Langmuir 2003, 19, 3619-3627
3619
Dielectric Spectroscopy Measurements on Latex Dispersions C. Chassagne, D. Bedeaux, and J. P. M. Van der Ploeg Leiden Institute of Chemistry, PO Box 9502, 2300 RA Leiden, The Netherlands
G. J. M. Koper* Laboratory of Physical Chemistry, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Received August 27, 2002. In Final Form: December 23, 2002 Experimental data on the dielectric response of latex dispersions for a large variety of particle sizes, surface charge densities, and ionic strengths are interpreted using an analytical theory developed previously (Chassagne, C.; Bedeaux, D.; Koper, G. J. M. J. Phys. Chem. B 2001, 105, 11743 and Physica A 2002, 317, 321). It is found that the conductivity increment is a crucial variable in the discussion about the need of a Stern layer conductance. Both the dielectric permittivity and the conductivitty increments are fitted using the zeta potential and a Stern layer conductance.
1. Introduction It has been established that colloidal latex particles exhibit anomalous electrokinetic behavior. In particular, the zeta potentials obtained from electrophoretic mobility measurements are usually different from the ones derived from conductivity or dielectric spectroscopy measurements.1-3 Furthermore the zeta potentials determined from the electrophoretic mobility often display a maximum in the region 1-10 mM of added salt. This is in disagreement with the classical theory, which predicts that at constant surface charge density, increasing the salt concentration decreases the zeta potential. These discrepancies have been attributed to Stern layer conduction, hairyness of the surface of the particle, or ion adsorption onto the surface and lead to complicated models.4,5 In two previous articles6,7 a new theory to calculate the dipolar coefficient arising from a particle surrounded by electrolyte subjected to an external electric field was developed. From this theory, the dielectric increment of the permittivity and the conductivity of the dispersion can be deduced. It has been applied to silica and hematite sol measurements from the literature.8 It was found that the theory predicted both the relaxation frequency and the dielectric enhancement at low frequency, with the zeta potential as the only free parameter, quite well. When the zeta potential was taken to be in agreement with the one found from the mobility, a Stern layer conductance had to be introduced and the data could also be fitted. From this study, it follows that the sols behave classically (1) Rosen, L. A.; Baygents, J. C.; Saville, D. A. J. Chem. Phys. 1993, 98 (5), 4183. (2) Kijlstra, J.; Van Leeuwen, H. P.; Lyklema, J. Langmuir 1993, 9 (7), 1625. (3) Rasmusson, M.; Wall, S. J. Colloid Interface Sci. 1999, 209, 312 and 327. (4) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 32 and 45. (5) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1997, 93 (17), 3145. (6) Chassagne, C.; Bedeaux, D.; Koper, G. J. M. J. Phys. Chem. B 2001, 105, 11743. (7) Chassagne, C.; Bedeaux, D.; Koper, G. J. M. Physica A 2002, 317, 321. (8) Chassagne, C.; Bedeaux, D.; Koper, G. J. M. J. Colloid Interface Sci. 2002, 255, 129.
Table 1. General Characteristics of the Nanospheres batch no.
radius (nm) from EM
radius (nm) from DLS
surface charge density (µC/cm2)
1412, 1 624, 1 465, 1 1063.C 1185 691, 1
9.5 ( 1.5 23.5 ( 3.5 55 ( 4 133.5 ( 16 133.5 ( 13 265 ( 13.5
13 ( 1 26 ( 4 67 ( 6 168 ( 15 168 ( 15 306 ( 12
1.8 4.4 0.9 1.1 3.0 3.0
in the sense that the zeta potential, found by both fitting procedures, decreases with increasing ionic strength or decreasing surface charge density. In this paper, results from dielectric spectroscopy measurements for sulfate latex spheres, of various sizes, from surface charge densities, and at different ionic strength will be presented. In the second section, the experimental conditions are presented. In the third section, the general results of the theory are recalled. In the fourth section, the fitting procedure is described. The results are discussed in the fifth section. In the last section, we present some conclusions. 2. Experimental Methods 2.1. Sulfate Latex Spheres. Monodisperse latex particles stabilized by sulfate groups with various surface charges were provided by IDC (Interfacial Dynamics Corp., Oregon, USA). Their characteristics are given in Table 1. Potassium chloride solutions were made using tridistilled water (Millipore) and pro analysis KCl from Merck. Early studies in our laboratory showed that extensively dialyzed samples exhibit the same behavior as fresh samples from the manufacturer. All samples were made just before the measurements, and manufacturer’s bottles were kept sealed and stored at 6 °C. We measured the hydrodynamic radius of the nanospheres listed in Table 1, using dynamic light scattering. Using the same technique, we furthermore verified that the thus obtained samples were stable against the formation of dimers. 2.2. Dielectric Spectroscopy. The dielectric measurements were performed with two different cells designed and built in our laboratory. A four-electrode cell,
10.1021/la0264771 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/21/2003
3620
Langmuir, Vol. 19, No. 9, 2003
Chassagne et al.
the characteristics of which are given in ref 9, was used to measure the low-frequency dielectric response (between 100 Hz and 20 kHz). Second, a cylindrical cell (also described elsewhere, ref 10) was used to measure the highfrequency response, from 10 kHz to 1 MHz. Therefore a whole frequency range (102-106 Hz) was scanned for all samples. Electrode polarization problems were eliminated in both cases, using procedures described in refs 9 and 10. The procedure used for the four-electrode cell did not allow the measurement of the conductivity increment of the dispersion, and therefore the conductivity data are presented only for the range (104-106 Hz), corresponding to the cylindrical cell measurements. The data are plotted according to the relation
∆ )
- 1 φ
(1)
and
∆K K - K1 ) K1 φK1 where and K are the measured permittivity and conductivity, respectively, and 1 and K1 are the electrolyte permittivity and conductivity, respectively. We did not use normalized increments, as did previous authors, and the benefit of this choice will become clear in the following. It was also verified that the increments are independent of volume fraction (verified for volume fractions of 0.25%, 0.5%, and 1% in most cases). The results presented here are for 1% (265 and 133.5 nm) and for 0.5% (55, 23.5, and 9.5 nm).
and 17 for a detailed discussion and mathematical justification. Incorporating these new boundary conditions in the set of equations, the following expression for the complex dipolar coefficient β is derived
β ) (K ˜2 - K ˜ 1 + 2K ˜ |St/a + K ˜ ⊥St/a + 2K ˜ |d/a + K ˜ ⊥d/a +
/
˜ ⊥D/a) (K ˜ 2 + 2K ˜ 1 + 2K ˜ |St/a - 2K ˜ ⊥St/a + 2K ˜ |D/a + K d
2K ˜ | /a - 2K ˜ ⊥d/a + 2K ˜ |D/a - 2K ˜ ⊥D/a) (2) where K ˜ 1 is the complex conductivity of the electrolyte and K ˜ 2 the complex conductivity of the particle
K ˜ 1 ≡ K1 + iω01
(9) Van der Touw, F. Dielectric properties of aqueous polyelectrolyte solutions. Ph.D. Thesis, Leiden University 1975. (10) Van der Ploeg, J. P. M.; Mandel, M. Meas. Sci. Technol. 1991, 2 (4), 389. (11) Dukhin, S. S. Dielectric phenomena and the double layer in disperse systems and polyelectrolytes; John Wiley and Sons: New York, 1974. (12) DeLacey, E. H. B.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 2007. (13) Albano, A. M.; Bedeaux, D.; Vlieger, J. Physica 1979, 99A, 293. (14) Albano, A. M.; Bedeaux, D.; Vlieger, J. Physica 1980, 102A, 105. (15) Bedeaux, D.; Albano, A. M.; Mazur, P. Physica 1976, 82A, 438.
K ˜ 2 ≡ iω|2
(3)
K1 is the conductivity of the electrolyte, 1 and 2 are the relative dielectric permittivities of water and of the particle, respectively, and 0 is the dielectric permittivity of vacuum. K ˜ |St and K ˜ ⊥St are the excess complex conductances of the Stern layer along and normal to the surface, respectively. K ˜ |d and K ˜ ⊥d are the excess complex conductances due to the (diffuse part of the) double layer. Finally K ˜ |D and K ˜ ⊥D are the excess complex conductances due to the diffusion layer which has a thickness of the order (D/ω)1/2, where D is a typical ionic diffusion coefficient. The expression for the dipolar coefficient is the same as the one found in a shell model with a core attributed to the particle, one shell to the Stern layer, one shell for the double layer, and one for the diffusion layer. For a dilute suspension of particles, the complex conductivity may be written12
K ˜ (φ,ω) ) K ˜ 1(1 + 3φβ)
3. Theoretical Overview In refs 6 and 7, we start from the standard equations (conservation of ionic species, Poisson-Boltzmann) used for these electrokinetic problems.11,12 We consider a nonconducting spherical particle. As definition of the radius of the particle a, we take the distance between the center of the particle and the slip plane. An electric field E0 exp(iωt) oscillating with a radial frequency ω is imposed along the z axis. The field strength is weak compared to the electric fields in the double layer, which is generally the case in dielectric spectroscopy measurements. The standard equations can therefore be derived in terms of the first-order perturbation of the electric potential and the densities.11,12 At the surface of shear, the standard boundary conditions (no-flux condition, continuity of the normal component of the displacement field, continuity of the potential across the slip plane) are valid for the actual solution. The far field (i.e., beyond the double layer) variables satisfy new boundary conditions incorporating excess densities in the sense of Gibbs and excess fluxes and fields which account for the variation of the actual variables in the double layer. We refer to refs 6, 7, 13-15,
and
(4)
where φ is the volume fraction of the particles. The enhancement of the dielectric permittivity therefore becomes
∆ ≡
[
]
- 1 Im(β) ) 3 1 Re(β) + K1 φ ω0
(5)
and the enhancement of the conductivity can be written
∆K ≡
K - K1 ) 3[K1 Re(β) - 01ω Im(β)] φ
(6)
Below, we will give the explicit formulas for the complex conductances of the double layer and the diffusion layer for the simple case of a 1-1 electrolyte (z+ ) -z- ) 1) with equal diffusion coefficients (D- ) D+ ) D). A good example for such electrolyte is KCl, which is therefore widely used in experiments. In this case the conductivity of the electrolyte becomes K1 ) 2De2n∞/kT, where e is the absolute value of the electron charge, n∞ is the salt concentration in the solution, k is Boltzmann’s constant, and T is the temperature. In the derivation of the high zeta potential expression (e|ζ|/kT . 1), we used the fact that counterions play a major role in the polarization and we neglect the contribution of the co-ions. We obtained K ˜ |D ) K ˜ ⊥d ) 0 for an (16) Bedeaux, D.; van Dijk; M. A.; Joosten, J. G. H.; Levine, Y. K. J. Phys. Chem. 1989, 93, 2506. (17) Bedeaux, D.; Vlieger, J. Optical properties of surfaces; Imperial College Press: London, 2002.
Dielectric Response of Latex Dispersions
Langmuir, Vol. 19, No. 9, 2003 3621
arbitrary electrolyte. In the case of KCl the expressions of K ˜ ⊥D and K ˜ |d reduce to
K ˜ |d )
eD kT
[| |
q -
| |]
01 Ψd a
(7)
and
K ˜ ⊥D ) -2
1 + λna 2 2
2 + 2λna + λn a
K ˜ |d
The potential fitted, Ψd, is the potential at the beginning of the diffuse part of the double layer, which is therefore not necessarily equal to the zeta potential ζ, as is assumed by most authors, because the surface of shear could be located in the Stern layer. Similarly q is the charge density at the beginning of the diffuse part of the double layer. This charge density is given by the empirical expression18 (see for instance ref 19)
q)
[
( )
( )]
01kT eΨd eΨd 4 + κ 2 sinh tanh e 2kT κa 4kT
(8)
It is a formula valid within a few percent for all potentials values, for κa g 0.5. Most authors assume Ψd ) ζ, however. 4. Fitting Procedure The expression for the dipolar coefficient β, incorporating Stern layer conductances, as shown in eq 2, is used. The Stern layer is by definition the region between the surface of the particle and the beginning of the diffuse part of the double layer. In the literature,1,11,20 one defines a dimensionless real coefficient St by
˜ St ≡ St aK1 K ˜ ⊥St ) K
(9)
This coefficient is often called the Dukhin number. This definition implies that we, similar to the authors just cited, do not consider capacitive contributions to the Stern layer conductances. Taking K ˜ |St ) K ˜ ⊥St leaves the denominator of β unchanged and therefore gives the same local field as when no Stern layer is introduced. The Stern layer then only contributes to the numerator of β. As we argued previously,8 this is similar to what other authors do. We will come back to this choice in the conclusions. The two fit parameters we use are Ψd and St. In Figure 1, theoretical predictions are displayed, varying these two parameters. In Figure 1A the variations of the permittivity increment (left) and the conductivity increment (right) are displayed for St ) 1, 2, and 3 and e|Ψd|/kT ) 4 as a function of frequency. In Figure 1B the same increments are displayed for St ) 0 and e|Ψd|/kT ) 4, 5, and 6 as a function of frequency. The importance of not normalizing the conductivity curves is visible in these figures, as increasing the Stern layer parameter shifts the conductivity curves up while a change of Ψd only changes the value for larger frequencies. This trend has also been found by authors using a Stern layer conductance in their model; see for instance ref 1. It can also be seen from these figures that the permittivity enhancement can be fitted by adjusting either St or Ψd. As most authors (18) Loeb, A. L.; Overbeek, J. Th. G.; Wiersema, P. H. The electrical double-layer around a spherical colloid particle; M.I.T. Press: Cambridge, MA, 1961. (19) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (20) Kijlstra J, Van Leeuwen, H. P.; Lyklema J. J. Chem. Soc., Faraday Trans. 1992, 88 (23), 3441.
Table 2 a ) 306 nm Cs κa |Ψd| σ St KSt
0 9 5.50 0.8 3.7 1.34
1 33 5.50 3.0 1.15 5.7
2 45 5.80 4.8 0.75 6.1
3 55 5.65 5.4 0.5 6.1
4 63 5.50 5.8 0.32 5.8
7 83 4.42 5.3 0.13 4.1
10 100 4.45 5.3 0.045 2.0
plot their data in terms of Re(β) and Im(β),3,21 we also displayed the variations of β with St and |Ψd|. In Figure 1C the real (left) and the imaginary (right) part of the dipolar coefficient β are displayed for St ) 1, 2, and 3 and e|Ψd|/kT ) 4 as a function of frequency. In Figure 1D the same quantities are displayed for St ) 0 and e|Ψd|/kT ) 4, 5, and 6 as a function of frequency. In the fits, we used the hydrodynamic radii given in Table 1 obtained by dynamic light scattering (DLS). We expect these radii to be closer to those of the slipping plane than the radii given by the manufacturer, which were obtained using electron microscopy (EM). These were about 10-20% smaller, as can be seen from the same table. The surface charge densities given by the manufacturer are also given in Table 1. As the bulk diffusion coefficients of the potassium and the chloride ion differ by about 6%, an average value of 2.0 × 10-9 m2/s was used for each ion. All the values of |Ψd| in the tables are reported in 25.8 mV (=kT/e at room temperature) units. The salt concentrations (Cs) are reported in mM (10-3mol/L), the electrokinetic charges calculated from |Ψd| using eq 8 are in µC/cm2, and the Stern layer conductance KSt is in nS‚m. 5. Results and Discussions 5.1. 306 nm Latex. From Figures 2 and 3 and Table 2, it can be seen that the amplitude of the dielectric and the conductivity response could be fitted over the whole range of ionic strength. The relaxation frequency, however, was not in good agreement with the data. This discrepancy has the tendency to reduce with increasing ionic strength. Moreover, as can most clearly be seen in Figure 3, measurements made with the two different apparatus do not always overlap in that region. One could fit the relaxation frequency by increasing the particle radius about 20%. For this latex, this would correspond to an increase of about 50 nm in radius, and although the presence of hairy layers on the latex has been suggested, such a layer was in general found to be 10 nm thick.3,22 As the thickness we find is an order of magnitude larger, we consider such a large increase unrealistic and chose not to fit this parameter. It should be noted that in the cases of silica and hematite studied previously,8 we did not encounter such a discrepancy and that the relaxation frequencies were well fitted. It is also important to recall that we verified, using DLS, that no aggregation occurred, which could also lead to such an increase. The increments were independent of volume fraction. From Table 2, it is seen that, to fit the dielectric permittivity enhancement, a large potential is required. Increasing ionic strength decreases both St and |Ψd|. The latter agrees with the classical picture. Arroyo et al.23 studied the same latex dispersion (265 nm from IDC) in KCl but only for 0.1 and 1 mM of added salt. In their fits, obtained from the numerical model of Mangelsdorf and (21) Russell, A. S.; Scales, P. J.; Mangelsdorf, C. S.; White, L. R. Langmuir 1995, 11 (5), 1553. (22) Vanderhoff, J. W. ACS Symp. Ser. 1981, No. 165, 61. (23) Arroyo, F. J.; Carrique, F.; Bellini, T.; Delgado, A. V. J. Colloid Interface Sci. 1999, 210, 194.
3622
Langmuir, Vol. 19, No. 9, 2003
Chassagne et al.
Figure 1. Theoretical predictions as a function of frequency, a ) 200 nm, Cs ) 1 mM (A and C) and e|Ψd|/kT ) 4 (B and D): (1) e|Ψd|/kT ) 4; (2) e|Ψd|/kT ) 5; (3) e|Ψd|/kT ) 6.
White,5 they used as fit parameters the zeta potential, which they do not distinguish from |Ψd|, and the ratio between the drag coefficient of the ith ionic species (i ) + here as K+ is the counterion) in the bulk solution and in the Stern layer. The latter corresponds to the inverse of the Dukhin number St. They therefore find in our notations for Cs ) 0.1 mM |Ψd|/kT ) 5.0 and St ) 1 and for Cs ) 1 mM |Ψd|/kT ) 5.4 and St ) 1.6. As can be seen in Table 2, for 1 mM of added KCl this is close to the values we find. When we plot the predictions with the parameters used by Arroyo et al., we find the dashed lines
in Figure 2B. Their predictions for the permittivity increment differ somewhat from our prediction, because they used the EM radius while we use the DLS radius. If we would also use the EM radius, the two curves would be indistinguishable. Arroyo et al.’s conductivity increment lies higher than ours, irrespective of the choice of the radius. As Arroyo et al. do not display the conductivity increments in their article, we cannot compare the precise choice of |Ψd| and St. It is nevertheless clear from the two curves in Figure 2B that the conductivity increment is crucial for the fit.
Dielectric Response of Latex Dispersions
Langmuir, Vol. 19, No. 9, 2003 3623
Figure 2. ∆/1000 and ∆K/K1 (left and right figures, respectively) as a function of frequency for various ionic strengths, a ) 306 nm, and added salt: (A) 0 mM; (B) 1 mM; (C) 2 mM; (D) 3 mM. Solid lines are fits; see text. For broken line in part B, see also the text.
The electrokinetic charge corresponding to |Ψd| is calculated and given in Table 2. It is found that this charge is of the order of 2.26 µC/cm2 when no salt is added. This is somewhat lower than the titrated charge given by the manufacturer (see Table 1). When the ionic strength is increased, this charge first dramatically increases and
then slightly decreases, following the behavior of the permittivity increment. Rasmusson et al.3 also found such a large increase in the electrokinetic charge (in their case more than 1 order of magnitude over the range 0.1-100 mM), but in their case, no slight decrease was visible around 1 mM. Arroyo et al., however, performed electro-
3624
Langmuir, Vol. 19, No. 9, 2003
Chassagne et al.
Figure 3. ∆/1000 and ∆K/K1 (left and right figures, respectively) as a function of frequency for various ionic strengths, a ) 306 nm, and added salt: (E) 4 mM; (F) 7 mM; (G) 10 mM. Solid lines are fits (see text). The broken line in part G represents the limiting value of -1.5, corresponding to an uncharged sphere.
phoretic mobility measurements on this latex and report that the mobility (and the zeta potential) displays a maximum when the concentration of KCl approaches 1 mM. 5.2. 168 nm Latex. The influence of surface charge density has been investigated, for two sets of dispersions. The particles have the same radii, but a different surface charge density (see Table 1). As can be seen from Figure 4, increasing the surface charge density shifts the conductivity increment up, whereas the shift in the permittivity increment is small. The difference in amplitude between the two permittivity increment increases with ionic strength. To fit these data, one requires a larger St, and a lower |Ψd| for the latex having the highest charge compared to the latex having the smallest charge; see Table 3. As a larger St means that a larger number of counterions are present in the Stern layer, this makes a smaller potential |Ψd| possible. 5.3. 67, 26, and 13 nm Latex. The increments of the dielectric permittivity and conductance are plotted in
Table 3 a ) 168 nm, low charge Cs κa |Ψd| σ St KSt
1 17 2.19 0.5 2.34 5.7
2 24 3.30 1.3 1.10 5.4
3 30 3.54 1.8 0.67 4.8
4 35 2.55 1.2 0.70 6.9
a ) 168 nm, high charge Cs κa |Ψd| σ St KSt
1 17 0.95 0.2 5.35 13.3
2 24 1.55 0.4 2.63 13.1
3 30 1.55 0.6 2.05 15.1
4 35 2.80 0.8 1.35 13.4
Figures 5 and 6. The same trend for St is observed as for the other latexes, although the values found (see Table 4) are higher. Given the definition in eq 9, it is reasonable to expect that St is inversely proportional to the radius. This indeed explains most of the increase. We will come back to this point in the conclusion. Contrary to what has been observed for the other latexes, the potential fitted is found to increase with ionic strength. The measurements for the 13 nm, which correspond to small κa could also be reasonably well fitted with our theory; see Figure 6B. Although the values of the potential found were low, we
Dielectric Response of Latex Dispersions
Langmuir, Vol. 19, No. 9, 2003 3625
Figure 4. ∆/1000 and ∆K/K1 (left and right figures, respectively) as a function of frequency, a ) 168 nm, and (A) 1 mM, (B) 2 mM, (C) 3 mM, and (D) 4 mM of added salt. Solid lines are fits (see text); “high” (“low”) refers to batch no. 1185 (1063.C).
used the high |Ψd| version of our theory, as it is clear that for these systems, the Stern layer conductance plays the dominant role. Arroyo et al.23 studied particles of 11.5 nm EM radius (also provided by IDC), for Cs ) 0.1 and 0.5 mM. In this case they find that using |Ψd|/kT ) 4.6 and 7.8, the numerical method of DeLacey and White (without Stern layer conductance), respectively, predict the same
order of magnitude of the dielectric increment. As we discussed in ref 7, the numerical methods predict higher values for the permittivity increment than we do for small κa. Without Stern layer conductance, using the high potential version of our theory, we could not have fitted their data or ours as the permittivity increments would have been too low, whatever the potential chosen. Our
3626
Langmuir, Vol. 19, No. 9, 2003
Chassagne et al.
Figure 5. ∆/1000 and ∆K/K1 (left and right figures, respectively) as a function of frequency, a ) 67 nm, and (A) 0.5 mM, (B) 1 mM, and (C) 2 mM of added salt. Solid lines are fits (see text). Table 4 a ) 67 nm Cs κa |Ψd| σ St KSt
0.5 4.4 0.55 0.09 9.6 4.6
1 6.2 2.0 0.49 3.36 3.35
a ) 26 nm 2 8.7 4.3 2.32 1.7 3.35
Cs κa |Ψd| σ St KSt
0.25 1.4 0.025 0.004 64 6.1
model also predicts a relaxation frequency of roughly 105 rad/s for the 67 nm spheres; see Figure 5. From Figure 6, it can furthermore be seen that our model predicts a shift of this relaxation frequency to higher values for the 26 and 13 nm particles. Arroyo et al. did not report the values of the conductivity increments. As can be seen from our figures, the conductivity increments are very high for the small particles. The standard model (i.e., without Stern layer conductance) does not predict such high increments (see for instance ref 1). We therefore would have liked to compare their conductivity increments with ours.
0.5 1.9 0.11 0.02 29.5 5.7
a ) 13 nm 1 2.6 0.5 0.13 12 4.7
2 3.8 3 1.29 6.5 4.9
Cs κa |Ψd| σ St KSt
0.5 0.94 3.5 0.09 63.5 6.0
2 1.89 39 0.16 14.5 5.6
6. Conclusions Dielectric spectroscopy measurements on various latexes have been considered. Although the permittivity increment could have been fitted in some cases (not shown) with only one parameter (|Ψd|), it is found that a Stern layer conductance was necessary to fit the corresponding conductivity increment correctly. Therefore both |Ψd| and the Stern layer conductance (related to the parameter St) were used in the fits. The Dukhin number St decreases strongly with ionic strength (see the tables). We find that for almost all latexes KSt gives an average value of about
Dielectric Response of Latex Dispersions
Langmuir, Vol. 19, No. 9, 2003 3627
Figure 6. ∆/1000 and ∆K/K1 (left and right figures, respectively) as a function of frequency for various ionic strengths: (A) a ) 26 nm; (B) a ) 13 nm. Solid and broken lines are fits (see text).
5.5 nS‚m, for all particle radii and all ionic strengths. Exceptions are the 168 nm latex with the highest charge, which gives a value that is of the order of 13.5 nS‚m, and the 67 nm latex, which gives a value around 3.5 nS‚m. We conclude that the Stern layer conductancesK|St and K⊥St are almost constant for all latexes and all ionic strengths. This is in particular the case for both large (306 nm) and small (13 nm) latexes. This supports the idea that a Stern layer conductance is indeed necessary to fit the data for the small latexes, contrary to what was found by Arroyo et al. using the numerical method of DeLacey and White. Arroyo et al. stated that “unlike the large particles, the dielectric constant of the [11.5 nm EM radius] suspensions is reasonably described by the DeLacey and White’s theory, despite the fact that it can be expected that the surfaces of both types of lattices will be quite similar ....” We conclude that the Stern layer conductance scales neither with the particle radius nor with the conductance of the electrolyte. It is more practical to fit KSt than the parameter St. One may ask whether choosing K|St and K⊥St unequal, would improve the fit of the relaxation frequency. Following our theory, the crossover from low- to highfrequency behavior is found to be around λna ) 1. Above that frequency K ˜ ⊥D goes to zero. This leads to a substantial change both of the numerator and of the denominator, which gives the crossover for all latexes. Choosing the
parallel and the normal conductivity of the Stern layer to br unequal does not change the relaxation frequency. Choosing K|St and K⊥St to be unequal changes the denominator in eq 2 for the dielectric coefficient. This denominator gives the local field correction.24 Unequal values of K|St and K⊥St would in fact lead to rather large changes of the local field. We consider this unlikely and therefore chose them to equal 8. The potential Ψd used in the fits was found to increase with ionic strength for κa < 10 and to decrease for κa > 10. In the region κa ) 10 where the 168 nm latexes were measured, the potential was found first to increase and then to decrease. This kind of behavior has been reported by others authors3,23 and seems to be a characteristic behavior of latex dispersions. Moreover the electrokinetic charge calculated from |Ψd| is found to follow the trend of the permittivity enhancement. For the latexes studied, it first strongly increases, and for the larger particles (306 and 168 nm), it then slightly levels off. The increase of the electrokinetic charge with increasing ionic strength has also already been reported (see ref 3), but, as for the behavior of the potential, it is still not well understood. LA0264771 (24) Bo¨ttcher, C. J. F. Theory of electric polarization; Elsevier Publishing Co.: Amsterdam, 1952.
