Determination of Stagnant Layer Conductivity in Polystyrene Suspensions: Temperature Effects

Journal of Colloid and Interface Science 281 (2005) 503–509 www.elsevier.com/locate/jcis

Determination of stagnant layer conductivity in polystyrene suspensions: temperature effects

M.L. Jiménez a,F.J.Arroyoa,∗,F.Carriqueb, U. Kaatze c, A.V. Delgado a

a Departamento de Física Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain b Departamento de Física Aplicada I, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain c Drittes Physikalisches Institut, Universität Göttingen, Germany Received 19 May 2004; accepted 11 August 2004 Available online 1 October 2004

Abstract In the classical theory of electrokinetic phenomena, it is admitted that the whole electrokinetic behavior of any colloidal system is fully determined by the zeta potential, ζ , of the interface. However, both experimental data and theoretical models have shown that this is an incomplete picture, as ions in the stagnant layer (the region between the solid surface and the slip plane—the plane where the equilibrium σ potential equals ζ ) may respond to the ﬁeld. In this paper, we aim at the evaluation of this contribution by the estimation of both KSL (the σ surface conductivity of the stagnant layer) and Kd (the conductivity associated with the diffuse layer). This will be done by measuring the high-frequency dielectric dispersion (HFDD) in polystyrene suspensions; here “high-frequency” means the frequency interval where Maxwell–Wagner–O’Konski relaxation takes place (typically at MHz frequencies). Prior to any conclusions, a treatment of electrode polarization effects in the measurements was needed: we used two methods, and both led to similar results. Simulating the existence of surface conductivity by bulk conductivity, we reached the conclusion that no consistent explanation can be given for our HFDD and additional elec- σ trophoresis data based on the existence of diffuse-layer conductivity alone. We thus show how KSL can be estimated and demonstrate that it can be explained by an ionic mobility very close to that characteristic of ions in the bulk solution. Such mobility, and hence also the values σ of KSL, increases with temperature as expected on simple physical grounds.  2004 Elsevier Inc. All rights reserved.

Keywords: Dielectric dispersion; Polystyrene; Stagnant-layer conductivity; Surface conductivity; Zeta potential

1. Introduction is, when in equilibrium, a constant-potential surface, and the potential on it is the electrokinetic or zeta potential, ζ [1–4]. In the classical or standard description of the electrical As mentioned, the standard model is characterized by zero σ = double layer surrounding a charged particle in an electrolyte stagnant layer conductivity (SLC), or KSL 0, so that ζ is solution, a surface is imagined (the electrokinetic or slip used for the full electrical characterization of the interface. plane) located somewhere in the liquid such that the portion The merit of this description has long been recognized, of the ionic atmosphere between the solid and that surface but so have its drawbacks. In particular, severe discrepan- is absolutely nonconducting. That region contains ions and cies have been found when zeta potentials calculated using liquid, but both are assumed to be immobile in the particle’s different techniques for the same interface were compared system of reference, even in the presence of external ﬁelds. [5–10]. For example, electrophoresis predicts zeta potentials systematically lower than those deduced from dc or ac Following Dukhin and Lyklema [1,2], we call that portion of conductivity measurements [11,12]. Let us ﬁnally mention the electrical double layer the stagnant layer. The slip plane the maxima often found in electrophoretic mobility vs ionic strength plots [13]. * Corresponding author. Fax: +34-958-24-32-14. In addition to this indirect evidences, direct determina- E-mail address: [email protected] (F.J. Arroyo). tions support these arguments too. It has been repeatedly

0021-9797/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2004.08.093 504 M.L. Jiménez et al. / Journal of Colloid and Interface Science 281 (2005) 503–509 shown that the surface charge density evaluated from the tion mechanism, occurring in the kHz frequency range and zeta potential (electrokinetic charge density) is considerably mainly controlled by the zeta potential. The latter is called lower than the titratable surface charge. This means that the the α-relaxation [20]. In contrast to this, the MWO relax- latter is mostly balanced by counterions located in the inner ation is not directly related to the zeta potential but rather to part of the double layer. If such ions can move in response the surface conductivity, Kσ , which can thus be measured to externally applied fields, then nonzero stagnant layer con- without any hypothesis about the zeta potential value and its ductivity must exist. relationship to surface electrical phenomena. It will be clear that more than one kind of experimental In this paper we will describe how HFDD determinations determination is required to fully characterize the ionic at- have been carried out with polystyrene suspensions at dif- mosphere from an electrical viewpoint. Particularly promis- ferent temperatures and discuss the experimental results in ing in this respect appears to be the combination of elec- terms of different contributions to the surface conductivity of trophoretic mobility and dielectric dispersion data in suspen- the particles. Our aim is to demonstrate that HFDD can yield σ sions of well-defined particles. In fact, it has been demon- reasonable values of KSL and ζ and to suggest procedures to strated that the dielectric constant of disperse systems is very perform this task. In addition, changing the temperature and sensitive to the existence of nonzero SLC [14–16].Previ- analyzing how the evaluated quantities behave for different ously, Simonova and Shilov [17] had theoretically found the temperatures will help in checking the internal coherence of effect of ionic mobility in the stagnant layer on the elec- the procedure and its physical feasibility. trophoresis of particles. The possibility of gaining information on SLC values by simultaneously fitting electrophoretic mobility and electric permittivity data has thus been ex- 2. Theoretical review plored in a number of works [11–13]. In the present work we propose an entirely different ap- 2.1. HFDD and surface conductivity σ proach, based on the independent determination of KSL by means of high-frequency dielectric dispersion (HFDD) mea- As mentioned above, at sufficiently high frequency, surements. As is well known, colloidal suspensions are het- only the Maxwell–Wagner–O’Konski relaxation will be ob- erogeneous systems which exhibit different dielectric be- served. For the case of a dilute suspension of spheres, the real havior depending on the frequency of the applied field part of the (complex) permittivity of the suspension, ε(ω), [2–4,18]. Maxwell and Wagner were the first to give a phys- will change with frequency very approximately as follows, ical explanation for such observation: they predicted a re- = + laxation phenomenon when the dielectric constant of a sus- ε (ω) ε∞ φε0ε (ω), pension of spheres was calculated for different permittivity/ εMWO ε (ω) = , (1) conductivity ratios of the solid particles and the suspending + 2 2 1 ω τMWO liquid. where ε∞ is the extrapolated high-frequency permittivity of The origin of this relaxation lies in the polarization of the suspension, ε is the permittivity of vacuum, and φ is the both the particle and the liquid surrounding it. Maxwell and 0 volume fraction of solids. The dielectric increment ε(ω) Wagner studied the polarization mechanism due to the accu- carries information on the relaxation, described by its ampli- mulation of ions in the particle–solution interface during the tude, ε , and its characteristic time, τ . Both quan- electromigration, which is caused by the different conduc- MWO MWO tities are related to the permittivity and conductivity ratios tivities of both media. At sufficiently high frequencies, such between the particles (conductivity K and permittivity ε ) movements cannot follow the fast changes of the external p p and the medium (K and ε ) [21], field, so the latter polarization cannot develop, and the only m m process fast enough to change with the field is the molecular 9ε /ε (K /K − ε /ε )2 = m 0 p m p m polarization of each phase. Hence, the overall dielectric con- εMWO 2 , (2) (εp/εm + 2)(Kp/Km + 2) stant is in this case due to the permittivities of particle and ε (ε /ε + 2) solution. = m p m τMWO + . (3) O’Konski [19] considered how the presence of finite sur- Km(Kp/Km 2) face conductivity would affect the characteristic frequency We will admit that the bulk conductivity of the particles of the relaxation. His procedure involves essentially the cal- is zero but will take into account the presence of their con- culation of the effective bulk conductivity of a spherical ducting ionic shell (the electrical double layer) by means of particle, equivalent to a heterogeneous one consisting of an equivalent conductivity, which is related to the surface an insulating core and a conducting shell. The characteris- conductivity, Kσ , by means of [19] tic frequency of the effect is named the Maxwell–Wagner– 2Kσ O’Konski relaxation frequency, ωMWO. Kp = , (4) The frequency interval around ωMWO (typically in the a MHz range) is often called “high frequency” to distinguish where a is the spherical particle radius. It is usual to refer to it from the relaxation due to the concentration polariza- the ratio Kp/Km in terms of the Dukhin number, Du,defined M.L. Jiménez et al. / Journal of Colloid and Interface Science 281 (2005) 503–509 505

where e is the elementary charge, NA is Avogadro’s number, 3 c is the concentration (mol/m ) of electrolyte, kB is Boltz- mann’s constant, T is the absolute temperature, and κ is the reciprocal double layer thickness: 2e2N c κ = A . (7) εmkBT

In Eq. (6), D+ (D−) is the diffusion coefficient of cations (anions) and m+ (m−) a dimensionless quantity, 2 2εm kBT m± = , (8) 3ηD± e where η is the dynamic viscosity. Finally, ζ is made dimen- Fig. 1. Effect of the surface conductivity (Eq. (5)), on the frequency depen- sionlessasusual: dence of the MWO dielectric increment, ε (ω) (Eq. (1)). a = 100 nm; [NaCl] = 1mM;εp = 2.5ε ; T = 298 K. eζ 0 ζ˜ = . (9) kBT by [2] σ The equation for KSL is simpler, but less rigorous than Kσ K the Bikerman solution. It is based on the assumptions that = = 1 p Du . (5) no liquid motion can take place in the inner part of the ionic Kma 2 Km atmosphere, and that only counterions can participate in the Fig. 1 illustrates the large effect of Du on the amplitude of charge transport. The former assumption is most often valid, the MWO relaxation (the effect on the relaxation frequency but the latter will be true mainly for highly charged colloidal ω ≡ 1/τ is much less significant). The increase in MWO MWO particles. The expression is [2] εMWO with Du can be justified on a physical basis: when the ac field is applied to the suspension, a buildup of ions σ = eDSLσSL KSL , (10) will take place on the high- and low-potential sides of each kBT particle, if its conductivity differs from that of the medium. where D is the diffusion coefficient of counterions ad- Cations (anions) will accumulate on the high- (low-) po- SL sorbed at the stagnant layer and σ is the surface charge tential side if K

3.2. Methods

Electrophoretic mobility measurements were performed on dilute suspensions (volume fraction φ ∼ 10−4)usinga Malvern Zetasizer 2000 (Malvern Instruments, UK). Dielectric spectra in the frequency range 300 kHz– 300 MHz were recorded with an open-ended cell working as a waveguide below its cutoff frequency. The cell was con- nected to an HP-8753A network analyzer via the HP-5044A reflection test set [25]. At each frequency of measurement the permittivity of the suspensions (volume fraction ∼1%) was obtained from the reflection coefficient of an electro- magnetic wave traveling through the cell containing the suspension. Although electrode polarization effects are small at Fig. 2. Experimental results on the dielectric increment of a suspension of these frequencies [26], the dielectric increments are also low, polystyrene latex spheres in 1 mM NaCl, at the temperatures indicated. The and hence care has to be taken to eliminate electrode effects lines correspond to the best fit of the two first terms of the r.h.s. of Eq. (15) as carefully as possible. In our case, this was done either by to the experimental results. simulating these effects by a power-law frequency depen- − dence, ε ∝ ω m, or by using the logarithmic derivative 4. Results and discussion method [27]. Two parameters were used to characterize the HFDD. 4.1. HFDD data The first one is the high-frequency limit of the spectrum (ε∞), related to the permittivities of particle and solution Fig. 2 shows our experimental results for the dielectric and to the volume fraction of particles (Maxwell mixture for- increment of polystyrene suspensions as a function of fre- mula). From it, we calculated the volume fraction of solids quency in the range 2 × 106 to 109 rad/s. The low-frequency in the suspension. The other one is the dielectric increment region of these data does not correspond to a constant (εMWO). Two procedures were used to estimate εMWO plateau, as expected if no relaxation takes place in a given from experimental ε (ω) data. One involved fitting Eq. (1) frequency interval. Two reasons can be given for this: to the data after substracting the previously fitted Aω−m contribution of the electrodes. Alternatively, the logarithmic • In this frequency range the α-relaxation has not been derivative of the dielectric constant was first calculated: completed and it is the decrease of the dielectric constant with frequency that one observes. π ∂ε • There is significant contribution of electrode polariza- ε (ω) =− . (12) D 2 ∂ ln ω tion, that manifests in an increase in ε (ω) when the frequency decreases. This quantity is in fact a good approximation to the imag- inary part of the complex permittivity of the suspension, but Both contributions will be superimposed, and they can be with the advantage that electrode polarization effects are bet- represented by a power law in addition to the relaxation ter separated from the true dielectric relaxation [27].The function proposed above, procedure involved fitting the function ε (ω) = ε∞ + εMWO + −m φ 2 Aω , (15) ε0 ε0 1 + ω2τ π ∂ε (HN) MWO ε (HN) =− , (13) D 2 ∂ ln ω where A and m are positive constants. The full lines in Fig. 2 were calculated from the two first terms of the right hand to the experimental εD data, after elimination of the electrode side of Eq. (15). The best-fit parameters of these terms are polarization contribution, as above described. In Eq. (13), included in Table 1. Obviously, ε∞ decreases with temper- ε (HN) is the real part of the Havriliak–Negami relaxation ature, indicating the normal behavior of the orientation po- function for a dilute suspension [28], larization of a dipolar liquid due to the increasing thermal energy. Furthermore, the ε∞ values of the suspensions are ε0εMWO systematically smaller than those of pure water, at the corre- = ∞ + ε (HN) Re ε φ a b , (14) [1 + (iωτMWO) ] sponding temperature [29]. Indeed, the presence of the electrolyte could partially explain that reduction, but it must be where a and b are parameters of the corresponding relax- mostly due to the small dielectric constant of the polystyrene ation distribution function. particles (εp ∼ 2.5), as explained by Maxwell’s mixture for- M.L. Jiménez et al. / Journal of Colloid and Interface Science 281 (2005) 503–509 507

Table 1 Best-ﬁt parameters of a Debye equation to experimental data in Fig. 2 ◦ T ( C) εMWO ε∞/ε0 τMWO (ns) 25 61.0 ± 0.876.6 ± 0.116.5 ± 0.5 35 58.7 ± 0.973.2 ± 0.123.0 ± 0.7 40 53.6 ± 1.171.5 ± 0.123.6 ± 1.0

Fig. 4. Logarithmic derivative (Eq. (12)) of the dielectric constant of a suspension of polystyrene latex in 1 mM NaCl at the temperatures indicated.

Table 2 Best-fit parameters of Eqs. (13) and (14) to data in Fig. 4 ◦ T ( C) εMWO τMWO (ns) ab 25 72.2 ± 0.55.8 ± 0.40.741 ± 0.008 2.27 ± 0.11 35 75.6 ± 1.925± 20.82 ± 0.03 1.07 ± 0.08 Fig. 3. Effect of temperature on the frequency dependent dielectric incre- 40 61 ± 316± 20.79 ± 0.04 1.41 ± 0.17 ment. ζ =−130 mV; [NaCl] = 1mM. mula: Table 3 − Values of the Dukhin number Du (Eq. (5)) and of the ζ potential (Eq. (6)) εp εm obtained from the data in Fig. 2 and Table 1 ε∞ = εm 1 + 3φ . (16) + ◦ εp 2εm T ( C) Du ζ (mV) From this formula, we find that the volume fraction of parti- 25 0.760 −145 cles in the suspension is 1.6%. 35 0.765 −150 40 0.735 −150 Note that εMWO also decreases with temperature. This result agrees qualitatively with the predictions of standard electrokinetic theories, as demonstrated in Fig. 3,where ior of the relaxation. Unless the tail of the α-relaxation is ε (ω) is plotted as a function of frequency for the three seriously affecting the data (which seems unlikely at these temperatures investigated, assuming −130 mV as zeta po- frequencies), the quality of the data with respect to their in- tential of the particles, according to the MWO model. terpolation and derivation does not seem to be sufficiently The following physical arguments can be given to explain high to allow for quantitative statements on the shape of the the decrease of ε with temperature: raising T will lead to relaxation function from the logarithmic derivative. There- faster ionic motions, so the out-of-phase component of the fore, we will use hereafter only the results in Table 1. current around the particle will decrease in favor of the in- phase component. In other words, the displacement current 4.2. Diffuse and stagnant layer conductivities decreases, while the conduction current increases, as T is raised. This is equivalent to a decrease in the permittivity of From the data in Fig. 2 and Table 1,thevalueofDu the suspension (a measure of the out-of-phase current), as (Eqs. (2) and (5)) can be calculated for each temperature. σ observed in Fig. 3. Knowing Km, if we assume that Du is entirely due to Kd σ Fig. 4 shows the results of the logarithmic derivative ap- (i.e., neglecting KSL), the ζ potential can be calculated using proach, and Table 2 includes the best-fit parameters of the Bikerman’s equation (6). Table 3 shows the results, which Havriliak–Negami logarithmic derivative to the experimen- indicate a rather high zeta potential, essentially temperature- tal data. Roughly, the decrease in εMWO found by fitting independent. Based on HFDD data alone, we have no clear the simple Debye function directly to ε(ω) data is repro- indication on the validity of this approach. An easy check duced using the derivative method. In addition, maxima in of this procedure is considering the electrophoretic mobil- εD (corresponding to the MWO characteristic relaxation ity data, ue, independently obtained; they can be used in two frequencies) are well detected and separated from the elec- ways. One involves calculating ue from the zeta potentials trode polarization effect. However, the data in Table 2 also given in Table 3 [(ue)ε]. The other is based on calculat- show that the parameters a and b are relatively far from the ing ζ from experimental ue and the use of these zeta values value of 1 expected from the practically Debye-like behav- (ζu) for the calculation of εMWO[(εMWO)u].Thetheory 508 M.L. Jiménez et al. / Journal of Colloid and Interface Science 281 (2005) 503–509

Table 4 5. Conclusions Experimental electrophoretic mobilities (ue) of polystyrene particles as a function of temperature, ζ potentials (ζu) deduced from them, and, finally, We have shown that high-frequency dielectric dispersion (ε ) estimated from ζ MWO u u measurements can help in estimating different contributions ◦ −1 −1 T ( C) ue (µm s /Vcm ) ζu (mV) (εMWO)u to the overall surface conductivity, Kσ , of colloidal particles. 25 −3.05 ± 0.10 −44.5 0.085 These data, used in conjunction with simpler electrophoretic − ± − 35 3.91 0.14 48.4 0.13 mobility determinations have allowed us to obtain the zeta 40 −4.43 ± 0.06 −51.2 0.18 potential and, from this, the diffuse–layer surface conductivity. HFDD data enabled us to calculate the total surface conductivity; this led us to an estimation of the stagnant- Table 5 σ σ σ σ layer contribution, K ,toK . It is demonstrated that the Zeta potential (ζ ), diffuse (Kd ) and stagnant-layer (KSL) conductivities, SL + diffusion coefficient of ions in the stagnant layer is similar diffusion coefficient of Na ions in the stagnant layer (DSL), and its ratio with respect to the bulk values, Dbulk, for polystyrene spheres in 1 mM to that in the bulk, in agreement with findings by other au- NaCl at different temperatures thors. σ σ T ζ Kd KSL DSL DSL/Dbulk ◦ − − −9 2 ( C) (mV) (10 9 S) (10 9 S) (10 m /s) 25 −29 0.13 2.50 1.71.29 Acknowledgments 35 −31 0.17 3.10 2.21.32 40 −33 0.19 3.22 2.41.27 Financial support for this work by MCyT, Spain (Projects MAT 2001-3803 and BFM 2000-1099), and FEDER funds is gratefully acknowledged. of O’Brien and White [22] was used for relating ue and ζ . The results of these calculations are displayed in Table 4,and they clearly indicate that the dielectric increments calculated References from ζu underestimate and predict an opposite tendency than the experimental ones, whereas the zeta potentials from Ta- [1] S.S. Dukhin, B.V. Derjaguin, in: E. Matijevic´ (Ed.), Surface and Col- ble 3 yield unphysically high electrophoretic mobilities. loid Science, in: Electrokinetic Phenomena, vol. 7, Wiley, New York, This suggests that a simple model, based on the assump- 1974. tion that all transport mechanisms in the stagnant layer are [2] J. Lyklema, Fundamentals of Interface and Colloid Science, in: Solid– Liquid Interfaces, vol. II, Academic Press, New York, 1995. frozen, does not provide complete description of electroki- [3] R.J. Hunter, Foundations of Colloid Science, Oxford Univ. Press, Ox- netics. We thus tried to improve our description of HFDD by ford, 2001. σ [4] A. Delgado, Interfacial Electrokinetics and Electrophoresis, in: Surfac- considering the fact that KSL could be nonzero. The procedure followed was as follows: tant Science Series, vol. 106, Dekker, New York, 2002, chs. 1 and 3. [5] M. Minor, H.P. van Leeuwen, J. Lyklema, J. Colloid Interface Sci. 206 (1998) 397. (i) Obtain Du from experimental HFDD data, using [6] C.F. Zukoski, D.A. Saville, J. Colloid Interface Sci. 114 (1986) 45. Eqs. (1)–(5). [7] B.R. Midmore, R.J. Hunter, J. Colloid Interface Sci. 122 (1988) 521. (ii) Calculate ζ from experimental electrophoretic mobil- [8] B.R. Midmore, D. Diggins, R.J. Hunter, J. Colloid Interface Sci. 129 ity by means of Eq. (11), together with the Du value (1989) 153. obtained in i. [9] M. Löbbus, H.P. van Leeuwen, J. Lyklema, Colloids Surf. A Physic- σ ochem. Eng. Aspects 161 (2000) 103. (iii) Estimate the diffuse-layer surface conductivity, Kd us- [10] J. Lyklema, M. Minor, Colloids Surf. A Physicochem. Eng. Aspects ing Bikerman’s equation (6). 140 (1998) 33. σ σ σ [11] F.J. Arroyo, F. Carrique, T. Bellini, A.V. Delgado, J. Colloid Interface (iv) Calculate KSL as the difference between K and Kd . Sci. 210 (1999) 194. (v) Obtain DSL using Eq. (10). First, estimate σSL = −(σ 0 + σ ζ ). σ 0 is the titratable surface charge, and [12] F.J. Arroyo, F. Carrique, A.V. Delgado, J. Colloid Interface Sci. 217 (1999) 411. ζ σ is the electrokinetic surface charge, which can be [13] J. Lyklema, H.P. van Leeuwen, M. Minor, Adv. Colloid Interface calculated from ζ [2]. Sci. 83 (1999) 33. [14] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 86 (1990) Table 5 summarizes the results. As observed, reasonable 2859. ζ [15] L.A. Rosen, J.C. Baygents, D.A. Saville, J. Chem. Phys. 98 (1993) values are obtained for both and the surface conductivities. 4183. σ The only effect of temperature is to slightly increase Kd ,an [16] D.A. Saville, Colloids Surf. A Physicochem. Eng. Aspects 92 (1994) expected result, given the increase of diffusion coefficients 29. with temperature. The same behavior was found for DSL. [17] T.S. Simonova, V.N. Shilov, Colloid J. 48 (1986) 319. Furthermore, the values of this quantity are similar to those [18] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and the Double Layer + in Disperse Systems and Polyelectrolytes, Wiley, New York, 1974. found for the bulk diffusion coefficients of Na ions in solu- [19] C.T. O’Konski, J. Phys. Chem. 64 (1960) 605. tion. This is a result also reported by the (scarce number of) [20] A. Delgado, Interfacial Electrokinetics and Electrophoresis, in: Sur- authors that have tried to estimate DSL [2]. factant Science Series, vol. 106, Dekker, New York, 2002, ch. 2. M.L. Jiménez et al. / Journal of Colloid and Interface Science 281 (2005) 503–509 509

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