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Anomalous Trends in the Electroviscous Effect of Polystyrene Latexes: Experimental and Theoretical Study M. J. Garcı´a-Salinas and F. J. de las Nieves* Complex Fluids Physics Group, Department of Applied Physics, University of Almerı´a, 04120 Almerı´a, Spain Received February 7, 2000. In Final Form: June 7, 2000 In this work the primary electroviscous effect was determined for two polystyrene latexes with different surface charge densities and functional groups (sulfonate and carboxylate). For each latex, the electroviscous coefficient (p) was obtained for several electrolyte (NaCl) concentrations; that is, as a function of the electrokinetic radius κa. In both cases, a maximum in the p against κa plot was found experimentally, which was not predicted by the theory. A study using Booth’s and Watterson-White’s theories was done to explain the initial differences between experimental and theoretical data at low κa values. These maximum values were theoretically explained as a consequence of a third ionic species present in the solution.
Introduction Those phenomena involving the modification of the hydrodynamic behavior of a suspension of charged particles due to the presence of the electrical double layer around the particles are known as electroviscous effects.1,2 This work deals with the primary electroviscous effect, which is basically an enhancement in viscosity due to a deformation of the electrical double layers around the particles. This effect is studied in dilute suspensions to avoid interactions between double layers of different particles, which would cause the secondary electroviscous effect to appear. When the fluid is sheared the double layers of the particles are distorted by the shear field. As a result of the resistance of the ionic atmosphere there is an extra-dissipation of energy, and the viscosity increases, which is taken into account as a correction factor "p" (called the primary electroviscous effect coefficient) to the Einstein equation:2,3
ηr ) 1 + k(1 + p)φ
(1)
where ηr is the relative viscosity (relation between the viscosity of the suspension and the viscosity of the solvent: η/η0), φ is the volume fraction of solid, and k ) 2.5 for spherical particles. Several authors have reported a theoretical estimation of the primary electroviscous effect: Smoluchowski,4 Krasny-Ergen,5 Booth,6 Russel,7 and Watterson and White,8 among others. Attempts to find the theoretically predicted electroviscous effect (p) have been made with several colloidal systems9 such as polymer latexes,10-16 * To whom correspondence should be addressed. (1) Hunter R. J. Zeta Potential in Colloid Science. Principles and Applications; Academic Press: London, 1981; Chapters 3, 5. (2) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997; p 145-192. (3) Einstein, A. Ann. Phys. (Leipzig) 1906, 19, 298. (4) Smoluchowski, M. Kolloid-Z. 1916, 18, 190. (5) Krasny-Ergen, B. Kolloid-Z. 1936, 74, 172. (6) Booth, F. Proc. R. Soc. Ser. A 1950, 203, 533. (7) Russel, W. B. J. Fluid Mech. 1978, 85, 673. (8) Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1980, 77, 1115. (9) Ise, N. Proc. Jpn. Acad., Ser. B 1998, 74, 192. (10) McDonogh, R. W.; Hunter, R. J. J. Rheol. 1983, 27(3), 189. (11) Ali, S. A.; Sengupta M. J. Colloid Interface Sci. 1986, 113(1), 172.
silica,16-17 Pyrex glass,18 or even clays.19 Discrepancies were usually found between the theoretical and experimental results, with the latter being up to 1 order of magnitude higher even with latexes considered as model systems. These discrepancies have not yet been explained. In this work we present an experimental study of the electroviscous effect for two polystyrene latexes with sulfonate and carboxylate functional groups. For each latex, the p coefficient of this effect has been obtained for several electrolyte (NaCl) concentrations; that is, as a function of the electrokinetic radius κa. In both cases the theoretical and experimental data were in very good agreement. At low electrolyte concentration, however, a maximum in the p against κa plot was found experimentally. These kinds of results have been scarcely reported,12,16 but at this time we have not found any theoretical study that explains the appearance of this maximum at low electrolyte concentration. The usual behavior of polystyrene latexes, which display a maximum in electrophoretic mobility against electrolyte concentration, cannot explain this result because the mobility maximum appears at higher electrolyte concentration (10-3-10-2 M NaCl for our latexes). In this paper the experimental results were analyzed and explained in the light of the theories developed by Booth6 and Watterson and White.8 Theory This section gives a brief description of the basic considerations involved in the well-known theories of the primary electroviscous effect developed by Booth6 and Watterson and White.8 In this paper, however, it will be shown how the inclusion of a third ionic species affects (12) Delgado, A.; Gonza´lez-Caballero, F.; Cabrerizo, M. A.; Alados I. Acta Polym. 1987, 38, 66. (13) Yamanaka, J.; Matsuoka, H.; Kitano, H.; Ise, N. J. Colloid Interface Sci. 1990, 134(1), 92. (14) Stone-Masui, J.; Watillon, A. J, Colloid Interface Sci, 1968, 28, 187. (15) Yamanaka, J.; Ise, N.; Miyoshi, H.; Yamaguchi, T. Phys. Rev., 1995, 51(2), 1276. (16) Zurita, L.; Carrique, F.; Delgado, A. V. Colloids Surf. A 1994, 92, 23. (17) Honig, E. P.; Pu¨nt, W. F. J.; Offermans, P. H. G. J. Colloid Interface Sci. 1990, 134, 169. (18) Delgado, A.; Gonza´lez-Caballero, F.; Salcedo, J.; Cabrerizo, M. A. J. Dispersion Sci. Technol. 1989, 10(2), 107. (19) Adachi, Y.; Nakaishi, K.; Tamaki, M. J. Colloid Interface Sci. 1998, 198, 100.
10.1021/la000165l CCC: $19.00 © 2000 American Chemical Society Published on Web 08/12/2000
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the results, causing a reversal of the trend in this effect due to counterion replacement in the low electrokinetic radius (κa) region (κ-1 is the Debye length and a is the particle radius). Both contributions6,8 use the same fundamental equations to describe the system and define the problem mathematically: the Navier-Stokes equation describes the hydrodynamic flow of the fluid, the Poisson-Boltzmann equation describes the electrostatic potential, and the movement equation (balance of hydrodynamic drag, and electrostatic and thermodynamic forces) describes the ions. Both theories also use the same assumptions. The differences appear in the method of resolution of the equations and in the calculation of the effective viscosity of the suspension, and thus the p coefficient. A review of Booth’s theory is given first, because it is a good approach of the exact calculations and serves to illustrate the reasons for the occurrence of the maximum value of the electroviscous effect. Booth’s Theory. In Booth’s theory, once the equations have been solved for the fluid velocity and electrostatic potential, the effective viscosity is calculated following a method devised by Fro¨hlich and Sack.20 The electroviscous coefficient p is then obtained as a power series in Q (particle charge number) or in ζ (zeta potential):
( )
∞
pBO )
eζ
n
∑ bn kT n)1
(2)
where e, k, and T represent the elementary charge, Boltzmann constant, and absolute temperature, respectively. Booth found the first two terms of the series: b1 ) 0 and b2 ) q* ‚ Z(κa) ‚ (1 + κa)2. Thus, the correction to viscosity due to the electroviscous effect was of order ζ2, with q* being a dimensionless quantity independent of the concentration for a given electrolyte, and Z(κa) being a complicated semianalytical function which increases with decreasing κa and reflects the deformability of the double layer. A study of this function and an experimental determination can be found in Yamanaka’s work.15 A previous contribution by Honig and co-workers17 gave a simpler form of the Z function, renamed as F in the following expression for the electroviscous coefficient, which was the one used throughout this work:
pBO )
6e 2 ζ (1 + κa)2G(µi)F(κa) η0kT
(3)
where and η0 represent the dielectric permittivity and bulk viscosity of the electrolyte solution, respectively. We are interested in G(µi): N
G(µi) )
nizi2µi-1 ∑ i)1 N
(4)
ni zi ∑ i)1
2
As it will be shown, this function yields a maximum value of p for a certain value of κa. In eq 4, µi is the ionic mobility of the i ionic species, zi is its valence, and ni is its concentration. For any electrolyte, G is a dimensionless coefficient which reflects the mobilities and composition of ions and does not depend on electrolyte concentration; (20) Fro¨hlich, H.; Sack, R. Proc. R. Soc., Ser. A 1946, 185, 415.
Figure 1. Functions for the calculation of the electroviscous effect by Booth’s theory. G* includes only electrolyte (NaCl) ions, G includes also protons, and F(κa) is defined in ref 17.
that is, it only depends on how fast the ions are able to move, and what are their proportions (electrolyte type 1-1, 2-1, ... etc.). However, if we consider a residual ionic species in the bulk, the G value is no longer constant, and in this case it does depend on concentrations of the ions. For the simplest case in which the third ionic species included is the pH-determinant ion (H+ or OH-), let us consider NaCl electrolyte and acid pH, taking into account H+ ions. In the following expressions, the numerical subscript i has been replaced by the atomic symbol of the ionic species:
G*NaCl ) GNaCl )
nNaµNa-1 + nClµCl-1 1 ) (µNa-1 + µCl-1) nNa + nCl 2
(5)
nNaµNa-1nClµCl-1nHµH-1 ) nNa + nCl + nH n(µNa + µCl-1) + nHµH-1 (6) 2n + nH
It is seen from eq 5 that G (renamed G* in this case) is constant for any electrolyte concentration when only ions from the electrolyte are considered. Equation 6 gives the G value including the contribution of H+ ions, and although the concentration of H+ ions, nH, is constant (for fixed pH), G depends on electrolyte concentration, n. Figure 1 shows the functions involved in the calculation of p for pH 5 as a function of κa. Different y-scales have been used to visualize F, G, and F‚G. It is clearly seen that the increase of F as κa decreases (which would cause an increase in p) is overcome by the decrease of G, causing a maximum to appear at low κa values. If pH is fixed, there is a minimum ion concentration initially that determines the maximum double layer thickness, that is, the maximum κ-1. The electroviscous effect can be calculated. When a small electrolyte quantity is added κa increases, so κ-1 or the double layer thickness decreases, and p should decrease too. However, new ions have been introduced. If we consider only the counterions, Na+ ions enter the double layer and replace H+ ions. Because Na+ ions have much lower mobility, an increase in p should be concluded. These two trends compete at low κa and at the beginning the second trend dominates. Once the electrolyte added is enough to produce similar concentrations of both kinds of counterions, the maximum is reached and for higher concentrations the presence of pH-ions is negligible.
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Figure 2. Booth’s electroviscous effect for an ideal system of radius a ) 100 nm and zeta potential ζ ) -100 mV for different pH values acid (open symbols) and basic (solid symbols).
Figure 2 shows the electroviscous coefficient p against κa for an ideal system of radius 100 nm and constant zeta potential of -100 mV for different pH values. A maximum is reached for each pH. It can be seen that p reaches the highest values for neutral pH and decreases when going to acid or basic pHs in a similar way, the small differences being due to the slightly different ionic mobilities21 of H+ and OH- ions. The maximum appears for higher electrolyte concentrations as pH is further from 7, and that confirms the physical description outlined before. To conclude this analysis of Booth’s theory on electroviscous effect, we have to point out two important deficiencies. Because zeta potential enters as zeta square and the valences of ions are squared too, no qualitative differences are found in the behavior of p when the sign of the charge changes. This makes the behavior of p versus κa be symmetrical with respect to acid or basic pH, which is not the real behavior that will be obtained in next section. Another point to consider is that zeta potential enters just by means of Debye’s approximation, which is inaccurate in most cases, because zeta potential is high enough. Watterson-White’s Theory. To solve the fundamental equations, Watterson and White introduced perturbation terms for the number densities and the electrostatic potential due to the shear field. Substituting these terms, neglecting products of first-order quantities and introducing new functions, the authors get a set of coupled ordinary differential equations for these new functions, with the boundary conditions given by the requirements of the problem. The set comprises one fourth-order homogeneous differential equation and N (as many as the number of ionic species) second-order inhomogeneous differential equations:
L4F(r) ) -
L2φi(r) )
2e2
( )∑ dψ0
r η0kT dr 2
( )(
N
ni0(r)zi2φi(r)
(7)
i)1
)
λi ezi dψ0 dφi + (r - 3F) kT dr dr ezi
(8)
where L2 and L4 are the second- and fourth-order differential operators; F and φi are the unknown functions related with the velocity field v(r) and the electrostatic potential, respectively; ψ0 is the electrostatic potential (21) Lyklema, J. Fundamentals of Interface and Colloid Science (I); Academic Press: New York, 1991; Chapter 6.
Figure 3. Watterson-White’s electroviscous effect for an ideal system of radius a ) 100 nm and zeta potential ζ ) -100 mV for different pH values: acid (open symbols), and basic (solid symbols).
when no shear field is applied (all of these unknowns are defined in ref 8) and λi is the drag coefficient of the i ionic species. To solve the coupled equations, they followed the method of O’Brien and White,22 which has also been followed in the present work. Finally, following Landau and Lifshitz,23 only the asymptotic behavior of the flow field v(r) is needed to calculate the effective viscosity from the averaged stress tensor, and the electroviscous coefficient is
pWW )
6 CN+1 - 1 5a3
(9)
where CN+1 is a constant that gives the asymptotic form of F
F(r) ≈
CN+1 r
2
+
CN+2 r4
(10)
From this brief summary of Watterson and White’s method to obtain the p coefficient, it is clear that the inclusion of any new contribution from ions means an additional coupled differential equation and a modification of the old equations. No analytical function is available for this type of analysis. A plot of pWW against κa is shown in Figure 3 for pH 4-9 for an ideal system of a ) 100 nm and ζ ) -100 mV. Comparison of Figures 2 and 3 elicits several remarks: •As predicted in Watterson-White’s paper, pBO is higher than the exact calculation pWW, being these differences lower as the p value decreases. •The most important qualitative difference between Figures 2 and 3 is the behavior for basic pH. In light of Watterson-White’s theory (Figure 3) no maximum is achieved. The maximum predicted by Booth (Figure 2) is fictitious and it is a consequence of the ζ2 dependence of pBO and also of the fact that no sign is considered for the ions in the double layer. When these misconceptions are removed, it is seen that replacement of co-ions is not a decisive phenomenon for the viscous contribution of the double layer. •Both theories predict a maximum value of the electroviscous effect for each acid pH, and this maximum is (22) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (23) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics ed. Pergamon Press, 1959.
Electroviscous Effect of Polystyrene Latexes
Figure 4. Watterson-White’s electroviscous effect for an ideal system of radius a ) 100 nm, pH ) 5, and different zeta potentials.
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Figure 6. Electrophoretic mobility (squares) and zeta potential (circles) obtained by Dukhin-Semenikhin26 theory for the sulfonate latex (open symbols) and carboxylate latex (solid symbols). Table 1. Particle Diameter and Surface Charge Density of the Polystyrene Latexes sulfonate carboxylate
Figure 5. Maximum value of Watterson-White’s electroviscous effect (squares) and κa value for which it is reached (circles) for each zeta potential. Data obtained from Figure 4.
reached for the same κa value in each case. This κa corresponds to an electrolyte concentration approximately above the concentration of the H+ ion fixed for each pH value. The position of this maximum can change for a fixed pH if the zeta potential changes. Figure 4 is a plot of pWW versus κa for different values of the zeta potential, being the radius constant. From this figure we can draw two interesting conclusions: first, for any fixed κa, p increases with increasing zeta potential until this trend reverses; this is the maximum in p found in the original contribution by Watterson and White.8 On the other hand, the maximum in p for each zeta potential is higher for higher values of zeta potential and appears for lower κa values (see Figure 5). Materials and Experimental Procedures All of the chemicals in this study were of analytical grade and were used without further purification. Ultrapure water (ATAPA S. A., Spain) with electrical conductivity less than 1 µS/cm was used in all experiments. The polystyrene latexes used in this work were synthesized by our group using an emulsifier-free emulsion polymerization method in a discontinuous reaction using 4,4′-azo-bis(4-cianopentanioc acid) as initiator for the carboxylate latex24 and using sodium styrene sulfonate as an ionic comonomer for the sulfonate latex.25 Both latexes were cleaned by serum replacement until the conductivity of the supernatant was similar to that of the water. The particle diameter (as determined by transmission electron (24) Bastos, D.; Ortega, J. L.; de las Nieves, F. J.; Hidalgo-Alvarez, R. J. Colloid Interface Sci. 1995, 176, 232. (25) de las Nieves, F. J.; Daniels, E. S.; EL-Aasser, M. S. Colloids Surf. 1991, 60, 107.
D/nm (TEM)
σ0/µC/cm2
289 ( 7 187 ( 7
22 ( 2 4 ( 1 (pH ) 5)
microscopy (TEM)) and the surface charge density (as determined by conductometric titration) are shown in Table 1. The electrophoretic mobility measurements were carried out with a Malvern Zetamaster S device. The zeta potentials were calculated using the Dukhin and Semenikhin26 theory. Both quantities are shown in Figure 6 for the carboxyl latex at pH ) 5 and the sulfonate latex at pH ) 5.5. It is remarkable that although the sulfonate latex has a much higher surface charge, the mobilities and potentials were not as high as could be expected. This was be taken into account when the electroviscous coefficients were compared. The concentration of the stock latex solutions was determined by evaporating to dryness at about 90 °C and the volume fractions of the suspensions were carefully calculated and prepared. The density of the samples was calculated considering the amount of latex and salt present.27,28 The viscosity of the samples was measured with Schott-Gera¨te equipment using Ubbelohde capillary viscometers in a thermostatic bath (with refrigeration and agitation) keeping a constant temperature of 25.0 ( 0.1 °C. The experimental electroviscous effect, p, is obtained from the slopes of the ηr versus φ plots. More details on experimental procedures were described previously.29,30 Note that for these measurements, very dilute suspensions were used (φ was always less than 0.001) to avoid as much as possible the presence of electroviscous effects of higher order than the first. Furthermore, for our viscosity data, no correlation was found between the reduced viscosity and the volume fraction, showing that the Huggins coefficient7,31 that gives a measure of the secondary electroviscous effect is zero or negligible.
Results For the carboxylate latex, a study for pH ) 5 as obtained by adding HCl without buffer solution was carried out. Figure 7 shows the experimental electroviscous coefficient versus κa. A maximum is observed for κa ) 2, and a continuous decrease is observed for higher κa values. This (26) Semenikhin, N. M.; Dukhin, S. S. Kolloidn. Zh. 1975, 37, 1127. (27) Millero, F. J. Chem. Rev.s 1971, 71(2), 147. (28) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions, 3rd ed.; Reinhold: New York, 1958; p 358. (29) Garcı´a-Salinas, M. J.; de las Nieves, F. J. Prog. Colloid Polym. Sci. 1998, 110, 134. (30) Garcı´a-Salinas, M. J.; de las Nieves, F. J. Macromol. Symp. 2000, 151, 435. (31) Huggins, M. L. J. Am. Chem. Soc. 1942, 64, 2716.
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Figure 7. Experimental electroviscous effect (squares) for the carboxyl latex and theoretical calculations: Booth’s theory (dotted line); Booth’s theory including the third ionic species (dotted-dashed line); Watterson-White’s theory (bold dashed line); and Watterson-White’s theory including the third ionic species (solid line).
maximum in p was not predicted by the theories under consideration (see dotted and dashed lines in Figure 7). However, the maximum found in the experimental data of p is also obtained theoretically when the HCl (used to adjust pH ) 5) is included in the calculations; that is, when a third ionic species, H+, is considered. Booth’s theory gives the right trend (dotted-dashed line) although it still overestimates the experimental data. Watterson-White’s calculations confirm the previous trends and the preliminary explanations indicated in the Theory section. In this case, the theoretical pWW values are in very good agreement with the experimental values (see solid line in Figure 7). The p coefficient was previously determined for this latex at different pH values.29 Inherent difficulties impede the experimental observation of the maximum. First, to obtain the experimental electroviscous effect coefficient, a fixed pH is needed, even more for this latex with pHdependent surface charge. This may be achieved by choosing a pH value that is easy to keep constant during the viscosity measurements or by using buffer solutions. The latter option has an undesirable consequence: new ions are introduced in the solution, thus complicating the theoretical analysis and imposing a minimum ionic strength. This also means that the most interesting region, for κa values lower than 3, where the electroviscous effect is higher and we have the best measurement conditions, is inaccessible. Figure 5 in ref 29 shows some results for pH 6 and 7. For neutral pH a minimum ionic strength of ∼1 mM (equivalent to κa ∼ 8) is given by the buffer solution, so no measurements below this limit could be done. For that reason, the preliminary measurements shown in ref 29 do not display the maximum in p, which occurs for κa < 3. For this reason, pH ) 5 was chosen here to determine the theoretically predicted maximum value of the electroviscous coefficient. For a negatively charged latex the maximum in p appears for pH neutral or acid (see Figure 3) and in this case pH around 5 is relatively stable without the use of buffer solution. Lower pH values would cause aggregation of the samples and neutral pH requires buffer solution, which means that the maximum p value reported theoretically in our work is in a forbidden region of low κa. A similar study was carried out with another system (the sulfonate latex) to test the universality of the results
Garcı´a-Salinas and de las Nieves
Figure 8. Experimental electroviscous effect (squares) for the sulfonate latex and theoretical calculations: Booth’s theory (dotted line); Booth’s theory including the third ionic species (dotted dashed line); Watterson-White’s theory (bold dashed line); and Watterson-White’s theory including the third ionic species (solid line).
found. In this second case the surface charge density was constant against pH25 and higher in comparison with the carboxyl latex. All of the experiments with this system were carried out at pH = 5.5. Figure 8 shows the experimental p values for the sulfonate latex. Again, a maximum is clearly observed which seems to confirm the appearance of this behavior for polystyrene latexes at low κa values (the maximum appears at κa = 2). Figure 8 also shows Booth’s theoretical values: again, those values obtained considering only Na+ and Cl - ions are shown by the dotted line, while the dotted-dashed line includes H+ ions. A bold-dashed line represents Watterson-White calculations considering only Na+ and Cl - ions, and finally, the solid line stands for Watterson-White calculations considering Na+, Cl -, and H+ ions. Although the trend is correct, the experimental values are higher than those calculated by the Watterson-White theory. The Booth’s theory still overestimates the p values. The agreement, therefore, is not as good as the one obtained for the carboxyl latex. Can we trust these calculations? If we look carefully, the predictions for both latexes are quite similar, with the sulfonate just slightly above the carboxyl, despite their differences in size and charge (see Table 1). These results could be striking, but a simple analysis of the variables involved (pH, radius, and zeta potential) confirms that the electroviscous effect is a little higher for the sulfonate latex due to the higher zeta-potential, a feature that overcomes its larger size. The quantitative discrepancies found for the sulfonate latex currently are being analyzed. The controversy held traditionally for these latexes related to the insensitivity of electrophoretic mobility to surface charge variations has been explained in several ways.25,32 The use of an ionic comonomer in the polymerization could lead to a “hairy” surface,25 and it would cause an increase in viscosity. Recently, ion condensation phenomena have been studied for these systems.32 Condensed ions would have a much lower ionic mobility, leading to an increase in the electroviscous effect. To conclude this section, we comment on previous contributions. Generally speaking, the residual ionic (32) Ferna´ndez-Nieves, A.; Ferna´ndez-Barbero, A.; de las Nieves, F. J. Langmuir 2000, 16, 4090. (33) Okubo, T. J. Chem. Phys. 1987, 87(11), 6733.
Electroviscous Effect of Polystyrene Latexes
species present in the system once it has been cleaned and before any electrolyte was added have not been taken into account. These residues (due to pH or ionic species from the synthesis that have not been eliminated) are considered negligible. They are indeed very small quantities, but in fact, it is when there are very few ions, that is, when the double layer is widely extended, that the electroviscous effect is significant. Okubo33 mentioned that the disagreement between the theoretical estimation of the effect and the experimental results was principally due to “experimental problems”, because “the complete deionization was very difficult”. Later, Ho¨nig and co-workers17 used the “residue electrolyte concentration present in the solution”, which was NaCl, as a parameter to adapt their experimental data to the Booth’s equation. They also used the particle charge as a parameter. The charge was found to be constant but the residual electrolyte was different for each set of experiences, with relatively high values: 0.6-1 mM. However, no exhaustive study was made on the implications of this residue. In a previous work, Ali and Sengupta11 estimated the residual electrolyte (KCl) concentration from specific conductance measurements, obtaining values from 0.2 mM to 3.8 mM. No maximum was found theoretically in any of these contributions. In their work with polystyrene latex, Delgado and coworkers12 found, using Watterson and White’s theory, a maximum in pWW against log [NaCl] at 10-4 M. This maximum was not explained as it has been here (in fact, they did not mention the inclusion of more ionic species in the calculations). The maximum was due to that reached by the zeta potential. The potential was calculated using the O’Brien and White22 theory, and the electrophoretic mobility maximum could not be removed.34,35 However, their experimental results, although higher than predicted, followed a similar trend. The experimental pexp versus log [NaCl] plot seemed to have a flat maximum for approximately the same electrolyte concentration (around 5‚10-4 M). Following our previous theoretical analysis, the maximum predicted for pH ) 6.3 and a zeta potential ζ = 100 mV (their experimental conditions), would happen for κa = 0.5, and this value corresponds to that electrolyte concentration. Thus, the maximum found theoretically was due to the usage of the O’Brien and White theory to obtain the zeta potential; the maximum found experimentally could be due to pH ions. Note that in our work the maximum electroviscous effect appears at a electrolyte concentration (10-5-5‚10-5 M (34) Moleo´n-Baca, J. A.; Rubio-Herna´ndez, F. J.; de las Nieves, F. J.; Hidalgo-A Ä lvarez, R. J. Non-Equilib. Thermdyn. 1991, 16, 187. (35) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1990, 86(16), 2859.
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NaCl) lower than that obtained for the maximum in the electrophoretic mobility. The zeta-potential is obtained by the Dukhin and Semenikhin26 theory, which gives a continuous decrease with increasing electrolyte concentration. Therefore, the maximum in the p coefficient cannot be explained by the behavior of the zeta potential. In a more recent paper, Zurita and co-workers16 studied the electroviscous effect in silica particles as a function of pH. They found experimentally a maximum value of the effect for pH ) 7. The authors claimed that it could be due to the solubility equilibrium of amorphous silica. However, their theoretical values (being 1 order of magnitude lower) increased with increasing pH. The reason for this increasing trend was the behavior of the estimated zeta potential (O’Brien-White22), which also increased with pH. The maximum was not obtained theoretically because the H+ or OH- ions were not included in the calculations, nor were those ions from the buffer solution if there were any. Our previous analysis shows if the zeta potential does not change significantly, that the higher values for the electroviscous effect are obtained for neutral pH. From this brief review, we can conclude that the maximum electroviscous effect has been scarcely reported and never explained suitably. In this work, the maximum electroviscous effect, which appears at low electrolyte concentration, has been theoretically justified by using the Booth and Watterson-White theories as a consequence of a third ionic species being present in the double layer. Experimental confirmation of this behavior has been obtained for two polystyrene latexes, showing that there is no dependence on specific experimental features. In support of this, we note the reinterpreted data12,16 discussed in the previous paragraph. Conclusions New experimental data have been shown for the primary electroviscous effect of two polystyrene latexes. A maximum in the plot of the electroviscous effect versus the electrokinetic radius (κa) has been found for both systems at low electrolyte concentration. An explanation has been outlined and a theoretical confirmation of this behavior has been obtained, being essentially a consequence of the presence in the double layer of a third ionic species with a much higher ionic mobility. Acknowledgment. The financial support provided by the Comisio´n Interministerial de Ciencia y Tecnologı´a (CICYT) under Project MAT2000-1550-C03-02 is greatly appreciated. LA000165L
