Measurement and Interpretation of Electrokinetic Phenomena ✩

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Journal of Colloid and Interface Science 309 (2007) 194–224 www.elsevier.com/locate/jcis Feature article International Union of Pure and Applied Chemistry, Physical and Biophysical Chemistry Division IUPAC Technical Report Measurement and interpretation of electrokinetic phenomena ✩

Prepared for publication by A.V. Delgado a, F. González-Caballero a, R.J. Hunter b, L.K. Koopal c, J. Lyklema c,∗

a University of Granada, Granada, Spain b University of Sydney, Sydney, Australia c Wageningen University, Wageningen, The Netherlands

With contributions from S. Alkafeef, College of Technological Studies, Hadyia, Kuwait; E. Chibowski, Maria Curie Sklodowska University, Lublin, Poland; C. Grosse, Universidad Nacional de Tucumán, Tucumán, Argentina; A.S. Dukhin, Dispersion Technology, Inc., New York, USA; S.S. Dukhin, Institute of Water Chemistry, National Academy of Science, Kiev, Ukraine; K. Furusawa, University of Tsukuba, Tsukuba, Japan; R. Jack, Malvern Instruments Ltd., Worcestershire, UK; N. Kallay, University of Zagreb, Zagreb, Croatia; M. Kaszuba, Malvern Instruments Ltd., Worcestershire, UK; M. Kosmulski, Technical University of Lublin, Lublin, Poland; R. Nöremberg, BASF AG, Ludwigshafen, Germany; R.W. O’Brien, Colloidal Dynamics Inc., Sydney, Australia; V. Ribitsch, University of Graz, Graz, Austria; V.N. Shilov, Institute of Biocolloid Chemistry, National Academy of Science, Kiev, Ukraine; F. Simon, Institut für Polymerforschung, Dresden, Germany; C. Werner, Institut für Polymerforschung, Dresden, Germany; A. Zhukov, University of St. Petersburg, Russia; R. Zimmermann, Institut für Polymerforschung, Dresden, Germany Received 4 December 2006; accepted 7 December 2006 Available online 21 March 2007

Abstract In this report, the status quo and recent progress in electrokinetics are reviewed. Practical rules are recommended for performing electrokinetic measurements and interpreting their results in terms of well-deﬁned quantities, the most familiar being the ζ -potential or electrokinetic potential. This potential is a property of charged interfaces and it should be independent of the technique used for its determination. However, often the ζ-potential is not the only property electrokinetically characterizing the electrical state of the interfacial region; the excess conductivity of the stagnant layer is an additional parameter. The requirement to obtain the ζ -potential is that electrokinetic theories be correctly used and applied within their range of validity. Basic theories and their application ranges are discussed. A thorough description of the main electrokinetic methods is given; special attention is paid to their ranges of applicability as well as to the validity of the underlying theoretical models. Electrokinetic consistency tests are proposed in order to assess the validity of the ζ -potentials obtained. The recommendations given in the report apply mainly to smooth and homogeneous solid particles and plugs in aqueous systems; some attention is paid to nonaqueous media and less ideal surfaces. © 2005 IUPAC. Keywords: Electrokinetics, symbols; Electrokinetics, deﬁnitions; Electrokinetics, measurements; Dielectric dispersion; Permittivity; Electroacoustics; Conductivity; Surface conductivity; Zeta potential; Aqueous and nonaqueous systems

✩ Republication of article in Pure Appl. Chem. 77 (2005) 1753–1805. © IUPAC. * Corresponding author. E-mail address: [email protected] (J. Lyklema). Membership of the Division Committee during preparation of this report (2004–2005) was as follows: President: R.D. Weir (Canada); Vice President: C.M.A. Brett (Portugal); Secretary: M.J. Rossi (Switzerland); Titular Members: G.H. Atkinson (USA); W. Baumeister (Germany); R. Fernández-Prini (Argentina); J.G. Frey (UK); R.M. Lynden-Bell (UK); J. Maier (Germany); Z.-Q. Tian (China); Associate Members: S. Califano (Italy); S. Cabral de Menezes (Brazil); A.J. Mc- Quillan (New Zealand); D. Platikanov (Bulgaria); C.A. Royer (France); National Representatives: J. Ralston (Australia); M. Oivanen (Finland); J.W. Park (Korea); S. Aldoshin (Russia); G. Vesnaver (Slovenia); E.L.J. Breet (South Africa).

Contents

1. Introduction ...... 196 1.1. Electrokineticphenomena...... 196 1.2. Definitions...... 196 1.3. Model of charges and potentials in the vicinity of a surface...... 197 1.4. Planeofshear,electrokineticpotentialandelectrokineticchargedensity...... 197 1.5. Basic problem: Evaluation of ζ -potentials...... 198 1.6. Purposeofthedocument...... 198 2. Elementary theory of electrokinetic phenomena ...... 199 3. Surface conductivity and electrokinetic phenomena ...... 200 4. Methods ...... 201 4.1. Electrophoresis ...... 201 4.1.1. Operational definitions; recommended symbols and terminology; relationship between the measured quantity and ζ -potential...... 201 4.1.2. How and under which conditions can the electrophoretic mobility be converted into ζ -potential?...... 201 4.1.3. Experimental techniques available: Samples ...... 203 4.2. Streaming current and streaming potential ...... 205 4.2.1. Operational definitions; recommended symbols and terminology; conversion of the measured quantities into ζ -potential205 4.2.2. Samplesthatcanbestudied...... 207 4.2.3. Samplepreparation...... 207 4.3. Electro-osmosis...... 207 4.3.1. Operational definitions; recommended symbols and terminology; conversion of the measured quantities into ζ -potential207 4.3.2. Samplesthatcanbestudied...... 208 4.3.3. Sample preparation and standard samples ...... 208 4.4. Experimental determination of surface conductivity ...... 208 4.5. Dielectricdispersion...... 208 4.5.1. Operational definitions; recommended symbols and terminology; conversion of the measured quantities into ζ -potential 208 4.5.2. Dielectric dispersion and ζ -potential:Models...... 209 4.5.3. Experimental techniques available ...... 211 4.5.4. Samples for LFDD measurements ...... 212 4.6. Electroacoustics ...... 212 4.6.1. Operational definitions; recommended symbols and terminology; experimentally available quantities ...... 212 4.6.2. Estimation of the ζ -potential from UCV, ICV,orAESA ...... 213 4.6.3. Experimental procedures ...... 215 4.6.4. Samplesforcalibration...... 215 5. Electrokinetics in nonaqueous systems ...... 215 5.1. Difference with aqueous systems: Permittivity ...... 215 5.2. Experimental requirements of electrokinetic techniques...... 216 5.3. Conversion of the electrokinetic data into ζ -potentials...... 216 6. Remarks on non-ideal surfaces ...... 217 6.1. Generalcomments...... 217 6.2. Hard surfaces ...... 217 6.2.1. Sizeeffects...... 217 6.2.2. Shapeeffects...... 218 6.2.3. Surface roughness ...... 218 6.2.4. Chemical surface heterogeneity ...... 218 6.3. Softparticles...... 218 6.3.1. Charged particles with a soft uncharged layer ...... 219 6.3.2. Uncharged particles with a soft charged layer ...... 219 6.3.3. Charged particles with a soft charged layer ...... 219 6.3.4. Ion-penetrable or partially penetrable particles...... 219 6.3.5. Liquid droplets and gas bubbles in a liquid ...... 219 7. Discussionandrecommendations...... 219 Acknowledgments...... 220 Appendix A. Calculation of the low-frequency dielectric dispersion of suspensions ...... 220 AppendixB. Listofsymbols...... 221 References...... 222 196 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

4 −1 −1 1. Introduction divided by the electric field strength, Qeo,E (m V s ), 3 −1 or divided by the electric current, Qeo,I (m C ). A re- 1.1. Electrokinetic phenomena lated concept is the electro-osmotic counter-pressure, peo (Pa), the pressure difference that must be applied across the Electrokinetic phenomena (EKP) can be loosely defined as system to stop the electro-osmotic volume flow. The value all those phenomena involving tangential fluid motion adjacent peo is considered to be positive if the high pressure is on to a charged surface. They are manifestations of the electri- the higher electric potential side. • cal properties of interfaces under steady-state and isothermal Streaming potential (difference), Ustr (V), is the potential conditions. In practice, they are often the only source of infor- difference at zero electric current, caused by the flow of liq- mation available on those properties. For this reason, their study uid under a pressure gradient through a capillary, plug, di- constitutes one of the classical branches of colloid science, elec- aphragm, or membrane. The difference is measured across trokinetics, which has been developed in close connection with the plug or between the ends of the capillary. Streaming po- the theories of the electrical double layer and of electrostatic tentials are created by charge accumulation caused by the surface forces [1–4]. flow of counter-charges inside capillaries or pores. From the point of view of nonequilibrium thermodynamics, • Streaming current, Istr (A), is the current through the plug EKP are typically cross phenomena, because thermodynamic when the two electrodes are relaxed and short-circuited. −2 forces of a certain kind create fluxes of another type. For in- The streaming current density, jstr (A m ), is the stream- stance, in electro-osmosis and electrophoresis, an electric force ing current per area. leads to a mechanical motion, and in streaming current (poten- • Dielectric dispersion is the change of the electric permittiv- tial), an applied mechanical force produces an electric current ity of a suspension of colloidal particles with the frequency (potential). First-order phenomena may also provide valuable of an applied alternating current (ac) field. For low and mid- information about the electrical state of the interface: for in- dle frequencies, this change is connected with the polariza- stance, an external electric field causes the appearance of a tion of the ionic atmosphere. Often, only the low-frequency surface current, which flows along the interfacial region and dielectric dispersion (LFDD) is investigated. is controlled by the surface conductivity of the latter. If the ap- • Sedimentation potential, Used (V), is the potential differ- plied field is alternating, the electric permittivity of the system ence sensed by two identical electrodes placed some verti- as a function of frequency will display one or more relaxations. cal distance L apart in a suspension in which particles are The characteristic frequency and amplitude of these relaxations sedimenting under the effect of gravity. The electric field may yield additional information about the electrical state of the generated, U /L, is known as the sedimentation field, interface. We consider these first-order phenomena as closely sed E (V m−1). When the sedimentation is produced by a related to EKP. sed centrifugal field, the phenomenon is called centrifugation potential. 1.2. Definitions • Colloid vibration potential, UCV (V), measures the ac potential difference generated between two identical relaxed Here follows a brief description of the main and related EKP electrodes, placed in the dispersion, if the latter is sub- [1–9]. jected to an (ultra)sonic field. When a sound wave travels through a colloidal suspension of particles whose density • Electrophoresis is the movement of charged colloidal par- differs from that of the surrounding medium, inertial forces ticles or polyelectrolytes, immersed in a liquid, under the induced by the vibration of the suspension give rise to a influence of an external electric field. The electrophoretic motion of the charged particles relative to the liquid, caus- velocity, v −1 e (m s ), is the velocity during electrophore- ing an alternating electromotive force. The manifestations sis. The electrophoretic mobility, u (m2 V−1 s−1), is the e of this electromotive force may be measured, depending on magnitude of the velocity divided by the magnitude of the the relation between the impedance of the suspension and electric field strength. The mobility is counted positive if that of the measuring instrument, either as U or as col- the particles move toward lower potential (negative elec- CV loid vibration current, I (A). trode) and negative in the opposite case. CV • Electrokinetic sonic amplitude (ESA) method provides the • Electro-osmosis is the motion of a liquid through an im- mobilized set of particles, a porous plug, a capillary, or a amplitude, AESA (Pa), of the (ultra)sonic field created by membrane, in response to an applied electric field. It is the an ac electric field in a dispersion; it is the counterpart of result of the force exerted by the field on the counter-charge the colloid vibration potential method. • in the liquid inside the charged capillaries, pores, etc. The Surface conduction is the excess electrical conduction moving ions drag the liquid in which they are embedded tangential to a charged surface. It will be represented −1 by the surface conductivity, Kσ (S), and its magnitude along. The electro-osmotic velocity, veo (m s ), is the uniform velocity of the liquid far from the charged interface. with respect to the bulk conductivity is frequently ac- Usually, the measured quantity is the volume flow rate of counted for by the Dukhin number, Du (see Eq. (12) be- liquid (m3 s−1) through the capillary, plug, or membrane, low). A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 197

part governed by purely electrostatic forces). The ﬁxed surface- charge density is denoted σ 0, the charge density at the IHP σ i, and that in the diffuse layer σ d. As the system is electroneu- tral

σ 0 + σ i + σ d = 0. (1)

Potentials. As isolated particles cannot be linked directly to an external circuit, it is not possible to change their surface potential at will by applying an external field. Contrary to mer- cury and other electrodes, the surface potential, ψ0, of a solid is therefore not capable of operational definition, meaning that it cannot be unambiguously measured without making model Fig. 1. Schematic representation of the charges and potentials at a positively assumptions. As a consequence, for disperse systems it is the charged interface. The region between the surface (electric potential ψ0;charge surface charge that is the primary parameter, rather than the sur- 0 density σ )andtheinner Helmholtz plane (distance β from the surface) is free face potential. The potential at the OHP, at distance d from the ψi σ i of charge. The IHP (electric potential ; charge density )isthelocusof surface, is called the diffuse-layer potential, ψd: it is the poten- specifically adsorbed ions. The diffuse layer starts at x = d (outer Helmholtz plane), with potential ψd and charge density σ d.Theslip plane or shear plane tial at the beginning of the diffuse part of the double layer. The is located at x = dek. The potential at the slip plane is the electrokinetic or potential at the IHP, located at distance β(0 β d) from the zeta-potential, ζ ;theelectrokinetic charge density is σ ek. surface, the IHP potential, is given the symbol ψi. All potentials are defined with respect to the potential in bulk solution. 1.3. Model of charges and potentials in the vicinity of a Concerning the ions in the EDL, some further comments can surface be of interest. Usually, a distinction is made between indifferent and specifically adsorbing ions. Indifferent ions adsorb through Charges. The electrical state of a charged surface is de- Coulomb forces only; hence, they are repelled by surfaces of termined by the spatial distribution of ions around it. Such a like sign, attracted by surfaces of opposite sign, and do not pref- distribution of charges has traditionally been called electrical erentially adsorb on an uncharged surface. Specifically adsorb- double layer (EDL), although it is often more complex than just ing ions possess a chemical or specific affinity for the surface two layers, and some authors have proposed the term “electri- in addition to the Coulomb interaction, where chemical or spe- cal interfacial layer.” We propose here to keep the traditional cific is a collective adjective, embracing all interactions other terminology, which is used widely in the field. The simplest than those purely Coulombic. It was recommended in [10], and picture of the EDL is a physical model in which one layer of is now commonly in use to restrict the notion of surface ions the EDL is envisaged as a fixed charge, the surface or titrat- to those that are constituents of the solid, and hence are present able charge, firmly bound to the particle or solid surface, while on the surface, and to proton and hydroxyl ions. The former the other layer is distributed more or less diffusely within the are covalently adsorbed. The latter are included because they solution in contact with the surface. This layer contains an ex- are always present in aqueous solutions, their adsorption can cess of counterions (ions opposite in sign to the fixed charge), be measured (e.g., by potentiometric titration) and they have, and has a deficit of co-ions (ions of the same sign as the fixed for many surfaces, a particularly high affinity. The term specif- charge). ically adsorbed then applies to the sorption of all other ions For most purposes, a more elaborate model is necessary having a specific affinity to the surface in addition to the generic [3,10]: the uncharged region between the surface and the lo- Coulombic contribution. Specifically adsorbed charges are locus of hydrated counterions is called the Stern layer, whereas cated within the Stern layer. ions beyond it form the diffuse layer or Gouy layer (also, Gouy– Chapman layer). In some cases, the separation of the EDL into 1.4. Plane of shear, electrokinetic potential and electrokinetic a charge-free Stern layer and a diffuse layer is not sufficient to charge density interpret experiments. The Stern layer is then subdivided into an inner Helmholtz layer (IHL), bounded by the surface and Tangential liquid flow along a charged solid surface can be the inner Helmholtz plane (IHP) and an outer Helmholtz layer caused by an external electric field (electrophoresis, electro- (OHL), located between the IHP and the outer Helmholtz plane osmosis) or by an applied mechanical force (streaming poten- (OHP). This situation is shown in Fig. 1 for a simple case. The tial, current). Experience and recent molecular dynamic simu- necessity of this subdivision may occur when some ion types lations [11] have shown that in such tangential motion usually a (possessing a chemical affinity for the surface in addition to very thin layer of fluid adheres to the surface: it is called the hy- purely Coulombic interactions), are specifically adsorbed on drodynamically stagnant layer, which extends from the surface the surface, whereas other ion types interact with the surface to some specified distance, dek, where a so-called hydrody- charge only through electrostatic forces. The IHP is the locus namic slip plane is assumed to exist. For distances to the wall, of the former ions, and the OHP determines the beginning of xviscosity with roughly a step-function dependence [12].The ing modifications in the electrical state of the interface rather space charge for x>dek is hydrodynamically mobile and elec- than in obtaining accurate ζ -potentials. On the other hand, trokinetically active, and a particle (if spherical) behaves hydro- when the purpose is to compare the calculated values of ζ of dynamically as if it had a radius a + dek. The space charge for a system under given conditions using different electrokinetic xionic strength: at low ionic strength, the decay of the poten- signal. tial as a function of distance is small and ζ =∼ ψd;athighionic Otherwise, a correct quantitative explanation of EKP will strength, the decay is steeper and |ζ | |ψd|. A similar reason- require the additional estimation of the stagnant-layer conduc- ing applies to the electrokinetic charge, compared to the diffuse tivity. This requires more elaborate treatments [2,3,13–17] than charge. standard or classical theories, in which only conduction at the solution side of the plane of shear is considered. 1.5. Basic problem: Evaluation of ζ -potentials It should be noted that there is a number of situations where electrokinetic measurements, without further interpreta- The notion of slip plane is generally accepted in spite of the tion, provide extremely useful and unequivocal information, of fact that there is no unambiguous way of locating it. It is also great value for technological purposes. The most important of accepted that ζ is fully defined by the nature of the surface, these situations are its charge (often determined by pH), the electrolyte concentration in the solution, and the nature of the electrolyte and of the • identification of the isoelectric point (or point of zero ζ - solvent. It can be said that for any interface with all these para- potential) in titrations with a potential determining ion meters fixed, ζ is a well-defined property. (e.g., pH titration); Experience demonstrates that different researchers often • identification of the isoelectric point in titrations with other find different ζ -potentials for supposedly identical interfaces. ionic reagents such as surfactants or polyelectrolytes; and Sometimes, the surfaces are not in fact identical: the high spe- • identification of a plateau in the adsorption of an ionic cific surface area and surface reactivity of colloidal systems species indicating optimum dosage for a dispersing agent. make ζ very sensitive to even minor amounts of impurities in solution. This can partly explain variations in electrokinetic In these cases, the complications and digressions, which are determinations from one laboratory to another. Alternatively, discussed below, are essentially irrelevant. The electrokinetic since ζ is not a directly measurable property, it may be that an property (or the estimated ζ -potential) is then zero or constant inappropriate model has been used to convert the electrokinetic and that fact alone is of value. signal into a ζ -potential. The level of sophistication required (for the model) depends on the situation and on the particular 1.6. Purpose of the document phenomena investigated. The choice of measuring technique and of the theory used depends to a large extent on the purpose The present document is intended to deal mainly with the of the electrokinetic investigation. following issues, related to the role of the different electroki- There are instances in which the use of simple models can be netic phenomena as tools for surface chemistry research. justified, even if they do not yield the correct ζ -potential. For example, if electrokinetic measurements are used as a sort of • Describe and codify the main and related electrokinetic quality-control tool, one is interested in rapidly (online) detect- phenomena and the quantities involved in their definitions. A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 199

• Give a general overview of the main experimental tech- osmotic velocity), veo, is given by niques that are available for electrokinetic characterization. ε ε ζ • v =− rs 0 E, (4) Discuss the models for the conversion of the experimen- eo η tal signal into ζ -potential and, where appropriate, other where η is the dynamic viscosity of the liquid. This is the double-layer characteristics. Smoluchowski equation for the electro-osmotic slip velocity. • Identify the validity range of such models, and the way in From this, the electro-osmotic flow rate of liquid per current, which they should be applied to any particular experimental Q (m3 s−1 A−1), can be derived, situation. eo,I Qeo ε0εrsζ Qeo,I = =− , (5) The report first discusses the most widely used electrokinetic I ηKL phenomena and techniques, such as electrophoresis, streaming- −1 KL being the bulk liquid conductivity (S m ) and I the electric potential, streaming current, or electro-osmosis. Attention is current (A). also paid to the rapidly growing techniques based on dielectric It is impossible to quantify the distribution of the elec- dispersion and electro-acoustics. tric field and the velocity in pores with unknown or complex geometry. However, this fundamental difficulty is avoided for 2. Elementary theory of electrokinetic phenomena κa 1, when Eqs. (4) and (5) are valid [3]. Electrophoresis is the counterpart of electro-osmosis.Inthe All electrokinetic effects originate from two generic phe- latter, the liquid moves with respect to a solid body when an nomena, namely, the electro-osmotic flow and the convective electric field is applied, whereas during electrophoresis the liq- electric surface current within the EDL. For nonconducting uid as a whole is at rest, while the particle moves with respect to solids, Smoluchowski [18] derived equations for these generic the liquid under the influence of the electric field. In both phe- phenomena, which allowed an extension of the theory to all nomena, such influence on the double layer controls the relative other specific EKP. Smoluchowski’s theory is valid for any motions of the liquid and the solid body. Hence, the results ob- shape of a particle or pores inside a solid, provided the (local) tained in considering electro-osmosis can be readily applied for curvature radius a largely exceeds the Debye length κ−1, obtaining the corresponding formula for electrophoresis. The expression for the electrophoretic velocity, that is, the velocity κa 1, (2) of the particle with respect to a medium at rest, becomes, after changing the sign in Eq. (4), where κ is defined as ε ε ζ v = rs 0 E, (6) N 2 2 1/2 e = e z ni η κ = i 1 i (3) εrsε0kT and the electrophoretic mobility, ue, εrsε0ζ with e the elementary charge, zi , ni the charge number and ue = . (7) number concentration of ion i (the solution contains N ionic η species), εrs the relative permittivity of the electrolyte solution, This equation is known as the Helmholtz–Smoluchowski (HS) ε0 the electric permittivity of vacuum, k the Boltzmann con- equation for electrophoresis. stant, and T the thermodynamic temperature. Note that under Let us consider a capillary with circular cross-section of ra- condition (2), a curved surface can be considered as flat for any dius a and length L with charged walls. A pressure difference small section of the double layer. This condition is traditionally between the two ends of the capillary, p , is produced exter- called the “thin double-layer approximation,” but we do not rec- nally to drive the liquid through the capillary. Since the fluid ek ommend this language, and we rather refer to this as the “large near the interface carries an excess of charge equal to σ , its κa limit.” Many aqueous dispersions satisfy this condition, but motion will produce an electric current known as streaming cur- not those for very small particles in low ionic strength media. rent, Istr: Electro-osmotic flow is the liquid flow along any section of ε ε πa2 p I =− rs 0 ζ. (8) the double layer under the action of the tangential component str η L E of an external field E. In Smoluchowski’s theory, this field t The observation of this current is only possible if the ex- is considered to be independent of the presence of the double tremes of the capillary are connected through a low-resistance layer, i.e., the distortion of the latter is ignored.1 Also, because external circuit (short-circuit conditions). If this resistance is the EDL is assumed to be very thin compared to the particle high (open circuit), transport of ions by this current leads to radius, the hydrodynamic and electric field lines are parallel for the accumulation of charges of opposite signs between the two large κa. Under these conditions, it can be shown [3] that at ends of the capillary and, consequently, to the appearance of a large distance from the surface the liquid velocity (electro- a potential difference across the length of the capillary, the streaming-potential, Ustr. This gives rise to a conduction cur- 1 The approximation that the structure of the double layer is not affected by rent, Ic: the applied field is one of the most restrictive assumptions of the elementary 2 Ustr theory of EKP. I = K πa . (9) c L L 200 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

σ The value of the streaming-potential is obtained by the condi- surface (K ) and bulk (KL) conductivities tion of equality of the conduction and streaming currents (the Kσ net current vanishes) Du = , (12) KLa U ε ε ζ str = rs 0 . (10) where a is the local curvature radius of the surface. For a col- p ηKL loidal system, the total conductivity, K, can be expressed as the For large κa,Eq.(10) is also valid for porous bodies. sum of a solution contribution and a surface contribution. For As described, the theory is incomplete in mainly three as- instance, for a cylindrical capillary, the following expression re- pects: (i) it does not include the treatment of strongly curved sults: surfaces (i.e., surfaces for which the condition κa 1 does not = + σ = + apply); (ii) it neglects the effect of surface conduction both in K (KL 2K /a) KL(1 2Du). (13) the diffuse and the inner part of the electrical double layer; and The factor 2 in Eq. (13) applies for cylindrical geometry. For (iii) it neglects EDL polarization. Concerning the first point, the other geometries, this value may be different. By considering theoretical analysis described above is based on the assumption Ohm’s law, it becomes clear that the field strength is now related that the interface is flat or that its radius of curvature at any to K and hence to KL and Du. point is much larger than the double-layer thickness. When this As mentioned, HS theory does not consider surface con- condition is not fulfilled, the Smoluchowski theory ceases to be duction, and only the solution conductivity, KL, is taken into valid, no matter the existence or not of surface conduction of account to derive the tangential electric field within the double any kind. However, theoretical treatments have been devised layer. Thus, in addition to Eq. (2), the applicability of the HS to deal with these surface curvature effects. Roughly, in or- theory requires der to check if such corrections are needed, one should simply calculate the product κa, where a is a characteristic radius of Du 1. (14) curvature (e.g., particle radius, pore or capillary radius). When The surface conductivity can have contributions owing to the describing the methods below, we will give details about ana- diffuse-layer charge outside the plane of shear, Kσ d, and to the lytical or numerical procedures that can be used to account for charge in the stagnant layer Kσ i: this effect. With respect to surface conductivity, a detailed account is Kσ = Kσ d + Kσ i. (15) given in Section 3, and mention will be made to it where nec- Accordingly, Du can be written as essary in the description of the methods. Here, it may suffice to say that it may be important when the ζ -potential is moderately Kσ d Kσ i Du = + = Dud + Dui. (16) large (>50 mV, say). KLa KLa Finally, the polarization of the double layer implies accu- The Kσ d contribution is called the Bikerman surface con- mulation of excess charge on one side of the colloidal particle ductivity after Bikerman [19–21], who found a simple equation and depletion on the other. The resulting induced dipole is the for Kσ d (see below). The stagnant-layer conductivity (SLC) source of an electric field distribution that is superimposed on may include a contribution due to the specifically adsorbed the applied field and affects the relative solid/liquid motion. charge and another one due to the part of the diffuse-layer The extent of polarization depends on surface conductivity, and charge that may reside behind the plane of shear. The charge on its role in electrokinetics will be discussed together with the the solid surface is generally assumed to be immobile; it does methodologies. not contribute to Kσ . The conductivity in the diffuse double layer outside the plane 3. Surface conductivity and electrokinetic phenomena of shear, Kσ d, consists of two parts: a migration contribution, caused by the movement of charges with respect to the liq- Surface conduction is the name given to the excess electric uid; and a convective contribution, due to the electro-osmotic conduction that takes place in dispersed systems owing to the liquid flow beyond the shear plane, which gives rise to an ad- presence of the electric double layers. Excess charges in them ditional mobility of the charges and hence leads to an extra may move under the influence of electric fields applied tangen- contribution to Kσ d. For the calculation of Kσ d, the Bikerman tially to the surface. The phenomenon is quantified in terms of equation (Eq. (17)) can be used. This equation expresses Kσ d the surface conductivity, Kσ , which is the surface equivalent to as a function of the electrolyte and double-layer parameters. For the bulk conductivity, K . Kσ is a surface excess quantity just L a symmetrical z–z electrolyte, a convenient expression is as the surface concentration Γi of a certain species i. Whatever the charge distribution, Kσ can always be defined through the 2 2 σ d 2e NAz c −zeζ/2kT 3m+ K = D+(e − ) + two-dimensional analog of Ohm’s law, 1 1 2 kT κ z σ σ j = K E, (11) zeζ/2kT 3m− + D−(e − 1) 1 + , (17) z2 where j σ is the (excess) surface current density (A m−1). −3 A measure of the relative importance of surface conductiv- where c is the electrolyte concentration (mol m ), NA is the − ity is given by the dimensionless Dukhin number, Du, relating Avogadro constant (mol 1), and m+ (m−) is the dimensionless A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 201 mobility of the cations (anions) (b.1) Obtain the mobility ue for a range of indifferent elec- 2 trolyte concentrations. If ue decreases with increas- 2 kT εrsε0 m± = , (18) ing electrolyte concentration, use the HS formula, 3 e ηD± Eq. (7), to obtain ζ . − (b.1.1) If the ζ value obtained is low (ζ 50 mV, where D± (m2 s 1) are the ionic diffusion coefficients. The parameters m± indicate the relative contribution of electro- say), concentration polarization is negligible, osmosis to the surface conductivity. and one can trust the value of ζ . The extent to which Kσ influences the electrokinetic behav- (b.1.2) If ζ is rather high (ζ>50 mV, say), then ior of the systems depends on the value of Du. For the Bikerman HS theory is not applicable. One has to use part of the conductivity, Dud can be written explicitly. For a more elaborate models. The possibilities are: symmetrical z–z electrolyte and identical cation and anion dif- (i) the numerical calculations of O’Brien fusion coefficients so that m+ = m− = m: and White [22]; (ii) the equation derived by Dukhin and Semenikhin [5] for symmetrical Kσ d 2 3m zeζ z–z electrolytes Dud ≡ = 1 + cosh − 1 . (19) K a κa z2 2kT L 3 ηe d ue From this equation, it follows that Du is small if κa 1, and 2 εrsε0kT both m and ζ are small. Substitution of this expression for Dud 3yek in Eq. (16) yields = − 6 yek(1 + 3m/z2) 2 σi 2 3m zeζ K × sinh2(zyek/4) Du = 1 + cosh − 1 1 + . (20) κa z2 2kT Kσ d − + 2z 1 sinh(zyek/2) − 3my ek This equation shows that, in general, Du is dependent on the × ek ζ -potential, the ion mobility in bulk solution, and Kσ i/Kσ d. ln cosh(zy /4) κa Now, the condition Du 1 required for application of the HS + 8(1 + 3m/z2) sinh2(zyek/4) theory is achieved for κa 1, rather low values of ζ , and − (24m/z2) ln cosh(zyek/4) , (22) Kσ i/Kσ d < 1. where m was defined in Eq. (18) and 4. Methods eζ yek = (23) 4.1. Electrophoresis kT is the dimensionless ζ -potential. For aqueous 4.1.1. Operational definitions; recommended symbols and solutions, m is about 0.15. O’Brien [4] found terminology; relationship between the measured quantity and that Eq. (22) can be simplified by neglecting ζ -potential terms of order (κa)−1 as follows: Electrophoresis is the translation of a colloidal particle or polyelectrolyte, immersed in a liquid, under the action of an 3 ηe 3 ek ue = y externally applied field, E, constant in time and position- 2 εrsε0kT 2 ek independent. For uniform and not very strong electric fields, 6 y − ln 2 {1 − exp(−zyek)} a linear relationship exists between the steady-state elec- − 2 z ek . (24) 2 + κa exp − zy trophoretic velocity, ve (attained by the particle roughly a few 1+3m/z2 2 milliseconds after application of the field) and the applied field Note that both the numerical calculations and ve = ueE, (21) Eqs. (22) and (24) automatically account for diffuse-layer conductivity. where ue is the quantity of interest, the electrophoretic mobility. (b.2) If a maximum in ue (or in the apparent ζ -potential deduced from the HS formula) vs electrolyte concen- 4.1.2. How and under which conditions can the tration is found, the effect of stagnant-layer conduc- electrophoretic mobility be converted into ζ -potential? tion is likely significant. For low ζ -potentials, when As discussed above, it is not always possible to rigorously concentration polarization is negligible, the follow- obtain the ζ -potential from measurements of electrophoretic ing expression can be used [23]: mobility only. We give here some guidelines to check whether the system under study can be described with the standard elec- ε ε Du kT 2ln2 u = rs 0 ζ 1 + − 1 , (25) trokinetic models: e η 1 + Du e|ζ| z (a) Calculate κa for the suspension. with Du including both the stagnant- and diffuse- (b) If κa 1(κa > 20, say), we are in the large κa regime, layer conductivities. This requires additional infor- and simple analytical models are available. mation about the value of Kσ i (see Section 3). 202 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

(c) If κa is low, the O’Brien and White [22] numerical calculations remain valid, but there are also several analytical approximations. For κa < 1, the Hückel–Onsager (HO) equation applies [4]: 2 ε ε u = rs 0 ζ. (26) e 3 η (d) For the transition range between low and high κa, Hen- ry’s formula can be applied if ζ is presumed to be low (<50 mV; in such conditions, surface conductivity and concentration polarization are negligible). For a nonconducting sphere, Henry derived the expression

2 εrsε0 ue = ζf1(κa), (27) 3 η (a)

where the function f1 varies smoothly from 1.0, for low values of κa,to1.5asκa approaches inﬁnity. Henry [24] gave two series expansions for the function f1, one for small κa and one for large κa. Ohshima [25] has provided an approximate analytical expression which duplicates the Henry expansion almost exactly. Ohshima’s relation is − 1 2.5 3 f (κa) = 1 + 1 + . (28) 1 2 κa[1 + 2exp(−κa)] Equation (27) can be used in the calculation of the electrophoretic mobility of particles with non-zero bulk conductivity, Kp. With that aim, it can be modiﬁed to read [2] 2 ε ε u = rs 0 ζ × F (κa, K ), (29) e 3 η 1 p

2 − 2Krel F1(κa, Kp) = 1 + f1(κa) − 1 (30) 1 + Krel (b) with

Kp Krel = . (31) KL (e) If stagnant-layer conduction is likely for a system with low κa, then as discussed before, ζ ceases to be the only parameter needed for a full characterization of the EDL, and additional information on Kσ is required (see Section 3). Numerical calculations like those of Zukoski and Saville [13] or Mangelsdorf and White [14–16] can be used.

Fig. 2a allows a comparison to be established between the predictions of the different models mentioned, for the case of spheres. For the κa chosen, the curvature is enough for the HS theory to be in error, the more so the higher |ζ|.Ac- cording to Henry’s treatment, the electrophoretic mobility is (c) lower than predicted by the simpler HS equation. Note also that Fig. 2. (a) Electrophoretic mobility ue plotted as a function of the ζ -potential Henry’s theory fails for low-to-moderate ζ -potentials; this is a according to different theoretical treatments, all neglecting stagnant-layer con- consequence of neglecting concentration polarization. The full ductance: Helmholtz–Smoluchowski, O’Brien–White (full theory), Henry (no O’Brien and White theory demonstrates that as ζ increases, the surface conductance), for κa = 15. (b) Role of κa on the mobility–ζ -potential mobility is lower than predicted by either Henry’s or HS cal- relationship (O’Brien–White theory). (c) Effect of stagnant-layer conductance, SLC on the electrophoretic mobility–ζ -potential relationship for the same sus- culations. The existence of surface conduction can account for pensions as in part (a). The ratios between the diffusion coefficients of counte- this. In addition, for sufficiently high ζ -potential, the effect of rions in the stagnant layer and in the bulk electrolyte are indicated (the upper concentration polarization is a further reduction of the mobility, curve corresponds to zero SLC). A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 203 which goes through a maximum and eventually decreases with • Its main limitations are the bias and subjectivity of the the increase of ζ -potential. observer, who can easily select only a narrow range of The effect of κa on the ue(ζ ) relationship is depicted in velocities, which might be little representative of the Fig. 2b. Note that the maximum is more pronounced with the true average value of the suspension. Furthermore, mea- larger κa, and that the electrophoretic mobility increases (in the surements usually take a fairly long time, and this can range of κa shown) with the former. Finally, Fig. 2c demon- bring about additional problems such as Joule heating, strates the drastic change that can occur in the mobility–ζ -po- pH changes, and so on. Hence, some manufacturers of tential trends if SLC is present. This quantity always tends to commercial apparatus have modified their designs to in- decrease ue, as the total surface conductivity is increased, as clude automatic tracking by digital image processing. compared to the case of diffuse-layer conductivity alone. • Recall that electrophoresis is the movement of the particles with respect to the fluid, which is assumed to be 4.1.3. Experimental techniques available: Samples at rest. However, the observed velocity is in fact rela- (i) Earlier techniques, at present seldom used in colloid sci- tive to the instrument, and this is a source of error, as an ence: electro-osmotic flow of liquid is also induced by the ex- • Moving boundary [26]. In this method, a boundary is ternal field if the cell walls are charged, which is often mechanically produced between the suspension and its the case. If the cell is open, the velocity over its section equilibrium serum. When the electric field is applied, would be constant and equal to its value at the outer the migration of the solid particles provokes a dis- double-layer boundary. However, in almost all exper- placement of the solid/liquid separation whose velocity imental set-ups, the measuring cell is closed, and the is in fact proportional to ve. The traditional moving- electro-osmotic counter-pressure provokes a liquid flow boundary method contributed to a great extent to the of Poiseuille type. The resulting velocity profile for the knowledge of proteins and polyelectrolytes as well as case of a cylindrical channel is given by [4] of colloids. It inspired gel electrophoresis, presently es- r2 sential in such important fields as genetic analysis. v = v 2 − 1 , (32) L eo a2 • Mass transport electrophoresis [27]. The mass transport method is based on the fact the application of a known where veo is the electro-osmotic liquid velocity in the potential difference to the suspension causes the parti- channel, a is the capillary radius, and r is the radial dis- cles to migrate from a reservoir to a detachable collec- tance from the cylinder√ axis. From Eq. (32), it is clear tion chamber. The electrophoretic mobility is deduced that vL = 0ifr = a/ 2, so that the true electrophoretic from data on the amount of particles moved after a cer- velocity will be displayed only by particles moving in tain time, which can be determined by simply weighing a cylindrical shell placed at 0.292a from the channel the collection chamber or otherwise analyzing its con- wall. It is easy to estimate the uncertainties associated tents. with errors in the measuring position: if a ∼ 2 mm and (ii) Microscopic (visual) microelectrophoresis the microscope has a focus depth of ∼50 µm, then an er- Probably the most widespread method until the 1980s, ror of 2% in the velocity will be always present. A more microscopic (visual) microelectrophoresis is based on the accurate, although time-consuming method, consists in direct observation, with a suitable magnifying optics, of measuring the whole parabolic velocity profile to check individual particles in their electrophoretic motion. In fact, for absence of systematic errors. These arguments also it is not the particle that is seen, but a bright dot on a dark apply to electrophoresis cells with rectangular or square background, due to the Tyndall effect, that is the strong cross-sections. lateral light scattering of colloidal particles. Some authors (see, e.g., [28]) have suggested that a procedure to avoid this problem would be to cover the Size range of samples cell walls, whatever their geometry, with a layer of un- The ultramicroscope is necessary for particles smaller than charged chemical species, for instance, polyacrylamide. 0.1 µm. Particles about 0.5 µm can be directly observed us- However, it is possible that after some usage, the layer ing a traveling microscope illuminated with a strong (cold) gets detached from the walls, and this would mask the light source. electrophoretic velocity measured at an arbitrary depth, Advantages and prerequisites of the technique with an electro-osmotic contribution, the absence of • The particles are directly observed in their medium. which can only be ascertained by measuring ue of stan- • The suspensions to be studied should be stable and di- dard, stable particles, which in turn remains an open lute; if they are not, individual particles cannot be iden- problem in electrokinetics. tified under the microscope. However, in dilute systems; A more recent suggestion [29] is to perform the elec- the aggregation times are very long, even in the worst trophoresis measurements in an alternating field with conditions, so that velocities can likely be measured. frequency much larger than the reciprocal of the characteristic time τ for steady electro-osmosis (τ ∼ 1s),but Problems involved in the technique and proposed actions smaller than that of steady electrophoresis (τ ∼ 10−4 s). to solve them Under such conditions, no electro-osmotic flow can de- 204 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

velop and hence the velocity of the particle is indepen- of the sample will affect the particle mobility, and ther- dent of the position in the cell. mostating of the cell is not at all easy. Reduction of the Another way of overcoming the electro-osmosis prob- applied voltage decreases this effect, but will also re- lem is to place both electrodes providing the external duce the resolution obtainable from the measurement. field inside the cell, completely surrounded by the sus- The presence of some ions in the medium is recom- pension; since no net external field acts on the charged mended (e.g., 10−4 mol/L NaCl) as this will stabilize layer close to the cell walls, the associated electro- the field in the cell and will improve the repeatabil- osmotic flow will not exist [30]. ity of measurements. Furthermore, some salt is always (iii) Electrophoretic light scattering (ELS) needed anyway because otherwise the double layer be- These are automated methods based on the analysis of comes ill-defined. the (laser) light scattered by moving particles [31–34]. (d) Sample viscosity They have different principles of operation [35].The There is no particular limit as to the viscosity range most frequently used method, known as laser Doppler of samples that can be measured. But it must be em- velocimetry, is based on the analysis of the intensity au- phasized that increasing the viscosity of the medium tocorrelation function of the scattered light. The method will reduce the mobility of the particles and may re- of phase analysis light scattering (PALS) [36–38] has quire longer observation times, with the subsequent the advantage of being suited for particles moving very increased risk of Joule heating. slowly, for instance, close to their isoelectric point. The (e) Permittivity method is capable of detecting electrophoretic mobilities Measurements in a large variety of solvents are possi- − − − − − − as low as 10 12 m2 V 1 s 1, that is, 10 4 µm s 1/Vcm 1 ble, depending on the instrument configuration. in practical mobility units (note that mobilities typi- (f) Fluorescence cally measurable with standard techniques must be above Sample fluorescence results in a reduction in the − − − ∼10 9 m2 V 1 s 1). These techniques are rapid, and mea- signal-to-noise ratio of the measurement. In severe surements can be made in a few seconds. The results cases, this may completely inhibit measurements. obtained are very reproducible, with typical standard devi- ations less than 2%. A small amount of sample is required Sample preparation for analysis, often a few milliliters of a suitable disper- Many samples will be too concentrated for direct measure- sion. However, dilution of the sample may be required, ment and will require dilution. How this dilution is carried and therefore the sample preparation technique becomes out is critical. The aim of sample preparation is to preserve very important. the existing state of the particle surface during the process of dilution. One way to ensure this is by filtering or gen- Samples that can be studied tly centrifuging some clear liquid from the original sample (a) Sample composition and using this to dilute the original concentrated sample. Measurements can be made of any colloidal dispersion In this way, the equilibrium between surface and liquid is where the continuous phase is a transparent liquid and perfectly maintained. If extraction of a supernatant is not the dispersed phase has a refractive index which differs possible, then just letting a sample naturally sediment and from that of the continuous phase. using the fine particles left in the supernatant is a good (b) Size range of samples alternative method. The possibility also exists of dialyzing The lower size limit is dependent upon the sample con- the concentrate against a solution of the desired ionic com- centration, the refractive index difference between dis- position. Another method is to imitate the original medium perse and continuous phase, and the quality of the op- as closely as possible. This should be done with regard to tics and performance of the instrument. Particle sizes pH, concentration of each ionic species in the medium, and down to 5 nm can be measured under optimum condi- concentration of any other additive that might be present. tions. However, attention must be paid to the possible modifica- The upper size limit is dependent upon the rate of sedi- tion of the surface composition upon dilution, particularly mentation of particles (which is related to particle size when polymers or polyelectrolytes are in solution [39]. and density). ELS methods are inherently directional Also, if the particles are positively charged, care must be in their measurement plane. Hence, for a horizontal taken to avoid long storage in glass containers, as dissolu- field, samples can be measured while they are sedi- tion of glass can lead to adsorption of negatively charged menting. Measurement is possible so long as there are species on the particles. For emulsion systems, dilution is particles present in the detection volume. Typically, always problematic, because changing the phase volume measurements are possible for particles with diameters ratio may alter the surface properties due to differential below 30 µm. solubility effects. (c) Sample conductivity The conductivity of samples that can be measured Ranges of electrolyte and particle concentration that can ranges from that of particles dispersed in deionized be investigated water up to media containing greater than physiologi- Microelectrophoresis is a technique where samples must cal saline. In high salt concentration, the Joule heating be dilute enough for particles not to interfere with each A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 205

other. For any system under investigation, it is recom- Streaming current mended that an experiment should be done to check the The first quantity of interest is the streaming current per −1 effect of concentration on the mobility. The concentration pressure drop, Istr/p (SI units: A Pa ), where Istr is the mea- range which can be studied will depend upon the suit- sured current and p the pressure drop. The relation between ability of the sample (e.g., size, refractive index) and the Istr/p and ζ -potential has been found for a number of cases: optics of the instrument. By way of example, a 200-nm polystyrene latex standard (particle refractive index 1.59, (a) If κa 1(a is the capillary radius), the HS formula can be particle absorbance 0.001) dispersed in water (refractive used, index 1.33) can be measured at a solids concentration Istr εrsε0ζ Ac ranging from 2 × 10−3 to 1 × 10−6 g/cm3. =− , (33) p η L Standard samples for checking correct instrument opera- where Ac is the capillary cross-section and L its length. tion If instead of a single capillary, the experimental system is Microelectrophoresis ELS instruments are constructed a porous plug or a membrane, Eq. (33) remains approxi- from basic physical principles and as such need not be mately valid, provided that κa 1 everywhere in the pore calibrated. The correctness of their operation can only be walls. In the case of porous plugs, attention has to be paid to verified by measuring a sample of which the ζ -potential the fact that a plug is not a system of straight parallel capil- is known. A pioneering study in this direction was per- laries, but a random distribution of particles with a resulting formed in 1970 by a group of Japanese surface and colloid porosity and tortuosity, for which an equivalent capillary chemists, forming a committee under the Division of Sur- length and cross-section is just a simplified model. In addi- face Chemistry in the Japan Oil Chemists Society [6, tion, the use of Eq. (33) requires that the conduction current 39]. This group measured and compared ζ -potentials of in the system is determined solely by the bulk conductiv- samples of titanium dioxide, silver iodide, silica, micro- ity of the supporting solution. It often happens that surface capsules, and some polymer latexes. The study involved conductivity is important, and, besides that, the ions in the different devices in nine laboratories, and concluded that plug behave with a lower mobility than in solution. the negatively charged PSSNa (polystyrene-sodium p- Ac/L can be estimated experimentally as follows [40,41]. vinylbenzenesulfonate copolymer) particles prepared as Measure the resistance, R∞, of the plug or capillary wetted described in [40] could be a very useful standard, pro- by a concentrated (above 10−2 mol/L, say) electrolyte so- viding reliable and reproducible mobility data. Currently, ∞ lution, with conductivity KL . Since for such a high ionic there is no negative ζ -potential standard available from strength the double-layer contribution to the overall con- the U.S. National Institute of Standards and Technology ductivity is negligible, we may write (NIST). A positively charged sample available from NIST is Ac 1 = ∞ . (34) Standard Reference Material (SRM) 1980. It contains a L KL R∞ 500 mg/L goethite (α-FeOOH) suspension saturated with In addition, theoretical or semi-empirical models exist that / × −2 / 100 µmol g phosphate in a 5 10 mol L sodium per- relate the apparent values of Ac and L (external dimensions chlorate electrolyte solution at a pH of 2.5. When prepared of the plug) to the volume fraction, φ, of solids in the plug. according to the procedure supplied by NIST, the certified For instance, according to [42] value and uncertainty for the positive electrophoretic mo- − − ap bility of SRM1980 is 2.53 ± 0.12 µm s 1/Vcm 1.This Ac Ac = exp(Bφ), (35) will give a ζ -potential of +32.0 ± 1.5 mV if the HS equa- L Lap tion (Eq. (7)) is used. where B is an empirical constant that can be experimentally determined by measuring the electro-osmotic volume ap ap 4.2. Streaming current and streaming potential flow for different plug porosities. In Eq. (35), L and Ac are the apparent (externally measured) length and cross- 4.2.1. Operational definitions; recommended symbols and sectional area of the plug, respectively. An alternative ex- terminology; conversion of the measured quantities into pression was proposed in [43]: ζ -potential ap The phenomena of streaming current and streaming poten- Ac = Ac −5/2 φL , (36) tial occur in capillaries and plugs and are caused by the charge L Lap displacement in the electrical double layer as a result of an ap- where φL is the volume fraction of liquid in the plug (or plied pressure inducing the liquid phase to move tangentially void volume fraction). Other estimates of Ac/L can be to the solid. The streaming current can be detected directly by found in [44–46]. measuring the electric current between two positions, one up- For the case of a close packing of spheres, theoretical treat- stream and the other downstream. This can be carried out via ments are available involving the calculation of streaming nonpolarizable electrodes, connected to an electrometer of suf- current using cell models. No simple expressions can be ficiently low internal resistance. given in this case; see [3,47–52] for details. 206 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

(b) If κa is intermediate (κa ≈ 1–10, say), the HS equation is (b) The most frequent case (except for high ionic strengths, or σ not valid. For low ζ , curvature effects can be corrected by high KL) is that surface conductance, K , is signiﬁcant. means of the Burgreen and Nakache theory [4,49], Then the following equation should be used:

Istr εrsε0ζ Ac U ε ε ζ 1 = 1 − G(κa) , (37) str = rs 0 , (41) p η L p η KL(1 + 2Du) where where Du is given by Eqs. (12) and (20). tanh(κa) An empirical way of taking into account the existence of G(κa) = (38) κa surface conductivity is to measure the resistance R∞ of the plug or capillary in a highly concentrated electrolyte so- for slit-shaped capillaries (2a corresponds in this case to ∞ the separation of the parallel solid walls). In the case of lution of conductivity KL . As for such a solution, Du is cylindrical capillaries of radius a, the calculation was first negligible, one can write carried out by Rice and Whitehead [50]. They found that σ ∞ 2K the function G(κa) in Eq. (37) reads K R∞ = K + R , (42) L L a s 2I1(κa) G(κa) = , (39) where Rs is the resistance of the plug in the solution under κaI0(κa) study, of conductivity KL.Now,Eq.(41) can be approxi- where I0 and I1 are the zeroth-order and first-order mod- mated by ified Bessel functions of the first kind, respectively. Fig. 3 illustrates the importance of this curvature correction. Ustr εrsε0ζ Rs = ∞ . (43) (c) If the ζ -potential is not low and κa is small, no simple p η KL R∞ expression for Istr can be given, and only numerical pro- (c) If κa is intermediate (κa ∼ 1,...,10) and the ζ -potential cedures are available [52]. is low, Rice and Whitehead’s corrections are needed [50]. For a cylindrical capillary, the result is Streaming potential The streaming potential difference (briefly, streaming poten- 1 − 2I1(κa) Ustr εrsε0ζ Rs κaI0(κa) tial) Ustr can be measured between two electrodes, upstream = ∞ , (44) p η K R∞ 2I (κa) I 2(κa) and downstream in the liquid flow, connected via a high-input L 1 − β 1 − 1 − 1 κaI0(κa) I 2(κa) impedance voltmeter. The quantity of interest is, in this case, 0 the ratio between the streaming potential and the pressure drop, where −1 Ustr/p (V Pa ). The conversion into ζ -potentials can be re- 2 (εrsε0κζ) Rs alized in a number of cases. β = ∞ . (45) η KL R∞ (a) If κa 1 and surface conduction can be neglected, the HS Fig. 4 illustrates some results that can be obtained by using formula can be used: Eq. (44). (d) As in the case of streaming current, for high ζ -potentials, Ustr = εrsε0ζ 1 . (40) only numerical methods are available (see, e.g., [53] for p η KL details).

Fig. 3. Streaming current, Eqs. (37) and (39), relative to that of the Helm- holtz–Smoluchowski value, Eq. (33), plotted as a function of the product κa Fig. 4. Streaming potential, Eq. (44), relative to that of the Helmholtz–Smo- (a: capillary radius, or slit half-width) for slit- and cylindrical-shaped capillar- luchowski value, Eq. (40), as a function of the product κa (a: capillary radius), ies. for the ζ -potentials indicated. Surface conductance is neglected. A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 207

In practice, instead of potential or current measurements for materials have been frequently analyzed and may, therefore, just one driving pressure, the streaming potential and stream- serve as potential reference samples [56,57]. ing current are mostly measured at various pressure differences applied in both directions across the capillary system, and the Range of electrolyte concentrations From the operational standpoint, there is no lower limit to slopes of the functions Ustr = Ustr(p) and Istr = Istr(p) are used to calculate the ζ -potential. This makes it possible to de- the ionic strength of the systems to be investigated by these tect electrode asymmetry effects and correct for them. It is also methods, although in the case of narrow channels, very low advisable to verify that the p dependencies are linear and pass ionic strengths require effective electrical insulation of the set- through the origin. up in order to prevent short-circuiting. However, such low ionic strength values can only be attained if the solid sample is extremely pure and insoluble. The upper value of electrolyte con- 4.2.2. Samples that can be studied centration depends on the sensitivity of the electrometer and Streaming potential/current measurements can be applied to on the applied pressure difference; usually, solutions above study macroscopic interfaces of materials of different shape. 10−1 mol/L of 1–1 charge-type electrolyte are difficult to mea- Single capillaries made of flat sample surfaces (rectangular sure by the present techniques. capillaries) and cylindrical capillaries can be used to produce micro-channels for streaming potential/current measurements. Further, parallel capillaries and irregular capillary systems such 4.3. Electro-osmosis as fiber bundles, membranes, and particle plugs can also be studied. Recall, however, the precautions already mentioned 4.3.1. Operational definitions; recommended symbols and in connection with the interpretation of results in the case of terminology; conversion of the measured quantities into plugs of particles. Other effects, including temperature gradi- ζ -potential ents, Donnan potentials, or membrane potential can contribute In electro-osmosis, a flow of liquid is produced when an to the observed streaming potential or electro-osmotic flow. An electric field E is applied to a charged capillary or porous additional condition is the constancy of the capillary geometry plug immersed in an electrolyte solution. If κa 1 everywhere during the course of the experiment. Reversibility of the signal at the solid/liquid interface, far from that interface the liquid upon variations in the sign and magnitude of p is a criterion will attain a constant (i.e., independent of the position in the for such constancy. channel) velocity (the electro-osmotic velocity) veo, given by Most of the materials studied so far by streaming poten- Eq. (4). If such a velocity cannot be measured, the convenient tial/current measurements, including synthetic polymers and physical quantity becomes the electro-osmotic flow rate, Qeo 3 −1 inorganic non-metals, are insulating. Either bulk materials or (m s ), given by thin films on top of carriers can be characterized. In addition, x = in some cases, semiconductors [54] and even bulk metals [55] Qeo veo dS, (46) have been studied, proving the general feasibility of the experi- Ac ment. where dS is the elementary surface vector at the location in Note that streaming potential/current measurements on sam- the channel where the fluid velocity is veo. The counterparts of ples of different geometries (flat plates, particle plugs, fiber Qeo are Qeo,E (flow rate divided by electric field) and Qeo,I bundles, cylindrical capillaries, ...) each require their own set- (flow rate divided by current). These are the quantities that can up. be related to the ζ -potential. As before, several cases can be distinguished: 4.2.3. Sample preparation The samples to be studied by streaming potential/current (a) If κa 1 and there is no surface conduction, measurements have to be mechanically and chemically stable in Qeo εrsε0ζ the aqueous solutions used for the experiment. First, the geome- Qeo,E ≡ =− Ac, try of the plug must be consolidated in the measuring cell. This E η can be checked by rinsing with the equilibrium liquid through Qeo εrsε0ζ 1 Qeo,I ≡ =− . (47) repeatedly applying p in both directions until finding a con- I η KL stant signal. Another issue to consider is the necessity that the (b) With surface conduction, the expression for Q is as in solid has reached chemical equilibrium with the permeating liq- eo,E Eq. (47), and that for Q is uid; this may require making the plug from a suspension of eo,I the correct composition, followed by rinsing. Checking that the =−εrsε0ζ 1 experimental signal does not change during the course of mea- Qeo,I . (48) η KL(1 + 2Du) surement may be a good practice. The presence or formation of air bubbles in the capillary system has to be avoided. In Eq. (48), the empirical approach for the estimation of Du can be followed: Standard samples εrsε0ζ Rs No standard samples have been developed specifically so far Qeo,I =− ∞ . (49) for streaming potential/current measurements, although several η KL R∞ 208 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

(c) Low ζ -potential, ﬁnite surface conduction, and arbitrary works best if the two electrokinetic techniques have a capillary radius [46]: rather different sensitivity for surface conduction (such as ε ε ζ electrophoresis and LFDD). Q =− rs 0 A 1 − G(κa) , eo,E η c In many cases, it is found that the surface conductivity ob- ε ε ζ [1 − G(κa)] Q =− rs 0 tained in one of these ways exceeds Kσ d, sometimes by orders eo,I + η KL(1 2Du) of magnitude. This means that Kσ i is substantial. The proce- ε ε ζ R [1 − G(κa)] σ i σ d σ =∼ − rs 0 s dure for obtaining K consists of subtracting K from K . ∞ , (50) σ d η KL R∞ For K , Bikerman’s equation (Eq. (17)) can be used. The where the function G(κa) is given by Eq. (38). method is not direct because this evaluation requires the ζ -po- (d) When ζ is high and the condition κa 1 is not fulﬁlled, tential, which is one of the unknowns; hence, iteration is required. no simple expression can be given for Qeo.

As in the case of streaming potential and current, the pro- 4.5. Dielectric dispersion cedures described can be also applied to either plugs or membranes. If the electric field E is the independent variable (see 4.5.1. Operational definitions; recommended symbols and Eq. (47)), then Ac must be estimated. In that situation, the rec- terminology; conversion of the measured quantities into ommendations suggested in Section 4.2.1 can be used, since ζ -potential Eq. (47) can be written as The phenomenon of dielectric dispersion in colloidal suspensions involves the study of the dependence on the frequency Q ε ε ζ A eo,E =− rs 0 c , (51) of the applied electric field of the electric permittivity and/or Vext η L the electric conductivity of disperse systems. When dealing where Vext is the applied potential difference. with such heterogeneous systems as colloidal dispersions, these quantities are defined as the electric permittivity and conductiv- 4.3.2. Samples that can be studied ity of a sample of homogeneous material, that when placed be- The same samples as with streaming current/potential, see tween the electrodes of the measuring cell, would have the same Section 4.2.2. resistance and capacitance as the suspension. The dielectric investigation of dispersed systems involves determinations of 4.3.3. Sample preparation and standard samples their complex permittivity, ε∗(ω) (F m−1) and complex conduc- See Section 4.2.3, referring to streaming potential/current tivity K∗(ω) (S m−1) as a function of the frequency ω (rad s−1) determination. of the applied ac field. These quantities are related to the volume, surface, and geometrical characteristics of the dispersed 4.4. Experimental determination of surface conductivity particles, the nature of the dispersion medium, and also to the concentration of particles, expressed either in terms of volume Surface conductivities are excess quantities and cannot be fraction, φ (dimensionless) or number concentration N (m−3). directly measured. There are, in principle, three methods to es- ∗ It is common to use the relative permittivity εr (ω) (dimen- timate them. sionless), instead of the permittivity ∗ = ∗ (i) In the case of plugs, measure the plug conductivity Kplug ε (ω) εr (ω)ε0, (52) as a function of K . The latter can be changed by adjusting L ε being the permittivity of vacuum. K∗(ω) and ε∗(ω) are not the electrolyte concentration. The plot of K vs K has 0 plug L independent quantities: a large linear range which can be extrapolated to KL = 0 where the intercept represents Kσ . This method requires a ∗ ∗ ∗ K (ω) = KDC − iωε (ω) = KDC − iωε0ε (ω) (53) plug and seems relatively straightforward. r (ii) For capillaries, deduce Kσ from the radius dependence of or, equivalently,

the streaming potential, using Eq. (41) and the definition ∗ Re K(ω) = K + ωε Im ε (ω) , of Du (Eq. (12)). This method is rather direct, but requires DC 0 r ∗ =− ∗ a range of capillaries with different radii, but identical sur- Im K (ω) ωε0 Re εr (ω) , (54) face properties [58,59]. (iii) Utilize the observation that, when surface conductivity is where KDC is the direct-current (zero frequency) conductivity of the system. not properly accounted for, different electrokinetic tech- ∗ niques may give different values for the ζ -potential of the The complex conductivity K of the suspension can be ex- same material under the same solution conditions. Cor- pressed as rect the theories by inclusion of the appropriate surface ∗ ∗ K (ω) = K + δK (ω), (55) conductivity, and find in this way the value of Kσ that har- L monizes the ζ -potential. This laborious method requires where δK∗(ω) is usually called conductivity increment of the insight into the theoretical backgrounds [60,61], and it suspension. Similarly, the complex dielectric constant of the A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 209 suspension can be written in terms of a relative permittivity in- the characteristic value of the displacement current density ex- ∗ crement or, briefly, dielectric increment δεr (ω): ceeds that of conduction currents, and the space distribution of the local electric fields is determined by polarization of the ∗ = + ∗ εr (ω) εrs δεr (ω). (56) molecular dipoles, rather than by the distribution of ions. As in homogeneous materials, the electric permittivity is the macroscopic manifestation of the electrical polarizability 4.5.2. Dielectric dispersion and ζ -potential: Models of the suspension components. Mostly, more than one relax- (a) Middle-frequency range: Maxwell–Wagner–O’Konski relaxation ation frequency is observed, each associated with one of the There are various mechanisms for the polarization of a het- various mechanisms contributing to the system’s polarization. erogeneous material, each of which is always associated Hence, the investigation of the frequency dependence of the with some property that differs between the solid, the liq- electric permittivity or conductivity allows us to obtain infor- uid, and their interface. The most widely known mecha- mation about the characteristics of the disperse system that are nism of dielectric dispersion, the Maxwell–Wagner disper- responsible for the polarization of the particles. sion, occurs when the two contacting phases have different The frequency range over which the dielectric dispersion of conductivities and electric permittivities. If the ratio ε /K suspensions in electrolyte solutions is usually measured extends rp p is different from that of the dispersion medium, i.e., if between 0.1 kHz and several hundred MHz. In order to define εrp ε in this frame the low-frequency and high-frequency ranges, it = rs , (61) is convenient to introduce an important concept dealing with a Kp KL point in the frequency scale. This frequency corresponds to the the conditions of continuity of the normal components of reciprocal of the Maxwell–Wagner–O’Konski relaxation time the current density and the electrostatic induction on both τMWO, sides of the surface are inconsistent with each other. This results in the formation of free ionic charges near the sur- 1 (1 − φ)Kp + (2 + φ)KL ωMWO ≡ ≡ , (57) face. The finite time needed for the formation of such a τ ε [(1 − φ)ε + (2 + φ)ε ]ε MWO 0 rp rs rs free charge is in fact responsible for the Maxwell–Wagner and it is called the Maxwell–Wagner–O’Konski relaxation fre- dielectric dispersion. quency. In Eq. (57), Kp and εrp are, respectively, the conduc- In the Maxwell–Wagner model, no specific properties are tivity and dielectric constant of the dispersed particles. For low assumed for the surface, which is simply considered as volume fractions and low permittivity of the particles (εrp a geometrical boundary between homogeneous phases. εrs), this expression reduces to The Maxwell–Wagner model was generalized by O’Kon- ski [62], who first took the surface conductivity Kσ ex- ε ε 1 τ ≈ rs 0 ≈ , plicitly into account. In his treatment, the conductivity of MWO 2 (58) KL 2Dκ the particle is modified to include the contributions of both where D is the mean diffusion coefficient of ions in solution. the solid material and the excess surface conductivity. This The last term in Eq. (58) suggests that τMWO roughly corre- effective conductivity will be called Kpef: sponds in this case to the time needed for ions to diffuse a 2Kσ distance of the order of one Debye length. In fact, in such con- K = K + . (62) pef p a ditions τMWO equals τel, the so-called relaxation time of the electrolyte solution. It is a measure of the time required for Both the conductivity and the dielectric constant can be charges in the electrolyte solution to recover their equilibrium considered parts of a complex electric permittivity of any of distribution after ceasing an external perturbation. Its role in the system’s components. Thus, for the dispersion medium, κ−1 the time domain is similar to the role of in assessing the ∗ = − KL εrs εrs i , (63) double-layer thickness. ωε0 The low-frequency range can be defined by the inequality and for the particle, K ω<ωMWO. (59) ∗ = − pef εrp εrp i . (64) For these frequencies, the characteristic value of the conduction ωε0 current density in the electrolyte solution significantly exceeds In terms of these quantities, the Maxwell–Wagner–O’Kon- the displacement current density, and the spatial distribution of ski theory gives the following expression for the complex the local electric fields in the disperse system is mainly de- dielectric constant of the suspension: termined by the distribution of ionic currents. The frequency ∗ + ∗ + ∗ − ∗ ∗ ∗ εrp 2εrs 2φ(εrp εrs) dependence shown by the permittivity of colloidal systems in ε = ε . (65) r rs ε∗ + 2ε∗ − φ(ε∗ − ε∗ ) this frequency range is known as low-frequency dielectric dis- rp rs rp rs persion or LFDD. (b) Low-frequency range: dilute suspensions of nonconducting In the high-frequency range, determined by the inequality spherical particles with κa 1, and negligible Kσ i At moderate or high ζ -potentials, mobile counterions are ω>ωMWO (60) more abundant than coions in the electrical double layer. 210 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

∗ ∗ Fig. 5. Real and imaginary parts of the dielectric increment δεr (divided by the Fig. 6. Real part of the dielectric increment δεr (per volume fraction) as a func- volume fraction φ) for dilute suspensions of 100-nm particles in KCl solution, tion of the frequency of the applied field for dilute suspensions of 100-nm par- as a function of the frequency of the applied field for κa = 10 and ζ = 100 mV. ticles in KCl solution. The ζ -potentials are indicated, and in all cases, κa = 10. The arrows indicate the approximate location of the α (low frequency) and Maxwell–Wagner–O’Konski relaxations. electrokinetic technique can provide such a clear account of the double-layer relaxation processes. The effect of ζ - Therefore, the contribution of the counterions and the potential on the frequency dependence of the dielectric con- coions to surface currents in the EDL differs from their constant is plotted in Fig. 6: the dielectric increment always tribution to currents in the bulk solution. Such difference increases with ζ , as a consequence of the larger concentra- gives rise to the existence of a field-induced perturbation of tion of counterions in the EDL: all processes responsible the electrolyte concentration, δc(r), in the vicinity of the for LFDD are amplified for this reason. polarized particle. The ionic diffusion caused by δc(r) pro- The procedure for obtaining ζ from LFDD measurements vokes a low-frequency dependence of the particle’s dipole ∗ is somewhat involved. It is advisable to determine experi- coefficient, C0 (see below). This is the origin of the low- mentally the dielectric constant (or, equivalently, the con- frequency dielectric dispersion (α-dispersion) displayed by ductivity) of the suspension over a wide frequency range, colloidal suspensions. Recall that the dipole coefficient re- ∗ and use Eqs. (A.2) and (A.3) (see Appendix A) to esti- lates the dipole moment d to the applied field E.Forthe mate the LFDD curve that best fits the data. A simpler case of a spherical particle of radius a, the dipole coeffi- routine is to measure only the low-frequency values, and cient is defined through the relation deduce ζ using the same equations, but substituting ω = 0. ∗ ∗ d = 4πε ε a3C E. (66) However, the main experimental problems occur at low fre- rs 0 0 quencies. The calculation of this quantity proves to be essential for (c) Dilute suspensions of nonconducting spherical particles evaluation of the dielectric dispersion of the suspension with arbitrary κa, and negligible Kσ i [63–65]. A model for the calculation of the low-frequency In this situation, there are no analytical expressions relating conductivity increment δK∗(ω) and relative permittivity LFDD measurements to ζ . Instead, numerical calculations ∗ ∗ increment δεr (ω) from the dipole coefficient C0 when based on DeLacey and White’s treatment [66] are recom- κa 1 is described in Appendix A. There it is shown that, mended. As before, the computing routine should be con- in the absence of stagnant-layer conductivity, the only para- structed to perform the calculations a number of times with meter of the solid/liquid interface that is needed to account different ζ -potentials as inputs, until agreement between forLFDDistheζ -potential. theory and experiment is obtained over a wide frequency The overall behavior is illustrated in Fig. 5 for a dilute dis- range (or, at least, at low frequencies). persion of spherical nonconducting particles (a = 100 nm, (d) Dilute suspensions of nonconducting spherical particles −3 εrp = 2)ina10 mol/L KCl solution (εrs = 78.5), and with κa 1 and stagnant-layer conductivity with negligible ionic conduction in the stagnant layer. In The problem of generalizing the theory by taking into ac- this figure, we plot the variation of the real and imaginary count surface conduction caused by ions situated in the ∗ parts of δεr with frequency. Note the very significant ef- hydrodynamically stagnant layer has been dealt with in [60, fect of the double-layer polarization on the low-frequency 61,64]. In theoretical treatments, stagnant-layer conduction dielectric constant of a suspension. The variation with fre- is equated to conduction within the Stern layer. quency is also noticeable, and both the α- (around 105 s−1) According to these models, the dielectric dispersion is de- and Maxwell–Wagner (∼2 × 107 s−1) relaxations are ob- termined by both ζ and Kσ i. This means that, as discussed served, the amplitude of the latter being much smaller than before, additional information on Kσ i (see the methods that of the former for the conditions chosen. No other described in Section 4.4) must accompany the dielectric A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 211

or, in terms of the particle volume fraction − 1 1/2 L = a 1 + . (68) D (φ−1/3 − 1)2 From these expressions, the simplified model allows us to obtain the dielectric increment at low frequency as follows. Let us call Re[δε∗(0)] ε(0) ≡ r (69) φ the specific (i.e., per unit volume fraction) dielectric increment (for ω → 0). In the case of dilute suspensions, this quantity is a constant (independent of φ), that we denote εd(0): Fig. 7. Real part of the dielectric increment (per volume fraction) of dilute sus- [ ∗ ] Re δεr (0) pensions as in Fig. 5. The curves correspond to increasing importance of SLC; εd(0) ≡ . (70) the ratios between the diffusion coefficients of counterions in the stagnant layer φ φ→0 and in the bulk electrolyte are indicated (the lower curve corresponds to zero SLC). The model allows us to relate Eq. (69) with Eq. (70) through the volume-fraction dependence of LD: − dispersion measurements. Using dielectric dispersion data 1 3/2 ε(0) = ε (0) 1 + . (71) alone can only yield information about the total surface d (φ−1/3 − 1)2 conductivity [66]. (e) Dilute suspensions of nonconducting spherical particles A similar relationship can be established between dilute with arbitrary κa and stagnant-layer conductivity (SLC) and concentrated suspensions in the case of the relaxation frequency ωα = 1/τα: Only numerical methods are available if this is the physical nature of the system under study. The reader is referred to 1 ωα = ωα 1 + , (72) [13–16,61,62,67–69]. Fig. 7 illustrates how important the d (φ−1/3 − 1)2 effect of SLC on Re[ε∗(ω)] can be for the same conditions r where ω is the reciprocal of the relaxation time for a di- as in Fig. 5. Roughly, the possibility of increased ionic mo- αd lute suspension. Using this model, the dielectric increment bilities in the stagnant layer brings about a systematically and characteristic frequency of a concentrated suspension larger dielectric increment of the suspension: surface cur- can be related to those corresponding to the dilute case rents are larger for a conducting stagnant layer, and hence which, in turn, can be related, as discussed above, to the the electrolyte concentration gradients, ultimately respon- ζ -potential and other double-layer parameters. A general sible for the dielectric dispersion, will also be increased. treatment of the problem, valid for arbitrary values of κa (f) Nondilute suspensions of nonconducting spherical parti- and ζ can be found in [74]. cles with large κa and negligible Kσ i Considering that many suspensions of technological inter- Summing up, we can say that the dielectric dispersion of est are rather concentrated, the possibilities of LFDD in the suspensions is an interesting physical phenomenon, extremely characterization of moderately concentrated suspensions sensitive to the characteristics of the particles, the solution, and have also received attention. This requires establishing a their interface. It can provide invaluable information on the dy- theoretical basis relating the dielectric or conductivity in- namics of the EDL and the processes through which it is altered crements of such systems to the concentration of particles by the application of an external field. However, because of [70–72]. Here, we focus on a simplified model [73] that al- the experimental difficulties involved in its determination, it is lows the calculation of the volume-fraction dependence of unlikely that dielectric dispersion measurements alone can be ∗ both the low-frequency value of the real part of δεr , and useful as a tool to obtain the ζ -potential of the dispersed parti- of the characteristic time τα of the α-relaxation. The start- cles. ing point is the assumption that LD, the length scale over which ionic diffusion takes place around the particle, can 4.5.3. Experimental techniques available be averaged in the following way between the values of a One of the most usual techniques for measuring the dielec- very dilute (LD ≈ a) and a very concentrated (LD ≈ b − a; tric permittivity and/or the conductivity of suspensions as a b is half the average distance between the centers of neigh- function of the frequency of the applied field is based on the use boring particles) dispersion, of a conductivity cell connected to an impedance analyzer. This technique has been widely employed since it was first proposed − by Fricke and Curtis [75]. In most modern set-ups, the distance 1 1 1/2 LD = + , (67) between electrodes can be changed (see, e.g., [69,76–79]). The a2 (b − a)2 need for variable electrode separation stems from the problem 212 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 of electrode polarization at low frequencies, since at sufficiently Colloid vibration potential/current may be considered as the low frequencies the electrode impedance dominates over that ac analog of sedimentation potential/current. Similarly, ESA of the sample. The method makes use of the assumption that may be considered as the ac analog of classical electrophore- electrode polarization does not depend on their distance. A so- sis. The relationships between electroacoustics and classical called quadrupole method has been recently introduced [80] in electrokinetic phenomena may be used for testing modern elec- which the correction for electrode polarization is optimally car- troacoustic theories, which should at least provide the correct ried out by proper calibration. Furthermore, the method based limiting transitions to the well-known and well-established re- on the evaluation of the logarithmic derivative of the imaginary sults of the classical electrokinetic theory. A very important part of raw ε∗(ω) data also seems to be promising [81]. advantage of the electroacoustic techniques is the possibility These are not, however, the only possible procedures. they offer to be applied to concentrated dispersions. A four-electrode method has also been employed with success Experimentally available quantities [60,61,68]: in this case, since the sensing and current-supplying electrodes are different, polarization is not the main problem, Colloid vibration potential (UCV) but the electronics of the experimental set-up is rather compli- If a standing sound wave is established in a suspension, cated. a voltage difference can be measured between two different points in the standing wave. If measured at zero current flow, 4.5.4. Samples for LFDD measurements it is referred to as colloid vibration potential. The measured There are no particular restrictions to the kind of colloidal voltage is due to the motion of the particles: it alternates at particles that can be studied with the LFDD technique. The ob- the frequency of the sound wave and is proportional to another vious precautions involve avoiding sedimentation of the parti- measured value, P , which is the pressure difference between cles during measurement, and control of the stability of the sus- the two points of the wave. The kinetic coefficient, equal to the pensions. LFDD quantities are most sensitive to particle size, ratio particle concentration, and temperature. Hence, the constancy UCV UCV mσ i of the latter is essential. Another important concern deals with = (ω,φ,ρ/ρ,K ,ζ,η,a), (73) p p the effect of electrode polarization. Particularly at low frequencies, electrode polarization can be substantial and completely characterizes the suspension. In Eq. (73), ρ is the difference invalidate the data. This fact imposes severe limitations on the in density between the particles and the suspending fluid, of electrolyte concentrations that can be studied; it is very hard to density ρ. obtain data for ionic strengths in excess of 1 to 5 mmol L−1. Colloid vibration current (ICV) If the measurements of the electric signal caused by the 4.6. Electroacoustics sound wave in the suspension, are carried out under zero UCV conditions (short-circuited), an alternating current ICV can be 4.6.1. Operational definitions; recommended symbols and measured. Its value, measured between two different points terminology; experimentally available quantities in the standing wave, is also proportional to the pressure dif- Terminology ference between those two points, and the kinetic coefficient The term “electroacoustics” refers to two kinds of closely ICV/p characterizes the suspension and is closely related to related phenomena: UCV/p:

ICV ICV ∗ UCV • Colloid vibration current (I ) and colloid vibration po- = (ω,φ,ρ/ρ,Kσ i,ζ,η,a)= K . (74) CV p p p tential (UCV) are two phenomena in which a sound wave is passed through a colloidal dispersion and, as a result, elec- Electrokinetic sonic amplitude (ESA) trical currents and fields arise in the suspension. When the This refers to the measurement of the sound wave amplitude, wave travels through a dispersion of particles whose den- which is caused by the application of an alternating electric field sity differs from that of the surrounding medium, inertial to a suspension of particles of which the density is different forces induced by the vibration of the suspension give rise from that of the suspending medium. The ESA signal (i.e., the to a motion of the charged particles relative to the liquid, amplitude AESA of the sound pressure wave generated by the causing an alternating electromotive force. The manifesta- applied electric field) is proportional to the field strength E, and tions of this electromotive force may be measured in a way the kinetic coefficient AESA/E can be expressed as a function depending on the relation between the impedance of the of the characteristics of the suspension suspension and the properties of the measuring instrument, AESA AESA either as ICV (for small impedance of the meter) or as UCV = (ω,φ,ρ/ρ,Kσ i,ζ). (75) (for large one). E E • The reciprocal effect of the above two phenomena is the Measurement of ICV or AESA rather than UCV has the oper- electrokinetic sonic amplitude (ESA), in which an alternat- ational advantage that it enables measurement of the kinetic ing electric field is applied to a suspension and a sound characteristics, ICV/p and AESA/E, which are independent wave arises as a result of the motion of the particles caused of the complex conductivity K∗ of the suspension, and thus by their ac electrophoresis. knowledge of K∗ is not a prerequisite for the extraction of the A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 213

ζ -potential from the interpretation of the electroacoustic measured at any particle concentration, there is some difficulty (see ∗ surements. Note, however, that if SLC is significant, as with the below) in the conversion of ud to a ζ -potential, except for the other electrokinetic phenomena, additional measurements will simplest case of spherical particles in fairly dilute suspensions be needed, as both ζ and Kσ i are required to fully characterize (up to a few vol%). In this respect, the situation is similar the interface. to that for the more conventional electrophoretic procedures. There are, though, some offsetting advantages. In particular, the 4.6.2. Estimation of the ζ -potential from UCV, ICV,orAESA ability to operate on concentrated systems obviates the prob- There are two recent methods for the theoretical interpre- lems of contamination which beset some other procedures. It tation of the data of electroacoustic measurements and ex- also makes possible meaningful measurements on real systems tracting from them a value for the ζ -potential. One is based without the need for extensive dilution, which can compromise on the symmetry relation proposed in [82,83] to express both the estimation of ζ -potential, especially in emulsions systems. kinds of electroacoustic phenomena (colloid vibration poten- ∼ tial/current and ESA) in terms of the same quantity, namely Dilute systems (up to 4 vol%) the dynamic electrophoretic mobility, u∗, which is the complex, d frequency-dependent analog of the normal direct current (dc) 1. For a dilute suspension of spherical particles with κa electrophoretic mobility. The second method [105] is based on 1 and arbitrary ζ -potential, the following equation can be used [84], which relates the dynamic mobility with the ζ - the direct evaluation of ICV without using the symmetry relations, and hence it is not necessarily based on the concept of potential and other particle properties: dynamic electrophoretic mobility. Both methods for ζ -potential ∗ 2εrsε0ζ determination from electroacoustic measurements are briefly u = (1 + f)G(α). (79) d 3η described below. Using the dynamic mobility method has some advantages: The restriction concerning double-layer thickness requires ∗ in practice that κa > ∼20 for reliable results, although the (i) the zero frequency limiting value of ud is the normal elec- ∗ error is usually tolerable down to about κa = 13. trophoretic mobility, and (ii) the frequency dependence of ud can be used to estimate not only the ζ -potential, but also (for The function f is a measure of the tangential electric field particle radius > 40 nm) the particle size distribution. Since around the particle surface. It is a complex quantity, given the calculation of the ζ -potential in the general case requires by a knowledge of κa, it is helpful to have available the most ap- 1 + iω −[2Du + iω (εrp/εrs)] propriate estimate of the average particle size (most colloidal f = , + +[ + ] (80) dispersions are polydisperse, and there are many possible “aver- 2(1 iω ) 2Du iω (εrp/εrs) age” sizes which might be chosen). Although the calculation of where ω ≡ ωεrsε0/KL is the ratio of the measurement fre- the ζ -potential from the experimental measurements would be quency, ω, to the Maxwell–Wagner relaxation frequency of a rather laborious procedure, the necessary software for effect- the electrolyte. If it can be assumed that the tangential cur- ing the conversion is provided as an integral part of the available rent around the particle is carried essentially by ions outside measuring systems, for both dilute and moderately concentrated the shear plane, then Du isgivenbytheDud (Eqs. (16)– sols. The effects of SLC can also be eliminated in some cases, (19)); see also [85].2 without access to alternative measuring devices, simply by un- The function G(α) is also complex and given by dertaking a titration with an indifferent electrolyte. √ In the case of methods based on the direct evaluation of ICV, 1 + (1 + i) α/2 G(α) = √ . (81) the use of different frequencies, if available, or of acoustic at- 1 + (1 + i) α/2 + i α (3 + 2ρ/ρ) tenuation measurements, allows the determination of particle 9 size distributions. It is a direct measure of the inertia effect. The dimensionless parameter α is defined as Method based on the concept of dynamic electrophoretic mobility ωa2ρ α = , (82) The symmetry relations lead to the following expressions, η relating the different electroacoustic phenomena to the dynamic electrophoretic mobility, u∗ [82,83]: so G is strongly dependent on the particle size. G varies d monotonically from a value of unity, with zero phase angle, ∗ U ρ u when a is small (less than 0.1 µm typically) to a minimum CV ∝ φ d , (76) p ρ K∗ of zero and a phase lag of π/4 when a is large (say, a> I ρ ∗ 10 µm). For an illustration, see Fig. 8. CV ∝ φ u , (77) p ρ d A ρ ∗ 2 ESA ∝ φ u . (78) If this assumption breaks down, Du must be estimated by reference to a E ρ d suitable surface conductance model or, preferably, by direct measurement of the conductivity over the frequency range involved in the measurement (normally Although the kinetic coefficients on the right-hand side of from about 1 to 40 MHz) [86]. Another procedure involved the analysis of the these relations (both magnitude and phase) can readily be mea- results of a salt titration (see previous subsection). 214 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

for cylindrical particles with zero permittivity and low ζ - potentials, but for arbitrary κa. The results are consistent with those of [93,94].

Concentrated systems The problem of considering the effect on the electroacoustic signal of hydrodynamic or electrostatic interactions between particles in concentrated suspensions was ﬁrst theoretically tackled [96] by using the cell model of Levine and Neale [97,98] to provide a solution that was claimed to be valid for UCV measurements on concentrated systems (see [87] for a discussion on the validity of such approach in the high-frequency range). The limitations of the cell models for treating concentrated (a) systems at high frequency can be overcome in the case of near neutrally buoyant systems (where the relative density compared to water is in the range 0.9–1.5) using a procedure developed by O’Brien [99,100]. In that case, the interparticle interactions can be treated as pairwise additive and only nearest-neighbor interactions need to be taken into account. An alternative approach is to estimate the effects of particle concentration considering in detail the behavior of a pair of particles in all possible orientations [101,102]. Empirical relations have been developed that appear to represent the interaction effects for more concentrated suspensions up to volume fractions of 30% at least. In [103],an example can be found where the dynamic mobility was analyzed assuming the system to be dilute. The resulting value of the ζ -potential (ζapp) was then corrected for concentration using the semi-empirical relation (b) 1 ζ = ζ exp 2φ 1 + s(φ) , with s(φ) = . corr app + ( . /φ)4 Fig. 8. Modulus (a) and phase angle (b) of the dynamic mobility of spherical 1 0 1 particles in a KCl solution with κa = 20 as a function of frequency for different (83) ζ -potentials; cf. Eqs. (80)–(82). Parameters: particle radius 100 nm; dielectric Finally, O’Brien et al. [104] have recently developed an ef- constant of the particles (dispersion medium): 2 (78.54); density of the particles − (dispersion medium): 5 × 103 (1 × 103)kgm 3. fective analytical solution to the concentration problem for the dynamic mobility and have incorporated it into the software of Equations (80)–(82) are, as mentioned above, applicable to their measuring device.

systems of arbitrary ζ -potential, even when conduction oc- Method based on the direct evaluation of the ICV curs in the stagnant layer, in which case Du in Eq. (80) This approach [105] applies a “coupled phase model” [106– must be properly evaluated. They have been amply con- 109] for describing the speed of the particle relative to the firmed by measurements on model suspensions [86], and liquid. The Kuwabara cell model [110] yields the required hy- have proved to be of particular value in the estimation drodynamic parameters, such as the drag coefficient, whereas of high ζ -potentials [87]. The maximum that appears in the cell model by Shilov et al. [111] was used for the general- the plot of dc mobility against ζ -potential ([22], see also ization of the Kuwabara cell model to the electrokinetic part of Fig. 2) gives rise to an ambiguity in the assignment of the the problem. ζ -potential, which does not occur when the full dynamic The method allows the study of polydisperse systems with- mobility spectrum is available. out using any superposition assumption. It is important in con- 2. For double layers with κa < ∼20, there are approximate centrated systems, where superposition does not work because analytical expressions [88,89] for dilute suspensions of of the interactions between particles. spheres, but they are valid only for low ζ -potentials. They An independent exact expression for ICV in the quasi- have been checked against the numerical solutions for gen- stationary case of low frequency, using Onsager’s relationship eral κa and ζ [90,91], and those numerical calculations and the HS equation, and neglecting the surface conductivity have been subjected to an experimental check in [92]. effect (Du 1), is [105] 3. The effect of particle shape has been studied in [93,94], − again in the limit of dilute systems with thin double layers. εrsε0ζKDCφ (ρp ρs) dp ICV|ω→0 = , (84) This analysis has been extended in [95] to derive formulae ηKL ρs dx A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 215 where x is the coordinate in the direction of propagation of the In ICV measurements, calibration is often carried out using TM pressure wave, ρs is the density of the suspension (not of the a colloid with known ζ -potential, such as Ludox (commer- dispersion medium: this is essential for concentrated suspen- cially available colloidal silica), which is diluted to a mass con- sions; see [112]). Equation (84) is the analog of the HS equation centration of 0.1 g/cm3 with 10−2 mol L−1 KCl. for stationary electrophoretic mobility. It has been claimed to be valid over the same wide range of real dispersions, with parti- 5. Electrokinetics in nonaqueous systems cles of any shape and size, and any concentration. 5.1. Difference with aqueous systems: Permittivity 4.6.3. Experimental procedures In the basic experimental procedures, an ac voltage is applied to a transducer that produces a sound wave which trav- The majority of the investigations of electrokinetic phenom- els down a delay line and passes into the suspension. This ena are devoted to aqueous systems, for which the main charg- acoustic excitation causes a very small periodic displacement ing mechanisms of solid/water interfaces have been established of the fluid. Although the particles tend to stay at rest be- and the EDL properties are fairly well known. All general the- cause of their inertia, the ionic cloud surrounding them tends ories of electrokinetic phenomena assume that the equations to move with the fluid to create a small oscillating dipole mo- relating them to the ζ -potential are applicable to any liquid ment. The dipole moment from each particle adds up to create medium characterized by two bulk properties: electric permit- an electric field that can be sensed by the receiving transducer. tivity εrsε0 and viscosity η.Thevalueofεrs is an important The voltage difference that then appears between the electrodes parameter of the liquid phase, a.o. because it determines the dissociation of the electrolytes embedded in it. Most nonaque- (measured at zero current flow) is proportional to UCV. The current, measured under short-circuit conditions, is a measure of ous solvents are less polar than water, and hence εrs is lower. All I . Alternatively, an ac electric field can be applied across the liquids can be classified roughly as nonpolar (εrs 5), weakly CV electrodes. A sound wave is thereby generated near the elec- polar (5 <εrs 12), moderately polar (12 <εrs 40), and po- trodes in the suspension, and this wave travels along the delay lar (εrs > 40). line to the transducer. The transducer output is then a measure For εrs > 40, dissociation of most dissolved electrolytes is of the ESA effect. For both techniques, measurements can be complete, and all equations concerning EDL or EKP remain performed for just one frequency or for a set of frequencies unmodified, except that a lower value for εrs has to be used. For ranging between 1 and 100 MHz, depending on the instru- moderately polar solvents, the electrolyte dissociation is incom- ment. plete, which means that the concentration of charged species Recently, the term electroacoustic spectrometry has been (i.e., the species that play a role in EDL or EKP) may be coined to refer to the measurement of the dynamic mobility lower than the corresponding electrolyte concentration. More- as a function of frequency. The plot of magnitude and phase over, the charge number of the charge carriers can become ∗ of u over a range of (MHz) frequencies may yield informa- lower. For instance, in solutions of Ba(NO3)2, we may find not d 2+ − + tion about the particle size as well as the ζ -potential [86,113], (only) Ba and NO3 ions, but (also) Ba(NO3) complexes. provided the particles are in a suitable size range (roughly The category of weakly polar solvents is a transition to the 0.05 µm

The very slow decay of the potential with distance has two 5.3. Conversion of the electrokinetic data into ζ -potentials consequences. First, as the decay between surface and slip plane is negligible, ψ0 ≈ ζ . This simplifies the analysis. Second, The first step in interpreting the electrokinetic behavior of the slow decay implies that colloid particles “feel” each oth- nonaqueous systems must be to check whether the system be- er’s presence at long range. So, even dilute sols may behave haves as a dilute or as a concentrated system (see Section 5.1). physically as “concentrated” and formation of small clusters of In the dilute regime, all theories described for aqueous systems coagulated particles is typical. can be used, provided one can find the right values of the essen- σ Interestingly, electrophoresis can help in making a distinc- tial parameters κa, Kp, and K . tion between concentrated and dilute systems by studying the The calculation of κ requires knowledge of the ionic strength dependence of the electrophoretic mobility on the concentration of the solution; this, in turn, can be estimated from the measure- of dispersed particles. If there is no dependence, the behavior is ment of the dialysate conductivity, KL, and knowledge of the that of a dilute system. In this case, equations devised for dilute mobilities and valences of the ionic species. systems can be used. Otherwise, the behavior is effectively that The effective conductivity of the solid particle, Kpef, including its surface conductivity3 can be calculated from the experi- of a concentrated system, and ESA or UCV measurements are more appropriate [86]. mental values of conductivities of the liquid KL, and of a dilute suspension of particles, K, with volume fraction φ using either Street’s equation [5] 5.2. Experimental requirements of electrokinetic techniques K − Kpef 2 1 = 1 + KL (85) KL 3 φ The methods described are also applicable to electrokinetic measurements in nonaqueous systems, but some pre- or Dukhin’s equation [5] cautions need to be taken. Microelectrophoresis cells are of- − − K + Kpef (1 φ) (1 φ/2) ten designed for aqueous and water-like media whose con- = 2 KL (86) KL (1 − φ) K − (1 + 2φ) ductivity is high relative to that of the material from which KL the cell is made. A homogeneous electric field between the which accounts for interfacial polarization. electrodes of such cells filled with well- or moderately con- For the estimation of the ζ -potential in the case of elec- ducting liquids is readily achieved. However, when the cell is trophoresis, considering that low κa values are not rare, it is filled with a low-conductivity liquid, the homogeneity of the suggested to use Henry’s theory, after substitution of Kpef es- electric field can be disturbed by the more conducting cell timated as described above, for the particle conductivity. In walls and/or by the surface conduction due to adsorption of formulas, it is suggested to employ Eqs. (29) and (30). traces of water dissolved from the solvents of low polarity Concerning streaming potential/current or electro-osmotic on the more hydrophilic cell walls. Special precautions (coat- flow measurements, all the above-mentioned features of non- ing the walls with hydrophobic layers) are necessary to im- aqueous systems must be taken into account to obtain correct prove the field characteristics. The electric field in cells of values of ζ -potential from this sort of data in either single cap- regular geometrical shape is calculated from the measured cur- illaries, porous plugs, or diaphragms. In view of the low κa rent density and the conductivity of the nonaqueous liquid. values normally attained by nonaqueous systems, the Rice and Because of the low ionic strength of the latter, electrode po- Whitehead curvature corrections are recommended [50];see larization may occur and it can sometimes be observed as bub- Eqs. (37)–(39), (44), and (50). ble formation. Hence, an additional pair of electrodes is often Finally, electroacoustic characterization of ζ -potential in used for the measurements of the voltage gradient across the nonaqueous suspensions requires subtraction of the background cell. arising from the equilibrium dispersion medium. This is imper- For the correct measurement of the streaming potential in ative because the electro-acoustic signal generated by particles nonaqueous systems, attention must be paid to ensure that the in low- or nonpolar media is very weak. It is recommended to resistance of the capillary, plug, or diaphragm filled with liq- make measurements at several volume fractions in order to en- uid is at least 100 times less than the input impedance of sure that the signal comes, in fact, from the particles. the electrical measuring device. The usual practice is to use millivoltmeter-electrometers with input resistance higher than 3 The low conductivity of nonaqueous media with respect to aqueous solu- 1014 and platinum gauze electrodes placed in contact with tions is the main reason why often the finite bulk conductivity of the dielectric the ends of the capillary or plug for both resistance and stream- solids cannot be neglected. In contrast, if one can assume that no water is adsorbed at the interface, any stagnant-layer conduction effect can be neglected. ing potential measurements. The resistance is usually measured Furthermore, the joint adsorption of the ionic species of both signs or (and) the with an ac bridge in the case of polar solvents (typical frequen- adsorption of such polar species as water at the solid/liquid interface can pro- cies used are around a few kHz) and by dc methods in the case duce an abnormally high surface conduction. This is not due to the excess of σ of nonpolar or weakly polar liquids. The use of data record- free charges in the EDL, commonly taken into account in the parameter K of ing on the electrometer output is common practice to check the the generalized theories of electrokinetic phenomena. Rather, it is conditioned by the presence of thin, highly conducting adsorption layers. Therefore, the ra- rate of attainment of equilibrium and possible polarization ef- tio Kpef/KL is an important parameter that has to be estimated for nonaqueous fects. systems. A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 217

Note that there is one more issue that complicates the for- (like ue), apply some equation (like HS) to compute what we mulation of a rigorous electrokinetic theory in concentrated can call an “effective” ζ -potential, but the physical interpreta- nonaqueous systems, namely, the conductivity of a dispersion tion of such a ζ -potential is ambiguous. We must keep in mind of particles in such media becomes position-dependent, as the that the obtained value has only a relative meaning and should double layers of the particles may occupy most of the vol- not be confused with an actual electrostatic potential close to ume of the suspension. This is a significant theoretical ob- the surface. Nevertheless, such an “effective ζ ” can help us in stacle in the elaboration of the theory. In the case of EKP practice, because it may give some feeling for the sign and mag- based on the application of nonstationary fields (dielectric dis- nitude of the part of the double-layer charge that controls the persion, electroacoustics), this problem can be overcome. This interaction between particles. When the purpose of the mea- is possible because one of the peculiarities of low- and non- surement is to obtain a realistic value of the ζ -potential, there polar liquids compared to aqueous systems is the very low is no general recipe. It may be appropriate to use more than one value of the Maxwell–Wagner–O’Konski frequency, Eqs. (57) electrokinetic method and to take into account the specific de- and (58). This means that in the modern electroacoustic meth- tails of the non-ideality as well as possible in each model for the ods based on the application of electric or ultrasound fields calculation of the ζ -potential. If the ζ -potentials resulting from in the MHz frequency region, all effects related to conduc- both methods are similar, physical meaning can be assigned to tivity can be neglected, because that frequency range is well this value. above the Maxwell–Wagner characteristic frequency of the liq- Below, we will discuss different forms of non-ideality in uid. This makes electroacoustic techniques most suitable for somewhat more detail. We will mainly point out what the dif- the electrokinetic characterization of suspensions in nonaque- ficulties are, how these can be dealt with and where relevant ous media. literature can be found.

6. Remarks on non-ideal surfaces 6.2. Hard surfaces

6.1. General comments Some typical examples of non-ideal particles that still can be considered as hard are discussed below. Attention is paid The general theory of electrokinetic phenomena described to size and shape effects, surface roughness, and surface het- so far strictly applies to ideal, nonporous, and rigid surfaces erogeneity. For hard non-ideal surfaces, both the stagnant-layer or particles. By ideal, we mean smooth (down to the scale of concept and the ζ -potential remain locally defined and exper- molecular diameters) and chemically homogeneous, and we use imental data provide some average electrokinetic quantity that rigid to describe those particles and surfaces that do not deform will lead to an average ζ -potential. The kind of averaging may under shear. We will briefly indicate interfaces that are non- depend on the electrokinetic method used, therefore, different porous and rigid as “hard” interfaces. For hard particles and methods may give different average ζ -potentials. surfaces, there is a sharp change of density in the transition between the particle or surface and the surrounding medium. 6.2.1. Size effects Hard particles effectively lead to stacking of water (solvent, For rigid particles that are spherical with a homogeneous in general) molecules; that is, only for hard surfaces the no- charge density, but differ in size, a rigorous value of the ζ - tion of a stagnant layer close to the real surface is concep- potential can be found if the electrokinetic quantity measured is tually simple. Not many surfaces fulfill these conditions. For independent of the particle radius, a. In general, this will be the instance, polystyrenesulfonate latex colloid has, in fact, a het- case for κa 1 (HS limit). In the case of electrophoresis, the erogeneous interface: the hydrophilic sulfate end-groups of its mobility is also independent of a for κa < 1 (Hückel–Onsager polymer chains occupy some 10% of the total surface of the par- limit). For other cases, the particle size will come into play and ticles, the remainder being hydrophobic styrene. Moreover, the most of the time an average radius has to be used to calculate sulfate groups will protrude somewhat into the solution. Gener- an average ζ -potential [3,114]. The type of average radius used ally speaking, many interfaces are far from molecularly smooth, will, in principle, affect the average ζ -potential. Furthermore, as they may contain pores or can be somewhat deformable. different electrokinetic phenomena may require different aver- Such interfaces can briefly be indicated as “soft” interfaces. ages. For instance, in [115] it was found that a simple number For soft interfaces, such as, for instance, rigid particles with average is suitable for analysis of electrophoresis in polydis- “soft” or “hairy” polymer layers, gel-type penetrable particles perse systems; a volume average, on the other hand, was found and water–air or water–oil interfaces, the molecular densities to be the best choice when discussing dielectric dispersion. vary gradually across the phase boundary. A main problem in In principle, the best solution would be found if the full size such cases is the description of the hydrodynamics, and in some distribution is measured, but even in such a case the signal- cases it is even questionable if a discrete slip plane can be de- processing procedure must be taken into account. For instance, fined operationally. in ELS methods, the scattering of light by particles of differ- The difficulties encountered when interpreting experimen- ent sizes is determinant for the average ue value found by the tal results obtained for non-ideal interfaces depend on the type instrument. In this context, measuring the dynamic mobility and magnitude of the non-idealities and on the aim of the mea- spectrum may be useful in providing information on both the surements. In practice, one can always measure some quantity size and the ζ -potential from the same signal. 218 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

6.2.2. Shape effects sumed to lead to one (smeared-out) surface charge density and For nonspherical particles (cylinders, discs, rods, etc.) of ho- also one (averaged) electrokinetic potential and electrokinetic mogeneous ζ -potential, the induced dipole moment is different charge density at given solution conditions; and (2) a patch- for different orientations with respect to the applied field (i.e., it wise heterogeneity with large patches. The patches will lead to acquires a tensorial character). Only when the systems are suf- an inhomogeneous charge distribution. In this case, it is usu- ficiently simple (cylinders, rods) is it sufficient to distinguish ally assumed that each patch has its own smeared-out charge between a parallel and a normal orientation of the dipoles to the density at given solution conditions. The characteristic size of applied field [116]. In spite of these additional difficulties, some the patches should be at least of the order of the Debye length, approximate approaches to either the electrophoresis [117–120] otherwise the surface may be considered as regularly hetero- or the permittivity [121,122] of suspensions of nonspherical geneous with one smeared-out charge density over the entire particles are available. surface. Particles with random heterogeneity can be treated in A more complicated situation arises if the particles are poly- the same way as particles with a homogeneous charge density, disperse in shape and size. Except if κa 1 (HS equation so it does not present additional problems. valid) or if κa 1 (HO limit), the only (approximate) approach The patchwise heterogeneity case applies, for instance, to is to define an “equivalent spherical particle” (for instance, one surfaces with different crystal faces. The electrokinetic theory having the same volume as the particle under study) and use for this type of surface has been considered by several au- theories developed for spheres. In that case, the “average” ζ - thors [118,119,125,126]. Anderson et al. [127–129] have also potential that is obtained depends on the type(s) of polydisper- performed experimental checks on the validity of some theoret- sity of the sample and the definition used for the “equivalent ical predictions. In the case of spherical particles, it has been sphere.” demonstrated [130] that their motion depends on the first, second, and third moments of the ζ -potential distribution along the 6.2.3. Surface roughness surface of the particle. It is important to note that the second Most theories described so far assume that the interface is moment (dipole moment) brings about a rotational motion su- smooth down to a molecular scale. However, even the surfaces perimposed on the translational (when present) electrophoretic prepared with the strictest precautions show some degree of motion. Clay particles [126,131], with anionic charges at the roughness, which will be characterized by R, the typical dimen- plate surfaces and pH-dependent charges at the edges, are typ- sion of the mountains or valleys present on the surface. Surface ical examples of systems with patchwise heterogeneous sur- roughness (with R as a measure of the roughness size) affects the position of the plane of shear except in conditions where faces. κR 1 (HS limit) or κR 1 (roughness not seen). If it is A rather specific situation may appear in the case of sparse assumed that the outer parts of the asperities determine the po- polyelectrolyte adsorption onto oppositely charged surfaces. sition of the slip plane, there will be diffusely bound or even This may lead to a mosaic-type charge distribution [132,133]. free ions in the valleys. This will inevitably lead to large sur- Not only the very concepts of slip plane and hence ζ -potential face conductivities behind the apparent slip plane, and to an are doubtful here, but also electrokinetic data alone can lead additional mechanism of stagnancy [6,116,123]. Due care must us to erroneous conclusions. For instance, one can find a high | be taken to measure this conductivity and to take it into ac- |ue value for particles with attached polyelectrolytes and from count in evaluating the ζ -potentials. Experiments with surfaces this predict a high colloidal stability. This might not be found, of well-defined roughness are required to gain further insight in since the patchwise nature of the charges might induce floccula- the complications. tion even in systems that have considerable average ζ -potential. We will not proceed with a more thorough description of Such instability will be due to attraction between oppositely these highly specialized—and mostly unsolved—topics. As a charged patches (corresponding to regions with and without ad- rule of thumb, we recommend that the readers deal with an in- sorbed polyelectrolyte). terface of known, high roughness by using a simple approach based on the calculation of an effective ζ -potential obtained un- 6.3. Soft particles der the assumption that the interface is smooth, but to refrain from using this effective ζ for further calculations. This is particularly true if the estimation of interaction energies between Some familiar examples in which the interface must be con- the particles is sought, as it has been shown [124] that asperi- sidered as soft are discussed below. The examples refer to two ties have a considerable effect on both electrostatic and van der different groups. The first consists of hard particles with hairy, Waals forces between colloid particles. grafted, or adsorbed layers and of particles that are (partially) penetrable. The hydrodynamic permeability and the conductiv- 6.2.4. Chemical surface heterogeneity ity in the permeable layer make the interpretation of the data In general, chemical surface heterogeneity will also lead to complicated. The second group are the water–oil or water–air surface charge heterogeneity. When the charges do not pro- interfaces. Droplets and bubbles comprise a specific class of trude into the solution, the position of the slip plane is unaf- “soft” particles, for which the definition of the slip plane is an fected. With charge heterogeneity, often two extreme cases are academic question. When and how this issue can be solved de- considered: (1) a random or regular heterogeneity that is as- pends on the surface active components present at the interface. A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 219

6.3.1. Charged particles with a soft uncharged layer 6.3.4. Ion-penetrable or partially penetrable particles For a good understanding of charged hard particles covered Some proteins, many biological cells, and other natural par- by nonionic surfactants or uncharged polymers, the position of ticles are penetrable for water and ions. The most important the slip plane and conduction behind the slip plane must be con- complications for the description of the electrokinetic behavior sidered. It is very useful to also investigate the bare particles. of such particles are associated with the conductivity, the di- Neutral adsorbed layers reduce the tangential fluid flow and the electric constant, and the liquid transport inside the particles. tangential electric current near a particle surface, but to a dif- An additional complication occurs when the particles are able ferent extent [134]. Both reductions have to be considered for a to swell depending on the solution conditions. correct interpretation of the electrokinetic results. By doing this, When the particles are only partially penetrable, we may the comparison of the results between bare and covered parti- consider them as hard with a gel-like corona. This situation cles may give information about the net particle potential and is very similar to hard particles coated with a polyelectrolyte effective particle charge, the surface charge adjustment, and the layer. In the limit of a very small particle with a very thick adsorbed-layer thickness [135]. Let us mention that electroa- corona, the penetrable particle limit results. The simplest mod- coustic studies of the dynamic mobility spectrum of particles els assume that the electrical potential inside the gel layer is coated with adsorbed neutral polymer can give information on the Donnan potential, whereas the hindered motion of liquid the thickness of the adsorbed layer and the shift of the slip plane in it is represented by a friction parameter incorporated in the due to the polymer [136]. Navier–Stokes equation [143–146]. Ohshima’s theory of electrophoresis of soft spheres basically ranges from hard particles 6.3.2. Uncharged particles with a soft charged layer to penetrable particles. References to his work can be found in The surface of many latex particles can be considered as his review on electrokinetic phenomena [141]. being itself uncharged, but with charged groups at the end of Systems with biological or medical relevance that have re- oligomeric or polymeric “hairs” that protrude into the solution. ceived some systematic attention are protein-coated latex par- The extension of the “hairs” from the surface into the liquid de- ticles [147], electrophoresis of biological cells [148], lipo- termines the position of the apparent slip plane. Due to the ion somes [149], and bacteria [150]. penetrability of the stagnant layer, the ionic strength will affect the distance between the surface and this (apparent) slip plane, 6.3.5. Liquid droplets and gas bubbles in a liquid and the electric conduction in the stagnant layer. This will lead The electrophoresis of uncontaminated liquid droplets and to a more complex ionic strength dependence of the ζ -potential gas bubbles in a liquid show quite different behavior from that than with rigid particles. Some studies account for these effects of rigid particles. The main reason is that flow may also oc- through the introduction of a “softness factor” [137,138] as a cur within the droplets or bubbles because there is momentum fitting parameter. Stein [139] discusses the electrokinetics of transfer across the interface. The classical notion of a slip plane polystyrene latex particles in more detail. Hidalgo-Álvarez et loses its meaning. Due to the flow inside the droplet or bub- al. [140] discuss the anomalous electrokinetic behavior of poly- ble, the tangential velocity of the liquid surrounding the droplet mer colloids in general. does not have to become zero at the surface of the particle. As a result, the electrophoretic mobility is higher than for a cor- 6.3.3. Charged particles with a soft charged layer responding rigid body. However, to model this situation and to Charged particles with adsorbed or grafted polyelectrolyte arrive at a conversion of the mobility to a ζ -potential is not a or ionic surfactant layers fall in this class. The complications trivial task. As there is no slip plane, even the HS model cannot arising with these systems are similar to those mentioned for be applied. the uncharged surfaces with charged layers. Adsorbed layers In general, however, droplets or bubbles are not stable with- impede tangential fluid flow; therefore, in the presence of the out an adsorbed layer of a surface active component (surfactant, bound layer the hydrodynamic particle radius increases and the polymer, polyelectrolyte, protein). Most layers will make the apparent slip plane is moved outwards. This affects the tan- surface inextensible (rigid) if (nearly) completely covered. In gential electric current. In most cases, the particle surface and this case, the surface behaves as rigid, and it is possible to use the bound layers will have opposite charges, and, therefore, the the treatments for rigid particles [3]. Even the presence of the electrostatic potential profile is rather complicated. The elec- double layer itself, through the primary electroviscous effect, trokinetic phenomena will be dominated by conduction within makes momentum transfer to the liquid drop very slight [151]. the slip plane and the potential at the hydrodynamic boundary, For partially covered bubbles (as in flotation), the situation which is relatively far away from the surface [114,141,142]. is more complex; surface inhomogeneities may occur under the To unravel the situation, a systematic investigation is required influence of shear, and Marangoni effects become important. that considers the electrokinetic behavior of both uncovered and Problems arise with regard to the definition of the slip plane, its covered particles, the shift of the apparent slip plane and the location, and the charge inhomogeneity [152]. conduction behind the slip plane. When qualitative information is required with respect to the net charge of the particle plus ad- 7. Discussion and recommendations sorbed layer, the sign of the ζ -potential is important, as it will indicate whether the particle charge is overcompensated or not From the analyses given above, it will be clear that the com- (within the plane of shear). putation of the ζ -potential from experimental data is not always 220 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 a trivial task. For each method, the fundamental requirement If these approaches are not possible, there is no other way but for the conversion of experimental data is that models should employing different electrokinetic techniques on the same sam- be correctly used within their range of applicability. The bot- ple and performing the consistency test. In this respect, the best tom line is that the ζ -potential does exist as a characteristic of procedure would be using one technique in which neglecting a charged surface under fixed conditions. This is fortunate, be- Kσ i underestimates ζ (e.g., electrophoresis, streaming poten- cause otherwise, how could two authors compare their results tial), and another in which ζ is overestimated (dc conductivity, with the same material using two electrokinetic techniques, or dielectric dispersion) [3]. even two different versions of the same technique? For practical reasons—not every worker has available the This means that we must focus on the correct use of the numerical routines required for nonanalytical theories—another existing theories, and in their improvement, if necessary. The question emerges: “When can we use with confidence HS equa- sometimes-asked question: “Is the computed ζ -potential inde- tions for the different types of electrokinetic data?” pendent of the technique used?” must be answered with yes, Although most published data on ζ -potentials are based on the various versions of the HS equation for electrokinetic because ζ -potentials are unique characteristics for the charge phenomena, let us stress that this approach is correct only if state of interfaces under given conditions. Whether by different (Eq. (2)) κa 1, where a is a characteristic dimension of techniques identical ζ -potentials are indeed observed depends the system (curvature radius of the solid particle, capillary on the quality of both the technique and the interpretation. radius, equivalent pore radius of the plug, ...), and further- Measuring different electrokinetic properties for a given ma- more, the surface conductance of any kind must be low, i.e., terial in a given liquid medium, and checking the equality of Du(ζ, Kσ i) 1. Thus, in the absence of independent infor- ζ -potentials using appropriate models, is what Lyklema calls mation about Kσ i, additional electrokinetic determinations can an electrokinetic consistency test [3], and what Dukhin and Der- only be avoided for sufficiently large particles and high elec- jaguin called integrated electrosurface investigation [5]. trolyte concentrations. Another related question that could be asked is “Is the ζ - Another caveat can be given, even if the previously men- potential the only property characterizing the electrical equilib- tioned conditions on dimensions and Dukhin number are met. rium state of the interfacial region?” The answer is “most often For concentrated systems, the possibility of the overlap of the it is not.” Most recent models and experimental results demon- double layers of neighboring particles cannot be neglected if the strate that in the majority of cases the conductivity behind the concentration of the dispersed solids in a suspension or a plug slip plane must also be taken into account. This implies that for is high. In such cases, the validity of the HS equation is also a correct evaluation, the Du number should also be measured. doubtful and cell models for either electrophoresis, streaming Using Du only to characterize the interface has, in general, no potential, or electro-osmosis are required. Use of the latter two practical interest either. One experimental electrokinetic tech- kinds of experiments or of electroacoustic or LFDD measure- nique suffices only if Kσ i/Kσ d is small (<2–5, say), because ments is recommended. In all cases, a proper model accounting then Du(ζ, Kσ i) ≈ Du(ζ, 0) = Du(ζ ). The problem now is how for interparticle interaction must be available. does one know that Kσ i is in fact that small? We have some possibilities: Acknowledgments

• If we have also access to the value of the potentiometrically Financial assistance from IUPAC is gratefully acknowl- measured surface charge, σ 0, the values of σ ek (calculated edged. The Coordinators of the MIEP working party wish to from the value of the ζ -potential obtained by neglecting thank all members for their effort in preparing their contribu- conductance behind the slip plane) and σ 0 can be com- tions, suggestions, and remarks. pared. This comparison allows us to reach a first estimate of σ i.Fromσ i, Kσ i could be estimated assuming—as appears Appendix A. Calculation of the low-frequency dielectric to be the case, at least for monovalent counterions—that dispersion of suspensions the mobilities of ions in the inner layer are comparable to Neglecting stagnant-layer conduction, the complex conduc- those in the bulk. When Du(ζ, Kσ i) ≈ Du(ζ, 0) = Du(ζ ), tivity increment is related to the dipole coefficient as follows the calculated value of ζ is correct, otherwise, it has to be [62–64]: recalculated taking the inner-layer conduction into account. • In the case of capillaries or plugs, the total conductivity and ∗ 3φ ∗ ∗ δK (ω) = K C = δK | → the streaming potential can be measured. An initial value a3 L 0 ω 0 of Kσ d can be obtained from ζ using Bikerman’s equation, (R+ − R−)H iωτ + 9φK √ √ α . and Kσ i can be calculated. The improved value of ζ can L 2AS 1 + S iωτα + iωτα now be obtained using a suitable model including inner- (A.1) σ σ layer conduction, and again K d and K i can be obtained. Its low-frequency value is a real quantity: If the difference between ζ computed without and with fi- − + − − nite Kσ i is high, we should go back to the calculation of ζ ∗ 2Du(ζ ) 1 3(R R )H δK |ω→0 = 3φKL − . (A.2) and iterate till the difference between two steps is small. 2Du(ζ ) + 2 2B A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 221

The dielectric increment of the suspension can be calculated b(m) half distance between neighboring particles ∗ −3 from δK |ω→0 as follows: c(mol m ) electrolyte concentration − c+(c−)(mol m 3) concentration of cations (anions) ∗ i ∗ ∗ ∗ δε (ω) =− δK − δK |ω=0 C dipole coefficient of particles r ωε 0 0 d(m) distance between the surface and the outer Helmholtz 9 τ (R+ − R−)H 1 = α √ √ plane φεrs . ∗ 2 τel AS 1 + S iωτα + iωτα d (Cm) complex dipole moment (A.3) dek (m) distance between the surface and the slip plane Here D(m2 s−1) diffusion coefficient of counterions 2 −1 ek ek D+(D−)(m s ) diffusion coefficient of cations (anions) zy zy ek ± exp ∓ − 1 ± exp ∓ − 1 zy R = 4 2 + 6m 2 ± z Du Dukhin number κa κa κa Dud Dukhin number associated with diffuse-layer conduc- (A.4) tivity and Dui Dukhin number associated with stagnant-layer conductivity a2 1 τα = (A.5) e(C) elementary charge 2Def S E (Vm−1) applied electric field −1 is the value of the relaxation time of the low-frequency disper- Esed (Vm ) sedimentation field −1 sion. It is assumed that the dispersion medium is an aqueous Et (Vm ) tangential component of external field solution of an electrolyte of z–z charge type. The definitions F(Cmol−1) Faraday constant of the other quantities appearing in Eqs. (A.1)–(A.5) and the f1(κa), F1(κa, Kp) Henry’s functions ζ -potential are as follows: I(A) electric current intensity 2D+D− I0,I1 zeroth- and first-order modified Bessel functions of the Def = , (A.6) first kind D+ + D− I (A) conduction current A = 4Du(ζ ) + 4, (A.7) c + − + − + − − + ICV (A) colloid vibration current B = (R + 2)(R + 2) − U − U − (U R + U R )/2, Istr (A) streaming current (A.8) j σ (Am−1) surface current density B −2 S = , (A.9) jstr (Am ) streaming current density A k(JK−1) Boltzmann constant + − + − + − − − 2 2 − + + + −1 = (R R )(1 z ) U U z(U U ) K(Sm ) total conductivity of a colloidal system H , −1 A KDC (Sm ) direct current conductivity of a suspension −1 (A.10) KL (Sm ) conductivity of dispersion medium ± ek ∞ − ± 48m zy K (Sm 1) conductivity of a highly concentrated ionic solu- U = ln cosh , (A.11) L κa 4 tion − + −1 D − D Kp (Sm ) conductivity of particles = . (A.12) −1 − + + Kplug (Sm ) conductivity of a plug of particles z(D D ) −1 Kpef (Sm ) effective conductivity of particles The factor 1/S is of the order of unity, and comparison of Krel ratio between particle and liquid conductivities Eqs. (58) and (A.5) leads to the important conclusion that K∗ (Sm−1) complex conductivity of a suspension σ τα ≈ 2 K (S) surface conductivity (κa) . (A.13) σ d τMWO K (S) diffuse-layer surface conductivity σ i This means that for the case of κa 1, the characteristic fre- K (S) stagnant-layer surface conductivity quency for the α-dispersion is (κa)2 times lower than that of L(m) capillary length, characteristic dimension the Maxwell–Wagner dispersion. Lapp (m) apparent (externally measured) capillary length LD (m) ionic diffusion length Appendix B. List of symbols m dimensionless ionic mobility of counterions m+(m−) dimensionless ionic mobility of cations (anions) − Note: SI base (or derived) units are given in parentheses for n(m 3) number concentration of particles −1 all quantities, except dimensionless ones. NA (mol ) Avogadro constant −3 ni (m ) number concentration of type i ions 3 −1 a(m) particle radius, local curvature radius, capillary radius Qeo (m s ) electro-osmotic flow rate 2 4 −1 −1 Ac (m ) capillary cross-section Qeo,E (m s V ) electro-osmotic flow rate per electric field 2 3 −1 Ac,app (m ) apparent (externally measured) capillary cross- Qeo,I (m C ) electro-osmotic flow rate per current section r(m) spherical or cylindrical radial coordinate AESA (Pa) electrokinetic sonic amplitude R(m) capillary or pore radius, roughness of a surface 222 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

Rs () electrical resistance of a capillary or porous plug in an τMWO (s) characteristic time of the Maxwell–Wagner–O’Kon- arbitrary solution ski relaxation R∞ () electrical resistance of a capillary or porous plug in a τα (s) relaxation time of the low-frequency dispersion concentrated ionic solution φ volume fraction of solids T(K) thermodynamic temperature φL volume fraction of liquid in a plug ∗ 2 −1 −1 d ud (m s V ) dynamic electrophoretic mobility ψ (V) diffuse-layer potential i UCV (V) colloid vibration potential ψ (V) inner Helmholtz plane potential 2 −1 −1 0 ue (m s V ) electrophoretic mobility ψ (V) surface potential −1 Used (V) sedimentation potential ω(s ) angular frequency of an ac electric field −1 Ustr (V) streaming potential ωMWO (s ) Maxwell–Wagner–O’Konski characteristic fre- −1 ve (ms ) electrophoretic velocity quency −1 −1 veo (ms ) electro-osmotic velocity ωα (s ) characteristic frequency of the α-relaxation −1 −1 vL (ms ) liquid velocity in electrophoresis cell ωαd (s ) characteristic frequency of the α-relaxation for a di- yek dimensionless ζ -potential lute suspension z common charge number of ions in a symmetrical electrolyte References zi charge number of type i ions α relaxation of double-layer polarization, degree of elec- [1] S.S. Dukhin, Adv. Colloid Interface Sci. 61 (1995) 17. [2] R.J. Hunter, Foundations of Colloid Science, Oxford Univ. Press, Ox- trolyte dissociation, dimensionless parameter used in ford, 2001, Chap. 8. electroacoustics [3] J. Lyklema, Fundamentals of Interfaces and Colloid Science, vol. II, Aca- β(m) distance between the solid surface and the inner demic Press, New York, 1995, Chaps. 3, 4. Helmholtz plane (see also Eq. (45) for another use of [4] R.J. Hunter, Zeta Potential in Colloid Science, Academic Press, New this symbol) York, 1981. −2 [5] S.S. Dukhin, B.V. Derjaguin, in: E. Matijevic´ (Ed.), Surface and Colloid Γi (m ) surface concentration of type i ions Science, vol. 7, Wiley, New York, 1974, Chap. 2. −3 δc (mol m ) field-induced perturbation of electrolyte amount [6] A.V. Delgado (Ed.), Interfacial Electrokinetics and Electrophoresis, concentration Dekker, New York, 2001. δK∗ (Sm−1) conductivity increment of a suspension [7] Manual of symbols and terminology for physicochemical quantities and δε∗ relative dielectric increment of a suspension units, Appendix II: Definitions, terminology and symbols in colloid and r surface chemistry (Part I), Pure Appl. Chem. 31 (1972) 577. p (Pa) applied pressure difference [8] I. Mills, T. Cvitaš, K. Homann, N. Kallay, K. Kuchitsu, Quantities, Units peo (Pa) electro-osmotic counter-pressure and Symbols in Physical Chemistry, second ed., International Union of Vext (V) applied potential difference Pure and Applied Chemistry, Blackwell Science, Oxford, 1993, Sect. ε(0) low-frequency dielectric increment per volume frac- 2.14. tion [9] A.D. McNaught, A. Wilkinson, Compendium of Chemical Terminology. IUPAC Recommendations, second ed., International Union of Pure and εd(0) value of ε(0) for suspensions with low volume frac- Applied Chemistry, Blackwell Science, Oxford, 1997. tions [10] J. Lyklema, Pure Appl. Chem. 63 (1995) 885. − ρ (kg m 3) density difference between particles and disper- [11] J. Lyklema, S. Rovillard, J. de Coninck, Langmuir 14 (1998) 5659. sion medium [12] J.W. Lorimer, J. Membr. Sci. 14 (1983) 275. ε∗ (Fm−1) complex electric permittivity of a suspension [13] C.F. Zukoski, D.A. Saville, J. Colloid Interface Sci. 114 (1986) 32. ∗ [14] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 86 (1990) εr complex relative permittivity of a suspension 2859. εrp relative permittivity of the particle [15] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 94 (1998) ∗ εrp complex relative permittivity of a particle 2441. [16] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 94 (1998) εrs relative permittivity of the dispersion medium ∗ 2583. εrs complex relative permittivity of the dispersion medium −1 [17] J. Lyklema, M. Minor, Colloids Surf. A 140 (1998) 33. ε0 (Fm ) electric permittivity of vacuum [18] M. von Smoluchowski, in: Handbuch der Electrizität und des Mag- ζ(V) electrokinetic or ζ -potential netismus (Graetz), vol. II, Barth, Leipzig, 1921, p. 366. ζapp (V) electrokinetic or ζ -potential not corrected for the ef- [19] J.J. Bikerman, Z. Phys. Chem. A 163 (1933) 378. fect of particle concentration [20] J.J. Bikerman, Kolloid Z. 72 (1935) 100. η(Pa s) dynamic viscosity [21] J.J. Bikerman, Trans. Faraday Soc. 36 (1940) 154. −1 [22] R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. II 74 (1978) κ(m ) reciprocal Debye length 1607. −3 ρ(kg m ) density of dispersion medium [23] R.W. O’Brien, R.J. Hunter, Can. J. Chem. 59 (1981) 1878. −3 ρp (kg m ) density of particles [24] D.C. Henry, Proc. R. Soc. London A 133 (1931) 106. −3 ρs (kg m ) density of a suspension [25] H. Ohshima, J. Colloid Interface Sci. 168 (1994) 269. σ d (Cm−2) diffuse charge density [26] R.P. Tison, J. Colloid Interface Sci. 60 (1977) 519. [27] A. Homola, A.A. Robertson, J. Colloid Interface Sci. 51 (1975) 202. σ ek ( −2) Cm electrokinetic charge density [28] K. Furusawa, Y. Kimura, T. Tagawa, in: E.D. Goddard, B. Vincent (Eds.), i −2 σ (Cm ) charge density at the inner Helmholtz plane Polymer Absorption Dispersion Stability, in: ACS Symposium Series, − σ 0 (Cm 2) primary surface charge density vol. 240, Am. Chem. Soc., Washington, DC, 1984, p. 131. A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224 223

[29] M. Minor, A.J. van der Linde, H.P. van Leeuwen, J. Lyklema, J. Colloid [77] M.M. Springer, A. Korteweg, J. Lyklema, J. Electroanal. Chem. 153 Interface Sci. 189 (1997) 370. (1983) 55. [30] E.E. Uzgiris, Rev. Sci. Instrum. 45 (1974) 74. [78] K. Lim, E.I. Frances, J. Colloid Interface Sci. 110 (1986) 201. [31] R. Ware, W.H. Flygare, J. Colloid Interface Sci. 39 (1972) 670. [79] C. Grosse, A.J. Hill, K.R. Foster, J. Colloid Interface Sci. 127 (1989) [32] E.E. Uzgiris, Opt. Commun. 6 (1972) 55. 167. [33] E. Malher, D. Martin, C. Duvivier, Studia Biophys. 90 (1982) 33. [80] C. Grosse, M.C. Tirado, Mater. Res. Soc. Symp. Proc. 430 (1996) 287. [34] E. Malher, D. Martin, C. Duvivier, B. Volochine, J.F. Stolz, Biorheology [81] M.L. Jiménez, F.J. Arroyo, J. van Turnhout, A.V. Delgado, J. Colloid 19 (1982) 647. Interface Sci. 249 (2002) 327. [35] K. Oka, K. Furusawa, in: H. Ohshima, K. Furusawa (Eds.), Electrical [82] R.W. O’Brien, J. Fluid Mech. 212 (1990) 81. Phenomena at Interfaces, Dekker, New York, 1998, Chap. 8. [83] R.W. O’Brien, B.R. Midmore, A. Lamb, R.J. Hunter, Faraday Discuss. [36] J.F. Miller, K. Schätzel, B. Vincent, J. Colloid Interface Sci. 143 (1991) Chem. Soc. 90 (1990) 301. 532. [84] R.W. O’Brien, J. Fluid Mech. 190 (1988) 71. [37] F. Manerwatson, W. Tscharnuter, J. Miller, Colloids Surf. A 140 (1988) [85] R.W. O’Brien, D. Cannon, W.N. Rowlands, J. Colloid Interface Sci. 173 53. (1995) 406. [38] J.F. Miller, O. Velev, S.C.C. Wu, H.J. Ploehn, J. Colloid Interface Sci. [86] R.J. Hunter, Colloids Surf. A 141 (1998) 37. 174 (1995) 490. [87] R.J. Hunter, R.W. O’Brien, Colloids Surf. A 126 (1997) 123. [39] Japanese Surface and Colloid Chemical Group (Japan Oil Chemists [88] R.W. O’Brien, U.S. Patent 5,059,909, 1991. Soc.), Yukagaku 25 (1975) 239. [89] R.P. Sawatsky, A.J. Babchin, J. Fluid Mech. 246 (1993) 321. [40] F. Fairbrother, H. Mastin, J. Chem. Soc. (1924) 2319. [90] C.S. Mangelsdorf, L.R. White, J. Colloid Interface Sci. 160 (1993) 275. [41] D.R. Briggs, J. Phys. Chem. 32 (1928) 641. [91] C.S. Mangelsdorf, L.R. White, J. Chem. Soc. Faraday Trans. 88 (1992) [42] M. Chang, A.A. Robertson, Can. J. Chem. Eng. 45 (1967) 66. 3567. [43] G.J. Biefer, S.G. Mason, Trans. Faraday Soc. 55 (1959) 1234. [92] S.E. Gibb, R.J. Hunter, J. Colloid Interface Sci. 224 (2000) 99. [44] J. Happel, P.A. Ast, Chem. Eng. Sci. 11 (1960) 286. [93] M. Loewenberg, R.W. O’Brien, J. Colloid Interface Sci. 150 (1992) 158. [45] J. Schurz, G. Erk, Prog. Colloid Polym. Sci. 71 (1985) 44. [94] M. Loewenberg, J. Fluid Mech. 278 (1994) 149. [46] S. Levine, G. Neale, N. Epstein, J. Colloid Interface Sci. 57 (1976) 424. [95] H. Ohshima, J. Colloid Interface Sci. 185 (1997) 131. [47] J.L. Anderson, W.H. Koh, J. Colloid Interface Sci. 59 (1977) 149. [96] B.J. Marlow, D. Fairhurst, H.P. Pendse, Langmuir 4 (1988) 611. [48] S. Levine, J.R. Marriot, G. Neale, N. Epstein, J. Colloid Interface Sci. 52 [97] S. Levine, G. Neale, J. Colloid Interface Sci. 47 (1974) 520. (1975) 136. [98] S. Levine, G. Neale, J. Colloid Interface Sci. 49 (1974). [49] D. Burgreen, F.R. Nakache, J. Phys. Chem. 68 (1964) 1084. [99] R.W. O’Brien, W.N. Rowlands, R.J. Hunter, in: S.B. Malghan (Ed.), [50] C.L. Rice, P. Whitehead, J. Phys. Chem. 69 (1965) 4017. Electro-acoustics for Characterization of Particulates and Suspensions, [51] R.W. O’Brien, W.T. Perrins, J. Colloid Interface Sci. 99 (1984) 20. National Institute of Standards and Technology, Washington, DC, 1993, [52] D. Erickson, D. Li, J. Colloid Interface Sci. 237 (2001) 283. pp. 1–22, NIST Special Publication 856. [53] A. Szymczyk, B. Aoubiza, P. Fievet, J. Pagetti, J. Colloid Interface [100] R.J. Hunter, in: J. Sjoblom (Ed.), Encyclopedic Handbook of Emulsion Sci. 192 (2001) 105. Technology, Dekker, New York, 2001, Chap. 7. [54] N. Spanos, P.G. Koutsoukos, J. Colloid Interface Sci. 214 (1999) 85. [101] P. Rider, R.W. O’Brien, J. Fluid Mech. 257 (1993) 607. [55] M. Giesbers, J.M. Kleijn, M.A. Cohen-Stuart, J. Colloid Interface [102] J. Ennis, A.A. Shugai, S.L. Carnie, J. Colloid Interface Sci. 223 (2000) Sci. 248 (2002) 88. 21. [56] O. El-Gholabzouri, M.A. Cabrerizo, R. Hidalgo-Álvarez, J. Colloid In- [103] S.B. Johnson, A.S. Russell, P.J. Scales, Colloids Surf. A 141 (1998) 119. terface Sci. 261 (2003) 386. [104] R.W. O’Brien, A. Jones, W.N. Rowlands, Colloids Surf. A 218 (2003) [57] A. Pettersson, J.B. Rosenholm, Langmuir 18 (2002) 8447. 89. [58] J.Th.G. Overbeek, in: H.R. Kruyt (Ed.), Colloid Science, vol. I, Elsevier, [105] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Langmuir 15 (1999) Amsterdam, 1952, Chap. V. 6692. [59] C. Werner, R. Zimmermann, T. Kratzmüller, Colloids Surf. A 192 (2001) [106] A.H. Harker, J.A.G. Temple, J. Phys. D Appl. Phys. 21 (1988) 1576. 205. [107] R.L. Gibson, M.N. Toksoz, J. Acoust. Soc. Am. 85 (1989) 1925. [60] J. Kijlstra, H.P. van Leeuwen, J. Lyklema, Langmuir 9 (1993) 1625. [108] A.S. Dukhin, P.J. Goetz, Langmuir 12 (1996) 4987. [61] J. Kijlstra, H.P. van Leeuwen, J. Lyklema, J. Chem. Soc. Faraday [109] A.S. Ahuja, J. Appl. Phys. 44 (1973) 4863. Trans. 88 (1992) 3441. [110] S. Kuwabara, J. Phys. Soc. Jpn. 14 (1959) 527. [62] C.T. O’Konski, J. Phys. Chem. 64 (1960) 605. [111] V.N. Shilov, N.I. Zharkikh, Y.B. Borkovskaya, Colloid J. 43 (1981) 434. [63] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and the Double Layer [112] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, in: A.V. Delgado in Disperse Systems and Polyelectrolytes, Wiley, Jerusalem, 1974. (Ed.), Interfacial Electrokinetics and Electrophoresis, Dekker, New York, [64] C. Grosse, V.N. Shilov, J. Phys. Chem. 100 (1996) 1771. 2001, Chap. 17. [65] B. Nettleblad, G.A. Niklasson, J. Colloid Interface Sci. 181 (1996) 165. [113] R.J. Hunter, R.W. O’Brien, in: A. Hubbard (Ed.), Encyclopedia of Col- [66] E.H.B. DeLacey, L.R. White, J. Chem. Soc. Faraday Trans. II 77 (1981) loid and Surface Science, Dekker, New York, 2002, p. 1722. 2007. [114] H. Ohshima, in: H. Ohshima, K. Furusawa (Eds.), Electrical Phenomena [67] V.N. Shilov, A.V. Delgado, F. González-Caballero, C. Grosse, Colloids at Interfaces. Fundamentals, Measurements and Applications, Dekker, Surf. A 192 (2001) 253. New York, 2002, Chap. 2. [68] L.A. Rosen, J.C. Baygents, D.A. Saville, J. Chem. Phys. 98 (1993) 4183. [115] F. Carrique, F.J. Arroyo, A.V. Delgado, J. Colloid Interface Sci. 206 [69] F.J. Arroyo, F. Carrique, T. Bellini, A.V. Delgado, J. Colloid Interface (1998) 206. Sci. 210 (1999) 194. [116] S.S. Dukhin, Adv. Colloid Interface Sci. 44 (1993) 1. [70] E. Vogel, H. Pauli, J. Chem. Phys. 89 (1988) 3830. [117] R.W. O’Brien, D.N. Ward, J. Colloid Interface Sci. 121 (1988) 402. [71] Yu.B. Borkovskaja, V.N. Shilov, Colloid J. 54 (1992) 173. [118] M.C. Fair, J.L. Anderson, J. Colloid Interface Sci. 127 (1989) 388. [72] V.N. Shilov, Yu.B. Borkovskaja, Colloid J. 56 (1994) 647. [119] D. Velegol, J.L. Anderson, Y. Solomentsev, in: A.V. Delgado (Ed.), In- [73] A.V. Delgado, F.J. Arroyo, F. Gonzalez-Caballero, V.N. Shilov, Y. Bor- terfacial Electrokinetics and Electrophoresis, Dekker, New York, 2002, kovskaja, Colloids Surf. A 140 (1998) 139. Chap. 6. [74] F. Carrique, F.J. Arroyo, M.L. Jiménez, A.V. Delgado, J. Chem. Phys. [120] J.Y. Kim, B.J. Yoon, in: A.V. Delgado (Ed.), Interfacial Electrokinetics 118 (2003) 1945. and Electrophoresis, Dekker, New York, 2002, Chap. 7. [75] H. Fricke, H.J. Curtis, J. Phys. Chem. 41 (1937) 729. [121] C. Grosse, V.N. Shilov, J. Colloid Interface Sci. 193 (1997) 178. [76] F.J. Arroyo, F. Carrique, A.V. Delgado, J. Colloid Interface Sci. 217 [122] C. Grosse, S. Pedrosa, V.N. Shilov, J. Colloid Interface Sci. 220 (1999) (1999) 411. 31. 224 A.V. Delgado et al. / Journal of Colloid and Interface Science 309 (2007) 194–224

[123] S.S. Dukhin, R. Zimmermann, C. Werner, Colloids Surf. A 195 (2001) [140] R. Hidalgo-Álvarez, A. Martín, A. Fernández, D. Bastos, F. Martínez, 103. F.J. de las Nieves, Adv. Colloid Interface Sci. 67 (1996) 1. [124] J.Y. Walz, Adv. Colloid Interface Sci. 74 (1998) 119. [141] H. Ohshima, in: A.V. Delgado (Ed.), Interfacial Electrokinetics and Elec- [125] S.A. Allison, Macromolecules 29 (1996) 7391. trophoresis, Dekker, New York, 2002, Chap. 5. [126] M. Teubner, J. Chem. Phys. 76 (1982) 5564. [142] H.J. Keh, in: A.V. Delgado (Ed.), Interfacial Electrokinetics and Elec- [127] M.C. Fair, J.L. Anderson, Langmuir 8 (1992) 2850. trophoresis, Dekker, New York, 2002, Chap. 15. [128] D. Velegol, J.L. Anderson, S. Garoff, Langmuir 12 (1996) 675. [143] C. Tanford, J.A. Reynolds, Biochim. Biophys. Acta 457 (1976) 133. [129] J.L. Anderson, D. Velegol, S. Garoff, Langmuir 16 (2000) 3372. [144] D. Stigter, Cell Biophys. 11 (1987) 139. [130] J.L. Anderson, J. Colloid Interface Sci. 105 (1985) 45. [145] D. Stigter, J. Phys. Chem. B 104 (2000) 3402. [131] H. van Olphen, An Introduction to Clay Colloid Chemistry, Wiley, New [146] R.J. Hill, D.A. Saville, W.B. Russel, J. Colloid Interface Sci. 258 (2003) York, 1977. 56. [132] S. Akari, W. Schrepp, D. Horn, Langmuir 12 (1996) 857. [147] A. Martín-Rodríguez, J.L. Ortega-Vinuesa, R. Hidalgo-Álvarez, in: A.V. [133] K.E. Bremmell, G.J. Jameson, S. Biggs, Colloids Surf. A 139 (1998) Delgado (Ed.), Interfacial Electrokinetics and Electrophoresis, Dekker, 199. New York, 2002, Chap. 22. [134] M.A. Cohen-Stuart, F.H.W.H. Waajen, S.S. Dukhin, Colloid Polym. [148] E. Lee, F.Y. Yen, J.P. Hsu, in: A.V. Delgado (Ed.), Interfacial Electroki- Sci. 262 (1984) 423. netics and Electrophoresis, Dekker, New York, 2002, Chap. 23. [135] L.K. Koopal, V. Hlady, J. Lyklema, J. Colloid Interface Sci. 121 (1988) [149] R. Barchini, H.P. van Leewen, J. Lyklema, Langmuir 16 (2000) 49. 8238. [136] L. Kong, J.K. Beattie, R.J. Hunter, Phys. Chem. Chem. Phys. 3 (2001) [150] A. Van der Wal, M. Minor, W. Norde, A.J.B. Zehnder, J. Lyklema, Lang- 87. muir 13 (1997) 165. [137] H. Ohshima, J. Colloid Interface Sci. 163 (1995) 474. [151] S.A. Nespolo, M.A. Bevan, D.Y.C.Chan, F. Grieser, G.W. Stevens, Lang- [138] H. Ohshima, K. Makino, Colloids Surf. A 109 (1996) 71. muir 17 (2001) 7210. [139] H.N. Stein, in: A.V. Delgado (Ed.), Interfacial Electrokinetics and Elec- [152] C. Yang, T. Dabros, D. Li, J. Czarnecki, J.H. Masliyah, J. Colloid Inter- trophoresis, Dekker, New York, 2002, Chap. 21. face Sci. 243 (2001) 128.