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The zeta-potential of cement
CEMENT and CONCRETE RESEARCH. Vol. 17, pp. 573-580, 1987. Printed in the USA. 0008-8846/87 $3.00+00. Copyright (c) 1987 Pergamon Journals, Ltd.
THE Part
III:
Dept.
ZETA-POTENTIAL
The N o n - E q u i l i b r i u m
OF C E M E N T Double
Layer
E. N ~ g e l e of C i v i l E n g i n e e r i n g , U n i v e r s i t y M~nchebergstr. 7, D-3500 Kassel,
on C e m e n t
of Kassel FRG
(Communicated by F.H. Wittmann) (Received March 5, 1987)
ABSTRACT The theory of the n o n - e q u i l i b r i u m d o u b l e layer is a p p l i e d to the i n t e r f a c e b e t w e e n an oxidic m a t e r i a l and an a q u e o u s e l e c t r o l y t e . It is shown, that the theory also applies, w h e n slow c h e m i c a l r e a c t i o n s o c c u r at the solidliquid interface. The t h e o r y is a p p l i e d to c e m e n t and the r e s u l t s are c o m p a r e d to e q u i l i b r i u m models.
I.
Introduction
C e m e n t h y d r a t i o n is a c h e m i c a l r e a c t i o n i n v o l v i n g ions at n e a r l y e v e r y stage /1, 2/. The r e a c t i o n s occur in a t w o - p h a s e S y s t e m i n v o l v i n g both, the solid and the liquid phase. If such a s y s t e m c o n t a i n s ions, an e l e c t r i c a l d o u b l e layer is formed at the s u r f a c e of the solid /3, 4/, w h i c h a f f e c t s the p r o p e r t i e s of the s y s t e m s i g n i f i c a n t l y . For example, b l e e d i n g and f l o c c u l a t i o n of the c e m e n t p a r t i c l e s s h o u l d o c c u r w i t h c e m e n t p a s t e s of normal c o m p o s i t i o n , if the d o u b l e layer w e r e absent. In p a r t I of this w o r k /5/ the s i g n i f i c a n c e of the z e t a - p o t e n t i a l of c e m e n t has b e e n d i s c u s s e d and in p a r t II /6/ m e t h o d s h a v e b e e n d e s c r i b e d to o b t a i n z e t a - p o t e n t i a l data from m i c r o e l e c t r o p h o r e s i s experiments. The h y d r a t i n g cement, however, is in a n o n - e q u i l i b r i u m state, h e n c e n o n - e q u i l i b r i u m d o u b l e layer t h e o r y m u s t be applied. It is thus the p u r p o s e of this p a p e r to p r e s e n t a s u r v e y on the s t r u c t u r e and the p r o p e r t i e s of the e l e c t r i c a l d o u b l e layer on cement. S t a r t i n g w i t h a d i s c u s s i o n of the i n t e r f a c e b e t w e e n nonr e a c t i n g o x i d i c m a t e r i a l s and a q u e o u s e l e c t r o l y t e s , the theory of the n o n - e q u i l i b r i u m d o u b l e layer is a p p l i e d to this s y s t e m in the following. S u b s e q u e n t l y it is e x t e n d e d to r e a c t i n g m a t e r i a l s like h y d r a t i n g c e m e n t and the p r o p e r t i e s of the c o r r e s p o n d i n g z e t a - p o t e n t i a l are derived.
573
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Vol. 17, No. 4 E. N~gele
2.
The D o u b l e
Layer
on O x i d i c
Materials
On o x i d i c m a t e r i a l s the e s t a b l i s h m e n t of s u r f a c e occur by two d i s t i n c t b u t e q u i v a l e n t m e c h a n i s m s : i)
the a d s o r p t i o n site
of p r o t o n s
H+ MOH2 +
÷ +
or h y d r o x y l - i o n s
charge
may
on to an a m p h o t e r i c
OH-
MOH
~
MO
+ H~O
or
ii) the f o r m a t i o n in s o l u t i o n of h y d r o x y l a t e d species M(OH) ~-x)+ w h i c h d e p o s i t on the s u r f a c e or a l t e r n a t i v e l y the f o r m a t i o n X o f ' these species d i r e c t l y on the s u r f a c e /7/. The c h a r g e carriers are i m m o b i l e and the s u r f a c e charge is l o c a l i z e d on s u r f a c e atoms, ions or m o l e c u l e s or on s t r u c t u r a l defects. Thus, local field s t r e n g t h s can be very high, w h i c h results in a s t r u c t u r i n g process in the inner d o u b l e layer and an o r i e n t a t i o n of the w a t e r d i p o l e s in the layers a d j a c e n t to the surface. As a c o n s e q u e n c e , o x i d i c m a t e r i a l s h a v e high s u r f a c e charges up to 0.7 A s / m 2 (70 ~ C / c m 2) but only low z e t a - p o t e n t i a l s . T h e r e is only a m o d e r a t e a g r e e m e n t b e t w e e n e x p e r i m e n t and theory if c o n v e n t i o n a l G r a h a m e - m o d e l s are used, w h i c h treat the inner d o u b l e layer as some layers of a d s o r b e d species on to a s u r f a c e w i t h the same p o t e n t i a l everywhere. Bockris, D e v a n a t h a n and M u l l e r /8/ s u c c e e d e d in a c c o u n t i n g for the d i p o l e o r i e n t a t i o n and o b t a i n e d a good a g r e e m e n t w i t h e x p e r m e n t a l data for their model. A b e t t e r a p p r o a c h was s u g g e s t e d by L y k l e m a /9/, w h o a s s u m e d that the p o t e n t i a l d e t e r m i n i n g ions could p e n e t r a t e the s u r f a c e layers of the oxide to some e x t e n t and react there w i t h the a m p h o t e r i c sites. In this way large a m o u n t s of s u r f a c e c h a r g e could d e v e l o p and, if c o u n t e r - i o n s w e r e p e r m i t t e d to enter this porous layer too, the net e l e c t r i c a l p o t e n t i a l at the outer edge of this porous layer w o u l d be r e d u c e d c o n s i d e r a b l y in m a g n i t u d e . Thus high s u r f a c e charge could be r e c o n c i l e d w i t h m o d e s t values of the z e t a - p o t e n t i a l . This model was t r e a t e d in its m o s t g e n e r a l form by P e r r a m et a!./10/. The z e t a - p o t e n t i a l in this model is i d e n t i c a l w i t h the p o t e n t i a l at the j u n c t i o n b e t w e e n the gel and the electrolyte. The c o n c e n t r a t i o n s of ions in the g e l - l a y e r , ci, are r e l a t e d to the bulk c o n c e n t r a t i o n s ci° by e x p r e s s i o n s of the form o c i = c i exp
(- (F~ + ~i)/RT)
w h e r e ~i(r) is the s p e c i f i c a d s o r p t i o n Eq. (I) may be r e a r r a n g e d to give c i = ci°ex p(-F~/RT)
exp(-~i/RT)
(I) energy
of species
= ci °A.i (r) exp (-F~/RT)
i /7,9/.
(la)
w h e r e A i(r) = exp (-~i(r)/RT)- If this is c o m p a r e d to a G r a h a m e model, the charge in the gel layer is a p p r o x i m a t e l y o =
6"z-F-ci°exp
(-zF~/RT)
(2)
Vol. 17, No. 4
575 ZETA-POTENTIAL, NON-EQUILIBRIUM, DOUBLE-LAYER
where
6 =
2r Zi Ai(r)
(3)
has the d i m e n s i o n of a length and is i n t e r p r e t e d as the "thickness" of the inner d o u b l e layer, which, in this case, is a s s u m e d i n s i d e the gel. In c o n t r a s t to the oxidic m a t e r i a l s d i s c u s s e d thus far, c e m e n t reacts w i t h the s u r r o u n d i n g w a t e r and the equil i b r i u m models d e s c r i b e d a b o v e m u s t be replaced by a n o n - e q u i l i b r i u m d o u b l e layer. In the next chapter, n o n - e q u i l i b r i u m double layer theory w i l l be d e v e l o p p e d for n o n - r e a c t i n g materials. S u b s e q u e n t l y it will be e x t e n d e d to r e a c t i v e systems.
3. T h e o r y of the Non-Equi.librium D o u b l e L a y e r The theory of the n o n - e q u i l i b r i u m double layer p r e s e n t e d in the f o l l o w i n g is a special case of a g e n e r a l theory p r o p o s e d by D u k h i n and D e r j a g u i n /11/. ~ Under n o n - e q u i l i b r i u m c o n d i t i o n s ionic flows in the double layer do not vanish. Thus, the B o l t z m a n n d i s t r i b u t i o n , c o r r e l a t i n g ionic c o n c e n t r a t i o n s in the d o u b l e layer to those in the bulk electrolyte, no longer holds. But if we r e s t r i c t ourselves to s t a t i o n a r y conditions, the P o i s s o n e q u a t i o n w r i t t e n in the form A~ = F - I
g -I o
(z+c + - z-c-)
(4)
+ is still valid. In eq. (4) z and ~ are the v a l e n c i e s of the cations and anions respectively, c and c- are their c o n c e n t r a tions in the double layer. Although transferof ions occurs t h r o u g h o u t the d o u b l e layer, the law of this transfer under s t a t i o n a r y cond i t i o n s is such, that the ions are t r a n s f e r r e d through any elem e n t of the d o u b l e layer v o l u m e w i t h o u t a c c u m u l a t i n g in it /11/. The charge of the p a r t i c l e surface is thus u n a l t e r e d in time, that is, the ionic flows Ji for anions and cations at any surface e l e m e n t should be equal to zero. Ji ,
x
=
o
= O
(5)
0
(6)
For the same reasons div
J
=
everywhere. In eq. (6) J is the v e c t o r of the ionic flow. This e q u a t l o n together w i t h b o u n d a r y c o n d i t i o n s eq. (5) may now be used i n s t e a d of the B o l t z m a n n - d i s t r i b u t i o n to solve eq. (4). A c c o r d i n g to /11/ the p o t e n t i a l - d i s t r i b u t i o n in the n o n - e q u i l i b r i u m d o u b l e layer is for s y m m e t r i c a l e l e c t r o l y t e s (x)
=
~n(tanh
(c(x)
c o
(Z~o/4)/tanh
-I
) z -I sinh
(z~oeq/4))-
(Z~o/2) I/cosh
(z~oeq/2)]
(7)
576
Vol. 17, No. 4 E. Nggele
In eq. (7) z is the valency of the ions, c o is their bulk concentration, ~o is the dimensionless potential at the surface and ~ eq is the G o u y - C h a p m a n - e q u i l i b r i u m pQtentia! at the surface e~pressed dimensionless as F~/RT like ~o and ~(x). The zeta-potential of the n o n - e q u i l i b r i u m system may be calculated from the electrophoretic m o b i l i t y u by using /11/ J(4z) -I = 1,5 ~ -
z2~ 2 (2kR) -I - 3 eeo(RT)2(16~qDF2) -I (kR)-1~ 3 (8)
where ~ = u-6~qF(Ee ERT) - I u n d ~ = F~/RT are the dimensionless mobility and the dimensionless zeta-potential respectively, q is the viscosity of the electrolyte and D the diffusion coefficient of the ions. R is the particle radius and I/k is the Gouy-Chapmanthickness of the diffuse double layer. From eq. (8) it follows, that the non-equilibrium zeta-potential is always greater than the equilibrium value. Hence, anything disturbing the equilibrium increases the zeta-potential. According to /11/ a particle with a thin double layer under non-equilibrium conditions may be characterized as follows: Tangential ionic flows arise under stationary conditions because of ionic flows from the bulk to the surface of the particle and in the opposite direction. The conditions of continuity of the system of bulk and surface flows for both kinds of ions can be satisfied only of diffusion flows arise beyond the double layer and, particularly, along its outer boundary. Since locally, that is, between the given double layer area and the adjoining volume of electrolyte, equilibrium should be preserved, the change in concentration along the outer boundary of the double layer leads to a change in its thickness and, consequently, to a change in the zeta-potential. On increasing the electrolyte concentration the diffuse part of the double layer contracts. This, together with a constant surface charge leads to a decrease in the zetapotential. In this respect, the n o n - e q u i l i b r i u m system behaves like the system in equilibrium.
4. The Double Layer on Cement Cement reacts with water to a considerable extent. However, the reaction is slow enough to allow for the development of a t i m e - d e p e n d e n t zeta-potential /5, 6/. Thus, except for the very first minutes, we can assume stationary conditions at the cement surface, where the reaction takes place. The reaction itself generates no excess charge, thus Z.c.rz. ll
=
0
(9)
1
where the superscript "r" indicates, that only ions generated by the reaction are considered. Eq. (5) also holds and thus the surface charge is unaltered in time. The theory developped in the previous chapter may then be applied to cement, if the first minutes and the very late stages of the hydration are excluded. In the first minutes the ions accumulate in the double layer and
Vol. 17, No. 4
577 ZETA-POTENTIAL, NON-EQUILIBRIUM, DOUBLE-LAYER
a stationary state does not exist. At the later stages, the reaction rate is so slow, that equilibrium models can be applied, w i t h o u t introducing serious errors. During the induction period of the cement h y d r a t i o n the ions generated behind the double layer p e n e t r a t e through it, without accumulating there. The ionic flows inside the double layer result in structural changes of the double layer and hence the zeta-potential of cement varies with time, even if no ions accumulate in the double layer. The flows act in both directions, to an off the surface. As a consequence, the bulk electrolyte exerts the same effects on the zetapotential of cement as would be expected for a conventional GouyChapman theory. In particular, the reaction results in an increasing ionic strength w i t h time in the bulk. Therefore, the diffuse part of the double layer contracts and the absolute values of the zeta-potential of hydrating cement decrease slowly to zero irrespective of sign. Zeta-potentials of cement measured at d i f f e r e n t times m u s t therefore be corrected for this effect before they can be compared to each other. The reaction at the surface prevents any adsorption. Thus, any effects due to specific adsorption vanish for cement. Furthermore, no d i s t i n c t i o n must be made between specific adsorbed ions and p o t e n t i a l - d e t e r m i n i n g ions, because the first do not exist. On the other hand, every kind of ions taking part in the reaction or passing the double layer, either from or to the surface, affects the potential distribution and must be considered as a p o t e n t i a l - d e t e r m i n i n g ion. Consequently, the number of p o t e n t i a l - d e t e r m i n i n g ions is extremely high for cement. For example, rK+ > rNa+ and thus DK+ < DNa+. Replacing Na + by K + should thus result in a significant change in the zeta-potential of cement. The zeta-potential of cement is comparatively small, rarely exceeding 20mV in m a g n i t u d e /5, 6/. Under this condition ne and eq. lution: ~(x)
(7) simplifies 4RT (zF) -I artanh
=
eq
to the w e l l - k n o w n
(tanh(ZF~oeq/4RT)
exact Gouy-Chapman
exp(-kx))
so-
(10)
This may again be compared to the G r a h a m e - s o l u t i o n to obtain eq.'s (2) and (3) in full generality, that is, w i t h o u t the assumption of a porous gel layer on the surface. Using eq. (3) the thickness of the inner double layer may then be estimated from electrokinetic measurements. Such a possibility has been predicted also in /11/. Eq. (3) can be obtained from a conventional Grahame-model by replacing "2r", the Grahame-thickness of the inner Helmholtz-plane, by 6, the time-dependent "thickness" of the inner layer /12/. Thus we call this approximation the "modified Grahame-model".
5.
Application
of the Theory
The theory outlined in the previous chapter will now be applied to hydrating cement. Because m i c r o e l e c t r o p h o r e s i s is the
578
Vol. 17, No. 4 E. N~gele
b e s t m e t h o d to d e t e r m i n e the e l e c t r o p h o r e t i c m o b i l i t y u and the z e t a - p o t e n t i a l of h y d r a t i n g cement, the f o l l o w i n g e v a l u a t i o n is based on data reported in /5, 6/. V i s c o s i t i e s and d i f f u s i o n c o e f f i c i e n t s of ions w e r e taken from h a n d b o o k s like /13/. The plot of the z e t a - p o t e n t i a l of c e m e n t versus time may be considered to c o n s i s t of three parts: The first p a r t concerns the very first m i n u t e s of the hydration, w h e r e a z e t a - p o t e n t i a l does not exist, b e c a u s e the r e a c t i o n is too vigorous. At the end of this p e r i o d the double layer is established, w h e n the r e a c t i o n has slowed down sufficiently. Then a p e r i o d follows, w h e r e ions a c c u m u l a t e in the d o u b l e layer, b e c a u s e the p r o d u c t i o n rate of ions at the surface is h i g h e r than the p e r m e a t i o n rate through the d o u b l e layer. In this period, beg i n n i n g about two m i n u t e s after the c o n t a c t b e t w e e n cement and water the z e t a - p o t e n t i a l d e v e l o p s as the d o u b l e layer is stabilized. Due to the a c c u m u l a t i o n of ions, d~/dt is c o m p a r a t i v e l y high in this period, and c o n s e q u e n t l y the z e t a - p o t e n t i a l varies m u c h w i t h time. Thus, no z e t a - p o t e n t i a l has been reported in /5, 6/ b e f o r e 5 m i n u t e s of hydration, a l t h o u g h it is p o s s i b l e to o b t a i n data at earlier times. In the next period s t a t i o n a r y conditions are e s t a b l i s h e d at the s u r f a c e and the n o n - e q u i l i b r i u m double layer theory applies. In Fig.1 the e l e c t r o p h o r e t i c mobility u, the e q u i l i b r i u m z e t a - p o t e n t i a l a c c o r d i n g to S m o l u c h o w s k i /7/ ceq, the n o n - e q u i l i b r i u m z e t a - p o t e n t i a l from eq. (8) ~ne, and the true z e t a - p o t e n t i a l ~t w h i c h includes a c o r r e c t i o n for the i n c r e a s i n g ionic s t r e n g t h in the bulk liquid, are shown for 10 mg/l p o r t l a n d cement in d i s t i l l e d w a t e r /6/. As the a b s o l u t e values of the zeta-potential, except for the first value, are always s m a l l e r than 20 mV, ~ ne = ~eq as predicted by eq. (8). It should be m e n t i o n e d here, that eq. (8) is not very s e n s i t i v e to the v a l u e of k, as long as kR >> I. For cement roughly 250 ~ kR ~ 1 5 0 0 . For ~t (t=0) has been chosen as the reference state in Fig. I, b e c a u s e ~ (0) does not exist and a theoretical v a l u e cannot be estimated. From Fig. I it may be concluded, that the v a r i a t i o n s of the z e t a - p o t e n t i a l w i t h time for h y d r a t i o n times above 180 m i n u t e s are e n t i r e l y due to the inc r e a s i n g ionic s t r e n g t h in the liquid phase, w h e r e a s the variau o o
,ovl
o
0
c,
~ne
Fig.
E l e c t r o p h o r e t i c mobility u, e q u i l i b r i u m z e t a - p o t e n t i a l ~eq, n o n - e q u i l i v r i u m zetap o t e n t i a l ~ n e and true z e t a - p o t e n t i a l ~t for 10mg/l p o r t l a n d cement in d i s t i l l e d water
-lol \ ......
I- 10/~
.........
. ~
-
-o
.
. . . . . . . . . . .
.
a
-20
0
bO
120
180
2/,0
300
360
/,20
~,80
I:
%0
t Irninl
Vol. 17, No. 4
579 ZETA-POTENTIAL,
NON-EQUILIBRIUM, DOUBLE-LAYER
tions for s h o r t e r h y d r a t i o n times are due to the c h e m i c a l reactions. Thus, the z e t a - p o t e n t i a l can be used to trace the h y d r a tion reaction, if it is c o r r e c t e d for the i n c r e a s i n g i o n i c strength. The t r a n s i t i o n b e t w e e n v a r i a t i o n s in the z e t a - p o t e n t i a l g o v e r n e d by the c h e m i c a l r e a c t i o n and those due to the i n c r e a s i n g ion c o n c e n t r a t i o n s d e p e n d s on the c e m e n t c o n c e n t r a t i o n in the s u s p e n s i o n s t u d i e d /6/. If the z e t a - p o t e n t i a l is m e a s u r e d in sol u t i o n s w h e r e the p H - v a l u e has b e e n a d j u s t e d in some way, the i n c r e a s e d ionic s t r e n g t h due to the a d d e d m a t e r i a l s , for e x a m p l e a 0,01 m N a O H for pH=12, c o m p r e s s e s the d i f f u s e d o u b l e layer too. As a result, the v a r i a t i o n s of the z e t a - p o t e n t i a l w i t h time dec r e a s e s i g n i f i c a n t l y c o m p a r e d to the values o b t a i n e d in d i s t i l l e d or d e i o n i z e d water. In Fig. 2 £eq, ~ne and 6t are s h o w n for Ig/l p o r t l a n d c e m e n t at pH= 10 a d j u s t e d w i t h N a O H and KOH respectively. ~eq and ~ne are always i d e n t i c a l u n d e r these c o n d i t i o n s , b e c a u s e the z e t a - p o t e n t i a l s are v e r y low. ct does not d i f f e r m u c h from ~eq, b e c a u s e the ionic s t r e n g t h is n e a r l y c o n s t a n t w i t h the time.
6.
Conclusions
Due to the n o n - e q u i l i b r i u m s t a t e of the c e m e n t / w a t e r - s y s t e m , c e m e n t has a t i m e - d e p e n d e n t z e t a - p o t e n t i a l . It is low in m a g n i tude, r a r e l y e x c e e d i n g 20 mV. Such low values are c o m m o n for all o x i d i c m a t e r i a l s . The t h e o r y of the n o n - e q u i l i b r i u m d o u b l e layer p r o v i d e s f o r m u l a e for the c a l c u l a t i o n of both, the e l e c t r i c a l p o t e n t i a l as a f u n c t i o n of the d i s t a n c e from the p a r t i c l e surface and the z e t a - p o t e n t i a l u n d e r n o n - e q u i l i b r i u m c o n d i t i o n s . S i n c e the z e t a - ~ o t e n t i a l of c e m e n t is small, in g e n e r a l the rel a t i o n %ne = ~e~ m a y be a p p l i e d and the g e n e r a l solution, eq. (7) c o n v e r g e s t o w a r d s the w e l l k n o w n e x a c t G o u y - C h a p m a n s o l u t i o n eq. (10). If the r e s u l t s o b t a i n e d from the n o n - e q u i l i b r i u m theory are c o m p a r e d to the G r a h a m e - t h e o r y an e x p r e s s i o n for the a p p r o x i m a t e " t h i c k n e s s " of the i n n e r d o u b l e layer is obtained. At the r e a c t i n g s u r f a c e no a d s o r p t i o n is p o s s i b l e . Hence, no s p e c i f i c a d s o r b e d ions e x i s t and the t h e o r y is s i m p l i f i e d c o n s i d e r a b l y . On the o t h e r hand, the n o n - v a n i s h i n g ionic flows in the d o u b l e layer r e s u l t in a large n u m b e r of p o t e n t i a l - d e t e r m i n i n g ions for
[mV]
KOH NoOH . • ~eq= ~ne
o
Fig.
2:
o
~t
20
Equilibrium zeta-pot e n t i a l ~eq, n o n - e q u i librium zeta-potential %ne and t r u e z e t a - p o t e n t i a l 6t for p o r t land c e m e n t (Ig/l) in w a t e r at pH=10, adj u s t e d w i t h KOH and NaOH respectively
t[min] 50
-20
100
liE
150
580
Vol. 17, No. 4 E. N~gele
cement. T~ne plot o£ ~ versus time can be subdivided into three parts: In the first part, corresponding to the vigorous initial reaction, no zeta-potential exists. In the second part, beginning about 5 minutes after the contact between cement and water, depending on the cement concentration, the n o n - e q u i l i b r i u m theory applies and the variations of the zeta-potential are due to the chemical reactions. In the third part, they depend solely on the increasing ionic strength in the bulk liquid.
7. References /I/
/2/ /3/
/4/
/5/ /6/ /7/ /8/ /9/ /10/ /11/
/12/ /13/
J. Skalny, J.F. Young: "Mechanisms of Portland Cement Hydration" in: 7th Int. Symp. Chem. Cem., Paris, 1980, ppII/1-3 to II/I-45. F.W.Locher, W. Richartz, S.Sprung: Zement-Kalk-Gips, 29, 1976, pp. 435. S.M.Ahmed: "The Zeta-Potential at the Oxide-Solution Interface" in: Oxides and Oxide Films, J.W.Diggle, Ed. Marcel Dekker, New York, 1972, pp.486. H.Gatos, J.Lagowski: "Space Charge Layers" in: Surface Effects in Crystal Plasticity, R.M.Latanision, J.T.Fourie, Editors, Noordhoff, Leyden, NL, 1977, pp. 221. E.N~gele: Cem. Concr. Res.15, 1985, pp. 453. E.N~gele: Cem. Concr. Res.16, 1986, pp.853. R.J.Hunter: "Zeta-Potenti-al in Colloid Science" Academic Press, New York, 1981. J.O'M. Bockris, M.A.Devanathan, K.Muller, Proc. Roy. Soc. London, 274, 1963, pp. 55. J.Lyklema: J.Electroanal. Chem. I_8, 1968, pp. 341. J.Perram, R.J.Hunter, H.Wright: Austr. J.Chem. 27,1974,pp.461.__ S.S.Dukhin, B . V . D e r j a g u i n : " N o n - E q u i l i b r i u m Double Layer and Electrokinetic Phenomena" in: Surface and Colloid Science, E.Matijevic Editor, 7, 1974, J.Wiley & Sons, New York. D.C.Grahame: Chem. Rev. 41 , 1947, pp. 44. Handbook of C h e m i s t r y an-d Physics, CRC-Press, New York, 1983.
THE Part
III:
Dept.
ZETA-POTENTIAL
The N o n - E q u i l i b r i u m
OF C E M E N T Double
Layer
E. N ~ g e l e of C i v i l E n g i n e e r i n g , U n i v e r s i t y M~nchebergstr. 7, D-3500 Kassel,
on C e m e n t
of Kassel FRG
(Communicated by F.H. Wittmann) (Received March 5, 1987)
ABSTRACT The theory of the n o n - e q u i l i b r i u m d o u b l e layer is a p p l i e d to the i n t e r f a c e b e t w e e n an oxidic m a t e r i a l and an a q u e o u s e l e c t r o l y t e . It is shown, that the theory also applies, w h e n slow c h e m i c a l r e a c t i o n s o c c u r at the solidliquid interface. The t h e o r y is a p p l i e d to c e m e n t and the r e s u l t s are c o m p a r e d to e q u i l i b r i u m models.
I.
Introduction
C e m e n t h y d r a t i o n is a c h e m i c a l r e a c t i o n i n v o l v i n g ions at n e a r l y e v e r y stage /1, 2/. The r e a c t i o n s occur in a t w o - p h a s e S y s t e m i n v o l v i n g both, the solid and the liquid phase. If such a s y s t e m c o n t a i n s ions, an e l e c t r i c a l d o u b l e layer is formed at the s u r f a c e of the solid /3, 4/, w h i c h a f f e c t s the p r o p e r t i e s of the s y s t e m s i g n i f i c a n t l y . For example, b l e e d i n g and f l o c c u l a t i o n of the c e m e n t p a r t i c l e s s h o u l d o c c u r w i t h c e m e n t p a s t e s of normal c o m p o s i t i o n , if the d o u b l e layer w e r e absent. In p a r t I of this w o r k /5/ the s i g n i f i c a n c e of the z e t a - p o t e n t i a l of c e m e n t has b e e n d i s c u s s e d and in p a r t II /6/ m e t h o d s h a v e b e e n d e s c r i b e d to o b t a i n z e t a - p o t e n t i a l data from m i c r o e l e c t r o p h o r e s i s experiments. The h y d r a t i n g cement, however, is in a n o n - e q u i l i b r i u m state, h e n c e n o n - e q u i l i b r i u m d o u b l e layer t h e o r y m u s t be applied. It is thus the p u r p o s e of this p a p e r to p r e s e n t a s u r v e y on the s t r u c t u r e and the p r o p e r t i e s of the e l e c t r i c a l d o u b l e layer on cement. S t a r t i n g w i t h a d i s c u s s i o n of the i n t e r f a c e b e t w e e n nonr e a c t i n g o x i d i c m a t e r i a l s and a q u e o u s e l e c t r o l y t e s , the theory of the n o n - e q u i l i b r i u m d o u b l e layer is a p p l i e d to this s y s t e m in the following. S u b s e q u e n t l y it is e x t e n d e d to r e a c t i n g m a t e r i a l s like h y d r a t i n g c e m e n t and the p r o p e r t i e s of the c o r r e s p o n d i n g z e t a - p o t e n t i a l are derived.
573
574
Vol. 17, No. 4 E. N~gele
2.
The D o u b l e
Layer
on O x i d i c
Materials
On o x i d i c m a t e r i a l s the e s t a b l i s h m e n t of s u r f a c e occur by two d i s t i n c t b u t e q u i v a l e n t m e c h a n i s m s : i)
the a d s o r p t i o n site
of p r o t o n s
H+ MOH2 +
÷ +
or h y d r o x y l - i o n s
charge
may
on to an a m p h o t e r i c
OH-
MOH
~
MO
+ H~O
or
ii) the f o r m a t i o n in s o l u t i o n of h y d r o x y l a t e d species M(OH) ~-x)+ w h i c h d e p o s i t on the s u r f a c e or a l t e r n a t i v e l y the f o r m a t i o n X o f ' these species d i r e c t l y on the s u r f a c e /7/. The c h a r g e carriers are i m m o b i l e and the s u r f a c e charge is l o c a l i z e d on s u r f a c e atoms, ions or m o l e c u l e s or on s t r u c t u r a l defects. Thus, local field s t r e n g t h s can be very high, w h i c h results in a s t r u c t u r i n g process in the inner d o u b l e layer and an o r i e n t a t i o n of the w a t e r d i p o l e s in the layers a d j a c e n t to the surface. As a c o n s e q u e n c e , o x i d i c m a t e r i a l s h a v e high s u r f a c e charges up to 0.7 A s / m 2 (70 ~ C / c m 2) but only low z e t a - p o t e n t i a l s . T h e r e is only a m o d e r a t e a g r e e m e n t b e t w e e n e x p e r i m e n t and theory if c o n v e n t i o n a l G r a h a m e - m o d e l s are used, w h i c h treat the inner d o u b l e layer as some layers of a d s o r b e d species on to a s u r f a c e w i t h the same p o t e n t i a l everywhere. Bockris, D e v a n a t h a n and M u l l e r /8/ s u c c e e d e d in a c c o u n t i n g for the d i p o l e o r i e n t a t i o n and o b t a i n e d a good a g r e e m e n t w i t h e x p e r m e n t a l data for their model. A b e t t e r a p p r o a c h was s u g g e s t e d by L y k l e m a /9/, w h o a s s u m e d that the p o t e n t i a l d e t e r m i n i n g ions could p e n e t r a t e the s u r f a c e layers of the oxide to some e x t e n t and react there w i t h the a m p h o t e r i c sites. In this way large a m o u n t s of s u r f a c e c h a r g e could d e v e l o p and, if c o u n t e r - i o n s w e r e p e r m i t t e d to enter this porous layer too, the net e l e c t r i c a l p o t e n t i a l at the outer edge of this porous layer w o u l d be r e d u c e d c o n s i d e r a b l y in m a g n i t u d e . Thus high s u r f a c e charge could be r e c o n c i l e d w i t h m o d e s t values of the z e t a - p o t e n t i a l . This model was t r e a t e d in its m o s t g e n e r a l form by P e r r a m et a!./10/. The z e t a - p o t e n t i a l in this model is i d e n t i c a l w i t h the p o t e n t i a l at the j u n c t i o n b e t w e e n the gel and the electrolyte. The c o n c e n t r a t i o n s of ions in the g e l - l a y e r , ci, are r e l a t e d to the bulk c o n c e n t r a t i o n s ci° by e x p r e s s i o n s of the form o c i = c i exp
(- (F~ + ~i)/RT)
w h e r e ~i(r) is the s p e c i f i c a d s o r p t i o n Eq. (I) may be r e a r r a n g e d to give c i = ci°ex p(-F~/RT)
exp(-~i/RT)
(I) energy
of species
= ci °A.i (r) exp (-F~/RT)
i /7,9/.
(la)
w h e r e A i(r) = exp (-~i(r)/RT)- If this is c o m p a r e d to a G r a h a m e model, the charge in the gel layer is a p p r o x i m a t e l y o =
6"z-F-ci°exp
(-zF~/RT)
(2)
Vol. 17, No. 4
575 ZETA-POTENTIAL, NON-EQUILIBRIUM, DOUBLE-LAYER
where
6 =
2r Zi Ai(r)
(3)
has the d i m e n s i o n of a length and is i n t e r p r e t e d as the "thickness" of the inner d o u b l e layer, which, in this case, is a s s u m e d i n s i d e the gel. In c o n t r a s t to the oxidic m a t e r i a l s d i s c u s s e d thus far, c e m e n t reacts w i t h the s u r r o u n d i n g w a t e r and the equil i b r i u m models d e s c r i b e d a b o v e m u s t be replaced by a n o n - e q u i l i b r i u m d o u b l e layer. In the next chapter, n o n - e q u i l i b r i u m double layer theory w i l l be d e v e l o p p e d for n o n - r e a c t i n g materials. S u b s e q u e n t l y it will be e x t e n d e d to r e a c t i v e systems.
3. T h e o r y of the Non-Equi.librium D o u b l e L a y e r The theory of the n o n - e q u i l i b r i u m double layer p r e s e n t e d in the f o l l o w i n g is a special case of a g e n e r a l theory p r o p o s e d by D u k h i n and D e r j a g u i n /11/. ~ Under n o n - e q u i l i b r i u m c o n d i t i o n s ionic flows in the double layer do not vanish. Thus, the B o l t z m a n n d i s t r i b u t i o n , c o r r e l a t i n g ionic c o n c e n t r a t i o n s in the d o u b l e layer to those in the bulk electrolyte, no longer holds. But if we r e s t r i c t ourselves to s t a t i o n a r y conditions, the P o i s s o n e q u a t i o n w r i t t e n in the form A~ = F - I
g -I o
(z+c + - z-c-)
(4)
+ is still valid. In eq. (4) z and ~ are the v a l e n c i e s of the cations and anions respectively, c and c- are their c o n c e n t r a tions in the double layer. Although transferof ions occurs t h r o u g h o u t the d o u b l e layer, the law of this transfer under s t a t i o n a r y cond i t i o n s is such, that the ions are t r a n s f e r r e d through any elem e n t of the d o u b l e layer v o l u m e w i t h o u t a c c u m u l a t i n g in it /11/. The charge of the p a r t i c l e surface is thus u n a l t e r e d in time, that is, the ionic flows Ji for anions and cations at any surface e l e m e n t should be equal to zero. Ji ,
x
=
o
= O
(5)
0
(6)
For the same reasons div
J
=
everywhere. In eq. (6) J is the v e c t o r of the ionic flow. This e q u a t l o n together w i t h b o u n d a r y c o n d i t i o n s eq. (5) may now be used i n s t e a d of the B o l t z m a n n - d i s t r i b u t i o n to solve eq. (4). A c c o r d i n g to /11/ the p o t e n t i a l - d i s t r i b u t i o n in the n o n - e q u i l i b r i u m d o u b l e layer is for s y m m e t r i c a l e l e c t r o l y t e s (x)
=
~n(tanh
(c(x)
c o
(Z~o/4)/tanh
-I
) z -I sinh
(z~oeq/4))-
(Z~o/2) I/cosh
(z~oeq/2)]
(7)
576
Vol. 17, No. 4 E. Nggele
In eq. (7) z is the valency of the ions, c o is their bulk concentration, ~o is the dimensionless potential at the surface and ~ eq is the G o u y - C h a p m a n - e q u i l i b r i u m pQtentia! at the surface e~pressed dimensionless as F~/RT like ~o and ~(x). The zeta-potential of the n o n - e q u i l i b r i u m system may be calculated from the electrophoretic m o b i l i t y u by using /11/ J(4z) -I = 1,5 ~ -
z2~ 2 (2kR) -I - 3 eeo(RT)2(16~qDF2) -I (kR)-1~ 3 (8)
where ~ = u-6~qF(Ee ERT) - I u n d ~ = F~/RT are the dimensionless mobility and the dimensionless zeta-potential respectively, q is the viscosity of the electrolyte and D the diffusion coefficient of the ions. R is the particle radius and I/k is the Gouy-Chapmanthickness of the diffuse double layer. From eq. (8) it follows, that the non-equilibrium zeta-potential is always greater than the equilibrium value. Hence, anything disturbing the equilibrium increases the zeta-potential. According to /11/ a particle with a thin double layer under non-equilibrium conditions may be characterized as follows: Tangential ionic flows arise under stationary conditions because of ionic flows from the bulk to the surface of the particle and in the opposite direction. The conditions of continuity of the system of bulk and surface flows for both kinds of ions can be satisfied only of diffusion flows arise beyond the double layer and, particularly, along its outer boundary. Since locally, that is, between the given double layer area and the adjoining volume of electrolyte, equilibrium should be preserved, the change in concentration along the outer boundary of the double layer leads to a change in its thickness and, consequently, to a change in the zeta-potential. On increasing the electrolyte concentration the diffuse part of the double layer contracts. This, together with a constant surface charge leads to a decrease in the zetapotential. In this respect, the n o n - e q u i l i b r i u m system behaves like the system in equilibrium.
4. The Double Layer on Cement Cement reacts with water to a considerable extent. However, the reaction is slow enough to allow for the development of a t i m e - d e p e n d e n t zeta-potential /5, 6/. Thus, except for the very first minutes, we can assume stationary conditions at the cement surface, where the reaction takes place. The reaction itself generates no excess charge, thus Z.c.rz. ll
=
0
(9)
1
where the superscript "r" indicates, that only ions generated by the reaction are considered. Eq. (5) also holds and thus the surface charge is unaltered in time. The theory developped in the previous chapter may then be applied to cement, if the first minutes and the very late stages of the hydration are excluded. In the first minutes the ions accumulate in the double layer and
Vol. 17, No. 4
577 ZETA-POTENTIAL, NON-EQUILIBRIUM, DOUBLE-LAYER
a stationary state does not exist. At the later stages, the reaction rate is so slow, that equilibrium models can be applied, w i t h o u t introducing serious errors. During the induction period of the cement h y d r a t i o n the ions generated behind the double layer p e n e t r a t e through it, without accumulating there. The ionic flows inside the double layer result in structural changes of the double layer and hence the zeta-potential of cement varies with time, even if no ions accumulate in the double layer. The flows act in both directions, to an off the surface. As a consequence, the bulk electrolyte exerts the same effects on the zetapotential of cement as would be expected for a conventional GouyChapman theory. In particular, the reaction results in an increasing ionic strength w i t h time in the bulk. Therefore, the diffuse part of the double layer contracts and the absolute values of the zeta-potential of hydrating cement decrease slowly to zero irrespective of sign. Zeta-potentials of cement measured at d i f f e r e n t times m u s t therefore be corrected for this effect before they can be compared to each other. The reaction at the surface prevents any adsorption. Thus, any effects due to specific adsorption vanish for cement. Furthermore, no d i s t i n c t i o n must be made between specific adsorbed ions and p o t e n t i a l - d e t e r m i n i n g ions, because the first do not exist. On the other hand, every kind of ions taking part in the reaction or passing the double layer, either from or to the surface, affects the potential distribution and must be considered as a p o t e n t i a l - d e t e r m i n i n g ion. Consequently, the number of p o t e n t i a l - d e t e r m i n i n g ions is extremely high for cement. For example, rK+ > rNa+ and thus DK+ < DNa+. Replacing Na + by K + should thus result in a significant change in the zeta-potential of cement. The zeta-potential of cement is comparatively small, rarely exceeding 20mV in m a g n i t u d e /5, 6/. Under this condition ne and eq. lution: ~(x)
(7) simplifies 4RT (zF) -I artanh
=
eq
to the w e l l - k n o w n
(tanh(ZF~oeq/4RT)
exact Gouy-Chapman
exp(-kx))
so-
(10)
This may again be compared to the G r a h a m e - s o l u t i o n to obtain eq.'s (2) and (3) in full generality, that is, w i t h o u t the assumption of a porous gel layer on the surface. Using eq. (3) the thickness of the inner double layer may then be estimated from electrokinetic measurements. Such a possibility has been predicted also in /11/. Eq. (3) can be obtained from a conventional Grahame-model by replacing "2r", the Grahame-thickness of the inner Helmholtz-plane, by 6, the time-dependent "thickness" of the inner layer /12/. Thus we call this approximation the "modified Grahame-model".
5.
Application
of the Theory
The theory outlined in the previous chapter will now be applied to hydrating cement. Because m i c r o e l e c t r o p h o r e s i s is the
578
Vol. 17, No. 4 E. N~gele
b e s t m e t h o d to d e t e r m i n e the e l e c t r o p h o r e t i c m o b i l i t y u and the z e t a - p o t e n t i a l of h y d r a t i n g cement, the f o l l o w i n g e v a l u a t i o n is based on data reported in /5, 6/. V i s c o s i t i e s and d i f f u s i o n c o e f f i c i e n t s of ions w e r e taken from h a n d b o o k s like /13/. The plot of the z e t a - p o t e n t i a l of c e m e n t versus time may be considered to c o n s i s t of three parts: The first p a r t concerns the very first m i n u t e s of the hydration, w h e r e a z e t a - p o t e n t i a l does not exist, b e c a u s e the r e a c t i o n is too vigorous. At the end of this p e r i o d the double layer is established, w h e n the r e a c t i o n has slowed down sufficiently. Then a p e r i o d follows, w h e r e ions a c c u m u l a t e in the d o u b l e layer, b e c a u s e the p r o d u c t i o n rate of ions at the surface is h i g h e r than the p e r m e a t i o n rate through the d o u b l e layer. In this period, beg i n n i n g about two m i n u t e s after the c o n t a c t b e t w e e n cement and water the z e t a - p o t e n t i a l d e v e l o p s as the d o u b l e layer is stabilized. Due to the a c c u m u l a t i o n of ions, d~/dt is c o m p a r a t i v e l y high in this period, and c o n s e q u e n t l y the z e t a - p o t e n t i a l varies m u c h w i t h time. Thus, no z e t a - p o t e n t i a l has been reported in /5, 6/ b e f o r e 5 m i n u t e s of hydration, a l t h o u g h it is p o s s i b l e to o b t a i n data at earlier times. In the next period s t a t i o n a r y conditions are e s t a b l i s h e d at the s u r f a c e and the n o n - e q u i l i b r i u m double layer theory applies. In Fig.1 the e l e c t r o p h o r e t i c mobility u, the e q u i l i b r i u m z e t a - p o t e n t i a l a c c o r d i n g to S m o l u c h o w s k i /7/ ceq, the n o n - e q u i l i b r i u m z e t a - p o t e n t i a l from eq. (8) ~ne, and the true z e t a - p o t e n t i a l ~t w h i c h includes a c o r r e c t i o n for the i n c r e a s i n g ionic s t r e n g t h in the bulk liquid, are shown for 10 mg/l p o r t l a n d cement in d i s t i l l e d w a t e r /6/. As the a b s o l u t e values of the zeta-potential, except for the first value, are always s m a l l e r than 20 mV, ~ ne = ~eq as predicted by eq. (8). It should be m e n t i o n e d here, that eq. (8) is not very s e n s i t i v e to the v a l u e of k, as long as kR >> I. For cement roughly 250 ~ kR ~ 1 5 0 0 . For ~t (t=0) has been chosen as the reference state in Fig. I, b e c a u s e ~ (0) does not exist and a theoretical v a l u e cannot be estimated. From Fig. I it may be concluded, that the v a r i a t i o n s of the z e t a - p o t e n t i a l w i t h time for h y d r a t i o n times above 180 m i n u t e s are e n t i r e l y due to the inc r e a s i n g ionic s t r e n g t h in the liquid phase, w h e r e a s the variau o o
,ovl
o
0
c,
~ne
Fig.
E l e c t r o p h o r e t i c mobility u, e q u i l i b r i u m z e t a - p o t e n t i a l ~eq, n o n - e q u i l i v r i u m zetap o t e n t i a l ~ n e and true z e t a - p o t e n t i a l ~t for 10mg/l p o r t l a n d cement in d i s t i l l e d water
-lol \ ......
I- 10/~
.........
. ~
-
-o
.
. . . . . . . . . . .
.
a
-20
0
bO
120
180
2/,0
300
360
/,20
~,80
I:
%0
t Irninl
Vol. 17, No. 4
579 ZETA-POTENTIAL,
NON-EQUILIBRIUM, DOUBLE-LAYER
tions for s h o r t e r h y d r a t i o n times are due to the c h e m i c a l reactions. Thus, the z e t a - p o t e n t i a l can be used to trace the h y d r a tion reaction, if it is c o r r e c t e d for the i n c r e a s i n g i o n i c strength. The t r a n s i t i o n b e t w e e n v a r i a t i o n s in the z e t a - p o t e n t i a l g o v e r n e d by the c h e m i c a l r e a c t i o n and those due to the i n c r e a s i n g ion c o n c e n t r a t i o n s d e p e n d s on the c e m e n t c o n c e n t r a t i o n in the s u s p e n s i o n s t u d i e d /6/. If the z e t a - p o t e n t i a l is m e a s u r e d in sol u t i o n s w h e r e the p H - v a l u e has b e e n a d j u s t e d in some way, the i n c r e a s e d ionic s t r e n g t h due to the a d d e d m a t e r i a l s , for e x a m p l e a 0,01 m N a O H for pH=12, c o m p r e s s e s the d i f f u s e d o u b l e layer too. As a result, the v a r i a t i o n s of the z e t a - p o t e n t i a l w i t h time dec r e a s e s i g n i f i c a n t l y c o m p a r e d to the values o b t a i n e d in d i s t i l l e d or d e i o n i z e d water. In Fig. 2 £eq, ~ne and 6t are s h o w n for Ig/l p o r t l a n d c e m e n t at pH= 10 a d j u s t e d w i t h N a O H and KOH respectively. ~eq and ~ne are always i d e n t i c a l u n d e r these c o n d i t i o n s , b e c a u s e the z e t a - p o t e n t i a l s are v e r y low. ct does not d i f f e r m u c h from ~eq, b e c a u s e the ionic s t r e n g t h is n e a r l y c o n s t a n t w i t h the time.
6.
Conclusions
Due to the n o n - e q u i l i b r i u m s t a t e of the c e m e n t / w a t e r - s y s t e m , c e m e n t has a t i m e - d e p e n d e n t z e t a - p o t e n t i a l . It is low in m a g n i tude, r a r e l y e x c e e d i n g 20 mV. Such low values are c o m m o n for all o x i d i c m a t e r i a l s . The t h e o r y of the n o n - e q u i l i b r i u m d o u b l e layer p r o v i d e s f o r m u l a e for the c a l c u l a t i o n of both, the e l e c t r i c a l p o t e n t i a l as a f u n c t i o n of the d i s t a n c e from the p a r t i c l e surface and the z e t a - p o t e n t i a l u n d e r n o n - e q u i l i b r i u m c o n d i t i o n s . S i n c e the z e t a - ~ o t e n t i a l of c e m e n t is small, in g e n e r a l the rel a t i o n %ne = ~e~ m a y be a p p l i e d and the g e n e r a l solution, eq. (7) c o n v e r g e s t o w a r d s the w e l l k n o w n e x a c t G o u y - C h a p m a n s o l u t i o n eq. (10). If the r e s u l t s o b t a i n e d from the n o n - e q u i l i b r i u m theory are c o m p a r e d to the G r a h a m e - t h e o r y an e x p r e s s i o n for the a p p r o x i m a t e " t h i c k n e s s " of the i n n e r d o u b l e layer is obtained. At the r e a c t i n g s u r f a c e no a d s o r p t i o n is p o s s i b l e . Hence, no s p e c i f i c a d s o r b e d ions e x i s t and the t h e o r y is s i m p l i f i e d c o n s i d e r a b l y . On the o t h e r hand, the n o n - v a n i s h i n g ionic flows in the d o u b l e layer r e s u l t in a large n u m b e r of p o t e n t i a l - d e t e r m i n i n g ions for
[mV]
KOH NoOH . • ~eq= ~ne
o
Fig.
2:
o
~t
20
Equilibrium zeta-pot e n t i a l ~eq, n o n - e q u i librium zeta-potential %ne and t r u e z e t a - p o t e n t i a l 6t for p o r t land c e m e n t (Ig/l) in w a t e r at pH=10, adj u s t e d w i t h KOH and NaOH respectively
t[min] 50
-20
100
liE
150
580
Vol. 17, No. 4 E. N~gele
cement. T~ne plot o£ ~ versus time can be subdivided into three parts: In the first part, corresponding to the vigorous initial reaction, no zeta-potential exists. In the second part, beginning about 5 minutes after the contact between cement and water, depending on the cement concentration, the n o n - e q u i l i b r i u m theory applies and the variations of the zeta-potential are due to the chemical reactions. In the third part, they depend solely on the increasing ionic strength in the bulk liquid.
7. References /I/
/2/ /3/
/4/
/5/ /6/ /7/ /8/ /9/ /10/ /11/
/12/ /13/
J. Skalny, J.F. Young: "Mechanisms of Portland Cement Hydration" in: 7th Int. Symp. Chem. Cem., Paris, 1980, ppII/1-3 to II/I-45. F.W.Locher, W. Richartz, S.Sprung: Zement-Kalk-Gips, 29, 1976, pp. 435. S.M.Ahmed: "The Zeta-Potential at the Oxide-Solution Interface" in: Oxides and Oxide Films, J.W.Diggle, Ed. Marcel Dekker, New York, 1972, pp.486. H.Gatos, J.Lagowski: "Space Charge Layers" in: Surface Effects in Crystal Plasticity, R.M.Latanision, J.T.Fourie, Editors, Noordhoff, Leyden, NL, 1977, pp. 221. E.N~gele: Cem. Concr. Res.15, 1985, pp. 453. E.N~gele: Cem. Concr. Res.16, 1986, pp.853. R.J.Hunter: "Zeta-Potenti-al in Colloid Science" Academic Press, New York, 1981. J.O'M. Bockris, M.A.Devanathan, K.Muller, Proc. Roy. Soc. London, 274, 1963, pp. 55. J.Lyklema: J.Electroanal. Chem. I_8, 1968, pp. 341. J.Perram, R.J.Hunter, H.Wright: Austr. J.Chem. 27,1974,pp.461.__ S.S.Dukhin, B . V . D e r j a g u i n : " N o n - E q u i l i b r i u m Double Layer and Electrokinetic Phenomena" in: Surface and Colloid Science, E.Matijevic Editor, 7, 1974, J.Wiley & Sons, New York. D.C.Grahame: Chem. Rev. 41 , 1947, pp. 44. Handbook of C h e m i s t r y an-d Physics, CRC-Press, New York, 1983.