i o n the computerized HDC was used to monitor the p a r t i c l e growth pattern during the latex polymerization. The data generated from t h i s monitoring technique showed that very s i g n i f i c a n t departures from the controlled particle-growth pattern did occur i n d i f f e r e n t types of polymerizations. Both agglomeration and new p a r t i c l e generation were detected as deviations i n t h i s work. Literature Cited 1. 2.
H. Small, J . Colloid Interfac. S c i . . 48 (1974) 147. H. Small, F. L . Saunders, J . Solc., Advances in Colloid and Interfac S c i . , 6 (1976) 237-266. 3. G. R. McGowan and M. A. Langhorst, J . Colloid Interfac. S c i . , 89, No. 1 (1982). 4. J . W. Vanderhoff, J . F. Vitkuske, Ε. B. Bradford and T. Alfrey, J r . , J . Polym. Sci. 20, 225 (1956). 5. C. I. Kao, D. P. Gundlach, and R. T. Nelsen, J . Polym. S c i . ; Polym. Chem. Ed. 22, 3499 (1984). 6. W. E . Brown, J . Appl., Phys. 18, 273 (1947). 7. E . A. Willson, J . R. Miller and E . H. Rowe, J . Phys. Colloid Chem. 53, 357 (1949). 8. Ε. B. Bradford and J. W. Vanderhoff, J . Colloid S c i . 14, 543 (1959). 9. J . Gregory, Trans. Faraday Soc. 65, 2260 (1969). 10. J . Gregory, J . Colloid Interface S c i . , 42, 448 (1973). 11. R. Van Gilder, D. I. Lee, R. Purfeerst, and J. Allswede, Tappi J. 60, 11 (1983). RECEIVED September 18, 1986
Provder; Particle Size Distribution ACS Symposium Series; American Chemical Society: Washington, DC, 1987.