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A radiometric method for the characterization of particulate processes in colloidal suspensions Part 2. Experimental verification of the method
P o w d e r T e c h n o l o g y , 24 (1979) 41 - 47
41
© Elsevier Sequoia S_A., Lausanne - - Printed in the Netherlands
A Radiometric Method for the Characterization of Particulate Processes in Colloidal Suspensions P a r t 2. E x p e r i m e n t a l V e r i f i c a t i o n o f t h e M e t h o d
B O R I S SUBOTIC I n s t i t u t e " R u d e r B o s k o v i c " 4 1 0 0 1 Z a g r e b , P. B o x 1 0 1 6 ( Y u g o s l a v i a )
(Received March 1, 1979)
SUMMARY A radiometric method for the characterizat i o n o f p a r t i c u l a t e p r o c e s s e s is v e r i f i e d u s i n g stable hydroso!s of silver iodide. Silver iodide hydrosols satisfy the conditions required for the applications of the proposed method. Comparison shows that the values for the change of particle size measured in silver iodide hydrosols by the proposed method are in excellent agreement with the values obtained by other methods on the same systems (electron microscopy, sedimentation analysis, light scattering). This shows that the proposed m e t h o d is s u i t a b l e f o r t h e c h a r a c t e r i z a t i o n o f p a r t i c u l a t e p r o c e s s e s in c o l l o i d a l s u s p e n s i o n s .
INTRODUCTION The interrelation between the mean particle size F of an examined system and the exchange fraction F attained by the self-diffusiongoverned heterogeneous exchange process during an exchange time t E can be expressed by Wagner's solution of the differential equation of diffusion or/and its graphic r e p r e s e n t a t i o n ( t h e d i f f e r e n c e 1 - - F is p l o t t e d as a f u n c t i o n o f ~ 2 D s t z / F 2 , D s b e i n g t h e s e l f diffusion coefficient of exchangeable ions inside the solid phase) [1]. This interrelation is s t r i c t l y v a l i d o n l y i f D~ a n d F r e m a i n u n changed during the exchange process, and if the electrostatic part of the activation energy for the self-diffusion-governed exchange proc e s s , Es, is z e r o . H o w e v e r , i n r e a l c o l l o i d a l systems, particles change in size during particulate processes (aggregation, reerystallization). On the other hand, since particles of stable s o l s a r e c h a r g e d , t h e v a l u e o f E~ c a n n o t a t t a i n zero. These difficulties appearing in the analysis of real colloidal systems can be removed
by a suitable choice of the experimental procedure and by a modification of the mathematical treatment of experimental data. Es c a n b e r e d u c e d t o z e r o b y t h e e l e c t r o l y t i c coagulation of sols. Under such experimental conditions, the mean particle size determined by the graphical form of Wagner's solution of the differential equation for diffusion does not take a real value r but an imaginary value *F. T h e i m a g i n a r y v a l u e *~ is l a r g e r t h a n t h e r e a l v a l u e (F)o a t t h e b e g i n n i n g o f t h e c o a g u l a tion process (and the simultaneous exchange p r o c e s s ) a n d is l o w e r t h a n t h e r e a l v a l u e (F)t E at the end of the exchange process. However, f o r t h e c a s e w h e n r- c h a n g e s d u r i n g t h e h e t e r o geneous exchange process, mathematical a n a l y s i s o f t h e f u n c t i o n F = f(DstE/~ 2) s h o w s t h a t *r--b = q(Fb)0 a n d *ro = q ( ~ o ) o a n d t h u s (*~b/*FO)tE = (~b)o/(r-o)o- Here, (Fo)0 a n d (rb)0 are real values of particle size of stable sols at times tA = tA(a ) a n d tA = tA(b), respectively, before the beginning of the coagulation a n d exchange processes. Hence, the radiometric m e t h o d for characterization of particulate processes proposed in the first part of this series is based o n Rajagopal's relation [2] a n d o n the mathematical analysis of F versus D~'zfF z functions (Part I). Since in m o s t cases q at given time tz can be determined neither theoretically nor experimentally, it is evident that the proposed m e t h o d can be used only to determine the relative change of particle size during the ageing of systems, but does not allow the determination of the absolute change. This shortcoming is associated with modification of the original experimental procedure [3, 4] a n d with the mathematical treatment n e e d e d to reduce Es to zero (Part i). H o w e v e r , the relative change of the m e a n particle size during the ageing of the system is usually sufficient for the characterization of
42 p a r t i c u l a t e p r o c e s s e s in a s t a b l e s u s p e n s i o n . T h e a i m o f t h e p r e s e n t w o r k is t o v e r i f y t h e proposed method experimentally, by measuri n g t h e relative c h a n g e o f t h e m e a n p a r t i c l e size in s t a b l e h y d r o s o l s o f silver i o d i d e .
t h e s o l i d p h a s e . T h e initial a n d t h e e q u i l i b r i u m radioactivities of the liquid phase were d e t e r m i n e d as d e s c r i b e d p r e v i o u s l y [ 5 ] . All the systems examined were prepared 9 times a n d the results r e p o r t e d here represent t h e a v e r a g e values.
EXPERIMENTAL
RESULTS AND DISCUSSION Analar grade (Merck) AgNO3, NaI, and Mg(NOa)2" 6H20 chemicals were used throughout the experiments. Radioiodine, iodine-131, was o b t a i n e d f r o m t h e "Boris Kidric" I n s t i t u t e o f N u c l e a r S c i e n c e s , B e l g r a d e , Y u g o s l a v i a , as a s o l u t i o n o f c a r r i e r f r e e i o d i n e - 1 3 1 in t h e Na13ZI f o r m . D o u b l e - d i s t i l l e d w a t e r w a s u s e d as a s o l v e n t . S a m p l e s o f silver i o d i d e sols w e r e prepared by pipetting 5 ml of 0.02 and 0.004 molar AgNO 3 solutions and placing them into e r i e n m e y e r flasks c o n t a i n i n g e q u a l v o l u m e s o f 0.04 and 0.008 molar NaI solutions, respect i v e l y . T h e s o l u t i o n s in t h e flasks w e r e c o n t i n u o u s l y s t i r r e d w i t h a m a g n e t i c stirrer. I n t h e sols t h u s o b t a i n e d , t h e m o l a r r a t i o o f i o d i d e i o n s in t h e s o l i d p h a s e (n s) v e r s u s i o d i d e i o n s in t h e l i q u i d p h a s e (n L) w a s 1 ( a = n S / n L = l ) f o r e a c h sol. I n a n a t t e m p t t o d e t e r m i n e the heterogeneous exchange fractions F, and t h u s t h e r e l a t i v e p a r t i c l e sizes o f d i f f e r e n t l y a g e d s t a b l e sols, 1 0 m l o f t h e sol a g e d f o r a p r e d e t e r m i n e d p e r i o d o f t i m e t A ~,vas c o a g u lated with 0.05 ml of saturated Mg(NO3)2 solution. The stirred suspension was labelled w i t h 0 . 0 5 m l o f 13zI- s o l u t i o n 1 5 s e c o n d s after the beginning of coagulation. In order to stop the heterogeneous exchange process, a quantitative separation o f the phases was performed by centrifuging an aliquot of the s y s t e m s a t a p r e d e t e r m i n e d e x c h a n g e t i m e tE_ 0.2 ml o f clear s u p e r n a t a n t was d r a w n a n d used to determine the radioactivity AT. During the precipitation, the ageing process, and the heterogeneous exchange process, the s y s t e m s w e r e k e p t a t 2 0 +- 0 . 2 °C i n a H a a k e ultrathermostat. Using the radioactivities of t h e l i q u i d p h a s e ( t h e r a d i o a c t i v i t y A L a t rE), t h e initial r a d i o a c t i v i t y A o , a n d t h e e q u i l i b r i u m r a d i o a c t i v i t y AL~, t h e h e t e r o g e n e o u s e x c h a n g e fractions F were calculated by the relation F = ASIA s = (A o --AL)/(Ao
--A~)
(1)
w h e r e A s is t h e r a d i o a c t i v i t y o f t h e s o l i d phase attained during the exchange time ts a n d A s is t h e e q u i l i b r i u m r a d i o a c t i v i t y o f
Silver i o d i d e h y d r o s o l s are p a r t i c u l a r l y suitable systems for the verification of the proposed method. T h e e x c h a n g e p r o c e s s in f r e s h l y c o a g u l a t e d silver i o d i d e h y d r o s o l s is g o v e r n e d b y t h e selfdiffusion o f e x c h a n g e a b l e ions inside the crystal space of the solid phase during a relatively long interval of the process. This c a n easily b e p r o v e d b y p l o t t i n g F v e r s u s ( r e ) I/2. F o r a s e l f - d i f f u s i o n - g o v e r n e d h e t e r o g e n e o u s e x c h a n g e p r o c e s s , F is a l i n e a r f u n c tion of the square root of the exchange time [ 1 , 6, 7 ] . A s s h o w n in Figs. 1 a n d 2, f r a c t i o n s o f e x c h a n g e d e t e r m i n e d in A g I h y d r o s o l s b y t h e d e s c r i b e d p r o c e d u r e are l i n e a r f u n c t i o n s o f t h e (rE) 1/2. T h i s is a d i r e c t e x p e r i m e n t a l verification of the above statement. The mechanmm and kinetics of particle a g g r e g a t i o n in s t a b l e silver i o d i d e sols h a v e b e e n d e f i n e d p r e v i o u s l y [8, 9 ] . U s i n g v a r i o u s experimental methods (radiometric, optic, and sedimentation methods), we have found t h a t silver i o d i d e h y d r o s o l s p r e p a r e d as d e s c r i b e d in t h e p r e c e d i n g s e c t i o n age t h r o u g h two different processes: a rapid process which Co = [ A g I ] l r n o | din-3 =0 0 1 :
~Ncll]/tool
d i n - 3 =Q 0 1
°6i ~V-~--~._.....~0-3
tA/,--,.
I
o, • a
I0 100
0-2 t
l
Z
~
I
4
!
5
!
6
I
7
I
I
I
8
9
10
F i g . 1. E x c h a n g e f r a c t i o n F p l o t t e d a g ~ i r m t t h e s q u a r e
r o o t o £ t h e e x c h a n g e t i m e ( t E ) "1" f o r 0 . 0 1 molar silver i o d i d e h y d r o s o l a g e d f o r 1 , 1 0 a n d 1 0 0 m i n u t e s b e f o r e c o a g u l a t i o n a n d l a b e l l e d 1 5 s e c o n d s after initiation of the coagulation.
43
: o
os
f
[,,O,mo, d.,-~ :o.oo~~ [.Na!]/mo!dm-3 =0-00
=
(r)o
. /
o
tA/rnmn
02 I
I
l
t
+ •2(r2)o]
I(nl + n2)
(5)
w h e r e n 2 = (n ° - - n l ) / 2 is t h e n u m b e r o f s e c o n d a r y particles, and (rl)o a n d (r2) 0 d e n o t e t h e size o f p r i m a r y a n d s e c o n d a r y p a r t i c l e s , r e s p e c t i v e l y , o f t h e s t a b l e sols. I f t h e size r a t i o o f s e c o n d a r y a n d p r i m a r y p a r t i c l e s is ( r 2 ) o / ( r l ) o = p ( 2 , 1 ) , t h e n s u b s t i t u t i n g p ( 2 , 1 ) (r~) 0 f o r (r2) 0 a n d c o m b i n i n g e q n s . ( 3 ) , ( 4 ) a n d (5) gives
o s • SO0 I
= [nx(rl)o
I
Fig. 2. Exchange fraction F plotted against the square r o o t o f t h e e x c h a n g e t i m e ( r E ) lr2 f o r 0 . 0 0 2 m o l a r silver iodide hydrosol aged for 5 and 500 minutes before coagulation and labelled 15 seconds after initiation of the coagulation.
(F)0 = ( r l ) 0 1 2 + C o k ~ t A P ( 2 , 1 ) ] / ( 2
+ CokntA)
and hence
(6)
C o k = t A = 2 [ ( F ) o - - ( r , )0 ] / [ p ( 2 , 1 ) ( r l )o
(~)o1
-
-
(7)
starts i m m e d i a t e l y after t h e mixing o f the precipitating components and the duration of w h i c h is a p p r o x i m a t e l y i n v e r s e l y p r o p o r t i o n a l t o t h e m o l a r c o n c e n t r a t i o n o f silver i o d i d e , and a slow process which starts after the rapid process has been c o m p l e t e d . Electron-microscope and sedimentation analysis show that p r i m a r i l y f o r m e d p a r t i c l e s o f silver i o d i d e a r e monodisperse and that they aggregate to monodisperse secondary particles during the r a p i d a g e i n g p r o c e s s . I t is i n t e r e s t i n g t h a t t h e r e a c t i o n r a t e b e t w e e n s e c o n d a r y p a r t i c l e s , as well as t h e r e a c t i o n r a t e b e t w e e n p r i m a r y a n d s e c o n d a r y p a r t i c l e s , is s m a l l in c o m p a r i s o n with the reaction rate between primary p a r t i c l e s . T h u s t h e d e c r e a s e in nth-nber o f primary particles during the ageing time t A can be expressed b y t h e differential e q u a t i o n for a second-order kinetic process:
The fraction of exchange, F, attained for a g i v e n e x c h a n g e t i m e rE iS a f u n c t i o n o f (F)0 ( P a r t 1). H e n c e t h e size o f p r i m a r y p a r t i c l e s a n d t h e m e a n size o f p a r t i c l e s are e x p r e s s e d by the relations
dn x/dt A = kznZl
eqns. (8) a n d (9) c a n be rewritten as
(2)
(3)
w h e r e n l is t h e n u m b e r o f p r i m a r y p a r t i c l e s a t tA > 0,/Z 2 is t h e r a t e c o n s t a n t o f a s e c o n d o r d e r k i n e t i c p r o c e s s , a n d n ° is t h e n u m b e r o f p r i m a r y p a r t i c l e s a t t A = 0. D i v i d i n g e q n . (3) by n ° , one obtains the expression for the change of a fraction of primal- particles f~l:
f - z = n~ln° = l / ( C o l Z = t A + 1)
(4)
w h e r e Co = [ A g I ] is t h e m o l a r c o n c e n t r a t i o n o f silver i o d i d e , k~ = k x k 2 , a n d ka = C o / n °. T h e m e a n p a r t i c l e size (F)0 in s u c h a s y s t e m c a n b e e x p r e s s e d as
(8)
and (r}0 = *r/qt E
(9)
w h e r e *rx a n d *~ are i m a g i n a r y p a r t i c l e sizes calculated using the experimentally obtained F values, as d e s c r i b e d in ~he p r e v i o u s p a p e r o f this series ( P a r t 1). S i n c e 1/*~
= ~2Dstz/*r2
(10)
and 1 / ' 3 ~ = 7r2Dstz/*~l2
( r l ) o = *[3o~r Dx/-D-~-EstElqtz
T h e a n a l y t i c a l s o l u t i o n o f e q n . (2) is n l = I/(/Z2tA + 1 I n ° )
(rl)0 = *rl [qt E
(11)
(12)
and
(r )o = *[3tnVr-DstElqtE
(13)
and hence (r)o/(Fz)o = " 3 t / ' 3 o
(14)
H e r e 1 / * ~ is t h e n u m e r i c a l v a l u e o f = Z D s t F [ *F z c o r r e s p o n d i n g t o t h e m e a s u r e d 1 - - F v a l u e s , a n d 1[*/3~ is t h e n u m e r i c a l v a l u e o f u 2 D s t z / * r ~ c o ~ c n p o n d i n g to t h e 1 - - F v a l u e a t tA = 0, i.e. (F)o = (rx)o- S i n c e u n d e r g i v e n e x p e r i m e n t a l c o n d i t i o n s (Ds a n d tE are c o n s t a n t ) t h e v a l u e o f qtE is c o n s t a n t f o r all v a l u e s o f t A , e q n s . (6) a n d (7) c a n b e r e w r i t t e n as
44 (rrez)o = ( r ) 0 / ( r z ) o = */~t/*~0 = [ 2 + Cote=tAP(2,1)]/(2
(15)
Co [Agl]lrnoldrn-~ =0002 o-~5i,._, ~ q a l . ] / m o i d n n - I =0002
(16)
0.¢3L F o-t.l / --
=
+ ColZ=tA)
and
Cok~tA
= 2(*/~t - - * P 0 ) / [ P ( 2 , 1 ) * P o - - * P t ]
T h e p r o p o s e d m e t h o d c a n b e v e r i f i e d in t w o ways: First, the (~)o/(rl)0 versus tA functions calculated by eqn. (15) using known v a l u e s o f Co, k= a n d p ( 2 , 1 ) c a n b e c o r r e l a t e d with the functions calculated from the F values m e a s u r e d b y t h e p r o c e d u r e described. S e c o n d l y , t h e /z= v a l u e s c a l c u l a t e d b y e q n . (16} u s i n g *ft v a l u e s o b t a i n e d f r o m e x p e r i m e n t a l d a t a c a n b e c o r r e l a t e d w i t h t h e k~ v a l u e s m e a s u r e d in t h e s a m e s y s t e m s u s i n g o t h e r m e t h o d s . I n b o t h c a s e s * f t a n d *fo ave calculated f r o m the F values o b t a i n e d experimentally using a graphical form of Wagner's solution of the differential equation for d i f f u s i o n ( P a r t 1). F i g u r e s 3 a n d 4 s h o w t h e fractions of exchange F attained during a c o n s t a n t e x c h a n g e t i m e tE ( t z = 2 0 rain f o r 0 . 0 1 m o l a r A g I sol a n d t E = 3 0 r a i n f o r 0 . 0 0 2 molar A g I sol); the values are p l o t t e d against t h e age o f sols ( t a ) b e f o r e t h e b e g i n n i n g o f the coagulation process. Figures 5 and 6 show t h e c o r r e s p o n d i n g v a l u e s o f *fit- I n b o t h cases, t h e v a l u e o f *B0 w a s o b t a i n e d b y t h e e x t r a p o l a t i o n o f *fit v e r s u s t A c u r v e s a t t A = O. I n a c c o r d a n c e w i t h e q n . ( 1 5 ) , t h e r a t i o s *~t/*I~o o u g h t t o b e e q u a l t o t h e (r)o/(rz)o ratios if the theoretical approach of the proposed method is v a l i d ( P a r t 1). I n Figs. 7 a n d 8 t h e r a t i o s (F)o/(rz)0 d e t e r m i n e d u s i n g t h e p r o p o s e d m e t h o d in 0 . 0 1 a n d 0 . 0 0 2 m o l a r A g I sols a r e r e p r e s e n t e d as f u n c t i o n s o f t A ( d o t t e d c u r v e s ) . T h e e x p e r i m e n t a l d a t a are c o m p a r e d w i t h (r)o/(ri) o versus t A functions calculated by e q n . ( 1 5 ) u s i n g t h e v a l u e s o f (rz) 0 a n d p ( 2 , 1 )
tE/mm=20
0 37~-
0-35t" .
L
200
tAImln
t,~o
300
SO0
F i g . 4 . E x c h a n g e f r a c t i o n F , a t t a i n e d for exchange time t E = 30 minutes in 0.002 molar silver i o d i d e hyd:osol, plotted against t h e a g e o f sol (= tA) before
coagulation. Ii II L t
~l)t F /"---'~JI"J0 = 9 4S
.t
! 0
I 20
I~1
tE/rnln =:0 = I 30
I 20 ~0 tAIrnLrl
I 60
l 70
20
sol
i00
Fig. 5. Values of *fit corresponding to the F values measured by the experimental procedure described, in 0.01 molar silver iodide hydro~ol, plotted against t A
f
*13 t L ~ If
O
/
AfJo
-11-11
002 [Na|]lmol din-3 - 0002 tEImm=20
11 o 50 0-/.8
Co = [A~l~lrnol dm -3 = O O1 [Ncgl]lmol dm-3 = 0 01
' ~
0
0-46 0 .4
±
,"
~o
20
2o
~'o
6'o
tAImbl
,"
."
,'o
,~o
F i g . 3 . Exchange fraction F, attained for exchange t i m e tE = 2 0 m i n u t e s ill 0 . 0 1 m o l a r silver i o d i d e hydrosol, plotted against the age of sol (= tA ) before coagulation.
lO0
200 tA/mln 3
400
500
F i g . 6. V a l u e s o f *J~t corresponding to the -~ values measured by the experimental procedure described, i n 0 . 0 0 2 m o l a r silver i o d i d e h y d r o s o l , p l o t t e d ag~;n~t tA.
determined by electron microscopy [9] and w i t h t h e k= v a l u e s d e t e i x n i n e d b y s e d i m e n t a t i o n analysis (solid curves) and by the m e t a phase analysis method (dashed curves) [8 1 0 ] . T h e r a d i u s (rz) o d e t e r m i n e d b y e l e c t r o n
45 I
co: [~g~/~o, ~-'
d i f f e r e n c e s i n k,= v a l u e s d e t e r m i n e d b y d i f ferent methods and to a possible error in the determination of p(2,1) values by electron m i c r o s c o p y . P u t t i n g ( r l ) o = r~ef = 8 . 7 n m ( f o r 0 . 0 1 m o l a r A g I s o l ) a n d ( r z ) o = r~e~ = 8.4 nm (for 0.002 molar AgI sol), the change of the absolute particle size (F)0 can be calculated by the relation
o o 0,
r N o ~ / r n a l d m -3 = 0 01 I
( r ) o = ~ e l r r e f = ?-rel(rl )0 lu-O ,
10 . .
20 . .
30
t.0
50 6, , iA/~rlln
70,
80 ,
;0
:
Fig. 7. Change o f relative particle size (~el)O determ i n e d b y t h e p r o p o s e d m e t h o d ( d o t t e d curve) in 0.01 m o l a r silver i o d i d e h y d r o s o l , c o m p a r e d w i t h t h e change o f ( ~ e l ) 0 c a l c u l a t e d b y eqn. (15) using t h e value p ( 2 , 1 ) d e t e r m i n e d b y e l e c t r o n m i c r o s c o p y :~Ld t h e value o f kvr d e t e r m i n e d b y sedimentatio,-, analysis (solid curve) a n d b y m e t a p h a s e analy~_Z~ m e t h o d (dashed curve). m i c r o s c o p y is 8 . 7 a n d 8 . 4 n m f o r 0 . 0 1 a n d 0.002 molar AgI sol, respectively. The size r a t i o p ( 2 , 1 ) = ( r z ) o / ( r l ) o is 1 . 3 f o r 0 . 0 1 m o l a r AgI sol and 1.34 for 0.002 molar sol. The experimental data are in good agreement with the results calculated by eqn. (15) using the d a t a (/z=, ( r l ) o a n d p ( 2 , 1 ) ) d e t e r m i n e d b y other experimental methods. The small differences between the experimentally obtained (F)o/(rz)0 values and the (F)0/(rl) 0 values calculated by eqn. (15) are due to i
CO : [Ag~/mo!
dm-3 = 0 00Z
[
rNa~/mo[
d m -3 = 0 0 0 2
(17)
Figures 9 and 10 show the change of absolute particle size in 0.01 and 0.002 molar AgI sols as a function of ageing time. The values marked with crosses correspond to the values determined by electron microscopy [9]. The second way of verifying the proposed m e t h o d is a s f o l l o w s : T h e p ( 2 , 1 ) a n d k = v a l u e s calculated by eqn. (16) using the experim e n t a l l y d e t e r m i n e d *Bt v a l u e s ( F i g s . 6 a n d 7 ) axe compared with the p(2,1) and k= values determined by other methods (electron microscopy, sedimentation mualysis, metaphase analysis method, light scattering [8 1 0 l ). U s i n g e q n . ( 1 5 ) , w e c a n c a l c u l a t e t h e value of p(2,1) from the conditions Co/Z=tA(l) = 2(*ft, -- *fO)/[P(2,1)*P0 --*~t, ]
(z8) C0k~tA(2) = 2(*ft: -- *f0)/[P(2,1)*fo
--*ft: ] (19)
w h e r e *fit, a n d *~t= c o r r e s p o n d t o t h e *fit values determined for two different ageing t i m e s , n a m e l y f o r tA = t A ( 1 ) a n d t A = t A ( 2 ).
°l c= .~-
E ~o9
/ / "
Co = [ A ~ l l ] = 0 01 t o o l cirn ~2 [Nail = 0 O, rno, dm-~
..
to !
I
f
I t,nn
tA/mln
Fig. 8. Change o f relative particle size (~el)O d e t e r m i n e d b y t h e p r o p o s e d m e t h o d ( d o t t e d curve) in 0.002 m o l a r silver i o d i d e h y d r o s o l , c o m p a r e d w i t h t h e change o f (~el)O c a l c u l a t e d b y e q n . (15) using the value p ( 2 , 1 ) d e t e r m i n e d b y e l e c t r o n m i c r o s c o p y a n d the value o f k~ d e t e r m ; n e d b y sedlmaen~;ation nn:alysis (solid curve) a n d b y m e t a p h a s e analysis m e t h o d ( d a s h e d curve).
I tAImin
Fig. 9. Change o f a b s o l u t e particle size (~)o calculated b y eqn. (16) using t h e values o f (~el)O d e t e r m i n e d b y t h e p r o p o s e d m e t h o d in 0.01 m o l a r silver i o d i d e h y d r o s o l a n d t h e value o f (r 1 )o d e t e r m i n e d b y e l e c t r o n m i c r o s c o p y in t h e s a m e s y s t e m . T h e cro~-es d e n o t e t h e values o f (r--)0 d e t e r m i n e d b y e l e c t r o n microscopy.
46
"F Co= ~Ag~]/moldm-3 =001
~F
[~lat"Jlmo[ drn-3 = 00l plZ.'z) =128 9L
J ¢ "
_/o
0
200
100
tAImln
300
400
50(3
Fig. 10. Change of absolute partlcIe size (r-)0 caIeulated by eqn. (16) using the values o f (Frel) 0 determined by the proposed method in 0.002 molar silver iodide hydrosol and the vaIue o f (r 1 )o determined by electron microscopy in the same system. The crones denote the values o f CrY0 determined by electron microscopy.
D i v i d i n g e q n . ( 1 8 ) b y e q n . ( 1 9 ) r e s u l t s in a linear e q u a t i o n : ap(2,l) + b
=
0
(20)
where a
=
tA(Z)*~0*t3t2
tA(1)*~ 2
--
--
tA(2)*~O*~3t,
+
tA(2)*f~ b ~- t A ( 2 ) * ~ t
/,~
C O = [A91] =0-002 mot din-3 [No.l] : 0 002 me! din-3
x
*~t: --
tA(1)*/~t:*.B¢~
tA(2)*fo*fit=
--
+ t~(1)*/~o*ft,
U s i n g e q n . ( 2 0 ) , w e c a n easily c a l c u l a t e p ( 2 , 1 ) using o n l y the n u m e r i c a l values o f the c h o s e n ageing times tA(1 ) a n d tA(2 ) and t h e c o r r e s p o n d i n g v a l u e s o f *ft, a n d */3t~- F o r practical purposes we take an average value of p ( 2 , 1 ) c a l c u l a t e d u s i n g several c o m b i n a t i o n s of different values of tA(1) and tA(2) and c o r r e s p o n d i n g values o f *~t, a n d *~t=- F o r Co = 0 . 0 1 t o o l d m - a , p ( 2 , 1 ) --- 1 . 2 8 a n d f o r C o = 0 . 0 0 2 t o o l d m - 3 , p ( 2 , 1 ) ---- 1 . 2 7 , w h i c h is in excellent a g r e e m e n t with t h e p ( 2 , 1 ) values obtained by electron microscopy (p(2,1)1.30 and 1.34, respectively). If the theoretical considerations which underlie the proposed m e t h o d a r e valid, t h e v a l u e s o n t h e r i g h t - h a n d side o f e q n . ( 1 6 ) m u s t b e l i n e a r f u n c t i o n s o f
tA f o r a c o r r e s p o n d i n g value o f p ( 2 , 1 ) . Figures 11 a n d 12 s h o w (*~t - - */30)/[P(2,1)*/]o - analysed depend linearly t h a t t h e c o n s t a n t k~ c a n
t h a t t h e values o f * f t ] f o r t h e sols on ageing time, and be computed by
°i
O
I0
20
30
70
80
90
10~
tA/mTrt
F i g . 11, Values o f (*J3 t - - * f o ) i [ P ( 2 , 1 ) * f l o 0.01 molar silver iodide hyd~osol plotted ageing time t A.
- - *•t] of against the
d i v i d i n g t h e s l o p e o f t h e s t r a i g h t line b y Co/~ T a b l e 1 gives a c o m p a r i s o n o f t h e k~ v a l u e s f o r 0 . 0 1 a n d 0 . 0 0 2 m o l a r A g I sols o b t a i n e d b y t h e p r o p o s e d m e t h o d , a n d t h e h~ v a l u e s f o r t h e s a m e sols o b t a i n e d b y o t h e r m e t h o d s [8 - 10]. The validity of eqn. (16) for the *Bit v a l u e s c a l c u l a t e d f r o m t h e e x p e r i m e n t a l l 5 d e t e r m i n e d F values, a n d the g o o d a g r e e m e n b e t w e e n t h e k~ v a l u e s o b t a i n e d b y t h e p r o posed method and those obtained by other m e t h o d s ( T a b l e 1) s h o w t h a t t h e p r o p o s e d m e t h o d is s u i t a b l e f o r t h e c h a r a c t e r i z a t i o n oJ p a r t i c u l a t e p r o c e s s e s in s y s t e m s o f c o l l o i d a l hydrosols.
o
~ 2,
[ Agl]lm°l din-3 ~ 0 002 ~
p=
olf ~-
1
o 2
,
1
1
kit=6 9 too|-I dl-/i] mlrl-t
I
I.
i
1
.L.
....
Fig. 12. V.~ues o f (*ft - - *fo)l[P(2,1)*f0 - - * & ] o f 0.002 mola~r silver i o d i d e h y d c _ o s o l p l o t t e d a g a i n s t t h ageing time t A.
47 TABLE 1 C o m p a r i s o n o f kTr v a l u e s o b t a i n e d b y t h e p r o p o s e d m e t h o d w i t h kzr v a l u e s o b t a i n e d b y o t h e r methods for 0.01 and 0.002 molar AgI sols Co (tool dm -3 )
MethGd Proposed method
Metaphase analysis method
0.01
k n = 14.8
k n
0.002
k n =
6.9
= 8.8
k~ = 5 . 2
REFERENCES I K. E. Z i m e n s , Arklv K e m i Mineral. Geol., 2 0 A ( 1 9 4 5 ) 1. 2 E . S. R a j a g o p a l , K o l l o i d Z . , 1 6 7 ( 1 9 6 0 ) 1 7 . 3 R. Despotovi~ and V. Stengl, K o l l o i d Z. Z. Polym., 250 (1972) 950. 4 R . D e s p o t o v i ~ a n d B. S u h o t i ~ , C r o a t . C h e m . A e t a , 43 ( 1 9 7 1 ) 153. 5 R . D e s p o t o v i ~ a n d B. S u b o t i ~ , J . I n o r g . N u c l . C h e m . , 38 ( 1 9 7 6 ) 1 3 1 7 .
Sedimentation analysis method
Light scattering
k n --- 1 4 . 0
k.~ = 1 2 . 9
k~ = 6.0
kn = 5.0
6 R . H . H . W o l f , M. M i r n i k a n d B. T e z a k , K o l l o i d Z. Z. P o l y m . , 2 0 5 ( 1 9 6 5 ) 1 1 1 . 7 A. D y e r , G. G. Hayes, G. O. Phillips a n d R . P. T o w n s e n d , Adv. C h e m . Ser.; 121 ( 1 9 7 2 ) 299. 8 B. S u b o t i ~ , P r o e . I n t . C o n f . , C o l l o i d a n d I n t e r f a c ( Science, Vol. I (Ed. E. Wolfram), Akademiai K i a d o , B u d a p e s t , 1 9 7 5 , p. 2 5 7 . 9 B. S u b o t i ~ , P h . D . T h e s i s , U n i v . o f Z a g r e b , 1 9 7 6 . 1 0 R . D e s p o t o v i ~ e t a l . , C r o a t . C h e m . A c t a , 51 (1978) 113.
41
© Elsevier Sequoia S_A., Lausanne - - Printed in the Netherlands
A Radiometric Method for the Characterization of Particulate Processes in Colloidal Suspensions P a r t 2. E x p e r i m e n t a l V e r i f i c a t i o n o f t h e M e t h o d
B O R I S SUBOTIC I n s t i t u t e " R u d e r B o s k o v i c " 4 1 0 0 1 Z a g r e b , P. B o x 1 0 1 6 ( Y u g o s l a v i a )
(Received March 1, 1979)
SUMMARY A radiometric method for the characterizat i o n o f p a r t i c u l a t e p r o c e s s e s is v e r i f i e d u s i n g stable hydroso!s of silver iodide. Silver iodide hydrosols satisfy the conditions required for the applications of the proposed method. Comparison shows that the values for the change of particle size measured in silver iodide hydrosols by the proposed method are in excellent agreement with the values obtained by other methods on the same systems (electron microscopy, sedimentation analysis, light scattering). This shows that the proposed m e t h o d is s u i t a b l e f o r t h e c h a r a c t e r i z a t i o n o f p a r t i c u l a t e p r o c e s s e s in c o l l o i d a l s u s p e n s i o n s .
INTRODUCTION The interrelation between the mean particle size F of an examined system and the exchange fraction F attained by the self-diffusiongoverned heterogeneous exchange process during an exchange time t E can be expressed by Wagner's solution of the differential equation of diffusion or/and its graphic r e p r e s e n t a t i o n ( t h e d i f f e r e n c e 1 - - F is p l o t t e d as a f u n c t i o n o f ~ 2 D s t z / F 2 , D s b e i n g t h e s e l f diffusion coefficient of exchangeable ions inside the solid phase) [1]. This interrelation is s t r i c t l y v a l i d o n l y i f D~ a n d F r e m a i n u n changed during the exchange process, and if the electrostatic part of the activation energy for the self-diffusion-governed exchange proc e s s , Es, is z e r o . H o w e v e r , i n r e a l c o l l o i d a l systems, particles change in size during particulate processes (aggregation, reerystallization). On the other hand, since particles of stable s o l s a r e c h a r g e d , t h e v a l u e o f E~ c a n n o t a t t a i n zero. These difficulties appearing in the analysis of real colloidal systems can be removed
by a suitable choice of the experimental procedure and by a modification of the mathematical treatment of experimental data. Es c a n b e r e d u c e d t o z e r o b y t h e e l e c t r o l y t i c coagulation of sols. Under such experimental conditions, the mean particle size determined by the graphical form of Wagner's solution of the differential equation for diffusion does not take a real value r but an imaginary value *F. T h e i m a g i n a r y v a l u e *~ is l a r g e r t h a n t h e r e a l v a l u e (F)o a t t h e b e g i n n i n g o f t h e c o a g u l a tion process (and the simultaneous exchange p r o c e s s ) a n d is l o w e r t h a n t h e r e a l v a l u e (F)t E at the end of the exchange process. However, f o r t h e c a s e w h e n r- c h a n g e s d u r i n g t h e h e t e r o geneous exchange process, mathematical a n a l y s i s o f t h e f u n c t i o n F = f(DstE/~ 2) s h o w s t h a t *r--b = q(Fb)0 a n d *ro = q ( ~ o ) o a n d t h u s (*~b/*FO)tE = (~b)o/(r-o)o- Here, (Fo)0 a n d (rb)0 are real values of particle size of stable sols at times tA = tA(a ) a n d tA = tA(b), respectively, before the beginning of the coagulation a n d exchange processes. Hence, the radiometric m e t h o d for characterization of particulate processes proposed in the first part of this series is based o n Rajagopal's relation [2] a n d o n the mathematical analysis of F versus D~'zfF z functions (Part I). Since in m o s t cases q at given time tz can be determined neither theoretically nor experimentally, it is evident that the proposed m e t h o d can be used only to determine the relative change of particle size during the ageing of systems, but does not allow the determination of the absolute change. This shortcoming is associated with modification of the original experimental procedure [3, 4] a n d with the mathematical treatment n e e d e d to reduce Es to zero (Part i). H o w e v e r , the relative change of the m e a n particle size during the ageing of the system is usually sufficient for the characterization of
42 p a r t i c u l a t e p r o c e s s e s in a s t a b l e s u s p e n s i o n . T h e a i m o f t h e p r e s e n t w o r k is t o v e r i f y t h e proposed method experimentally, by measuri n g t h e relative c h a n g e o f t h e m e a n p a r t i c l e size in s t a b l e h y d r o s o l s o f silver i o d i d e .
t h e s o l i d p h a s e . T h e initial a n d t h e e q u i l i b r i u m radioactivities of the liquid phase were d e t e r m i n e d as d e s c r i b e d p r e v i o u s l y [ 5 ] . All the systems examined were prepared 9 times a n d the results r e p o r t e d here represent t h e a v e r a g e values.
EXPERIMENTAL
RESULTS AND DISCUSSION Analar grade (Merck) AgNO3, NaI, and Mg(NOa)2" 6H20 chemicals were used throughout the experiments. Radioiodine, iodine-131, was o b t a i n e d f r o m t h e "Boris Kidric" I n s t i t u t e o f N u c l e a r S c i e n c e s , B e l g r a d e , Y u g o s l a v i a , as a s o l u t i o n o f c a r r i e r f r e e i o d i n e - 1 3 1 in t h e Na13ZI f o r m . D o u b l e - d i s t i l l e d w a t e r w a s u s e d as a s o l v e n t . S a m p l e s o f silver i o d i d e sols w e r e prepared by pipetting 5 ml of 0.02 and 0.004 molar AgNO 3 solutions and placing them into e r i e n m e y e r flasks c o n t a i n i n g e q u a l v o l u m e s o f 0.04 and 0.008 molar NaI solutions, respect i v e l y . T h e s o l u t i o n s in t h e flasks w e r e c o n t i n u o u s l y s t i r r e d w i t h a m a g n e t i c stirrer. I n t h e sols t h u s o b t a i n e d , t h e m o l a r r a t i o o f i o d i d e i o n s in t h e s o l i d p h a s e (n s) v e r s u s i o d i d e i o n s in t h e l i q u i d p h a s e (n L) w a s 1 ( a = n S / n L = l ) f o r e a c h sol. I n a n a t t e m p t t o d e t e r m i n e the heterogeneous exchange fractions F, and t h u s t h e r e l a t i v e p a r t i c l e sizes o f d i f f e r e n t l y a g e d s t a b l e sols, 1 0 m l o f t h e sol a g e d f o r a p r e d e t e r m i n e d p e r i o d o f t i m e t A ~,vas c o a g u lated with 0.05 ml of saturated Mg(NO3)2 solution. The stirred suspension was labelled w i t h 0 . 0 5 m l o f 13zI- s o l u t i o n 1 5 s e c o n d s after the beginning of coagulation. In order to stop the heterogeneous exchange process, a quantitative separation o f the phases was performed by centrifuging an aliquot of the s y s t e m s a t a p r e d e t e r m i n e d e x c h a n g e t i m e tE_ 0.2 ml o f clear s u p e r n a t a n t was d r a w n a n d used to determine the radioactivity AT. During the precipitation, the ageing process, and the heterogeneous exchange process, the s y s t e m s w e r e k e p t a t 2 0 +- 0 . 2 °C i n a H a a k e ultrathermostat. Using the radioactivities of t h e l i q u i d p h a s e ( t h e r a d i o a c t i v i t y A L a t rE), t h e initial r a d i o a c t i v i t y A o , a n d t h e e q u i l i b r i u m r a d i o a c t i v i t y AL~, t h e h e t e r o g e n e o u s e x c h a n g e fractions F were calculated by the relation F = ASIA s = (A o --AL)/(Ao
--A~)
(1)
w h e r e A s is t h e r a d i o a c t i v i t y o f t h e s o l i d phase attained during the exchange time ts a n d A s is t h e e q u i l i b r i u m r a d i o a c t i v i t y o f
Silver i o d i d e h y d r o s o l s are p a r t i c u l a r l y suitable systems for the verification of the proposed method. T h e e x c h a n g e p r o c e s s in f r e s h l y c o a g u l a t e d silver i o d i d e h y d r o s o l s is g o v e r n e d b y t h e selfdiffusion o f e x c h a n g e a b l e ions inside the crystal space of the solid phase during a relatively long interval of the process. This c a n easily b e p r o v e d b y p l o t t i n g F v e r s u s ( r e ) I/2. F o r a s e l f - d i f f u s i o n - g o v e r n e d h e t e r o g e n e o u s e x c h a n g e p r o c e s s , F is a l i n e a r f u n c tion of the square root of the exchange time [ 1 , 6, 7 ] . A s s h o w n in Figs. 1 a n d 2, f r a c t i o n s o f e x c h a n g e d e t e r m i n e d in A g I h y d r o s o l s b y t h e d e s c r i b e d p r o c e d u r e are l i n e a r f u n c t i o n s o f t h e (rE) 1/2. T h i s is a d i r e c t e x p e r i m e n t a l verification of the above statement. The mechanmm and kinetics of particle a g g r e g a t i o n in s t a b l e silver i o d i d e sols h a v e b e e n d e f i n e d p r e v i o u s l y [8, 9 ] . U s i n g v a r i o u s experimental methods (radiometric, optic, and sedimentation methods), we have found t h a t silver i o d i d e h y d r o s o l s p r e p a r e d as d e s c r i b e d in t h e p r e c e d i n g s e c t i o n age t h r o u g h two different processes: a rapid process which Co = [ A g I ] l r n o | din-3 =0 0 1 :
~Ncll]/tool
d i n - 3 =Q 0 1
°6i ~V-~--~._.....~0-3
tA/,--,.
I
o, • a
I0 100
0-2 t
l
Z
~
I
4
!
5
!
6
I
7
I
I
I
8
9
10
F i g . 1. E x c h a n g e f r a c t i o n F p l o t t e d a g ~ i r m t t h e s q u a r e
r o o t o £ t h e e x c h a n g e t i m e ( t E ) "1" f o r 0 . 0 1 molar silver i o d i d e h y d r o s o l a g e d f o r 1 , 1 0 a n d 1 0 0 m i n u t e s b e f o r e c o a g u l a t i o n a n d l a b e l l e d 1 5 s e c o n d s after initiation of the coagulation.
43
: o
os
f
[,,O,mo, d.,-~ :o.oo~~ [.Na!]/mo!dm-3 =0-00
=
(r)o
. /
o
tA/rnmn
02 I
I
l
t
+ •2(r2)o]
I(nl + n2)
(5)
w h e r e n 2 = (n ° - - n l ) / 2 is t h e n u m b e r o f s e c o n d a r y particles, and (rl)o a n d (r2) 0 d e n o t e t h e size o f p r i m a r y a n d s e c o n d a r y p a r t i c l e s , r e s p e c t i v e l y , o f t h e s t a b l e sols. I f t h e size r a t i o o f s e c o n d a r y a n d p r i m a r y p a r t i c l e s is ( r 2 ) o / ( r l ) o = p ( 2 , 1 ) , t h e n s u b s t i t u t i n g p ( 2 , 1 ) (r~) 0 f o r (r2) 0 a n d c o m b i n i n g e q n s . ( 3 ) , ( 4 ) a n d (5) gives
o s • SO0 I
= [nx(rl)o
I
Fig. 2. Exchange fraction F plotted against the square r o o t o f t h e e x c h a n g e t i m e ( r E ) lr2 f o r 0 . 0 0 2 m o l a r silver iodide hydrosol aged for 5 and 500 minutes before coagulation and labelled 15 seconds after initiation of the coagulation.
(F)0 = ( r l ) 0 1 2 + C o k ~ t A P ( 2 , 1 ) ] / ( 2
+ CokntA)
and hence
(6)
C o k = t A = 2 [ ( F ) o - - ( r , )0 ] / [ p ( 2 , 1 ) ( r l )o
(~)o1
-
-
(7)
starts i m m e d i a t e l y after t h e mixing o f the precipitating components and the duration of w h i c h is a p p r o x i m a t e l y i n v e r s e l y p r o p o r t i o n a l t o t h e m o l a r c o n c e n t r a t i o n o f silver i o d i d e , and a slow process which starts after the rapid process has been c o m p l e t e d . Electron-microscope and sedimentation analysis show that p r i m a r i l y f o r m e d p a r t i c l e s o f silver i o d i d e a r e monodisperse and that they aggregate to monodisperse secondary particles during the r a p i d a g e i n g p r o c e s s . I t is i n t e r e s t i n g t h a t t h e r e a c t i o n r a t e b e t w e e n s e c o n d a r y p a r t i c l e s , as well as t h e r e a c t i o n r a t e b e t w e e n p r i m a r y a n d s e c o n d a r y p a r t i c l e s , is s m a l l in c o m p a r i s o n with the reaction rate between primary p a r t i c l e s . T h u s t h e d e c r e a s e in nth-nber o f primary particles during the ageing time t A can be expressed b y t h e differential e q u a t i o n for a second-order kinetic process:
The fraction of exchange, F, attained for a g i v e n e x c h a n g e t i m e rE iS a f u n c t i o n o f (F)0 ( P a r t 1). H e n c e t h e size o f p r i m a r y p a r t i c l e s a n d t h e m e a n size o f p a r t i c l e s are e x p r e s s e d by the relations
dn x/dt A = kznZl
eqns. (8) a n d (9) c a n be rewritten as
(2)
(3)
w h e r e n l is t h e n u m b e r o f p r i m a r y p a r t i c l e s a t tA > 0,/Z 2 is t h e r a t e c o n s t a n t o f a s e c o n d o r d e r k i n e t i c p r o c e s s , a n d n ° is t h e n u m b e r o f p r i m a r y p a r t i c l e s a t t A = 0. D i v i d i n g e q n . (3) by n ° , one obtains the expression for the change of a fraction of primal- particles f~l:
f - z = n~ln° = l / ( C o l Z = t A + 1)
(4)
w h e r e Co = [ A g I ] is t h e m o l a r c o n c e n t r a t i o n o f silver i o d i d e , k~ = k x k 2 , a n d ka = C o / n °. T h e m e a n p a r t i c l e size (F)0 in s u c h a s y s t e m c a n b e e x p r e s s e d as
(8)
and (r}0 = *r/qt E
(9)
w h e r e *rx a n d *~ are i m a g i n a r y p a r t i c l e sizes calculated using the experimentally obtained F values, as d e s c r i b e d in ~he p r e v i o u s p a p e r o f this series ( P a r t 1). S i n c e 1/*~
= ~2Dstz/*r2
(10)
and 1 / ' 3 ~ = 7r2Dstz/*~l2
( r l ) o = *[3o~r Dx/-D-~-EstElqtz
T h e a n a l y t i c a l s o l u t i o n o f e q n . (2) is n l = I/(/Z2tA + 1 I n ° )
(rl)0 = *rl [qt E
(11)
(12)
and
(r )o = *[3tnVr-DstElqtE
(13)
and hence (r)o/(Fz)o = " 3 t / ' 3 o
(14)
H e r e 1 / * ~ is t h e n u m e r i c a l v a l u e o f = Z D s t F [ *F z c o r r e s p o n d i n g t o t h e m e a s u r e d 1 - - F v a l u e s , a n d 1[*/3~ is t h e n u m e r i c a l v a l u e o f u 2 D s t z / * r ~ c o ~ c n p o n d i n g to t h e 1 - - F v a l u e a t tA = 0, i.e. (F)o = (rx)o- S i n c e u n d e r g i v e n e x p e r i m e n t a l c o n d i t i o n s (Ds a n d tE are c o n s t a n t ) t h e v a l u e o f qtE is c o n s t a n t f o r all v a l u e s o f t A , e q n s . (6) a n d (7) c a n b e r e w r i t t e n as
44 (rrez)o = ( r ) 0 / ( r z ) o = */~t/*~0 = [ 2 + Cote=tAP(2,1)]/(2
(15)
Co [Agl]lrnoldrn-~ =0002 o-~5i,._, ~ q a l . ] / m o i d n n - I =0002
(16)
0.¢3L F o-t.l / --
=
+ ColZ=tA)
and
Cok~tA
= 2(*/~t - - * P 0 ) / [ P ( 2 , 1 ) * P o - - * P t ]
T h e p r o p o s e d m e t h o d c a n b e v e r i f i e d in t w o ways: First, the (~)o/(rl)0 versus tA functions calculated by eqn. (15) using known v a l u e s o f Co, k= a n d p ( 2 , 1 ) c a n b e c o r r e l a t e d with the functions calculated from the F values m e a s u r e d b y t h e p r o c e d u r e described. S e c o n d l y , t h e /z= v a l u e s c a l c u l a t e d b y e q n . (16} u s i n g *ft v a l u e s o b t a i n e d f r o m e x p e r i m e n t a l d a t a c a n b e c o r r e l a t e d w i t h t h e k~ v a l u e s m e a s u r e d in t h e s a m e s y s t e m s u s i n g o t h e r m e t h o d s . I n b o t h c a s e s * f t a n d *fo ave calculated f r o m the F values o b t a i n e d experimentally using a graphical form of Wagner's solution of the differential equation for d i f f u s i o n ( P a r t 1). F i g u r e s 3 a n d 4 s h o w t h e fractions of exchange F attained during a c o n s t a n t e x c h a n g e t i m e tE ( t z = 2 0 rain f o r 0 . 0 1 m o l a r A g I sol a n d t E = 3 0 r a i n f o r 0 . 0 0 2 molar A g I sol); the values are p l o t t e d against t h e age o f sols ( t a ) b e f o r e t h e b e g i n n i n g o f the coagulation process. Figures 5 and 6 show t h e c o r r e s p o n d i n g v a l u e s o f *fit- I n b o t h cases, t h e v a l u e o f *B0 w a s o b t a i n e d b y t h e e x t r a p o l a t i o n o f *fit v e r s u s t A c u r v e s a t t A = O. I n a c c o r d a n c e w i t h e q n . ( 1 5 ) , t h e r a t i o s *~t/*I~o o u g h t t o b e e q u a l t o t h e (r)o/(rz)o ratios if the theoretical approach of the proposed method is v a l i d ( P a r t 1). I n Figs. 7 a n d 8 t h e r a t i o s (F)o/(rz)0 d e t e r m i n e d u s i n g t h e p r o p o s e d m e t h o d in 0 . 0 1 a n d 0 . 0 0 2 m o l a r A g I sols a r e r e p r e s e n t e d as f u n c t i o n s o f t A ( d o t t e d c u r v e s ) . T h e e x p e r i m e n t a l d a t a are c o m p a r e d w i t h (r)o/(ri) o versus t A functions calculated by e q n . ( 1 5 ) u s i n g t h e v a l u e s o f (rz) 0 a n d p ( 2 , 1 )
tE/mm=20
0 37~-
0-35t" .
L
200
tAImln
t,~o
300
SO0
F i g . 4 . E x c h a n g e f r a c t i o n F , a t t a i n e d for exchange time t E = 30 minutes in 0.002 molar silver i o d i d e hyd:osol, plotted against t h e a g e o f sol (= tA) before
coagulation. Ii II L t
~l)t F /"---'~JI"J0 = 9 4S
.t
! 0
I 20
I~1
tE/rnln =:0 = I 30
I 20 ~0 tAIrnLrl
I 60
l 70
20
sol
i00
Fig. 5. Values of *fit corresponding to the F values measured by the experimental procedure described, in 0.01 molar silver iodide hydro~ol, plotted against t A
f
*13 t L ~ If
O
/
AfJo
-11-11
002 [Na|]lmol din-3 - 0002 tEImm=20
11 o 50 0-/.8
Co = [A~l~lrnol dm -3 = O O1 [Ncgl]lmol dm-3 = 0 01
' ~
0
0-46 0 .4
±
,"
~o
20
2o
~'o
6'o
tAImbl
,"
."
,'o
,~o
F i g . 3 . Exchange fraction F, attained for exchange t i m e tE = 2 0 m i n u t e s ill 0 . 0 1 m o l a r silver i o d i d e hydrosol, plotted against the age of sol (= tA ) before coagulation.
lO0
200 tA/mln 3
400
500
F i g . 6. V a l u e s o f *J~t corresponding to the -~ values measured by the experimental procedure described, i n 0 . 0 0 2 m o l a r silver i o d i d e h y d r o s o l , p l o t t e d ag~;n~t tA.
determined by electron microscopy [9] and w i t h t h e k= v a l u e s d e t e i x n i n e d b y s e d i m e n t a t i o n analysis (solid curves) and by the m e t a phase analysis method (dashed curves) [8 1 0 ] . T h e r a d i u s (rz) o d e t e r m i n e d b y e l e c t r o n
45 I
co: [~g~/~o, ~-'
d i f f e r e n c e s i n k,= v a l u e s d e t e r m i n e d b y d i f ferent methods and to a possible error in the determination of p(2,1) values by electron m i c r o s c o p y . P u t t i n g ( r l ) o = r~ef = 8 . 7 n m ( f o r 0 . 0 1 m o l a r A g I s o l ) a n d ( r z ) o = r~e~ = 8.4 nm (for 0.002 molar AgI sol), the change of the absolute particle size (F)0 can be calculated by the relation
o o 0,
r N o ~ / r n a l d m -3 = 0 01 I
( r ) o = ~ e l r r e f = ?-rel(rl )0 lu-O ,
10 . .
20 . .
30
t.0
50 6, , iA/~rlln
70,
80 ,
;0
:
Fig. 7. Change o f relative particle size (~el)O determ i n e d b y t h e p r o p o s e d m e t h o d ( d o t t e d curve) in 0.01 m o l a r silver i o d i d e h y d r o s o l , c o m p a r e d w i t h t h e change o f ( ~ e l ) 0 c a l c u l a t e d b y eqn. (15) using t h e value p ( 2 , 1 ) d e t e r m i n e d b y e l e c t r o n m i c r o s c o p y :~Ld t h e value o f kvr d e t e r m i n e d b y sedimentatio,-, analysis (solid curve) a n d b y m e t a p h a s e analy~_Z~ m e t h o d (dashed curve). m i c r o s c o p y is 8 . 7 a n d 8 . 4 n m f o r 0 . 0 1 a n d 0.002 molar AgI sol, respectively. The size r a t i o p ( 2 , 1 ) = ( r z ) o / ( r l ) o is 1 . 3 f o r 0 . 0 1 m o l a r AgI sol and 1.34 for 0.002 molar sol. The experimental data are in good agreement with the results calculated by eqn. (15) using the d a t a (/z=, ( r l ) o a n d p ( 2 , 1 ) ) d e t e r m i n e d b y other experimental methods. The small differences between the experimentally obtained (F)o/(rz)0 values and the (F)0/(rl) 0 values calculated by eqn. (15) are due to i
CO : [Ag~/mo!
dm-3 = 0 00Z
[
rNa~/mo[
d m -3 = 0 0 0 2
(17)
Figures 9 and 10 show the change of absolute particle size in 0.01 and 0.002 molar AgI sols as a function of ageing time. The values marked with crosses correspond to the values determined by electron microscopy [9]. The second way of verifying the proposed m e t h o d is a s f o l l o w s : T h e p ( 2 , 1 ) a n d k = v a l u e s calculated by eqn. (16) using the experim e n t a l l y d e t e r m i n e d *Bt v a l u e s ( F i g s . 6 a n d 7 ) axe compared with the p(2,1) and k= values determined by other methods (electron microscopy, sedimentation mualysis, metaphase analysis method, light scattering [8 1 0 l ). U s i n g e q n . ( 1 5 ) , w e c a n c a l c u l a t e t h e value of p(2,1) from the conditions Co/Z=tA(l) = 2(*ft, -- *fO)/[P(2,1)*P0 --*~t, ]
(z8) C0k~tA(2) = 2(*ft: -- *f0)/[P(2,1)*fo
--*ft: ] (19)
w h e r e *fit, a n d *~t= c o r r e s p o n d t o t h e *fit values determined for two different ageing t i m e s , n a m e l y f o r tA = t A ( 1 ) a n d t A = t A ( 2 ).
°l c= .~-
E ~o9
/ / "
Co = [ A ~ l l ] = 0 01 t o o l cirn ~2 [Nail = 0 O, rno, dm-~
..
to !
I
f
I t,nn
tA/mln
Fig. 8. Change o f relative particle size (~el)O d e t e r m i n e d b y t h e p r o p o s e d m e t h o d ( d o t t e d curve) in 0.002 m o l a r silver i o d i d e h y d r o s o l , c o m p a r e d w i t h t h e change o f (~el)O c a l c u l a t e d b y e q n . (15) using the value p ( 2 , 1 ) d e t e r m i n e d b y e l e c t r o n m i c r o s c o p y a n d the value o f k~ d e t e r m ; n e d b y sedlmaen~;ation nn:alysis (solid curve) a n d b y m e t a p h a s e analysis m e t h o d ( d a s h e d curve).
I tAImin
Fig. 9. Change o f a b s o l u t e particle size (~)o calculated b y eqn. (16) using t h e values o f (~el)O d e t e r m i n e d b y t h e p r o p o s e d m e t h o d in 0.01 m o l a r silver i o d i d e h y d r o s o l a n d t h e value o f (r 1 )o d e t e r m i n e d b y e l e c t r o n m i c r o s c o p y in t h e s a m e s y s t e m . T h e cro~-es d e n o t e t h e values o f (r--)0 d e t e r m i n e d b y e l e c t r o n microscopy.
46
"F Co= ~Ag~]/moldm-3 =001
~F
[~lat"Jlmo[ drn-3 = 00l plZ.'z) =128 9L
J ¢ "
_/o
0
200
100
tAImln
300
400
50(3
Fig. 10. Change of absolute partlcIe size (r-)0 caIeulated by eqn. (16) using the values o f (Frel) 0 determined by the proposed method in 0.002 molar silver iodide hydrosol and the vaIue o f (r 1 )o determined by electron microscopy in the same system. The crones denote the values o f CrY0 determined by electron microscopy.
D i v i d i n g e q n . ( 1 8 ) b y e q n . ( 1 9 ) r e s u l t s in a linear e q u a t i o n : ap(2,l) + b
=
0
(20)
where a
=
tA(Z)*~0*t3t2
tA(1)*~ 2
--
--
tA(2)*~O*~3t,
+
tA(2)*f~ b ~- t A ( 2 ) * ~ t
/,~
C O = [A91] =0-002 mot din-3 [No.l] : 0 002 me! din-3
x
*~t: --
tA(1)*/~t:*.B¢~
tA(2)*fo*fit=
--
+ t~(1)*/~o*ft,
U s i n g e q n . ( 2 0 ) , w e c a n easily c a l c u l a t e p ( 2 , 1 ) using o n l y the n u m e r i c a l values o f the c h o s e n ageing times tA(1 ) a n d tA(2 ) and t h e c o r r e s p o n d i n g v a l u e s o f *ft, a n d */3t~- F o r practical purposes we take an average value of p ( 2 , 1 ) c a l c u l a t e d u s i n g several c o m b i n a t i o n s of different values of tA(1) and tA(2) and c o r r e s p o n d i n g values o f *~t, a n d *~t=- F o r Co = 0 . 0 1 t o o l d m - a , p ( 2 , 1 ) --- 1 . 2 8 a n d f o r C o = 0 . 0 0 2 t o o l d m - 3 , p ( 2 , 1 ) ---- 1 . 2 7 , w h i c h is in excellent a g r e e m e n t with t h e p ( 2 , 1 ) values obtained by electron microscopy (p(2,1)1.30 and 1.34, respectively). If the theoretical considerations which underlie the proposed m e t h o d a r e valid, t h e v a l u e s o n t h e r i g h t - h a n d side o f e q n . ( 1 6 ) m u s t b e l i n e a r f u n c t i o n s o f
tA f o r a c o r r e s p o n d i n g value o f p ( 2 , 1 ) . Figures 11 a n d 12 s h o w (*~t - - */30)/[P(2,1)*/]o - analysed depend linearly t h a t t h e c o n s t a n t k~ c a n
t h a t t h e values o f * f t ] f o r t h e sols on ageing time, and be computed by
°i
O
I0
20
30
70
80
90
10~
tA/mTrt
F i g . 11, Values o f (*J3 t - - * f o ) i [ P ( 2 , 1 ) * f l o 0.01 molar silver iodide hyd~osol plotted ageing time t A.
- - *•t] of against the
d i v i d i n g t h e s l o p e o f t h e s t r a i g h t line b y Co/~ T a b l e 1 gives a c o m p a r i s o n o f t h e k~ v a l u e s f o r 0 . 0 1 a n d 0 . 0 0 2 m o l a r A g I sols o b t a i n e d b y t h e p r o p o s e d m e t h o d , a n d t h e h~ v a l u e s f o r t h e s a m e sols o b t a i n e d b y o t h e r m e t h o d s [8 - 10]. The validity of eqn. (16) for the *Bit v a l u e s c a l c u l a t e d f r o m t h e e x p e r i m e n t a l l 5 d e t e r m i n e d F values, a n d the g o o d a g r e e m e n b e t w e e n t h e k~ v a l u e s o b t a i n e d b y t h e p r o posed method and those obtained by other m e t h o d s ( T a b l e 1) s h o w t h a t t h e p r o p o s e d m e t h o d is s u i t a b l e f o r t h e c h a r a c t e r i z a t i o n oJ p a r t i c u l a t e p r o c e s s e s in s y s t e m s o f c o l l o i d a l hydrosols.
o
~ 2,
[ Agl]lm°l din-3 ~ 0 002 ~
p=
olf ~-
1
o 2
,
1
1
kit=6 9 too|-I dl-/i] mlrl-t
I
I.
i
1
.L.
....
Fig. 12. V.~ues o f (*ft - - *fo)l[P(2,1)*f0 - - * & ] o f 0.002 mola~r silver i o d i d e h y d c _ o s o l p l o t t e d a g a i n s t t h ageing time t A.
47 TABLE 1 C o m p a r i s o n o f kTr v a l u e s o b t a i n e d b y t h e p r o p o s e d m e t h o d w i t h kzr v a l u e s o b t a i n e d b y o t h e r methods for 0.01 and 0.002 molar AgI sols Co (tool dm -3 )
MethGd Proposed method
Metaphase analysis method
0.01
k n = 14.8
k n
0.002
k n =
6.9
= 8.8
k~ = 5 . 2
REFERENCES I K. E. Z i m e n s , Arklv K e m i Mineral. Geol., 2 0 A ( 1 9 4 5 ) 1. 2 E . S. R a j a g o p a l , K o l l o i d Z . , 1 6 7 ( 1 9 6 0 ) 1 7 . 3 R. Despotovi~ and V. Stengl, K o l l o i d Z. Z. Polym., 250 (1972) 950. 4 R . D e s p o t o v i ~ a n d B. S u h o t i ~ , C r o a t . C h e m . A e t a , 43 ( 1 9 7 1 ) 153. 5 R . D e s p o t o v i ~ a n d B. S u b o t i ~ , J . I n o r g . N u c l . C h e m . , 38 ( 1 9 7 6 ) 1 3 1 7 .
Sedimentation analysis method
Light scattering
k n --- 1 4 . 0
k.~ = 1 2 . 9
k~ = 6.0
kn = 5.0
6 R . H . H . W o l f , M. M i r n i k a n d B. T e z a k , K o l l o i d Z. Z. P o l y m . , 2 0 5 ( 1 9 6 5 ) 1 1 1 . 7 A. D y e r , G. G. Hayes, G. O. Phillips a n d R . P. T o w n s e n d , Adv. C h e m . Ser.; 121 ( 1 9 7 2 ) 299. 8 B. S u b o t i ~ , P r o e . I n t . C o n f . , C o l l o i d a n d I n t e r f a c ( Science, Vol. I (Ed. E. Wolfram), Akademiai K i a d o , B u d a p e s t , 1 9 7 5 , p. 2 5 7 . 9 B. S u b o t i ~ , P h . D . T h e s i s , U n i v . o f Z a g r e b , 1 9 7 6 . 1 0 R . D e s p o t o v i ~ e t a l . , C r o a t . C h e m . A c t a , 51 (1978) 113.