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An Improved Method for Calculating Zeta-Potentials from Measurements of the Electrokinetic Sonic Amplitude
Journal of Colloid and Interface Science 229, 174–183 (2000) doi:10.1006/jcis.2000.6980, available online at http://www.idealibrary.com on
An Improved Method for Calculating Zeta-Potentials from Measurements of the Electrokinetic Sonic Amplitude Mario L¨obbus,∗,1 J¨urgen Sonnfeld,∗ Herman P. van Leeuwen,† Wolfram Vogelsberger,∗ and Johannes Lyklema† ∗ Institute of Physical Chemistry, Friedrich-Schiller-University Jena, Lessingstrasse 10, D-07743 Jena, Germany; and †Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB, Wageningen, Netherlands Received February 8, 2000; accepted May 16, 2000
The surface electric properties of the commercially available silica, Monospher 1000 (Fa. Merck), have been studied by conductivity and ESA (electrokinetic sonic amplitude) experiments. It could be shown that accounting for the contribution of the stagnant layer to surface conductivity is indispensable in the interpretation of electrokinetic data at low ionic strength. A general method has been put forward which allows to take into account the total, experimentally accessible surface conductivity in the evaluation of ESA data of moderately concentrated suspensions. This includes additional conductivity measurements which serve for the independent estimation of the total relative surface conductivity. The resulting zeta-potentials are clearly higher than those obtained after neglecting the contribution of the stagnant layer to surface conductivity. In addition, the ionic mobilities of potassium and magnesium in the hydrodynamically stagnant layer have been investigated in some detail. It has been found that the ionic mobility of potassium is of the same order of magnitude as in the bulk solution while the mobility of magnesium is significantly reduced. °C 2000 Academic Press Key Words: zeta-potential; surface conductivity; dynamic mobility; stagnant layer; electrokinetics; silica.
1. INTRODUCTION
The interaction between charged colloidal particles is mainly governed by the diffuse part of the electric double layer. The potential at the inner boundary of this double layer part, ψ d , cannot be directly measured. However, experience has shown instead that the electrokinetic zeta-potential, ζ , is an appropriate characteristic (1, 2). The zeta-potential has to be calculated from electrokinetic parameters measured by techniques such as microelectrophoresis, streaming potential, and electroacoustic methods like electrokinetic sonic amplitude (ESA) using some more or less sophisticated theory. A popular theoretical model is the so-called “standard electrokinetic model.” A fundamental principle of this model and other modern electrokinetic theories is that a relatively thin double layer with κa À 1 is nearly in equilibrium with the surrounding electrolyte and, hence, the 1 To whom correspondence should be addressed. E-mail: [email protected]. uni-jena.de.
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ion distribution in the double layer is given by the Boltzmann distribution (3). This important observation, which has become a landmark in the development of electrokinetics, dates back to Dukhin (4). The so-called electrokinetic equations comprise Poisson’s equation for electrostatics, the Navier–Stokes equation for the fluid transport and the Nernst–Planck equation for the ion fluxes. If these equations are solved simultaneously under appropriate boundary conditions double-layer polarization is automatically taken into account. Thus, the standard electrokinetic model includes the effect of surface conduction in the diffuse layer. However, the contribution of the hydrodynamically stagnant layer to surface conduction is ignored. This could be shown to be the reason for some methodical electrokinetic inconsistency, i.e., disparity between zeta-potentials for one and the same system but derived from different electrokinetic methods (5). The influence of surface conductivity is relatively high at low ionic strength, where the bulk conductivity itself is small. Moreover, accounting for surface conduction in the stagnant layer may result in methodically electrokinetic consistent zetapotentials (5, 6). A usable method for taking surface conduction into full account consists of combining different electrokinetic experiments (7–9). However, the most general approach is to rigorously consider the transport processes in the hydrodynamically stagnant layer in the development of electrokinetic theories, as it has been put forward by Mangelsdorf and White (10–12). Unfortunately, for such numerical analysis it is necessary to introduce parameters which describe the adsorption in the stagnant layer. A relatively new method to measure the electrokinetic properties of moderately concentrated suspensions is the ESA technique (13–20), for which well-designed commercial devices became available over the past decade (ESA-8000, Acoustosizer, Fa. Matec Applied Science). Such devices are normally equipped with automatic titration systems, which facilitate the pH-dependent measurement of electrokinetic properties of suspensions. However, so far only the Helmholtz–Smoluchowski equation has been used in the software of the ESA-8000 (21) and, hence, the theoretical interpretation is limited, since the influence of surface conductivity is neglected. The objective of this article is to present an improved method for the interpretation of ESA measurements that accounts for
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the total, experimentally accessible surface conduction. From an experimental point of view this may be achieved by combining conductivity and ESA measurements. To interpret the conductivity data, O’Brien’s theory developed for the electric conductivity in porous plugs (22) in connection with the numerical data of Kang and Sangani (23) has been used. The electroacoustic measurements were analyzed with O’Brien’s high κa double layer theory (24). The method has been tested with the monodisperse, commercially available silica, Monospher 1000 (Fa. Merck), in the presence of potassium chloride and magnesium nitrate solutions. It has been shown that for both KCl and Mg(NO3 )2 , neglecting the total surface conductivity results in zeta-potentials that are too small, especially at higher pH. The same statement still holds if the surface conductivity in the diffuse double layer is taken into account while neglecting the stagnant layer contribution. Therefore, the contribution of the surface conductivity in the stagnant layer has been investigated in some detail and an estimation has been made of the ionic mobility of potassium and magnesium ions behind the shear plane. 2. THEORY
Besides the zeta-potential, ζ , the total surface conductivity, K σ , is a useful parameter for the characterization of colloidal systems (5–9, 25). The principal advantage of K σ is that not only is the mobile charge in the diffuse layer included, but so is the mobile part of the hydrodynamically stagnant layer. Furthermore, for a proper interpretation of the measurement of a single electrokinetic phenomenon in terms of the zeta-potential, knowledge about K σ is indispensable. By definition surface conduction is an excess quantity and accounts for the excess conduction tangential to the charged surface. In general, using the considerations of Lyklema (1) the surface conductivity in the presence of a binary electrolyte may be written as K σ = FA ·
2 X i=1
Z |z i |
∞
[ci (x) − ci (∞)] · u i (x) dx,
[1]
0
where FA denotes Faraday’s constant. In Eq. [1] ci (x) and u i (x) are local ion concentrations and ionic mobilities, respectively, in infinitesimally small slabs parallel to the solid surfaces; ci (∞) and z i are the bulk ion concentrations and the valencies, respectively. In the sense of the Gouy–Stern theory, the total surface conductivity, K σ , may be divided into the contributions of the inner layer, K σ i , and the diffuse double layer, K σ d : K σ = K σ i + K σ d.
[2]
The quantity K σ d has generic character, whereas K σ i is systemspecific and, therefore, is pertinent to the system under investigation. The relative contribution of the total double layer in conducting current through a suspension can be expressed advanta-
geously by introducing the Dukhin number, Du (1). For highly charged surfaces (with high zeta-potential) the contribution of the coions compared to that of the counterions can be neglected. The Dukhin number due to counterions is generally related to the corresponding absolute surface conductivity, K 2σ , the particle radius, a, and the conductivity of the bulk electrolyte solution, K ∞ , by Du2 =
K 2σ . aK∞
[3]
In addition, it is possible to relate the counterion Dukhin number, Du2 , to such important double-layer characteristics as ζ , K σ i , and K σ d (1): 1 Du2 = κa
µ
3m 2 K σi 1 + 2 + 2σ d z2 K2
¶µ
¸ ¶ · z 2 eζ − 1 . [4] exp − 2kT
In Eq. [4] z 2 is the valency of the counterion and κ the Debye– H¨uckel parameter. The term 3m 2 /z 22 accounts for the fact that, besides conduction, the electric field also causes an electroosmotic fluid transport (26). The dimensionless ionic drag coefficient, m 2 , relates to the bulk viscosity, η, and the limiting ionic mobility in the bulk, u ∞ 2 , by m2 =
2z 2 ²kT , 3ηeu ∞ 2
[5]
where ² = ²r ²0 is the permittivity of the solvent, k the Boltzmann constant, and T the temperature. For flat surfaces the total surface conductivity can be measured with an experimental setup recently developed by Werner et al. (25). Moreover, the experimental estimation of the surface conductivity for a series of cylindrical glass capillaries has been carried out long before with the pioneering work of Rutgers (27, 28). However, for disperse systems it is far more complicated to experimentally obtain the total surface conductivity. A possible way of getting information about the counterion Dukhin number for disperse systems is the rigorous analysis of conductivity data with the O’Brien and Perrin theory for porous plugs (22) as presented in the following. The electric conductivity K ∗ of a plug or suspension may be related to the electric conductivity of the bulk electrolyte solution, K ∞ , and the dimensionless surface conductivity parameter, the Dukhin number, Du, (22). The result can be written in terms of the theory developed by O’Brien and Perrins (22) for a binary electrolyte as · ¸ K∗ u∞ 2 = 1 + 3φ f (0, φ) + ∞ { f (Du2 , φ) − f (0, φ)} . K∞ u2 + u∞ 1 [6] This equation was originally derived for close-packed porous plugs. However, Kang and Sangani (23) extended the analysis
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to suspensions and were able to show that it holds for suspensions as well. Eq. [6] is valid for smooth particles with large κa and sufficiently high zeta-potentials, where the contribution of the coions in the double layer may be neglected. In Eq. [6] φ means the volume fraction of the colloid material in the suspension, and u∞ i refers to the ionic mobility in infinite dilution of the bulk as before. The subscripts “2” and “1” represents counterions and coions, respectively. The dimensionless function f (Du2 , φ) accounts for the contribution of the surface conductivity to the total conductivity of the suspension. Besides on the surface conductivity and volume fraction, the function f (Du2 , φ) also depends on the way of packing. At low relative surface conductivity, i.e., Dukhin numbers Du2 < 0.5, the overall conductivity of the suspension is lower than that of the bulk because nonconducting particles are covered by a poorly conducting double layer. This leads to negative values for f (Du2 , φ). With increasing surface conductivity the contribution of double layers around the particles rises, leading eventually to positive values for f (Du2 , φ). In general, f (Du2 , φ) needs to be numerically computed. A collection of different numerical values for f (Du2 , φ) as a function of relative surface conductivity, Du2 , and the volume fraction is compiled in references (22, 23). Equation [6] and most other modern electrokinetic expressions were originally derived without considering the contribution of the stagnant layer to surface conductivity. However, it could be shown (29) that expressions like Eq. [6] for the conductivity of porous plugs or expressions for the electrophoretic mobility can be maintained under the condition that the Dukhin number is written as is in Eq. [4]. The ESA phenomenon results from the interaction of a relatively high frequency a.c. field (O(106 Hz)) with a dispersion of charged colloidal particles causing an oscillatory motion of the particles. In addition to the particles themselves the electric double layer around the particles is subject to changes by the a.c. field. If there is a density difference between particle and electrolyte solution this results in the formation of an acoustic dipole. The created sonic field may be detected using a piezo transducer. O’Brien was able to derive an expression which directly reˆ lates the ESA signal, ESA(ω), to the dynamic electrophoretic mobility, uˆ D (ω) (24). For dilute suspensions (φ < 5%) this relation is given by uˆ D (ω) =
ˆ ESA(ω) · A(ω), φ(1ρ/ρ)νs
[7]
where A(ω) is a calibration factor, ρ the density of the solvent, 1ρ the density difference between particle and solvent, and νs the rate of propagation of sound in the solvent. The factor A(ω) depends on the geometry of the measurement cell and is determined by calibration with the ESA signal of a well-characterized ˆ = 5.36 mPa m V−1 ). This procedure assample (Ludox, |ESA| sumes that the static electrophoretic mobility equals the dynamic one. With the concept of the polarized double layer O’Brien (24)
has shown that for a dilute dispersion (φ . 5%) of spherical particles with high κa the dynamic electrophoretic mobility may be related to the zeta-potential by the following expression provided that the wavelength of the sonic field substantially exceeds the particle size,
uˆ D (ω) =
2²ζ ˆ · G(ωa 2 /ν) · [1 + fˆ(Du)]. 3η
[8]
Here, ν is the kinematic viscosity of the solvent. The dimension2 ˆ /ν) contains the influence of inertia on the less function G(ωa dynamic mobility. It is determined by the particle mass and the frequency of the a.c. field according to à 2 ˆ G(ωa /ν) = 1 −
ωa 2 /ν · i(3 + 21ρ/ρ) p 9 · (1 + (1 − i) · ωa 2 /2ν)
!−1 . [9]
The factor 1 + fˆ(Du) in Eq. [8] is proportional to the component of the field tangential to the surface and takes the influence of surface conductivity into account. The higher the surface conductivity the more countercharge is moved back and forth by the external a.c. field. This oscillation leads to reduction of the applied field as a result of which the mobility decreases. This factor is given by O’Brien’s theory as fˆ(Du, ω0 ) =
1 − iω0 − [2Du − iω0 · (²p /²s )] , with 2 · (1 − iω0 ) + [2Du − iω0 · (²p /²s )]
ω0 =
ω² , K∞ [10]
where ²p and ²s are the permittivities of the particles and the solvent. To keep in line with the original literature, here, the symbol f is used both in Eq. [6] and Eq. [8], but the difference between f (Du2 , φ) (see Eq. [6]) and fˆ(Du, ω0 ) (see Eq. [8]) is highlighted. Unfortunately, with the ESA measurements it is impossible to experimentally determine the total mobile charge and, hence, the total surface conductivity of the double layer. In commercially available devices like the Acoustosizer and the ESA-8000 (both Matec Applied Science) either only the contribution of the diffuse layer to surface conductivity is taken into consideration or surface conductivity is completely neglected. Our solution to this problem is provided by the independent experimental determination of the counterion Dukhin number by conductivity measurements using Eq. [6]. Then, this Dukhin number is used in the interpretation of dynamic mobilities (see Eqs. [8] and [10]). In that way it is possible to account for the total surface conductivity while converting ESA signals to zeta-potentials.
ZETA-POTENTIALS FROM ESA MEASUREMENTS
177
3. EXPERIMENTAL
The electrokinetic investigations were carried out at 25◦ C using the commercially available, microporous, monodisperse silica sample Monospher 1000 (Fa. Merck). A suspension of the original product as obtained from the manufacturer showed a pH of around 9.5, indicating impurities. Because of this the sample was pretreated as follows. The sample was washed for 4 h with 1% HCl and then rinsed with deionized water and centrifuged till it did not show any more traces of chloride, dried at 110◦ C, and milled. The specific surface area, SBET , of this sample was determined by low temperature nitrogen adsorption with the BET method using the automatic surface analyzer AUTOSORB-1 (Fa. Quantachrome). It was found that SBET = 2.9 m2 g−1 , which agrees with the geometric surface area of 2.9 m2 g−1 . The density of Monospher 1000 was determined with a pycnometer, ρ = 2.04 g cm−3 . Nevertheless, the determination of the surface charge density by potentiometric titration suggests that micropores exist, although they are not detected by nitrogen adsorption (see Fig. 1). The titrable charge density far exceeds the theoretically possible surface charge densities (with an assumed surface silanol group concentration of Ns ≈ 4 nm−2 ). Hence, the conclusion is that beside a pure surface part of the measured specific charge density there must be a contribution of the micropores. This behavior is rather typical for some types of precipitated silica. It is likely that the micropores are so small that the nitrogen molecules are not able to penetrate and, hence, they are not “seen” by nitrogen adsorption. The radius of the Monospher 1000 particles has been determined by photon correlation spectroscopy to a = 481 nm, which is in close accordance to the statements of the manufacturer (500 nm). There is no indication for a pH-dependent agglomeration of the Monospher 1000 particles in 1 mM KCl in very dilute suspensions as used in photon correlation spectroscopy (see Fig. 2). The ESA measurements were carried out with Monospher suspensions for a volume fraction of φ = 0.025 with the ESA8000 apparatus from Matec Applied Science. The ionic strength of the electrolyte solutions (KCl, Mg(NO3 )2 ) was 1 mM. This
¯ and surface charge density, σ 0 , of MonoFIG. 1. Specific charge density, q, spher 1000 as a function of pH at 1 mM KCl. The data points depicted stem from averaging of at least three single determinations; σ 0 was calculated directly from titrable charge using the BET surface area.
¯ of a Monospher 1000 suspension as a FIG. 2. Mean particle radius, a, function of pH at 1 mM KCl as determined by photon correlation spectroscopy.
results in the value of κ · a ≈ 50, which is in the thin double layer region (20). Equations [7] and [8] have been derived for dilute suspensions. There is still some uncertainty about the upper limit of the volume fraction for the applicability of Eqs. [7] and [8]. In (20) the upper limit has been reported that the error in Eqs. [7] and [8] is below 2% up to φ ≤ 0.05. On the other hand, the results of Dukhin et al. (30) suggest that the upper limit of dilute suspensions is φ ≤ 0.02, which has been found for the interpretation of colloid vibration current (CVI) data. We estimated that the relative error due to concentration effects is less than 5% using the semiempirical method of O’Brien et al. (31). The ESA-8000 system measures the amplitude of the ESA signal and the phase angle relative to a suspension with known electrokinetic properties (Ludox, Fa. Aldrich) rather than absolutely. As phase reference a Monospher 1000 suspension (φ = 0.025, pH ≈ 8) without electrolyte has been used. It turned out that temperature fluctuations have a great influence on the phase angle. Temperature variations of approx. 2 K in the laboratory give rise to a change in the phase angle up to 15◦ (32). It is very likely that this behavior is caused by the temperature influence on the propagation speed of sonic waves in the delay rod of the ESA probe. After mounting a water jacket on the ESA probe this problem was avoided. With this additional equipment the phase angle for silica in the pH range 6 . pH . 10 usually varies by less than 5◦ . The total surface conductivity of the Monospher suspensions was determined with a conductivity cell Tetracon 96 (Fa. WTW, Weilheim) and a conductivity meter (Fa. WTW, Weilheim). To improve the signal/noise ratio a volume fraction (φ ≈ 0.3) higher than in the ESA measurements was used. The conductivities of the suspension and that of the corresponding electrolyte solution were determined with the following procedure: 1. A Monospher 1000 suspension with φ ≈ 0.3 (approx. 25 ml) is titrated with 1 N KOH to pH ≈ 6. (At pH < 6.0 sedimentation occurs, which is probably due to higher aggregation between the particles than in the very dilute suspension as used for photon correlation spectroscopy, see Fig. 2). 2. After 10 min sonification and adjusting the temperature with a thermostat the suspension conductivity, K ∗ , is
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determined. There was no time dependency in the conductivity results, indicating that under the chosen conditions the particles do not settle. 3. Next the suspension is centrifuged at 4500 rpm for 15 min, the clear supernatant electrolyte solution is removed with a micropipet, and after thermic equilibration the conductivity of this solution, K ∞ , is measured. 4. The Monospher 1000 particles are redispersed into the electrolyte solution again, and a new (higher) pH is adjusted by adding KOH and steps 2–4 are repeated. In that way the silica suspension has been titrated consecutively to higher pH with KOH. At each pH the suspension conductivity, K ∗ , and the conductivity of the supernatant electrolyte solution (after centrifugation) has been measured. This procedure gave the same results as those which result from the titration of different suspensions, i.e., where each suspension is titrated only once up to a certain pH. 4. RESULTS
4.1. Conductivity Mesaurements of Concentrated Silica Suspensions As previously stated, the conductivity measurements of silica suspensions with φ ≈ 0.3 were employed to determine the surface conductivity. First, the experimentally measured conductivity ratio K ∗ /K ∞ and the value f (Du = 0, φ) were used to calculate the function f (Du2 , φ) with Eq. [6] for the given volume fraction. To obtain the counterion Dukhin number Du2 we interpolated between the numerical values given by Kang and Sangani (23) using a two-dimensional, quadratic interpolation procedure. Hence, a counterion Dukhin number which includes surface conduction in the stagnant layer can be computed as a function of pH, volume fraction and electrolyte concentration. The results for a constant ionic strength, I = 1 mM, are depicted in Fig. 3. Changes in the volume fraction caused by added KOH are taken into consideration.
FIG. 4. Magnitude of the factor |1 + fˆ(Du)| (—) as a function of the relative surface conductivity for the given experimental system compared with the approximation in ESA-8000 software (– –).
It is evident from Fig. 3 that the surface conductivity depends on the valency of the counterion. A similar effect was also observed for latices (9). The experimentally determined surface conductivity is fairly high in the presence of either counterion. Unfortunately, the software of the ESA-8000 assumes the relative surface conductivity to be negligible, which results in |1 + fˆ(Du)| = 1.5. Figure 4 illustrates the result of this neglection compared with the proper behavior. For the highest Dukhin numbers the experimentally determined factor |1 + fˆ(Du)| is overestimated by about one-third by the ESA-8000 software. To evaluate the accuracy of the proposed method, error bars indicating the standard deviation, are included in Fig. 3. It is assumed that with the equipment used for the conductivity measurements the typical error made in the determination of the conductivity is about 1 µS/cm. Hence, applying the methods of error propagation, the error in f (Du2 ) is ¡ ¢± ∞ p ∞ u 2 · 1 + (K ∗ /K ∞ )2 . 1 f (Du2 ) = 1K /(3φ K ∞ ) · u ∞ 2 + u1 With it the error in Du2 = b · 1 f (Du2 ), where b equals 1.26 in the used range of volume fractions (0.25 ≤ φ ≤ 0.3). The value of b has been estimated from the numerical values of Kang and Sangani (23). 4.2. ESA Measurements of Dilute Silica Suspensions
FIG. 3. Counterion Dukhin number, Du2 , as determined from conductivity measurements of approx. 30 vol% silica suspensions as a function of pH at an ionic strength of 1 mM: (j) KCl, (×) Mg(NO3 )2 . The error bars result from error propagation assuming a mean error in the conductivity measurements 1K of 1 µS cm−1 .
The countercharge in the diffuse electric double layer may be determined by ESA measurements with O’Brien’s theory. The assumption in O’Brien’s theory coded in the software in the apparatus used, ESA-8000, assumes that particle–particle interactions are negligible and that there is no overlap of the double layers of neighboring particles. With the volume fraction used (φ = 0.025), the particle size of the Monospher 1000 particles (approx. 1 µm) and assuming a homogeneous distribution of the particles, the mean particle distance is approximately 1.7 µm, which is much larger than κ −1 . So, overlap can be ignored. The ESA measurements have been carried out in the pH range
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6.5 . pH . 10 since for lower pH sedimentation took place in the conductivity measurements of the concentrated suspensions. In the presence of a background electrolyte the measured ESA ˆ PE , phase angle αPE ) is the vector sum signal (magnitude |ESA| of signals of the colloid particle (“P”) and the background electrolyte (“E”). However, at the chosen electrolyte concentration, 1 mM, the background electrolyte signal is far smaller than the colloid signal and so the background electrolyte correction as proposed in reference (33) has been dropped. The ESA measurements have been carried out several times. The mean of the magnitude of the ESA signal, |ESAP |, and the phase angle, α¯ PE , result from complex addition of at least three (quadratically interpolated as a function of pH) single determinations |ESAP | =
p
x 2 + y2
y y α¯ P = y y
≤0∧x ≤0∧x >0∧x >0∧x
> 0, − arctan(|y/x|) ≤ 0, −[180 − arctan(|y/x|)] , > 0, arctan(|y/x|) ≤ 0, 180 − arctan(|y/x|)
[11]
P P with y = N1 iN |ESAP,i | sin αP,i and x = N1 iN |ESAP,i | cos αP,i The influence of dissolved silica may be estimated by a dissolution model recently proposed for solid oxides (34, 35). Under the given experimental conditions (sonification time, thermic equilibration, titration delay time) the total silicate concentration is smaller than 2 mmol L−1 , where at the highest pH approximately 22% of the orthosilicic acid exists in dissociated form (pK a = 9.8). That means that the influence of dissolved silica is even smaller than that of the background electrolyte and, therefore, may also be neglected in the background correction. The ESA-8000 automatically takes account of changes in the volume fraction due to added base. It is evident from Fig. 5 that in the presence of divalent cations the absolute value of the mobility is much lower than in the
presence of potassium. This difference is probably caused by more effective screening by the bivalent ion. 4.3. Calculation of Zeta-Potentials from Conductivity and ESA Measurements Properly established zeta-potentials can describe the electrostatic interactions in suspensions rather well. As previously outlined it is important to account for the total surface conductivity in calculating the zeta-potential from experimental data. The most general approach obviously consists in accounting for the transport processes in the hydrodynamically stagnant layer while solving the electrokinetic equations. Recently, some encouraging work has been published (10–12, 36). However, to apply these theories it is necessary either to know the experimental Stern adsorption isotherm or to introduce fitting parameters describing the extent of adsorption in the Stern layer in such numerical analyses. In the latter case there is a risk that the adjustable fitting parameters also contain deviations from the model and errors in the experimental data. So, in principle interpretation of electrokinetic data without fitting parameters should be preferred. For the analyses of the experimentally determined ESA signals we applied O’Brien’s theory, see Eqs. [8]–[10]. The inertia term Gˆ is automatically accounted for in the ESA-8000 software. For Monospher 1000 Eq. [9] yields Gˆ = 0.789 under the given experimental conditions. The sample investigated is a typical experimental system for which it can be shown (see Fig. 3) that here surface conductivity must be taken into consideration in the interpretation of electroacoustic data. By employing additional conductivity measurements it is possible to assess the total counterion Dukhin number, Du2 , and with that it is straightforward to calculate zeta-potentials, ζ (Du). First Eq. [10] is evaluated to obtain the factor |1 + fˆ(Du2 )| and then Eq. [8] is used. Since the total surface conductivity is accounted for, these combined zeta-potentials may be taken as the best available estimation of the effective electrostatic potential that governs the interaction between particles. The relation between the zeta-potentials calculated by the ESA-8000 software, ζ (K σ = 0), and zeta-potentials which include surface conduction, ζ (Du), in the electric double layer is then given by ζ (Du) =
FIG. 5. Dynamic electrophoretic mobilities of Monospher 1000 in presence of KCl (e) and Mg(NO3 )2 (j) at an ionic strength of 1 mM. The figure shows the magnitude of mobilities which result from averaging of, at least, three single curves. The error bars indicate the standard error of sample mean.
1.5 · ζ (Du = 0). |1 + fˆ(Du)|
[12]
In order to show that it is important to take into account the total surface conductivity rather than the contribution of the diffuse layer only, it is possible to calculate numerical values of ζ (Dud ) and Dud by numerical solution of Eqs. [4], [8], and [10] while neglecting K σ i . Hence, the zeta-potentials, ζ (Dud ), computed in this way presuppose that surface conduction does not take place in the stagnant layer; i.e., they are representative of the results obtainable with the software from the Acoustosizer (Matec Applied Science).
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¨ LOBBUS ET AL.
combination of conductivity and electroacoustic measurements approach the Smoluchowski result at lower pH. From all of this one may conclude that Eq. [6] seems quite generally applicable in estimating the total surface conductivity of highly dispersed colloidal systems. 4.4. Surface Conductivity in the Stagnant Layer
FIG. 6. Zeta-potentials for Monospher 1000 at 1 mM ionic strength in presence of KCl (left) and Mg(NO3 )2 (right) calculated by different methods as explained in the text: (d) ζ (Duσ = 0), (j) ζ (Dud ), (×) ζ (Du).
Compare, by way of illustration, the results of the different methods in the presence of potassium chloride which are summarized in the left-hand side plot in Fig. 6. The difference between the zeta-potentials ζ (Dud ) and ζ (Du), which exceeds 20 mV at pH 9 (see Fig. 6), is considerable. As the electrostatic contribution to the Helmholtz energy of colloidal interaction scales with ζ 2 the consequences for the interpretation of colloidal stability are substantial. There can be no doubt that for our experimental system it is indispensable to take the total surface conductivity into account. For smaller particles (see Eq. [3]), this effect is expected to be even more important. The physical reason for the divergence between ζ (Dud ) and ζ (Du) at high pH, i.e., high surface charge densities, is that in the former case the polarization of the double layer is not fully accounted for. That means that a part of the mobile countercharge is not taken into account. Hence, the electric counterfield created by the countercharge is underestimated, and so is the corresponding retarding force. Since an increase in the zeta-potential causes in principle higher polarization this eventually ends up in excessively small zeta-potentials. An additional argument to support these considerations has been found in the experiments in the presence of Mg(NO3 )2 . Here, stronger electrostatic interactions in the stagnant layer between counterions and surface sites are expected. However, also in this case, taking the total surface conductivity into account results in remarkable higher zeta-potentials compared with those provided by the ESA-8000 (see right-hand part of Fig. 6). Equation [6] has been derived under the assumption that the contribution of coions can be neglected which is a good approximation at sufficiently high zeta-potentials. For oxides this means working at high pH and low ionic strength. In fact, at lower pH the relative contribution of co-ions to the total surface conductivity rises, but the absolute magnitude of surface conductivity is in principle lower and, hence, the error made by application of Eq. [6] is relatively small. This is also supported by our experiments (see Fig. 6), where the zeta-potentials resulting from
In the preceding section the influence of surface conductivity on the interpretation of electrokinetic data has been discussed. It has been shown that the contribution of the stagnant layer should not be neglected. Quantitatively, the contribution of the stagnant layer is in the same order of magnitude as that of the diffuse layer. The relatively high inner layer surface conductivity is caused by the ability of the counterions to move in the stagnant layer and, hence, it is logical to address now the counterion mobility in the stagnant layer. Generally, the experimentally measured surface conductivity, K σ , consists of four parts, and for a binary electrolyte and a nonporous oxidic surface, it can be written as K σ = K 1σ i + K 2σ i + K 1σ d + K 2σ d ,
[13]
where the last two terms account for the contribution of the diffuse layer due to accumulation of counterions and depletion of co-ions. The first two terms refer to the stagnant layer part. Under conditions where the zeta-potential is sufficiently high the contributions of the coions may be neglected so that K σ approaches K 2σ . Then, the basic equation relating the total surface conductivity, the charge density in the stagnant and diffuse layer, σ2i and σ2d , and the ionic mobilities, u i2 and u ∞ 2 , reduces to µ ¶ 3m 2 d [14] K 2σ = u i2 σ2i + 1 + 2 u ∞ 2 · σ2 . z2 As before, the first term in Eq. [14] accounts for the contribution of the stagnant layer to surface conductivity, K σ i , while the second term refers to the same in the diffuse layer, K σ d . For Monospher 1000 in the presence of KCl, Fig. 7 illustrates
FIG. 7. Counterion Dukhin numbers for the total double layer (j) Du2 and the contribution of the diffuse double layer (d) Dud2 . The difference between these values corresponds to the contribution of the stagnant layer.
181
ZETA-POTENTIALS FROM ESA MEASUREMENTS
TABLE 1 Ionic Mobilities and Further Characteristic Double Layer Data for Monospher 1000 in the Presence of KCl at 1 mM
pH
ζ (mV)
σ0 (µC cm−2 )
σ2d (µC cm−2 )
7.4 7.6 7.8 8.0
−82.33 −87.02 −92.45 −99.48
−3.07 −3.51 −4.03 −4.64
0.743 0.823 0.936 1.100
Du2
K∞ (µS cm−1 )
Kσ (10−10 S)
K 2σ d (10−10 S)
u i2 /u ∞ 2
0.172 0.197 0.226 0.257
239.2 245.8 253.6 263.0
19.69 23.29 27.51 32.47
9.662 11.43 13.50 15.93
0.63 0.67 0.71 0.73
d 0 Note. The mobility ratio (last column) is given by −(K σ − K 2σ d )/[u ∞ 2 (σ − σ2 )].
the corresponding quantities Dui and Dud , where the relation Du = Dui + Dud holds. In Eqs. [13] and [14] the contribution of the counterions in micropores is completely omitted. Although titration data indicate that the micropores are accessible to counterions there are reasons to assume that the counterions inside the pores do not significantly contribute to the conductivity. Minor et al. (7, 37) concluded that the ionic mobility of the counterions in micropores of mp −3 a similar silica is in the range O(10−6 ) ≤ u 2 /u ∞ 2 ≤ O(10 ) because of the disordered, irregular pore structure. Recently published Molecular Dynamics simulation data of the tangential transport of counterions (38) confirm that the stagnant layer behaves as a two-dimensional gel in which the ions can move, albeit with mobilities lower than in the bulk of the solution. To obtain the mobility of the counterions in the stagnant layer, the surface charge density and the electrokinetic countercharge must be known. For oxides both parameters depend on pH and ionic strength. The electrokinetically active countercharge simply results from the zeta-potentials ζ (Du) using the relationships of static double-layer theory to relate ζ to σ2d . The surface charge density, σ 0 , is in principle accessible by potentiometric acid– base titration, but for microporous silica a specific charge den¯ is usually determined rather than a surface charge density sity, q, ¯ may be considered (see Fig. 1). The specific charge density, q, as resulting from titrating the outer plus the inner surface (that of the micropores) while the surface charge density, σ 0 , refers solely to the outer surface. The difference between the two is due to the ability of potential-determining ions [H+ ], [OH− ] and some of the counter- and coions to penetrate into the micropores, which gives rise to a volume charge density. De Keizer
and coworkers have shown that in principle a pure surface charge density may be distinguished from the specific charge density for St¨ober silica by investigating the particle size dependence of the specific charge density (39). However, they were unable to do so in the presence of alkali ions since the volume charge amounted to several times of the contribution of the surface. Furthermore, they could show that to a good approximation the pure surface charge density of microporous St¨ober silica is the same as that for Aerosil OX50. This is supported by the investigations of Zhuravlev (40) who, by using a deuterium-exchange method, found that the surface silanol group density of hundred microporous silica samples lies in the range 4 ≤ OH ≤ 6 per nm2 with a mean of 4.6 OH nm−2 . For this reason the surface charge density data of Aerosil OX50 are taken. With all this in mind we can straightforwardly compute the ionic mobilities of the counterions in the stagnant layer. The results of these calculations are listed in Tables 1 and 2. Equation [6] is valid for sufficiently high zeta-potentials and, hence, the most reliable mobility ratios for the corresponding counterion are found at high pH. It is clearly seen that qualitative differences between K+ and Mg2+ occur which are probably due to stronger electrostatic interactions of the divalent cation. In addition specific adsorption may play a role in the case of magnesium, whereas potassium is known not to specifically adsorb on silica. Although the mobility ratio for potassium (u iK+ /u ∞ K+ ≈ 0.73) is close to unity it is clearly lower than the values reported by Kijlstra et al. for a similar silica (u iK+ /u ∞ K+ ≈ 0.95) (41). The comparison between volume charge and surface charge density data for Monospher 1000 indicates a high extent of microporosity. With this in mind there is no
TABLE 2 Ionic Mobilities and Further Characteristic Double Layer Data for Monospher 1000 in the Presence of Mg(NO3 )2 at 1 mM Ionic Strength pH
ζ (mV)
σ0 (µC cm−2 )
σ2d (µC cm−2 )
Du2
K∞ (µS cm−1 )
Kσ (10−10 S)
K 2σ d (10−10 S)
u i2 /u ∞ 2
7.4 7.6 7.8 8.0
−58.05 −59.32 −60.56 −61.81
−5.31 −6.61 −8.32 −10.48
0.872 0.920 0.970 1.023
0.168 0.185 0.204 0.223
97.6 95.8 95.0 92.8
7.64 8.53 9.55 10.61
6.54 6.90 7.28 7.67
0.045 0.052 0.056 0.057
d 0 Note. The mobility ratio is given by −(K σ − K 2σ d )/[u ∞ 2 (σ − σ2 )].
¨ LOBBUS ET AL.
182
reason to assume the surface of the Monospher 1000 particles to be very flat; i.e., it is likely that the structure of micropores causes surface roughness up to certain point. In that way the observed somewhat lower mobility ratio (u iK+ /u ∞ K+ ) may be attributed to the special surface properties of Monospher 1000. A further reason is the fact that we actually used the surface charge density data of Aerosil OX50 to calculate the Stern layer charge density, σ i . Although the reasoning of de Keizer et al. (39) justifies this procedure in principle there remains a small uncertainty about the values of the σ i s to be used and, hence, of the calculated mobility ratio. At least from that point of view the difference between our u iK+ /u ∞ K+ and Kijlstra’s should not be overrated. More important is the fact that both mobility ratios are in the same order of magnitude. The conclusion is that in the case of monovalent, chemically indifferent counterions the mobility of these ions in the hydrodynamically stagnant layer is only somewhat reduced as compared to the bulk. On the other hand, similar electrolytes are quite often used in electrokinetic investigations of colloidal suspensions of practical interest. Thus, an independent estimation of surface conductivity contribution of the stagnant layer seems indispensable in the interpretation of, say, mobility data in the presence of such electrolytes. Counterions with higher charge are less mobile in the stagnant part of the double layer. Additionally, they tend to accumulate more strongly in the stagnant layer. For Mg2+ ions the mobility ratio u i2 /u ∞ 2 is found to be about 0.05 (Table 2). This value is again appreciably lower than what has been found for other systems (7, 41), which seems to underscore a general trend as explained above. So, we conclude that it is always recommendable to have independent conductivity data in the interpretation of electrokinetic measurements. 5. CONCLUSIONS
An important outcome of our investigations is that in principle the contribution of the stagnant layer to the total surface conductivity at low ionic strength must not be neglected. The lower the ionic strength the larger the effect is. The method proposed to account for total surface conductivity can be easily applied for large particles (r > 100 nm) and it does not contain any additional fitting parameters such as adsorption constants for the Stern layer. It could be shown that for Monospher particles in the presence of potassium chloride neglecting the stagnant layer conductivity underestimates the computed zetapotentials compared with those including total surface conductivity. The mobility of monovalent, nonspecific adsorbing counterions such as K+ is little reduced in the hydrodynamically stagnant part of the double layer. However, the mobility of magnesium is significantly lower than that in the bulk, which is due to stronger electrostatic interactions and, possibly, to additional specific interactions. On the other hand, because of these larger interactions a greater part of the countercharge resides in the stagnant layer and, hence, it is always recommendable to ex-
tend electroacoustic measurements by additional conductivity measurements. It could be demonstrated that the Smoluchowski equation used in the software of ESA-8000 is insufficient at low ionic strength. Furthermore, even if the surface conductivity in the diffuse part of the double layer is taken into account the electrostatic interactions are underestimated. The resulting magnitudes for the zeta-potentials are in principle too low. Zeta-potentials resulting from the proposed method are a very good characteristic to compute the electrostatic interactions between charged particles which is important for the colloid stability of electrostatically stabilized dispersions. Besides volume fraction and particle size, properly calculated zeta-potentials belong to the important colloidal properties, the knowledge of which is also indispensable in understanding the rheological behavior in terms of the underlying physicochemical properties. It therefore follows from our analysis that in many colloidal stability studies the electrostatic contribution to the interaction Gibbs energy may be substantially underestimated. During the process of revision of the present paper an article of Dukhin et al. has been published which also deals with the effect of surface conductivity on electroacoustic phenomena (42). The result of this theoretical investigation is a decrease of CVI with increasing Du at constant zeta-potential or a higher zetapotential calculated from a given CVI taking surface conductivity into account. This trend is in accordance with the outcome of our investigation. ACKNOWLEDGMENTS The authors thank the Deutsche Forschungsgemeinschaft, the Th¨uringer Ministerium f¨ur Wissenschaft Forschung und Kultur, and the Studienstiftung des deutschen Volkes for financial support. We became aware of References (30) and (42) by the comments of a referee, which is also acknowledged.
REFERENCES 1. Lyklema, J., “Fundamentals of Colloid and Interface Science,” Vol. 2, Chap. 4. Academic Press, London, 1995. 2. Hunter, R. J., “Zeta-Potential in Colloid Science.” Academic Press, London, 1981. 3. O’Brien, R. W., and White, L. R., J. Chem. Soc. Faraday Trans. II 74, 1607 (1978). 4. Dukhin, S. S., and Deryaguin, B. V., in “Surface and Colloid Science” (E. Matijevic, Ed.), Vol. 7. Wiley, London, 1974. 5. Kijlstra, J., van Leeuwen, H. P., and Lyklema, J., Langmuir 9, 1625 (1993). 6. L¨obbus, M., Ph.D. thesis, Friedrich-Schiller University, Jena, 1999. 7. Minor, M., van der Linde, A. J., van Leeuwen, H. P., and Lyklema, J., Colloids Surf. A 142, 165 (1998). 8. Spanos, N., Tsevis, A., Koutsoukos, P. G., Minor, M., van der Linde, A. J., and Lyklema, J., Colloids Surf. A 141, 101 (1998). 9. L¨obbus, M., van Leeuwen, H. P., and Lyklema, J., Colloids Surf. A 161, 103 (2000). 10. Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. II 88, 3567 (1992). 11. Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. 94, 2441 (1998). 12. Mangelsdorf, C. S., and White, L. R., J. Chem. Soc. Faraday Trans. 94, 2583 (1998).
ZETA-POTENTIALS FROM ESA MEASUREMENTS 13. O’Brien, R. W., Midmore, B. R., Lamb, A., and Hunter, R. J., Faraday Discuss. Chem. Soc. 90, 301 (1990). 14. O’Brien, R. W., Cannon, D. W., and Rowlands, W. N., J. Colloid Interface Sci. 173, 406 (1995). 15. Kosmulski, M., and Rosenholm, J. B., J. Phys. Chem. 100, 11681 (1996). 16. Rowlands, W. N., O’Brien, R. W., Hunter, R. J., and Patrick, V., J. Colloid Interface Sci. 188, 325 (1997). 17. Rasmussen, M., and Wall, S., Colloids Surf. A 122, 169 (1997). 18. Greenwood, R., and Bergstr¨om, L., J. Eur. Ceram. Soc. 17, 537 (1997). 19. Kn¨osche, C., Friedrich, H., and Stintz, M., Part. Syst. Charact. 14, 175 (1997). 20. Hunter, R. J., Colloids Surf. A 141, 37 (1998). 21. ESA-8000 Operational Manual, Matec Applied Sciences. 22. O’Brien, R. W., and Perrins, W. T., J. Colloid Interface Sci. 99, 20 (1984). 23. Kang, S.-Y., and Sangani, A. S., J. Colloid Interface Sci. 165, 195 (1994). 24. O’Brien, R. W., J. Fluid. Mech. 190, 71 (1988). 25. Werner, C., K¨orber, H., Zimmerman, R., Dukhin, S. S., and Jacobasch, H.-J., J. Colloid Interface Sci. 208, 329 (1998). 26. Bikermann, J. J., Z. Physik. Chem. A 136, 378 (1933). 27. Rutgers, A., and de Smet, M., Trans. Faraday Soc. 41, 758 (1945). 28. Rutgers, A., and de Smet, M., Trans. Faraday Soc. 43, 102 (1947).
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29. Minor, M., van der Linde, A. J., and Lyklema, J., J. Colloid Interface Sci. 203, 177 (1998). 30. Dukhin, A. S., Shilov, V. N., Oshima, H., and Goetz, P. J., Langmuir 15, 6692 (1999). 31. O’Brien, R. W., Rowlands, W. N., and Hunter, R. J., NIST Spec. Publ. 856, 1 (1993). 32. Sonnefeld, J., Postdoctoral thesis, Friedrich-Schiller University, Jena, 1999. 33. Desai, F. N., Hammad, H. R., and Hayes, K. F., Langmuir 9, 2888 (1993). 34. L¨obbus, M., Vogelsberger, W., Sonnefeld, J., and Seidel, A., Langmuir 14, 4386 (1998). 35. Vogelsberger, W., L¨obbus, M., Sonnefeld, J., and Seidel, A., Colloids Surf. A 159, 311 (2000). 36. Barchini, R., van Leeuwen, H. P., and Lyklema, J., submitted for publication. 37. Minor, M., Ph.D. thesis, Wageningen Agricultural University, 1998. 38. Lyklema, J., Rovillard, S., and de Coninck, J., Langmuir 14, 5659 (1998). 39. de Keizer, A., Ent, E. M. van der, and Koopal, L. K., Colloids Surf. A 142, 303 (1998). 40. Zhuravlev, L. T., Langmuir 3, 316 (1987). 41. Kijlstra, J., van Leeuwen, H. P., and Lyklema, J., Langmuir 9, 1625 (1993). 42. Dukhin, A. S., Shilov, V. N., Oshima, H., and Goetz, P. J., Langmuir 16, 2615 (2000).
An Improved Method for Calculating Zeta-Potentials from Measurements of the Electrokinetic Sonic Amplitude Mario L¨obbus,∗,1 J¨urgen Sonnfeld,∗ Herman P. van Leeuwen,† Wolfram Vogelsberger,∗ and Johannes Lyklema† ∗ Institute of Physical Chemistry, Friedrich-Schiller-University Jena, Lessingstrasse 10, D-07743 Jena, Germany; and †Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB, Wageningen, Netherlands Received February 8, 2000; accepted May 16, 2000
The surface electric properties of the commercially available silica, Monospher 1000 (Fa. Merck), have been studied by conductivity and ESA (electrokinetic sonic amplitude) experiments. It could be shown that accounting for the contribution of the stagnant layer to surface conductivity is indispensable in the interpretation of electrokinetic data at low ionic strength. A general method has been put forward which allows to take into account the total, experimentally accessible surface conductivity in the evaluation of ESA data of moderately concentrated suspensions. This includes additional conductivity measurements which serve for the independent estimation of the total relative surface conductivity. The resulting zeta-potentials are clearly higher than those obtained after neglecting the contribution of the stagnant layer to surface conductivity. In addition, the ionic mobilities of potassium and magnesium in the hydrodynamically stagnant layer have been investigated in some detail. It has been found that the ionic mobility of potassium is of the same order of magnitude as in the bulk solution while the mobility of magnesium is significantly reduced. °C 2000 Academic Press Key Words: zeta-potential; surface conductivity; dynamic mobility; stagnant layer; electrokinetics; silica.
1. INTRODUCTION
The interaction between charged colloidal particles is mainly governed by the diffuse part of the electric double layer. The potential at the inner boundary of this double layer part, ψ d , cannot be directly measured. However, experience has shown instead that the electrokinetic zeta-potential, ζ , is an appropriate characteristic (1, 2). The zeta-potential has to be calculated from electrokinetic parameters measured by techniques such as microelectrophoresis, streaming potential, and electroacoustic methods like electrokinetic sonic amplitude (ESA) using some more or less sophisticated theory. A popular theoretical model is the so-called “standard electrokinetic model.” A fundamental principle of this model and other modern electrokinetic theories is that a relatively thin double layer with κa À 1 is nearly in equilibrium with the surrounding electrolyte and, hence, the 1 To whom correspondence should be addressed. E-mail: [email protected]. uni-jena.de.
0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
ion distribution in the double layer is given by the Boltzmann distribution (3). This important observation, which has become a landmark in the development of electrokinetics, dates back to Dukhin (4). The so-called electrokinetic equations comprise Poisson’s equation for electrostatics, the Navier–Stokes equation for the fluid transport and the Nernst–Planck equation for the ion fluxes. If these equations are solved simultaneously under appropriate boundary conditions double-layer polarization is automatically taken into account. Thus, the standard electrokinetic model includes the effect of surface conduction in the diffuse layer. However, the contribution of the hydrodynamically stagnant layer to surface conduction is ignored. This could be shown to be the reason for some methodical electrokinetic inconsistency, i.e., disparity between zeta-potentials for one and the same system but derived from different electrokinetic methods (5). The influence of surface conductivity is relatively high at low ionic strength, where the bulk conductivity itself is small. Moreover, accounting for surface conduction in the stagnant layer may result in methodically electrokinetic consistent zetapotentials (5, 6). A usable method for taking surface conduction into full account consists of combining different electrokinetic experiments (7–9). However, the most general approach is to rigorously consider the transport processes in the hydrodynamically stagnant layer in the development of electrokinetic theories, as it has been put forward by Mangelsdorf and White (10–12). Unfortunately, for such numerical analysis it is necessary to introduce parameters which describe the adsorption in the stagnant layer. A relatively new method to measure the electrokinetic properties of moderately concentrated suspensions is the ESA technique (13–20), for which well-designed commercial devices became available over the past decade (ESA-8000, Acoustosizer, Fa. Matec Applied Science). Such devices are normally equipped with automatic titration systems, which facilitate the pH-dependent measurement of electrokinetic properties of suspensions. However, so far only the Helmholtz–Smoluchowski equation has been used in the software of the ESA-8000 (21) and, hence, the theoretical interpretation is limited, since the influence of surface conductivity is neglected. The objective of this article is to present an improved method for the interpretation of ESA measurements that accounts for
174
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ZETA-POTENTIALS FROM ESA MEASUREMENTS
the total, experimentally accessible surface conduction. From an experimental point of view this may be achieved by combining conductivity and ESA measurements. To interpret the conductivity data, O’Brien’s theory developed for the electric conductivity in porous plugs (22) in connection with the numerical data of Kang and Sangani (23) has been used. The electroacoustic measurements were analyzed with O’Brien’s high κa double layer theory (24). The method has been tested with the monodisperse, commercially available silica, Monospher 1000 (Fa. Merck), in the presence of potassium chloride and magnesium nitrate solutions. It has been shown that for both KCl and Mg(NO3 )2 , neglecting the total surface conductivity results in zeta-potentials that are too small, especially at higher pH. The same statement still holds if the surface conductivity in the diffuse double layer is taken into account while neglecting the stagnant layer contribution. Therefore, the contribution of the surface conductivity in the stagnant layer has been investigated in some detail and an estimation has been made of the ionic mobility of potassium and magnesium ions behind the shear plane. 2. THEORY
Besides the zeta-potential, ζ , the total surface conductivity, K σ , is a useful parameter for the characterization of colloidal systems (5–9, 25). The principal advantage of K σ is that not only is the mobile charge in the diffuse layer included, but so is the mobile part of the hydrodynamically stagnant layer. Furthermore, for a proper interpretation of the measurement of a single electrokinetic phenomenon in terms of the zeta-potential, knowledge about K σ is indispensable. By definition surface conduction is an excess quantity and accounts for the excess conduction tangential to the charged surface. In general, using the considerations of Lyklema (1) the surface conductivity in the presence of a binary electrolyte may be written as K σ = FA ·
2 X i=1
Z |z i |
∞
[ci (x) − ci (∞)] · u i (x) dx,
[1]
0
where FA denotes Faraday’s constant. In Eq. [1] ci (x) and u i (x) are local ion concentrations and ionic mobilities, respectively, in infinitesimally small slabs parallel to the solid surfaces; ci (∞) and z i are the bulk ion concentrations and the valencies, respectively. In the sense of the Gouy–Stern theory, the total surface conductivity, K σ , may be divided into the contributions of the inner layer, K σ i , and the diffuse double layer, K σ d : K σ = K σ i + K σ d.
[2]
The quantity K σ d has generic character, whereas K σ i is systemspecific and, therefore, is pertinent to the system under investigation. The relative contribution of the total double layer in conducting current through a suspension can be expressed advanta-
geously by introducing the Dukhin number, Du (1). For highly charged surfaces (with high zeta-potential) the contribution of the coions compared to that of the counterions can be neglected. The Dukhin number due to counterions is generally related to the corresponding absolute surface conductivity, K 2σ , the particle radius, a, and the conductivity of the bulk electrolyte solution, K ∞ , by Du2 =
K 2σ . aK∞
[3]
In addition, it is possible to relate the counterion Dukhin number, Du2 , to such important double-layer characteristics as ζ , K σ i , and K σ d (1): 1 Du2 = κa
µ
3m 2 K σi 1 + 2 + 2σ d z2 K2
¶µ
¸ ¶ · z 2 eζ − 1 . [4] exp − 2kT
In Eq. [4] z 2 is the valency of the counterion and κ the Debye– H¨uckel parameter. The term 3m 2 /z 22 accounts for the fact that, besides conduction, the electric field also causes an electroosmotic fluid transport (26). The dimensionless ionic drag coefficient, m 2 , relates to the bulk viscosity, η, and the limiting ionic mobility in the bulk, u ∞ 2 , by m2 =
2z 2 ²kT , 3ηeu ∞ 2
[5]
where ² = ²r ²0 is the permittivity of the solvent, k the Boltzmann constant, and T the temperature. For flat surfaces the total surface conductivity can be measured with an experimental setup recently developed by Werner et al. (25). Moreover, the experimental estimation of the surface conductivity for a series of cylindrical glass capillaries has been carried out long before with the pioneering work of Rutgers (27, 28). However, for disperse systems it is far more complicated to experimentally obtain the total surface conductivity. A possible way of getting information about the counterion Dukhin number for disperse systems is the rigorous analysis of conductivity data with the O’Brien and Perrin theory for porous plugs (22) as presented in the following. The electric conductivity K ∗ of a plug or suspension may be related to the electric conductivity of the bulk electrolyte solution, K ∞ , and the dimensionless surface conductivity parameter, the Dukhin number, Du, (22). The result can be written in terms of the theory developed by O’Brien and Perrins (22) for a binary electrolyte as · ¸ K∗ u∞ 2 = 1 + 3φ f (0, φ) + ∞ { f (Du2 , φ) − f (0, φ)} . K∞ u2 + u∞ 1 [6] This equation was originally derived for close-packed porous plugs. However, Kang and Sangani (23) extended the analysis
¨ LOBBUS ET AL.
176
to suspensions and were able to show that it holds for suspensions as well. Eq. [6] is valid for smooth particles with large κa and sufficiently high zeta-potentials, where the contribution of the coions in the double layer may be neglected. In Eq. [6] φ means the volume fraction of the colloid material in the suspension, and u∞ i refers to the ionic mobility in infinite dilution of the bulk as before. The subscripts “2” and “1” represents counterions and coions, respectively. The dimensionless function f (Du2 , φ) accounts for the contribution of the surface conductivity to the total conductivity of the suspension. Besides on the surface conductivity and volume fraction, the function f (Du2 , φ) also depends on the way of packing. At low relative surface conductivity, i.e., Dukhin numbers Du2 < 0.5, the overall conductivity of the suspension is lower than that of the bulk because nonconducting particles are covered by a poorly conducting double layer. This leads to negative values for f (Du2 , φ). With increasing surface conductivity the contribution of double layers around the particles rises, leading eventually to positive values for f (Du2 , φ). In general, f (Du2 , φ) needs to be numerically computed. A collection of different numerical values for f (Du2 , φ) as a function of relative surface conductivity, Du2 , and the volume fraction is compiled in references (22, 23). Equation [6] and most other modern electrokinetic expressions were originally derived without considering the contribution of the stagnant layer to surface conductivity. However, it could be shown (29) that expressions like Eq. [6] for the conductivity of porous plugs or expressions for the electrophoretic mobility can be maintained under the condition that the Dukhin number is written as is in Eq. [4]. The ESA phenomenon results from the interaction of a relatively high frequency a.c. field (O(106 Hz)) with a dispersion of charged colloidal particles causing an oscillatory motion of the particles. In addition to the particles themselves the electric double layer around the particles is subject to changes by the a.c. field. If there is a density difference between particle and electrolyte solution this results in the formation of an acoustic dipole. The created sonic field may be detected using a piezo transducer. O’Brien was able to derive an expression which directly reˆ lates the ESA signal, ESA(ω), to the dynamic electrophoretic mobility, uˆ D (ω) (24). For dilute suspensions (φ < 5%) this relation is given by uˆ D (ω) =
ˆ ESA(ω) · A(ω), φ(1ρ/ρ)νs
[7]
where A(ω) is a calibration factor, ρ the density of the solvent, 1ρ the density difference between particle and solvent, and νs the rate of propagation of sound in the solvent. The factor A(ω) depends on the geometry of the measurement cell and is determined by calibration with the ESA signal of a well-characterized ˆ = 5.36 mPa m V−1 ). This procedure assample (Ludox, |ESA| sumes that the static electrophoretic mobility equals the dynamic one. With the concept of the polarized double layer O’Brien (24)
has shown that for a dilute dispersion (φ . 5%) of spherical particles with high κa the dynamic electrophoretic mobility may be related to the zeta-potential by the following expression provided that the wavelength of the sonic field substantially exceeds the particle size,
uˆ D (ω) =
2²ζ ˆ · G(ωa 2 /ν) · [1 + fˆ(Du)]. 3η
[8]
Here, ν is the kinematic viscosity of the solvent. The dimension2 ˆ /ν) contains the influence of inertia on the less function G(ωa dynamic mobility. It is determined by the particle mass and the frequency of the a.c. field according to à 2 ˆ G(ωa /ν) = 1 −
ωa 2 /ν · i(3 + 21ρ/ρ) p 9 · (1 + (1 − i) · ωa 2 /2ν)
!−1 . [9]
The factor 1 + fˆ(Du) in Eq. [8] is proportional to the component of the field tangential to the surface and takes the influence of surface conductivity into account. The higher the surface conductivity the more countercharge is moved back and forth by the external a.c. field. This oscillation leads to reduction of the applied field as a result of which the mobility decreases. This factor is given by O’Brien’s theory as fˆ(Du, ω0 ) =
1 − iω0 − [2Du − iω0 · (²p /²s )] , with 2 · (1 − iω0 ) + [2Du − iω0 · (²p /²s )]
ω0 =
ω² , K∞ [10]
where ²p and ²s are the permittivities of the particles and the solvent. To keep in line with the original literature, here, the symbol f is used both in Eq. [6] and Eq. [8], but the difference between f (Du2 , φ) (see Eq. [6]) and fˆ(Du, ω0 ) (see Eq. [8]) is highlighted. Unfortunately, with the ESA measurements it is impossible to experimentally determine the total mobile charge and, hence, the total surface conductivity of the double layer. In commercially available devices like the Acoustosizer and the ESA-8000 (both Matec Applied Science) either only the contribution of the diffuse layer to surface conductivity is taken into consideration or surface conductivity is completely neglected. Our solution to this problem is provided by the independent experimental determination of the counterion Dukhin number by conductivity measurements using Eq. [6]. Then, this Dukhin number is used in the interpretation of dynamic mobilities (see Eqs. [8] and [10]). In that way it is possible to account for the total surface conductivity while converting ESA signals to zeta-potentials.
ZETA-POTENTIALS FROM ESA MEASUREMENTS
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3. EXPERIMENTAL
The electrokinetic investigations were carried out at 25◦ C using the commercially available, microporous, monodisperse silica sample Monospher 1000 (Fa. Merck). A suspension of the original product as obtained from the manufacturer showed a pH of around 9.5, indicating impurities. Because of this the sample was pretreated as follows. The sample was washed for 4 h with 1% HCl and then rinsed with deionized water and centrifuged till it did not show any more traces of chloride, dried at 110◦ C, and milled. The specific surface area, SBET , of this sample was determined by low temperature nitrogen adsorption with the BET method using the automatic surface analyzer AUTOSORB-1 (Fa. Quantachrome). It was found that SBET = 2.9 m2 g−1 , which agrees with the geometric surface area of 2.9 m2 g−1 . The density of Monospher 1000 was determined with a pycnometer, ρ = 2.04 g cm−3 . Nevertheless, the determination of the surface charge density by potentiometric titration suggests that micropores exist, although they are not detected by nitrogen adsorption (see Fig. 1). The titrable charge density far exceeds the theoretically possible surface charge densities (with an assumed surface silanol group concentration of Ns ≈ 4 nm−2 ). Hence, the conclusion is that beside a pure surface part of the measured specific charge density there must be a contribution of the micropores. This behavior is rather typical for some types of precipitated silica. It is likely that the micropores are so small that the nitrogen molecules are not able to penetrate and, hence, they are not “seen” by nitrogen adsorption. The radius of the Monospher 1000 particles has been determined by photon correlation spectroscopy to a = 481 nm, which is in close accordance to the statements of the manufacturer (500 nm). There is no indication for a pH-dependent agglomeration of the Monospher 1000 particles in 1 mM KCl in very dilute suspensions as used in photon correlation spectroscopy (see Fig. 2). The ESA measurements were carried out with Monospher suspensions for a volume fraction of φ = 0.025 with the ESA8000 apparatus from Matec Applied Science. The ionic strength of the electrolyte solutions (KCl, Mg(NO3 )2 ) was 1 mM. This
¯ and surface charge density, σ 0 , of MonoFIG. 1. Specific charge density, q, spher 1000 as a function of pH at 1 mM KCl. The data points depicted stem from averaging of at least three single determinations; σ 0 was calculated directly from titrable charge using the BET surface area.
¯ of a Monospher 1000 suspension as a FIG. 2. Mean particle radius, a, function of pH at 1 mM KCl as determined by photon correlation spectroscopy.
results in the value of κ · a ≈ 50, which is in the thin double layer region (20). Equations [7] and [8] have been derived for dilute suspensions. There is still some uncertainty about the upper limit of the volume fraction for the applicability of Eqs. [7] and [8]. In (20) the upper limit has been reported that the error in Eqs. [7] and [8] is below 2% up to φ ≤ 0.05. On the other hand, the results of Dukhin et al. (30) suggest that the upper limit of dilute suspensions is φ ≤ 0.02, which has been found for the interpretation of colloid vibration current (CVI) data. We estimated that the relative error due to concentration effects is less than 5% using the semiempirical method of O’Brien et al. (31). The ESA-8000 system measures the amplitude of the ESA signal and the phase angle relative to a suspension with known electrokinetic properties (Ludox, Fa. Aldrich) rather than absolutely. As phase reference a Monospher 1000 suspension (φ = 0.025, pH ≈ 8) without electrolyte has been used. It turned out that temperature fluctuations have a great influence on the phase angle. Temperature variations of approx. 2 K in the laboratory give rise to a change in the phase angle up to 15◦ (32). It is very likely that this behavior is caused by the temperature influence on the propagation speed of sonic waves in the delay rod of the ESA probe. After mounting a water jacket on the ESA probe this problem was avoided. With this additional equipment the phase angle for silica in the pH range 6 . pH . 10 usually varies by less than 5◦ . The total surface conductivity of the Monospher suspensions was determined with a conductivity cell Tetracon 96 (Fa. WTW, Weilheim) and a conductivity meter (Fa. WTW, Weilheim). To improve the signal/noise ratio a volume fraction (φ ≈ 0.3) higher than in the ESA measurements was used. The conductivities of the suspension and that of the corresponding electrolyte solution were determined with the following procedure: 1. A Monospher 1000 suspension with φ ≈ 0.3 (approx. 25 ml) is titrated with 1 N KOH to pH ≈ 6. (At pH < 6.0 sedimentation occurs, which is probably due to higher aggregation between the particles than in the very dilute suspension as used for photon correlation spectroscopy, see Fig. 2). 2. After 10 min sonification and adjusting the temperature with a thermostat the suspension conductivity, K ∗ , is
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determined. There was no time dependency in the conductivity results, indicating that under the chosen conditions the particles do not settle. 3. Next the suspension is centrifuged at 4500 rpm for 15 min, the clear supernatant electrolyte solution is removed with a micropipet, and after thermic equilibration the conductivity of this solution, K ∞ , is measured. 4. The Monospher 1000 particles are redispersed into the electrolyte solution again, and a new (higher) pH is adjusted by adding KOH and steps 2–4 are repeated. In that way the silica suspension has been titrated consecutively to higher pH with KOH. At each pH the suspension conductivity, K ∗ , and the conductivity of the supernatant electrolyte solution (after centrifugation) has been measured. This procedure gave the same results as those which result from the titration of different suspensions, i.e., where each suspension is titrated only once up to a certain pH. 4. RESULTS
4.1. Conductivity Mesaurements of Concentrated Silica Suspensions As previously stated, the conductivity measurements of silica suspensions with φ ≈ 0.3 were employed to determine the surface conductivity. First, the experimentally measured conductivity ratio K ∗ /K ∞ and the value f (Du = 0, φ) were used to calculate the function f (Du2 , φ) with Eq. [6] for the given volume fraction. To obtain the counterion Dukhin number Du2 we interpolated between the numerical values given by Kang and Sangani (23) using a two-dimensional, quadratic interpolation procedure. Hence, a counterion Dukhin number which includes surface conduction in the stagnant layer can be computed as a function of pH, volume fraction and electrolyte concentration. The results for a constant ionic strength, I = 1 mM, are depicted in Fig. 3. Changes in the volume fraction caused by added KOH are taken into consideration.
FIG. 4. Magnitude of the factor |1 + fˆ(Du)| (—) as a function of the relative surface conductivity for the given experimental system compared with the approximation in ESA-8000 software (– –).
It is evident from Fig. 3 that the surface conductivity depends on the valency of the counterion. A similar effect was also observed for latices (9). The experimentally determined surface conductivity is fairly high in the presence of either counterion. Unfortunately, the software of the ESA-8000 assumes the relative surface conductivity to be negligible, which results in |1 + fˆ(Du)| = 1.5. Figure 4 illustrates the result of this neglection compared with the proper behavior. For the highest Dukhin numbers the experimentally determined factor |1 + fˆ(Du)| is overestimated by about one-third by the ESA-8000 software. To evaluate the accuracy of the proposed method, error bars indicating the standard deviation, are included in Fig. 3. It is assumed that with the equipment used for the conductivity measurements the typical error made in the determination of the conductivity is about 1 µS/cm. Hence, applying the methods of error propagation, the error in f (Du2 ) is ¡ ¢± ∞ p ∞ u 2 · 1 + (K ∗ /K ∞ )2 . 1 f (Du2 ) = 1K /(3φ K ∞ ) · u ∞ 2 + u1 With it the error in Du2 = b · 1 f (Du2 ), where b equals 1.26 in the used range of volume fractions (0.25 ≤ φ ≤ 0.3). The value of b has been estimated from the numerical values of Kang and Sangani (23). 4.2. ESA Measurements of Dilute Silica Suspensions
FIG. 3. Counterion Dukhin number, Du2 , as determined from conductivity measurements of approx. 30 vol% silica suspensions as a function of pH at an ionic strength of 1 mM: (j) KCl, (×) Mg(NO3 )2 . The error bars result from error propagation assuming a mean error in the conductivity measurements 1K of 1 µS cm−1 .
The countercharge in the diffuse electric double layer may be determined by ESA measurements with O’Brien’s theory. The assumption in O’Brien’s theory coded in the software in the apparatus used, ESA-8000, assumes that particle–particle interactions are negligible and that there is no overlap of the double layers of neighboring particles. With the volume fraction used (φ = 0.025), the particle size of the Monospher 1000 particles (approx. 1 µm) and assuming a homogeneous distribution of the particles, the mean particle distance is approximately 1.7 µm, which is much larger than κ −1 . So, overlap can be ignored. The ESA measurements have been carried out in the pH range
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6.5 . pH . 10 since for lower pH sedimentation took place in the conductivity measurements of the concentrated suspensions. In the presence of a background electrolyte the measured ESA ˆ PE , phase angle αPE ) is the vector sum signal (magnitude |ESA| of signals of the colloid particle (“P”) and the background electrolyte (“E”). However, at the chosen electrolyte concentration, 1 mM, the background electrolyte signal is far smaller than the colloid signal and so the background electrolyte correction as proposed in reference (33) has been dropped. The ESA measurements have been carried out several times. The mean of the magnitude of the ESA signal, |ESAP |, and the phase angle, α¯ PE , result from complex addition of at least three (quadratically interpolated as a function of pH) single determinations |ESAP | =
p
x 2 + y2
y y α¯ P = y y
≤0∧x ≤0∧x >0∧x >0∧x
> 0, − arctan(|y/x|) ≤ 0, −[180 − arctan(|y/x|)] , > 0, arctan(|y/x|) ≤ 0, 180 − arctan(|y/x|)
[11]
P P with y = N1 iN |ESAP,i | sin αP,i and x = N1 iN |ESAP,i | cos αP,i The influence of dissolved silica may be estimated by a dissolution model recently proposed for solid oxides (34, 35). Under the given experimental conditions (sonification time, thermic equilibration, titration delay time) the total silicate concentration is smaller than 2 mmol L−1 , where at the highest pH approximately 22% of the orthosilicic acid exists in dissociated form (pK a = 9.8). That means that the influence of dissolved silica is even smaller than that of the background electrolyte and, therefore, may also be neglected in the background correction. The ESA-8000 automatically takes account of changes in the volume fraction due to added base. It is evident from Fig. 5 that in the presence of divalent cations the absolute value of the mobility is much lower than in the
presence of potassium. This difference is probably caused by more effective screening by the bivalent ion. 4.3. Calculation of Zeta-Potentials from Conductivity and ESA Measurements Properly established zeta-potentials can describe the electrostatic interactions in suspensions rather well. As previously outlined it is important to account for the total surface conductivity in calculating the zeta-potential from experimental data. The most general approach obviously consists in accounting for the transport processes in the hydrodynamically stagnant layer while solving the electrokinetic equations. Recently, some encouraging work has been published (10–12, 36). However, to apply these theories it is necessary either to know the experimental Stern adsorption isotherm or to introduce fitting parameters describing the extent of adsorption in the Stern layer in such numerical analyses. In the latter case there is a risk that the adjustable fitting parameters also contain deviations from the model and errors in the experimental data. So, in principle interpretation of electrokinetic data without fitting parameters should be preferred. For the analyses of the experimentally determined ESA signals we applied O’Brien’s theory, see Eqs. [8]–[10]. The inertia term Gˆ is automatically accounted for in the ESA-8000 software. For Monospher 1000 Eq. [9] yields Gˆ = 0.789 under the given experimental conditions. The sample investigated is a typical experimental system for which it can be shown (see Fig. 3) that here surface conductivity must be taken into consideration in the interpretation of electroacoustic data. By employing additional conductivity measurements it is possible to assess the total counterion Dukhin number, Du2 , and with that it is straightforward to calculate zeta-potentials, ζ (Du). First Eq. [10] is evaluated to obtain the factor |1 + fˆ(Du2 )| and then Eq. [8] is used. Since the total surface conductivity is accounted for, these combined zeta-potentials may be taken as the best available estimation of the effective electrostatic potential that governs the interaction between particles. The relation between the zeta-potentials calculated by the ESA-8000 software, ζ (K σ = 0), and zeta-potentials which include surface conduction, ζ (Du), in the electric double layer is then given by ζ (Du) =
FIG. 5. Dynamic electrophoretic mobilities of Monospher 1000 in presence of KCl (e) and Mg(NO3 )2 (j) at an ionic strength of 1 mM. The figure shows the magnitude of mobilities which result from averaging of, at least, three single curves. The error bars indicate the standard error of sample mean.
1.5 · ζ (Du = 0). |1 + fˆ(Du)|
[12]
In order to show that it is important to take into account the total surface conductivity rather than the contribution of the diffuse layer only, it is possible to calculate numerical values of ζ (Dud ) and Dud by numerical solution of Eqs. [4], [8], and [10] while neglecting K σ i . Hence, the zeta-potentials, ζ (Dud ), computed in this way presuppose that surface conduction does not take place in the stagnant layer; i.e., they are representative of the results obtainable with the software from the Acoustosizer (Matec Applied Science).
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combination of conductivity and electroacoustic measurements approach the Smoluchowski result at lower pH. From all of this one may conclude that Eq. [6] seems quite generally applicable in estimating the total surface conductivity of highly dispersed colloidal systems. 4.4. Surface Conductivity in the Stagnant Layer
FIG. 6. Zeta-potentials for Monospher 1000 at 1 mM ionic strength in presence of KCl (left) and Mg(NO3 )2 (right) calculated by different methods as explained in the text: (d) ζ (Duσ = 0), (j) ζ (Dud ), (×) ζ (Du).
Compare, by way of illustration, the results of the different methods in the presence of potassium chloride which are summarized in the left-hand side plot in Fig. 6. The difference between the zeta-potentials ζ (Dud ) and ζ (Du), which exceeds 20 mV at pH 9 (see Fig. 6), is considerable. As the electrostatic contribution to the Helmholtz energy of colloidal interaction scales with ζ 2 the consequences for the interpretation of colloidal stability are substantial. There can be no doubt that for our experimental system it is indispensable to take the total surface conductivity into account. For smaller particles (see Eq. [3]), this effect is expected to be even more important. The physical reason for the divergence between ζ (Dud ) and ζ (Du) at high pH, i.e., high surface charge densities, is that in the former case the polarization of the double layer is not fully accounted for. That means that a part of the mobile countercharge is not taken into account. Hence, the electric counterfield created by the countercharge is underestimated, and so is the corresponding retarding force. Since an increase in the zeta-potential causes in principle higher polarization this eventually ends up in excessively small zeta-potentials. An additional argument to support these considerations has been found in the experiments in the presence of Mg(NO3 )2 . Here, stronger electrostatic interactions in the stagnant layer between counterions and surface sites are expected. However, also in this case, taking the total surface conductivity into account results in remarkable higher zeta-potentials compared with those provided by the ESA-8000 (see right-hand part of Fig. 6). Equation [6] has been derived under the assumption that the contribution of coions can be neglected which is a good approximation at sufficiently high zeta-potentials. For oxides this means working at high pH and low ionic strength. In fact, at lower pH the relative contribution of co-ions to the total surface conductivity rises, but the absolute magnitude of surface conductivity is in principle lower and, hence, the error made by application of Eq. [6] is relatively small. This is also supported by our experiments (see Fig. 6), where the zeta-potentials resulting from
In the preceding section the influence of surface conductivity on the interpretation of electrokinetic data has been discussed. It has been shown that the contribution of the stagnant layer should not be neglected. Quantitatively, the contribution of the stagnant layer is in the same order of magnitude as that of the diffuse layer. The relatively high inner layer surface conductivity is caused by the ability of the counterions to move in the stagnant layer and, hence, it is logical to address now the counterion mobility in the stagnant layer. Generally, the experimentally measured surface conductivity, K σ , consists of four parts, and for a binary electrolyte and a nonporous oxidic surface, it can be written as K σ = K 1σ i + K 2σ i + K 1σ d + K 2σ d ,
[13]
where the last two terms account for the contribution of the diffuse layer due to accumulation of counterions and depletion of co-ions. The first two terms refer to the stagnant layer part. Under conditions where the zeta-potential is sufficiently high the contributions of the coions may be neglected so that K σ approaches K 2σ . Then, the basic equation relating the total surface conductivity, the charge density in the stagnant and diffuse layer, σ2i and σ2d , and the ionic mobilities, u i2 and u ∞ 2 , reduces to µ ¶ 3m 2 d [14] K 2σ = u i2 σ2i + 1 + 2 u ∞ 2 · σ2 . z2 As before, the first term in Eq. [14] accounts for the contribution of the stagnant layer to surface conductivity, K σ i , while the second term refers to the same in the diffuse layer, K σ d . For Monospher 1000 in the presence of KCl, Fig. 7 illustrates
FIG. 7. Counterion Dukhin numbers for the total double layer (j) Du2 and the contribution of the diffuse double layer (d) Dud2 . The difference between these values corresponds to the contribution of the stagnant layer.
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TABLE 1 Ionic Mobilities and Further Characteristic Double Layer Data for Monospher 1000 in the Presence of KCl at 1 mM
pH
ζ (mV)
σ0 (µC cm−2 )
σ2d (µC cm−2 )
7.4 7.6 7.8 8.0
−82.33 −87.02 −92.45 −99.48
−3.07 −3.51 −4.03 −4.64
0.743 0.823 0.936 1.100
Du2
K∞ (µS cm−1 )
Kσ (10−10 S)
K 2σ d (10−10 S)
u i2 /u ∞ 2
0.172 0.197 0.226 0.257
239.2 245.8 253.6 263.0
19.69 23.29 27.51 32.47
9.662 11.43 13.50 15.93
0.63 0.67 0.71 0.73
d 0 Note. The mobility ratio (last column) is given by −(K σ − K 2σ d )/[u ∞ 2 (σ − σ2 )].
the corresponding quantities Dui and Dud , where the relation Du = Dui + Dud holds. In Eqs. [13] and [14] the contribution of the counterions in micropores is completely omitted. Although titration data indicate that the micropores are accessible to counterions there are reasons to assume that the counterions inside the pores do not significantly contribute to the conductivity. Minor et al. (7, 37) concluded that the ionic mobility of the counterions in micropores of mp −3 a similar silica is in the range O(10−6 ) ≤ u 2 /u ∞ 2 ≤ O(10 ) because of the disordered, irregular pore structure. Recently published Molecular Dynamics simulation data of the tangential transport of counterions (38) confirm that the stagnant layer behaves as a two-dimensional gel in which the ions can move, albeit with mobilities lower than in the bulk of the solution. To obtain the mobility of the counterions in the stagnant layer, the surface charge density and the electrokinetic countercharge must be known. For oxides both parameters depend on pH and ionic strength. The electrokinetically active countercharge simply results from the zeta-potentials ζ (Du) using the relationships of static double-layer theory to relate ζ to σ2d . The surface charge density, σ 0 , is in principle accessible by potentiometric acid– base titration, but for microporous silica a specific charge den¯ is usually determined rather than a surface charge density sity, q, ¯ may be considered (see Fig. 1). The specific charge density, q, as resulting from titrating the outer plus the inner surface (that of the micropores) while the surface charge density, σ 0 , refers solely to the outer surface. The difference between the two is due to the ability of potential-determining ions [H+ ], [OH− ] and some of the counter- and coions to penetrate into the micropores, which gives rise to a volume charge density. De Keizer
and coworkers have shown that in principle a pure surface charge density may be distinguished from the specific charge density for St¨ober silica by investigating the particle size dependence of the specific charge density (39). However, they were unable to do so in the presence of alkali ions since the volume charge amounted to several times of the contribution of the surface. Furthermore, they could show that to a good approximation the pure surface charge density of microporous St¨ober silica is the same as that for Aerosil OX50. This is supported by the investigations of Zhuravlev (40) who, by using a deuterium-exchange method, found that the surface silanol group density of hundred microporous silica samples lies in the range 4 ≤ OH ≤ 6 per nm2 with a mean of 4.6 OH nm−2 . For this reason the surface charge density data of Aerosil OX50 are taken. With all this in mind we can straightforwardly compute the ionic mobilities of the counterions in the stagnant layer. The results of these calculations are listed in Tables 1 and 2. Equation [6] is valid for sufficiently high zeta-potentials and, hence, the most reliable mobility ratios for the corresponding counterion are found at high pH. It is clearly seen that qualitative differences between K+ and Mg2+ occur which are probably due to stronger electrostatic interactions of the divalent cation. In addition specific adsorption may play a role in the case of magnesium, whereas potassium is known not to specifically adsorb on silica. Although the mobility ratio for potassium (u iK+ /u ∞ K+ ≈ 0.73) is close to unity it is clearly lower than the values reported by Kijlstra et al. for a similar silica (u iK+ /u ∞ K+ ≈ 0.95) (41). The comparison between volume charge and surface charge density data for Monospher 1000 indicates a high extent of microporosity. With this in mind there is no
TABLE 2 Ionic Mobilities and Further Characteristic Double Layer Data for Monospher 1000 in the Presence of Mg(NO3 )2 at 1 mM Ionic Strength pH
ζ (mV)
σ0 (µC cm−2 )
σ2d (µC cm−2 )
Du2
K∞ (µS cm−1 )
Kσ (10−10 S)
K 2σ d (10−10 S)
u i2 /u ∞ 2
7.4 7.6 7.8 8.0
−58.05 −59.32 −60.56 −61.81
−5.31 −6.61 −8.32 −10.48
0.872 0.920 0.970 1.023
0.168 0.185 0.204 0.223
97.6 95.8 95.0 92.8
7.64 8.53 9.55 10.61
6.54 6.90 7.28 7.67
0.045 0.052 0.056 0.057
d 0 Note. The mobility ratio is given by −(K σ − K 2σ d )/[u ∞ 2 (σ − σ2 )].
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182
reason to assume the surface of the Monospher 1000 particles to be very flat; i.e., it is likely that the structure of micropores causes surface roughness up to certain point. In that way the observed somewhat lower mobility ratio (u iK+ /u ∞ K+ ) may be attributed to the special surface properties of Monospher 1000. A further reason is the fact that we actually used the surface charge density data of Aerosil OX50 to calculate the Stern layer charge density, σ i . Although the reasoning of de Keizer et al. (39) justifies this procedure in principle there remains a small uncertainty about the values of the σ i s to be used and, hence, of the calculated mobility ratio. At least from that point of view the difference between our u iK+ /u ∞ K+ and Kijlstra’s should not be overrated. More important is the fact that both mobility ratios are in the same order of magnitude. The conclusion is that in the case of monovalent, chemically indifferent counterions the mobility of these ions in the hydrodynamically stagnant layer is only somewhat reduced as compared to the bulk. On the other hand, similar electrolytes are quite often used in electrokinetic investigations of colloidal suspensions of practical interest. Thus, an independent estimation of surface conductivity contribution of the stagnant layer seems indispensable in the interpretation of, say, mobility data in the presence of such electrolytes. Counterions with higher charge are less mobile in the stagnant part of the double layer. Additionally, they tend to accumulate more strongly in the stagnant layer. For Mg2+ ions the mobility ratio u i2 /u ∞ 2 is found to be about 0.05 (Table 2). This value is again appreciably lower than what has been found for other systems (7, 41), which seems to underscore a general trend as explained above. So, we conclude that it is always recommendable to have independent conductivity data in the interpretation of electrokinetic measurements. 5. CONCLUSIONS
An important outcome of our investigations is that in principle the contribution of the stagnant layer to the total surface conductivity at low ionic strength must not be neglected. The lower the ionic strength the larger the effect is. The method proposed to account for total surface conductivity can be easily applied for large particles (r > 100 nm) and it does not contain any additional fitting parameters such as adsorption constants for the Stern layer. It could be shown that for Monospher particles in the presence of potassium chloride neglecting the stagnant layer conductivity underestimates the computed zetapotentials compared with those including total surface conductivity. The mobility of monovalent, nonspecific adsorbing counterions such as K+ is little reduced in the hydrodynamically stagnant part of the double layer. However, the mobility of magnesium is significantly lower than that in the bulk, which is due to stronger electrostatic interactions and, possibly, to additional specific interactions. On the other hand, because of these larger interactions a greater part of the countercharge resides in the stagnant layer and, hence, it is always recommendable to ex-
tend electroacoustic measurements by additional conductivity measurements. It could be demonstrated that the Smoluchowski equation used in the software of ESA-8000 is insufficient at low ionic strength. Furthermore, even if the surface conductivity in the diffuse part of the double layer is taken into account the electrostatic interactions are underestimated. The resulting magnitudes for the zeta-potentials are in principle too low. Zeta-potentials resulting from the proposed method are a very good characteristic to compute the electrostatic interactions between charged particles which is important for the colloid stability of electrostatically stabilized dispersions. Besides volume fraction and particle size, properly calculated zeta-potentials belong to the important colloidal properties, the knowledge of which is also indispensable in understanding the rheological behavior in terms of the underlying physicochemical properties. It therefore follows from our analysis that in many colloidal stability studies the electrostatic contribution to the interaction Gibbs energy may be substantially underestimated. During the process of revision of the present paper an article of Dukhin et al. has been published which also deals with the effect of surface conductivity on electroacoustic phenomena (42). The result of this theoretical investigation is a decrease of CVI with increasing Du at constant zeta-potential or a higher zetapotential calculated from a given CVI taking surface conductivity into account. This trend is in accordance with the outcome of our investigation. ACKNOWLEDGMENTS The authors thank the Deutsche Forschungsgemeinschaft, the Th¨uringer Ministerium f¨ur Wissenschaft Forschung und Kultur, and the Studienstiftung des deutschen Volkes for financial support. We became aware of References (30) and (42) by the comments of a referee, which is also acknowledged.
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