Langmuir 2000, 16, 5209-5212
Observation of the Mobility Maximum Predicted by the Standard Electrokinetic Model for Highly Charged Amidine Latex Particles Michal Borkovec* Department of Chemistry, Clarkson University, P.O. Box 5814, Potsdam, New York 13699 Sven H. Behrens The James Franck Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637 Micha Semmler Institute of Applied Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland Received December 14, 1999. In Final Form: February 9, 2000
Electrokinetic methods represent a central tool for the assessment of the charging properties of colloidal particles and interfaces. This information can often be correlated with colloid stability, deposition kinetics, and flotation phenomena.1-3 The most frequently used technique is electrophoresis, whereby the migration velocity of a charged colloidal particle in an applied electrical field is probed. While such experiments are rather straightforward, their interpretation is not. This problem was already tackled by Smoluchowski at the turn of the century,4 yet the exact calculation of the electrophoretic mobility of a spherical particle in an electrolyte solution was presented only 20 years ago by O’Brien and White.5 These authors have obtained the flow field in the presence of a charged particle within the Poisson-Boltzmann model. Their solution is now available as a commercial computer code. This so-called “standard electrokinetic model” evaluates the electrokinetic potential at the plane of shear, ζ, and neglects any contributions due to electrical conductance behind this plane. The standard electrokinetic model predicts a pronounced maximum in the electrophoretic mobility when plotted as a function of the electrokinetic potential ζ at fixed screening length. To observe this intriguing feature experimentally, some workers have systematically varied the surface potential.6,7 While the results were in general agreement with the standard electrokinetic model, the mobilities did not display any clear maxima. More frequently, however, the surface potential and the screening length are varied simultaneously; a typical example are mobility measurements as a function of electrolyte * Corresponding author. Phone: (315) 268-6621. Fax: (315) 268 6567. E-mail:
[email protected]. Web: http://www.clarkson. edu/∼borkovec. (1) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: New York, 1981. (2) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (3) Elimelech, M.; O’Melia, C. R. Environ. Sci. Technol. 1990, 24, 1528. (4) Smoluchowski, M.; 1921 Handbuch der Electrizita¨ t und des Magnetismus; Barth: Leipzig, Germany, 1921; Vol. II, p 366. (5) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 77, 1607. (6) Russel, A. S.; Scales, P. J.; Mangelsdorf, C. S.; Underwood, S. M. Langmuir 1995, 11, 1112. (7) Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566.
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Table 1. Properties of the Employed Amidine Latex Particlesa radiusa a (nm)
cond chargea σ (C m-2)
fit chargeb σ (C m-2)
distc d (nm)
Figure
34 60 97 235 450
0.048 0.081 0.083 0.159 0.223
0.017 0.030 0.045 0.030 0.032
0.95 0.60 0.25 1.00 0.85
3a 3b 1 2a 2b
a Obtained from the manufacturer by transmission electron microscopy and conductometric titration, respectively. b Surface charge density σ fitted by assuming the plane of shear to coincide with the plane of origin of the diffuse layer (d ) 0). c Distance to the plane of shear d fitted by using the conductometric surface charge.
concentration for particles with constant charge. In some experiments of this type, mobility was seen to go through a maximum at a certain concentration of added salt.8,10,11,17 This behavior was mostly taken as an indication of effects unaccounted for in the standard electrokinetic model, such as, specific ion binding,11 anomalous surface conduction,17 or the presence of a hairy surface layer.18 Due to the lack of a convincing experimental confirmation of the standard electrokinetic model, various authors have discussed extensions thereof (e.g., surface conductivity and porous surface layers), while others have even questioned the appropriateness of the model altogether.8-10,12-14 The latter conclusion is probably premature, since good agreement between electrokinetic potentials obtained from electrophoresis and dielectric spectroscopy with the standard electrokinetic model was obtained in some cases.15,16 In other situations, however, agreement was only found when additional effects were considered or not found at all.9,13 In this note, we present experimental evidence of the validity of the standard electrokinetic model based on electrophoretic mobility studies of highly charged amidine latex particles. In a plot against the salt concentration, we find a pronounced mobility maximum, which is seen to be a direct consequence of the standard electrokinetic model and is in good agreement with the theory. The positively charged amidine latex particles used in this study were purchased from Interfacial Dynamics Corp. (Portland, OR). Table 1 summarizes the particle radii and surface charge densities as reported by the manufacturer. The reliability of this information was verified for other types of particles.7 The stock particle suspensions were diluted with solutions of KCl (analytical grade) in deionized water down to volume fractions in the range 10-6-10-4 (depending on the particle size). The (8) Elimelech, M.; O’Melia, C. R. Colloids Surf. 1990, 44, 165. (9) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1985, 107, 322. (10) Midmore, B. R.; Hunter, R. J. J. Colloid Interface Sci. 1988, 122, 521. (11) Litton, G. M.; Olson, T. M. J. Colloid Interface Sci. 1994, 165, 522. (12) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. J. Chem. Soc., Faraday Trans. 1992, 88, 3441. (13) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. Langmuir 1993, 9, 1625. (14) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1988, 130, 281. (15) Gittings, M. R.; Saville, D. A. Langmuir 1995, 11, 798-800. (16) Russel, A. S.; Scales, P. J.; Mangelsdorf, C. S.; White, L. R. Langmuir 1995, 11, 1553. (17) Midmore, B. R.; Pratt, G. V.; Herrington, T. M. J. Colloid Interface Sci. 1996, 184, 170. (18) Rasmusson, M.; Wall, S. J. Colloid Interface Sci. 1999, 209, 312.
10.1021/la9916373 CCC: $19.00 © 2000 American Chemical Society Published on Web 04/07/2000
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Notes
suspension pH was around 5-6. The electrophoretic mobilities were measured on a standard commercial laser Doppler velocimetry electrophoresis apparatus. The mobility measurements were reproducible within 10-20%. Dialysis of the stock particle suspensions against deionized water had an insignificant effect on the results. The experimental results were compared with the standard electrokinetic model as implemented by O’Brien and White.5 As suggested by the ionization properties of an amidine group, we have assumed a fixed surface charge density σ and evaluated the diffuse layer potential φd using the inverse of the Gouy-Chapman relation2
φd )
( )
1 βeσ arcsinh βe 2κ0
(1)
where β denotes the inverse thermal energy, e the elementary charge, 0 the permittivity of water, and κ ) [2βe2c/(0)]1/2, the inverse Debye length (c being the concentration of monovalent electrolyte). We have assumed a finite constant distance d to the plane of shear from the plane of origin of the diffuse layer and calculated the potential at the plane of shear according to2
ζ)
4 arctanh[tanh(βeφd/4) exp(-κd)] βe
(2)
The electrophoretic mobility was calculated according to the procedure of O’Brien and White5 and combined with an accurate two-dimensional interpolation procedure. The calculation further required the particle radius a and the limiting conductances of 73.5 and 76.3 Ω-1 cm-2 mol-1 for K+ and Cl-1, respectively. In the limits of high and low ionic strengths, the electrophoretic mobility is proportional to ζ, and is given by1,2
u)γ
0 ζ η
(3)
where γ ) 1 for κa . 1 (Smoluchowski limit) and γ ) 2/3 for κa , 1 (Hu¨ckel limit). The shear viscosity of water is denoted as η. While eqs 1 and 2 refer to a planar interface, they are accurate in the present situation. We have compared these results with the corresponding solution of the Poisson-Boltzmann equation for an isolated sphere, but the deviations were insignificant in the range of parameters considered. The measured electrophoretic mobilities are shown as a function of the ionic strength for various particle sizes in Figures 1-3. All data sets display a maximum, which becomes increasingly pronounced with increasing particle size. Let us discuss the dependence on model parameters in more detail for the particles with 97 nm radius. Figure 1a illustrates the dependence on the surface charge density, assuming that the plane of shear coincides with the plane of origin of the diffuse layer (d ) 0). Interesting behavior is predicted by the theory at sufficiently high surface charge densities. With decreasing salt concentration, the mobility increases strongly initially, exhibits a maximum, goes through a minimum, and finally increases more gradually again. With decreasing charge density, the height of the maximum shifts toward lower salt concentrations, and progressively diminishes. The maximum disappears altogether below a certain threshold value (around 0.007 C/m2 in this case). At even lower charge densities, the mobility increases monotonically but more strongly at higher salt concentrations. Note that the Smoluchowski theory describes the faster initial rise
Figure 1. Electrophoretic mobility of highly charged amidine latex particles of 97 nm radius. Experimental data points are compared with the standard electrokinetic model for a fixed surface charge density σ: (a) variation of the surface charge density assuming the plane of shear to coincide with the plane of origin of the diffuse layer (d ) 0); (b) variation of the distance to the plane of shear with the conductometric surface charge σ ) 0.083 C m-2.
at high salt concentrations (κa . 1), while the Hu¨ckel theory describes the more gradual rise at low salt concentrations (κa , 1). Good agreement between theory and experiment is observed for a charge density of 0.045 C/m2, which is clearly lower than the value of 0.083 C/m2 obtained by the manufacturer by means of conductometric titration. Figure 1b shows the effect of increasing the distance d of the plane of shear from the plane of origin of the diffuse layer for the manufacturer’s charge density of 0.083 C/m2. Reasonably good agreement with theory is observed for d = 0.25 nm, a value that is comparable to a few layers of tightly bound water molecules on the surface. The same value was also reported to yield good agreement between electrophoretic mobilities and potentiometric titration data for two kinds of carboxylated latex particles.7 We note that increasing the distance of the plane of shear has an effect similar effect to that of decreasing the particle charge. Figures 2 and 3 show similar data for smaller and larger particles, respectively. The experimental data for all types of particles show a mobility maximum, which becomes increasingly pronounced with increasing particle size. This
Notes
Figure 2. Electrophoretic mobility of two kinds of highly charged amidine latex particles of larger radius. Experimental data points are compared with the standard electrokinetic model for a fixed surface charge density σ. Similar results are obtained when the surface charge is adjusted for the vanishing distance of the plane of shear and when the distance is adjusted with the known conductometric charge. Radii: (a) 235 nm; (b) 450 nm.
trend is in full accord with the standard electrokinetic model theory. Increasing the particle radius at fixed surface charge density has an effect similar to that of increasing the surface charge density at fixed particle radius. The lines in the figures correspond to best fits, which were obtained either by adjusting the surface charge density and assuming the shear plane to coincide with the plane of origin of the diffuse layer (d ) 0) or by taking the charge density from conductometric titrations and adjusting the distance to the plane of shear. The corresponding parameter values are summarized in Table 1. The agreement between experiment and model calculations is acceptable for the larger particles; for smaller particles, the agreement is less satisfactory. In all cases, we must assume either a surface charge density which is substantially lower than the ones found by conductometric titrations or a rather large distance to the plane of shear. The values for the distance of the plane of shear are too large to be rationalized by a few layers of water molecules. In both situations, the model predictions are very similar. The data could also be well described with intermediate parameter values by adjusting the surface charge density and the shear plane distance simultaneously. However, it is not possible to describe these data sets with the actual
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Figure 3. Electrophoretic mobility of two kinds of highly charged amidine latex particles of smaller radius. Experimental data points are compared with the standard electrokinetic model for a fixed surface charge density σ. Similar results are obtained when the surface charge is adjusted for the vanishing distance of the plane of shear and when the distance is adjusted with the known conductometric charge. Radii: (a) 34 nm; (b) 60 nm.
charge density and a realistic value for the distance to the plane of shear, as could be done for the latex of 97 nm radius. The rationalization of these discrepancies must remain tentative. A smaller surface charge, which is neutralized within the diffuse layer, could be explained either by specific adsorption of counterions or by ion-ion correlation effects. Since the charge densities of the latices used are rather high, we expect extensive specific adsorption of counterions, as discussed for metal oxides.19,20 Such information could possibly be obtained from adsorption or potentimetric titration studies. On the other hand, ionion correlation effects may set in at these charge densities, with the result of a smaller apparent charge on the particles.19,21 It is also conceivable that the discrepancies could be explained by surface conductance of the particles. To obtain a consistent interpretation of electrophoretic and dielectric spectroscopy data, surface conductance has to be included in some cases. With increasing surface (19) Borkovec, M.; Jo¨nsson, B.; Koper, G. J. M. In Surface and Colloid Science; Matijevic, E., Ed.; Plenum Press: New York, 2000; Vol. 16. (20) Hiemstra, T.; Venema, P.; van Riemsdijk, W. H. J. Colloid Interface Sci. 1991, 184, 680. (21) Lozada-Cassou, M.; Gonzalez-Tovar, E.; Olivares, W. Phys. Rev. E 1999, 60, R17-R20.
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conductance, the electrophoretic mobility tends to decrease, and the consideration of this effect might also lead to improved agreement between theory and experiment. However, the proper consideration of all these effects would require additional experimental and theoretical studies, which are beyond the scope of this note. In conclusion, the observed mobility maxima are in good accord with the standard electrokinetic model. We observe the same maxima for five different types of highly charged latex particles. With increasing particle size, the maximum becomes increasingly pronounced, a trend which is in full agreement with the theory. The experimental data can be quantitatively described with reasonable values of the surface charge and/or the distance to the plane of shear. For one kind of particles, the data could even be described with the expected parameter values, while for other particles, the surface charge density obtained from the mobility experiments was lower than values from conductometric titrations. Despite these minor difficulties, it is rather obvious that the mobility maxima displayed by the highly charged latex particles have the same origin as the maximum predicted by the standard electrokinetic model. Our findings are in line with various other reports where mobility as a function of salt concentration shows a maximum at higher salt concentration,8,10,11,17 as well as
Notes
a concentration minimum at low salt concentration.22 Our study further demonstrates that it is crucial to use the full solution of the standard electrokinetic model for the interpretation of such data. Interpretation on the basis of any simplified models, such as the Smoluchowski theory, or other approximate relations1 will lead to erroneous results. It is also essential to realize that the conversion of electrophoretic mobilities into electrostatic potentials is generally difficult due to the multivalued character of the inverse function. Note Added in Proof: A related study of negatively charged sulfate latex particles came to our attention.23 The presented data set covers a much smaller parameter range and the standard electrokinetic model is only used in an approximation restricted to large values of κa. Nevertheless, the reported results further corroborate the main conclusions of the present study. Acknowledgment. This work was supported by the Swiss National Science Foundation and the U.S. National Science Foundation (Grant CTS-9820795). LA9916373 (22) Deggelman, M.; Palberg, T.; Hagenbu¨chle, M.; Maier, E. E.; Krause, R.; Graf, C.; Weber, R. J. Colloid Interface Sci. 1991, 143, 318. (23) Antonietti, M.; Vorweg, L. Colloid Polym. Sci. 1997, 275, 883.
