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 polar- ization 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 poten- tials 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 informa- tion 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 .