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The primary electroviscous effect: Measurements on silica sols
The Primary Electroviscous Effect: Measurements on Silica Sols E. P. HONIG, 1 W. F. J. PlaINT, AND P. H. G. OFFERMANS Philips Research Laboratories, P.O. Box 80.000, 5600 JA Eindhoven, The Netherlands Received December 2, 1988; accepted April 4, 1989 The dependence of the viscosity of silica (ludox) sols on the electrolyte concentration of various aqueous solutions has been determined. At volume fractions below 5% the sols were Newtonian and the viscosity was proportional to the volume fraction of the particles. The (large) deviations from the Einstein equations can be explained very well with the effective increase of the particle volume due to the electrical double layer, as described by the Booth equation for the primary electroviscous effect. © 1990Academic Press, Inc.
INTRODUCTION
The presence of a surface charge on the particles in a suspension (or sol) leads to an increase in the suspension viscosity due to energy dissipation within the electrical double layers. There are three distinct effects, called the primary, secondary, and tertiary electroviscous effect, respectively. The primary electroviscous effect is due to the increase of the viscous drag forces on the particles as their electrical double layers are distorted by the shear field. The resulting contribution to the viscosity is, in first order, proportional to the volume fraction of the suspended particles. The secondary electroviscous effect results from the overlap of the electrical double layers of neighboring particles, and the resultant repulsion. The repulsive forces lead to a larger effective volume of the particles and hence to an increase of the viscosity. The leading contribution to the viscosity is proportional to the square of the volume fraction, because at least two particles are involved. The tertiary electroviscous effect arises from changes in the size and shape of (electrically charged) polyelectrolytes brought about by the applied shear field. I TO whom all correspondence should be addressed.
Although the secondary electroviscous effect depends on the square of particle concentration while the primary effect is proportional to the concentration, in most practical cases the secondary effect is much larger than the primary effect, even at low concentrations. Actually the primary effect is important only if the thickness of the double layer is of the same order of magnitude as the size of the particles. This implies that the particles must be very small in order for the primary electroviscous effect to be dominant. In relation to extended work on the viscoelasticity of concentrated silica sols, we first investigated the properties of dilute sols. A large deviation from the Einstein equation was observed, which could be attributed to the primary electroviscous effect, as shown in this paper. The silica particles in the ludox sols are small and of the same size. It turned out that such sols are good model systems for the primary electroviscous effect. THEORY OF THE PRIMARY ELECTROVISCOUS EFFECT
The primary electroviscous effect occurs in a suspension, in which the particles are electrically charged. Such particles are surrounded by an electrical double layer, that is a region around the particle, where there is an excess of ions of charge opposite to that of the par-
169 0021-9797/90 $3.00 Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
Copyright © 1990 by Academie Press, Inc. All tights of reproduction in any form reserved.
170
HONIG, PLINT, AND OFFERMANS
ticle. When such a suspension is sheared the double layer around the particles is distorted by the shear field, leading to an increased viscosity. The equation describing the primary electroviscous effect must thus contain the charge and the radius of the particles, and the thickness of the double layer in relation to the radius of the particles. The various equations describing the primary electroviscous effect of monodisperse suspensions can all be put in the form = ~/0[ 1 + 2.5~b L
×
1 + 2~r~oekTa2
,
where ~ is the viscosity of the suspension and 70 that of the solvent, q5is the volume fraction, a the radius of the suspended particles, Q is the number of elementary charges on each particle, NA is the Avogadro number, e is the dielectric constant of the solution, Xi are the conductivities of the various ionic species in the solution, b = Ka, and e, k, and T have their usual significance. The Debye length 1/K is defined with
K2
87re21 EkT '
-
[2]
where I is the ionic strength, defined with 1
I = ~ Z niz{,
[31
with zi and ni the valencies and the concentrations of the various ionic species in the solution, far away from any particle. The functions F(b) and G (~i) are listed in Table I. For simple electrolytes, the concentrations of the ionic species enter only through the function F(b). Usually the equations are presented in terms of the zeta-potential, rather than of the charge of the particles. One then assumes the Debye approximation to be valid (i.e., ~"< 25 mV):
Qe = a~'e(1 + b).
[4]
Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
Von Smoluchowski (1) was the first to present Eq. [ 1], without any derivation. Later Krasny-Ergen (2) derived the same equation with the factor F(b) 1.5 times as large. Booth (3) solved the complete hydrodynamic equations. Later Street (4) gave a very simple derivation, which was corrected for an algebraic error by Whitehead (5). However, there is a fundamental error in Street's derivation, because he started with the motion of a sphere in a quiescent liquid, rather than with the motion of a sphere in a shear field, as all other authors did. Russel (6) gave a similar derivation as Booth did with the same approximations, except that his results are valid only for large b. On the other hand his expression is valid also for high shear rates, and hence can be used in the region of non-Newtonian behavior for large Peclet numbers. Sherwood (7) extended the theory to high surface charge/ potential, but only for 1:1 electrolytes. Hinch and Sherwood (8) gave a general numerical solution; so did Watterson and White (9). In Table I the three analytic equations are presented. The Booth equation is presented in much simpler form than in the original paper. The Krasny-Ergen equation is valid only for large b. Indeed the Booth equation approaches the Krasny-Ergen equation for large values of b. The (corrected) Street equation approaches the Booth equation in no way: neither at large b, nor at small b. The approximations involved in the Booth equation (and in all the other theories, except for the sixth approximation) are: 1. laminar flow, neglect of inertia terms, no slip at the surface of the particles 2. small volume fraction of particles, which are nonconducting and spherical 3. uniformity of ionic conductivities, dielectric constant, and solvent viscosity throughout the electrolyte solution 4. uniform immobile charge density on the surface of the particles 5. no overlap of double layers of neighboring particles
171
ELECTROVISCOSITY OF SILICA SOLS TABLE I Equations for F(b) and G(~.i) Krasny-Ergen
Street
~, niz]
2 nizi2
G(~,i)
E ni]zi[~ki
Z niJzi[)ki
F(b)
1 b2(1 + b)2
2 15b2
Booth
ni[z~l/Xi
2 niz~ baH 2 + 6H(2/b + 2 + b ) - 1 + 1)/2 10b2(1 + b)2
with H = b{Es(b)
- E7(b)}exp(b) and D = b{-54Elt(2b) + 28Eg(2b) + 5Ev(2b) + E4(2b)}exp(2b). Note. En(x) is the exponential integral of order n. Summations are to be taken over all ionic species in solution.
6. the electroviscous effect must be less than the Einstein coefficient (2.5~b), i.e., low surface charge or low zeta-potential. Stone-Masui and Watillon ( t 0) found that the Booth equation could explain their experimental data, and the Street equation could not. Delgado et al. ( 11 ) could not fit their data to the Booth or the Watterson and White equations, although the authors claim that their experimental results reproduce the general trend of the Booth theory very well. The source of the discrepancy may be the fact that their particles were "hairy," whereas the theory assumes smooth particles with a uniform surface charge. EXPERIMENTAL
The silica used was ludox HS, a product of E. I. du Pont de Nemours and Co. It consists of spherical particles in an aqueous solution of NaC1 and Na2SO4, both at a concentration of approximately 6 m M . The silica content is about 40% by weight. The ludox was dialyzed against demineralized water in an Amicon dialysis apparatus for 6 days. Then dialysis was stopped because the viscosity became too high and too much silica was lost through the fibres of the Amicon apparatus. No CI- could be detected in the dialysate at this time. The dialyzed ludox dispersions were stored in polyethylene bottles. The viscosity of the ludox sols did not change significantly in the course of
time. By adequate dilutions of this stock, sols with different volume fractions and electrolyte concentrations were prepared. With laser beat spectroscopy the mean diameter of the silica spheres was found to be 35 +_ 7 nm. This value, however, is much too high, because even small aggregates do contribute very much to the scattered light intensity. We will therefore use the value given by the manufacturer, i.e. 14 n m for the diameter of the particles. With a pycnometer the density of the dried silica was found to be 2270 k g / m 3. We measured the viscosity of ludox sols at various volume fractions of silica and various concentrations of either of the electrolytes NaC1, BaC12, Na2SO4, and CeC13. For each of the electrolytes, a new sample of dialyzed sol was used. Volume fractions were calculated from the experimentally determined weight fraction using 2270 k g / m 3 as the density of the silica particles. The intrinsic viscosity of the silica sols was measured by means of a Mooney-Ewart type rheometer of Contraves (Ll $30) at 25.0 °C at shear rates between 0 and 1 0 0 S -1 .
RESULTS AND DISCUSSION
For volume fractions below 5% silica, the viscosity was found to be linearly dependent on the volume fraction of silica. The sols showed Newtonian behavior at these low volume fractions. The electrolyte concentration Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
172
HONIG, PONT, AND OFFERMANS
had a marked effect on the viscosity. Data are presented in Figs. 1 and 2 for a volume fraction of silica of 0.022. The relative viscosity is plotted as a function of the square root of the ionic strength of the added electrolyte. We tried to fit these data to the Street and to the Booth equations by adapting two parameters: the charge of the particles Qe and the (residue) electrolyte concentration present in the solution. That is, we assumed that two electrolytes were present: the electrolyte that was added of known concentration, and the electrolyte that was left in the sol after dialysis, assuming the latter to be NaC1. ( I f we had assumed this to be a mixture of NaC1 and Na2 SO4 the resultant curves of Figs. 1 and 2 would hardly be different, because the ionic conductivities of the various ionic species are not very different). The viscosity when no electrolyte was added is not the same for the four series of experiments. This can be explained by assuming that the residual electro-
I 1"5~N T//~/o 1.4
a2SO4(D)
1.3
1.2
1.1 Einstein 1.0 I 0
i t
t 2
k~-(mM)1/2
I 3
FIG. 1. Experimentaldata (O for NaCIand [] for BaC12) and theoretical curves (based on the Booth equation) for the viscosityas a function of the square root of the ionic Strength of added electrolyte. ~b = 0.022, a = 7 rim, Q = 260 (adapted). Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
T
1.5
~1/~1o 1.4 BaCI2 (D) 1.3
1.2
1.1 Einstein
I
1,00
i
1
i
2 ,~mM) 1/2
i
3 =
FIG. 2. Same as Fig. 1 (O for CeC13and [] for Na2SO4). lyte concentration was different in these four series of experiments. The ionic conductivities, needed for the calculations, were taken from Ref. (12). In no way could our data be fitted to the Street equation. Very reasonable fits were obtained with the Booth equation, using for the residual NaC1 concentration 0.70 m M for the NaCI and Na2SO4 series, 0.60 m M for the BaC12 series, and 1.00 m M for the CeC13 series. In other experiments, not treated in this paper, the electrolyte concentration of the dialysate solution was measured, and values of the same order of magnitude were found. F r o m the fit of all four curves the value of Q was found to be 260. We also titrated the sol with HC1, in which case all the negative charge of the ludox particles is compensated by H +. This way it was found that there was 162 m m o l e of O H - per kg of dry SIO2. For particles with a density of 2270 k g / m 3 and a radius of 7 n m this corresponds to 320 elementary charges per particle. This value is only a little larger than that obtained from the viscosity experiments. An explanation could be
ELECTROVISCOSITY OF SILICA SOLS that with the viscosity measurements one deals only with the charge on the outside of the particles, while with the titration experiment one determines the total area of the particles, including the area of the pores. The fitting of the data to the Booth equation was done, assuming the charge of the particles to be constant, i n d e p e n d e n t of nature and concentration the electrolyte. We also tried to fit our data to the Booth equation by assuming the zeta-potential to be constant, i.e., Eq. [4] was substituted into Eq. [ 1], and then we tried to a d a p t the zeta-potential and the residual electrolyte concentration. N o fit could be obtained in this way. Apparently we are dealing with " c o n s t a n t charge". It is to be noted that the fit could be m a d e down to values of the electrolyte concentration where the increase of the viscosity is m o r e than five times as large as the increase due to the " v o l u m e " alone, i.e. the Einstein equation. H e n c e it seems that the Booth equation has a larger range of validity than that for which it was derived. In accordance with our experimental results, the Booth equation predicts that it does not matter whether we deal with 2:1 or 1:2 electrolytes.
173
equation given by Booth. This correspondence was found, even b e y o n d the range for which this equation was derived. It is the electric charge of the particles, rather than the potential, that is independent of the nature and the concentration of the electrolyte in the aqueous solution. ACKNOWLEDGMENT We thank Mr. C. C. den Ouden, who performed most of the experiments. REFERENCES 1. 2. 3. 4. 5. 6. 7, 8. 9. 10.
CONCLUSIONS
11.
T h e p r i m a r y electroviscous effect in silica sols was found to c o n f o r m very well to the
12.
Von Smoluchowski, M., Kolloid-Z. 18, 190 (1916). Krasny-Ergen, W., Kolloid-Z. 74, 172 (1936). Booth, F., Proc. R. Soc. London A 203, 533 (1950). Street, N., J. ColloidSci. 13, 288 (1958). Whitehead, J. R., J. Colloid Interface Sci. 30, 424 (1969). Russel, W. B., J. FluidMech. 85, 209 (1978). Sherwood, J. D., J. FluidMech. 101, 609 (1980). Hinch, E. J., and Sherwood, J. D., J. Fluid Mech. 132, 337 (1983). Watterson, I. C., and White, L. R., J. Chem. Soc. Faraday Trans. 2 77, 1115 ( 1981 ). Stone-Masui, J., and Watillon, A., J. Colloid Interface Sci. 28, 187 (1968), and 34, 327 (1970). Delgado, A., Gonz~dez-Caballero, F., Cabrerizo, M. A., and Alados, I., Acta Polym. 38, 66 (1987). Korttim, G., "Lehrbuch der Elektrochemie, 4th ed.," p. 238. Verlag Chemic, Weinheim/Bergstr., 1966.
Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
INTRODUCTION
The presence of a surface charge on the particles in a suspension (or sol) leads to an increase in the suspension viscosity due to energy dissipation within the electrical double layers. There are three distinct effects, called the primary, secondary, and tertiary electroviscous effect, respectively. The primary electroviscous effect is due to the increase of the viscous drag forces on the particles as their electrical double layers are distorted by the shear field. The resulting contribution to the viscosity is, in first order, proportional to the volume fraction of the suspended particles. The secondary electroviscous effect results from the overlap of the electrical double layers of neighboring particles, and the resultant repulsion. The repulsive forces lead to a larger effective volume of the particles and hence to an increase of the viscosity. The leading contribution to the viscosity is proportional to the square of the volume fraction, because at least two particles are involved. The tertiary electroviscous effect arises from changes in the size and shape of (electrically charged) polyelectrolytes brought about by the applied shear field. I TO whom all correspondence should be addressed.
Although the secondary electroviscous effect depends on the square of particle concentration while the primary effect is proportional to the concentration, in most practical cases the secondary effect is much larger than the primary effect, even at low concentrations. Actually the primary effect is important only if the thickness of the double layer is of the same order of magnitude as the size of the particles. This implies that the particles must be very small in order for the primary electroviscous effect to be dominant. In relation to extended work on the viscoelasticity of concentrated silica sols, we first investigated the properties of dilute sols. A large deviation from the Einstein equation was observed, which could be attributed to the primary electroviscous effect, as shown in this paper. The silica particles in the ludox sols are small and of the same size. It turned out that such sols are good model systems for the primary electroviscous effect. THEORY OF THE PRIMARY ELECTROVISCOUS EFFECT
The primary electroviscous effect occurs in a suspension, in which the particles are electrically charged. Such particles are surrounded by an electrical double layer, that is a region around the particle, where there is an excess of ions of charge opposite to that of the par-
169 0021-9797/90 $3.00 Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
Copyright © 1990 by Academie Press, Inc. All tights of reproduction in any form reserved.
170
HONIG, PLINT, AND OFFERMANS
ticle. When such a suspension is sheared the double layer around the particles is distorted by the shear field, leading to an increased viscosity. The equation describing the primary electroviscous effect must thus contain the charge and the radius of the particles, and the thickness of the double layer in relation to the radius of the particles. The various equations describing the primary electroviscous effect of monodisperse suspensions can all be put in the form = ~/0[ 1 + 2.5~b L
×
1 + 2~r~oekTa2
,
where ~ is the viscosity of the suspension and 70 that of the solvent, q5is the volume fraction, a the radius of the suspended particles, Q is the number of elementary charges on each particle, NA is the Avogadro number, e is the dielectric constant of the solution, Xi are the conductivities of the various ionic species in the solution, b = Ka, and e, k, and T have their usual significance. The Debye length 1/K is defined with
K2
87re21 EkT '
-
[2]
where I is the ionic strength, defined with 1
I = ~ Z niz{,
[31
with zi and ni the valencies and the concentrations of the various ionic species in the solution, far away from any particle. The functions F(b) and G (~i) are listed in Table I. For simple electrolytes, the concentrations of the ionic species enter only through the function F(b). Usually the equations are presented in terms of the zeta-potential, rather than of the charge of the particles. One then assumes the Debye approximation to be valid (i.e., ~"< 25 mV):
Qe = a~'e(1 + b).
[4]
Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
Von Smoluchowski (1) was the first to present Eq. [ 1], without any derivation. Later Krasny-Ergen (2) derived the same equation with the factor F(b) 1.5 times as large. Booth (3) solved the complete hydrodynamic equations. Later Street (4) gave a very simple derivation, which was corrected for an algebraic error by Whitehead (5). However, there is a fundamental error in Street's derivation, because he started with the motion of a sphere in a quiescent liquid, rather than with the motion of a sphere in a shear field, as all other authors did. Russel (6) gave a similar derivation as Booth did with the same approximations, except that his results are valid only for large b. On the other hand his expression is valid also for high shear rates, and hence can be used in the region of non-Newtonian behavior for large Peclet numbers. Sherwood (7) extended the theory to high surface charge/ potential, but only for 1:1 electrolytes. Hinch and Sherwood (8) gave a general numerical solution; so did Watterson and White (9). In Table I the three analytic equations are presented. The Booth equation is presented in much simpler form than in the original paper. The Krasny-Ergen equation is valid only for large b. Indeed the Booth equation approaches the Krasny-Ergen equation for large values of b. The (corrected) Street equation approaches the Booth equation in no way: neither at large b, nor at small b. The approximations involved in the Booth equation (and in all the other theories, except for the sixth approximation) are: 1. laminar flow, neglect of inertia terms, no slip at the surface of the particles 2. small volume fraction of particles, which are nonconducting and spherical 3. uniformity of ionic conductivities, dielectric constant, and solvent viscosity throughout the electrolyte solution 4. uniform immobile charge density on the surface of the particles 5. no overlap of double layers of neighboring particles
171
ELECTROVISCOSITY OF SILICA SOLS TABLE I Equations for F(b) and G(~.i) Krasny-Ergen
Street
~, niz]
2 nizi2
G(~,i)
E ni]zi[~ki
Z niJzi[)ki
F(b)
1 b2(1 + b)2
2 15b2
Booth
ni[z~l/Xi
2 niz~ baH 2 + 6H(2/b + 2 + b ) - 1 + 1)/2 10b2(1 + b)2
with H = b{Es(b)
- E7(b)}exp(b) and D = b{-54Elt(2b) + 28Eg(2b) + 5Ev(2b) + E4(2b)}exp(2b). Note. En(x) is the exponential integral of order n. Summations are to be taken over all ionic species in solution.
6. the electroviscous effect must be less than the Einstein coefficient (2.5~b), i.e., low surface charge or low zeta-potential. Stone-Masui and Watillon ( t 0) found that the Booth equation could explain their experimental data, and the Street equation could not. Delgado et al. ( 11 ) could not fit their data to the Booth or the Watterson and White equations, although the authors claim that their experimental results reproduce the general trend of the Booth theory very well. The source of the discrepancy may be the fact that their particles were "hairy," whereas the theory assumes smooth particles with a uniform surface charge. EXPERIMENTAL
The silica used was ludox HS, a product of E. I. du Pont de Nemours and Co. It consists of spherical particles in an aqueous solution of NaC1 and Na2SO4, both at a concentration of approximately 6 m M . The silica content is about 40% by weight. The ludox was dialyzed against demineralized water in an Amicon dialysis apparatus for 6 days. Then dialysis was stopped because the viscosity became too high and too much silica was lost through the fibres of the Amicon apparatus. No CI- could be detected in the dialysate at this time. The dialyzed ludox dispersions were stored in polyethylene bottles. The viscosity of the ludox sols did not change significantly in the course of
time. By adequate dilutions of this stock, sols with different volume fractions and electrolyte concentrations were prepared. With laser beat spectroscopy the mean diameter of the silica spheres was found to be 35 +_ 7 nm. This value, however, is much too high, because even small aggregates do contribute very much to the scattered light intensity. We will therefore use the value given by the manufacturer, i.e. 14 n m for the diameter of the particles. With a pycnometer the density of the dried silica was found to be 2270 k g / m 3. We measured the viscosity of ludox sols at various volume fractions of silica and various concentrations of either of the electrolytes NaC1, BaC12, Na2SO4, and CeC13. For each of the electrolytes, a new sample of dialyzed sol was used. Volume fractions were calculated from the experimentally determined weight fraction using 2270 k g / m 3 as the density of the silica particles. The intrinsic viscosity of the silica sols was measured by means of a Mooney-Ewart type rheometer of Contraves (Ll $30) at 25.0 °C at shear rates between 0 and 1 0 0 S -1 .
RESULTS AND DISCUSSION
For volume fractions below 5% silica, the viscosity was found to be linearly dependent on the volume fraction of silica. The sols showed Newtonian behavior at these low volume fractions. The electrolyte concentration Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
172
HONIG, PONT, AND OFFERMANS
had a marked effect on the viscosity. Data are presented in Figs. 1 and 2 for a volume fraction of silica of 0.022. The relative viscosity is plotted as a function of the square root of the ionic strength of the added electrolyte. We tried to fit these data to the Street and to the Booth equations by adapting two parameters: the charge of the particles Qe and the (residue) electrolyte concentration present in the solution. That is, we assumed that two electrolytes were present: the electrolyte that was added of known concentration, and the electrolyte that was left in the sol after dialysis, assuming the latter to be NaC1. ( I f we had assumed this to be a mixture of NaC1 and Na2 SO4 the resultant curves of Figs. 1 and 2 would hardly be different, because the ionic conductivities of the various ionic species are not very different). The viscosity when no electrolyte was added is not the same for the four series of experiments. This can be explained by assuming that the residual electro-
I 1"5~N T//~/o 1.4
a2SO4(D)
1.3
1.2
1.1 Einstein 1.0 I 0
i t
t 2
k~-(mM)1/2
I 3
FIG. 1. Experimentaldata (O for NaCIand [] for BaC12) and theoretical curves (based on the Booth equation) for the viscosityas a function of the square root of the ionic Strength of added electrolyte. ~b = 0.022, a = 7 rim, Q = 260 (adapted). Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990
T
1.5
~1/~1o 1.4 BaCI2 (D) 1.3
1.2
1.1 Einstein
I
1,00
i
1
i
2 ,~mM) 1/2
i
3 =
FIG. 2. Same as Fig. 1 (O for CeC13and [] for Na2SO4). lyte concentration was different in these four series of experiments. The ionic conductivities, needed for the calculations, were taken from Ref. (12). In no way could our data be fitted to the Street equation. Very reasonable fits were obtained with the Booth equation, using for the residual NaC1 concentration 0.70 m M for the NaCI and Na2SO4 series, 0.60 m M for the BaC12 series, and 1.00 m M for the CeC13 series. In other experiments, not treated in this paper, the electrolyte concentration of the dialysate solution was measured, and values of the same order of magnitude were found. F r o m the fit of all four curves the value of Q was found to be 260. We also titrated the sol with HC1, in which case all the negative charge of the ludox particles is compensated by H +. This way it was found that there was 162 m m o l e of O H - per kg of dry SIO2. For particles with a density of 2270 k g / m 3 and a radius of 7 n m this corresponds to 320 elementary charges per particle. This value is only a little larger than that obtained from the viscosity experiments. An explanation could be
ELECTROVISCOSITY OF SILICA SOLS that with the viscosity measurements one deals only with the charge on the outside of the particles, while with the titration experiment one determines the total area of the particles, including the area of the pores. The fitting of the data to the Booth equation was done, assuming the charge of the particles to be constant, i n d e p e n d e n t of nature and concentration the electrolyte. We also tried to fit our data to the Booth equation by assuming the zeta-potential to be constant, i.e., Eq. [4] was substituted into Eq. [ 1], and then we tried to a d a p t the zeta-potential and the residual electrolyte concentration. N o fit could be obtained in this way. Apparently we are dealing with " c o n s t a n t charge". It is to be noted that the fit could be m a d e down to values of the electrolyte concentration where the increase of the viscosity is m o r e than five times as large as the increase due to the " v o l u m e " alone, i.e. the Einstein equation. H e n c e it seems that the Booth equation has a larger range of validity than that for which it was derived. In accordance with our experimental results, the Booth equation predicts that it does not matter whether we deal with 2:1 or 1:2 electrolytes.
173
equation given by Booth. This correspondence was found, even b e y o n d the range for which this equation was derived. It is the electric charge of the particles, rather than the potential, that is independent of the nature and the concentration of the electrolyte in the aqueous solution. ACKNOWLEDGMENT We thank Mr. C. C. den Ouden, who performed most of the experiments. REFERENCES 1. 2. 3. 4. 5. 6. 7, 8. 9. 10.
CONCLUSIONS
11.
T h e p r i m a r y electroviscous effect in silica sols was found to c o n f o r m very well to the
12.
Von Smoluchowski, M., Kolloid-Z. 18, 190 (1916). Krasny-Ergen, W., Kolloid-Z. 74, 172 (1936). Booth, F., Proc. R. Soc. London A 203, 533 (1950). Street, N., J. ColloidSci. 13, 288 (1958). Whitehead, J. R., J. Colloid Interface Sci. 30, 424 (1969). Russel, W. B., J. FluidMech. 85, 209 (1978). Sherwood, J. D., J. FluidMech. 101, 609 (1980). Hinch, E. J., and Sherwood, J. D., J. Fluid Mech. 132, 337 (1983). Watterson, I. C., and White, L. R., J. Chem. Soc. Faraday Trans. 2 77, 1115 ( 1981 ). Stone-Masui, J., and Watillon, A., J. Colloid Interface Sci. 28, 187 (1968), and 34, 327 (1970). Delgado, A., Gonz~dez-Caballero, F., Cabrerizo, M. A., and Alados, I., Acta Polym. 38, 66 (1987). Korttim, G., "Lehrbuch der Elektrochemie, 4th ed.," p. 238. Verlag Chemic, Weinheim/Bergstr., 1966.
Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990