Substantia. An International Journal of the History of Chemistry 1(2): 75-93, 2017 Firenze University Press www.fupress.com/substantia ISSN 2532-3997 (online) | DOI: 10.13128/substantia-28 Citation: J.(H.) Lyklema (2017) Interfa- cial Potentials: Measuring the Immeas- urable? Substantia 1(2): 75-93. doi: 10.13128/substantia-28 Copyright: © 2017 J.(H.) Lyklema. This is an open access, peer-reviewed article published by Firenze University Press (http://www.fupress.com/substan- tia) and distribuited under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All rel- evant data are within the paper and its Supporting Information files. Competing Interests: The author declared that no competing interests exist. Feature Article Interfacial Potentials: Measuring the Immeasurable? Johannes (Hans) Lyklema Physical Chemistry and Soft Matter, Wageningen University, Stippeneng 4, 6708 WE Wageningen, The Netherlands. E-mail: hans.lyklema@wur.nl Abstract. The scientific background of measuring and interpreting potentials at inter- faces is discussed in a historical setting. Various types of potentials have to be distin- guished. Some are measurable, others not. Static and dynamic aspects will be covered and, as applications, the interaction between electric double layers and the slip process in electrokinetics will be considered. In several cases it is expedient to interpret results in terms of charges rather than in terms of potentials. Keywords. Interfacial potential, surface potential, zeta potential, colloid interaction, electrokinetic slip. 1. INTRODUCTION – Zum Unterschied von Δψ, das wenigstens prinzipiell immer messbar ist, sind Abso- lutwerte von Δφ bis heute noch nicht zugänglich1 – ... the value of the potential of a single electrode is not amenable to direct experi- mental determination2 – We measured the electrosurface potential electrophoretically3 – The keyword “surface potential” occurred in a recent paper on modern AFM – like techniques4 – Surface potentials can be evaluated by second harmonic generation and similar non-linear optical techniques5 These five statements all relate to the measuring of surface potentials, but that seems to be the sole binding element between them. For the rest these quotes differ in many respects, even regarding the definitions and even with respect to the basic tenet whether or not these potentials are experimentally accessible at all. It sounds as a cacophony. Do all quotes refer to the same quantity? It is noted that between the oldest and the more recent of these quotes almost a century elapsed. One would wonder if over such a long time span the improvement of the experimental techniques and the development of better defined systems would have contributed so much to our physical insight that quantities that could not be measured long ago are now with- in reach of our technical achievements. Or, are there certain generic laws of principle precluding that? 76 Johannes (Hans) Lyklema The present paper intends to help solving this basic question and considers how our understanding devel- oped historically. Prior to that it is necessary to agree on the mean- ings of basic concepts, their terminology and definitions. In trying that, it is realized that in certain scientific cir- cles group-specific nomenclature is used, that does not necessarily match with that of other teams. Some peo- ple state that they have studied potentials whereas it appears that they are dealing with something else. One typical illustration is the notion of pair potential, rou- tinely encountered in simulation science. This dubbing is a generally adopted habit, which everybody involved in this field appears to understand. Nevertheless, it is basi- cally incorrect because the quantity under discussion is an energy and not a potential. Routinely, interaction potentials are expressed in units of kT or eV, that is in units of energy whereas potentials ought to be expressed in Volts. Another illustration is that of measuring so- called electrokinetic or zeta potentials (ζ), a daily routine for every colloid scientist. The procedure is straightfor- ward: one measures, say the electrophoretic mobility, of a colloidal particle and interprets it in terms of, say the Helmholtz-Smoluchovski law, which immediately pro- duces ζ in Volts. This looks like a direct potential meas- urement. However, in reality this is not what happened; an applied electric field can lead to the displacement of free charges, not to the displacement of potentials. So, in the derivation of the Helmholtz-Smoluchovski equation some conversion of a charge into a potential must have taken place. Not everybody is conscious of that. We shall come back to this in sec. 6. In the present paper we shall reserve the notion of “potential” to the electric potential, expressing it in Volts. With this in mind, it is realized that, before deal- ing with the issue of measuring interfacial electric potentials, it is mandatory to agree on terms and defi- nitions. To that end we shall heed the recommenda- tions made over the past decennia by the International Union of Pure and Applied Chemistry (IUPAC). This organization is appreciated for its attempts to recom- mend internationally accepted rules for definitions of standards and nomenclature. The obvious reason is to “decacaphone” the literature from incorrect statements. Some of its recommendations refer directly to the measurement and interpretation of interfacial poten- tials.2,6-10 and below we have heeded them maximally. In this way it is hoped that at least the text is under- standable for a wide readership. This might be particu- larly useful when addressing the older literature, stem- ming from before the time that such recommendations were available. 2. EARLY 20TH CENTURY There are good reasons for starting the historical considerations with the first part of the 20th century. The centuries before that saw the development of rigorous theories for electric fields, some of these resulting in field vector equations like Gauss’ and Stokes’ theorems and general phenomenological laws derived from these, such as Poisson’s law for the relation between charges and potentials and Smoluchovski’s laws for electrophoretic mobilities and streaming potentials. These theories did not address the basic question what the origin is of the spontaneous charging of colloidally dispersed particles; it was just accepted that (surface) charges were appar- ently present and that hence the particles would be the seats of an accompanying (surface) potential. The advent of understanding these origins, together with the devel- opment of options for their measurements take us to the beginning of the 20th century, and to the central theme of the present study. What was the state of understand- ing electrified interfaces around ca. 1920? In that year theory of diffuse double layers was already available, thanks to the pioneering work of Gouy11 and Chapman.12 Their model is based on the Poisson-Boltzmann (PB) distribution for the counter- charge. Underlying the PB theory are two assumptions. In the first place, ions are considered point charges, i.e. as volume-less charges. As a result, the theory is generic: dependent on the valences of the ions but not on their sizes. The second, mostly tacit, assumption is that the potential occurring in the Poisson equation is the same as that in the Boltzmann equation. This is not necessar- ily correct: the former is an average potential whereas the latter is a potential of the mean force. This distinc- tion is rather esoteric, and considered in the domain of statistical thermodynamics. If needed we shall return to this issue later; but let it for the moment be accepted that the difference between the two types of potential are negligible when the potentials are low and when the potentials are very high.13 Notwithstanding these quan- titative limitations, the Gouy-Chapman (GC) theory was an important leap forward because it was one of the first relevant attempts to quantify some surface potential. However, with respect to our main question about the origin and measurability of the interfacial potential, GC theory does not help. The reason was that this theory was developed as an attempt to interpret electrocapillary curves, in the measurement of which the surface poten- tial across a mercury–water interface is simply applied externally. This leaves unanswered why isolate colloidal particles can also carry a charge, let alone what their surface potentials are. 77Interfacial Potentials: Measuring the Immeasurable? Apart from this central issue in the present con- text, let us recall another piece of wisdom of the GC model. It predicted the screening of charges to scale with the square root of the electrolyte concentration, a law observed often thereafter, but which was difficult to understand otherwise. Still another feature of diffuse double layer theo- ry was the prediction of expulsion of electrolyte by a charged diffuse double layer, a phenomenon that was independently measured and called the Donnan effect.14 The basic phenomenon is the electrostatic expulsion of co-ions, which is a purely electrostatic phenomenon. However, phenomenologically it is observed as the expulsion of electroneutral electrolyte by charged sur- faces, colloids and polyelectrolytes. The insight that this phenomenon found its basic origin in diffuse double lay- er theory grew only several decades later. What was not yet known around 1920 included (what is now known as) Stern theory for non-diffuse layers15 and the Debye-Hückel theory16 for the activity of strong electrolytes. Nowadays it is common practice to consider as a first approximation the aqueous side of double layers as consisting of two parts, an inner part, or Stern layer and an outer diffuse part, or Gouy layer. The inner part is the seat of ion specificity, meaning that the sizes and non-electric binding Gibbs energies are differ- ent between different ionic species. The impact of Stern theory is in recognizing that ions can also adsorb with- out electric attraction and hence spontaneously create or reduce an electric potential difference. The idea itself of a molecular condensor was familiar around 1920 though, thanks to the work of Helmholtz in the 19th century. Such layers were mostly called Helmholtz double layers and considered as an alternative for diffuse double layers, rather than as an addendum to them. However, around 1920 this essential step in the spontaneous creation of interfacial potentials by specific adsorption in the Stern, or Helmholtz part of a double layer was not yet so clear. 3. FREUNDLICH With this in mind it is historically interesting to read the pertinent literature from that period. A very rich source for obtaining this information is found in the standard books by Herbert Freundlich (1890-1941), one of the most versatile and all-round colloid scientists of that era. He wrote a sequel of books under the main (German) title Kapillarchemie, where the subtitle Eine Darstellung der Chemie der Kolloide und verwandter Gebiete explained what he understood by the title and subtitle. Nowadays one would call it “Colloid and related science”. Later, also English translations appeared under the name Capillary Chemistry. For the present purpose we shall now consider the second print of 1922.17 The problem that Freundlich was facing was match- ing the interfacial electrochemistry of large flat surfaces and colloidally dispersed systems. Are there common principles in the creation of electric potentials at the sur- faces of dispersed particles and macroscopic amounts of surface? It is particularly interesting to read how Freun- dlich interprets disparate measurements, some even for ill-defined systems. In chapter 6 of that book he starts with comparing measurements on Nernst-type poten- tials for macroscopic surfaces with those obtained elec- trokinetically for colloids.17 At that time it was by no means clear to which extent measured potentials would be identical under identical ambient conditions. Trans- lated into modern symbols, his question was by how much ψo (the electric potential at the phase boundary) and the electrokinetic potential ζ differ. At that time for colloids no other independent experimental evidence (like titrations) was available. It was even not yet clear whether there was a difference of principle between transversal and tangential potentials jumps. Of course, potentials being scalars, their absolute values are only determined by the position in the double layer, hence this question can be translated into that of the spatial variation of ψ and ζ along a surface. Given the scarcity of experimental evidence, Freun- dlich approached this problem by investigating how the addition of certain substances affected ψo and ζ. Of the many reported facts involved he states that, following Nernst, oxide and carbon electrodes may be considered as hydrogen or oxygen electrodes. With a bit of surprise he also refers to glass electrodes in the sense that their ”potential jumps” with respect to electrolyte solutions behave as if the glass were a swollen gel layer containing a fixed concentration of H+ and/or OH- ions. He reports that the ζ potential for this system, in agreement with earlier measurements, is very sensitive to the presence of low concentrations of additives. For example, H+ ions can reduce ζ without changing its negative sign, whereas AlCl3 and crystal violet do invert the sign of ζ already at concentrations as low as 1-2 and 50 μM, respectively, without having any substantial influence on the Nernst potential, ψo. The only effect AlCl3 had on ψo was chang- ing the spontaneous acidity of the solution, because it is a weak electrolyte. If this effect was suppressed by carry- ing out the experiment in an (acetate) buffer, the influ- ence on ψo disappeared completely. Freundlich had to conclude that ψo and ζ are different quantities. Supposed that ζ would be measurable that would not yet mean that ψo would be measurable as well. 78 Johannes (Hans) Lyklema His qualitative explanation for this difference is essentially the same as our present idea, namely the existence of a slip plane, or a thin slip range. His argu- ment was that upon tangential shear of the water the forces involved are not strong enough to move the water by shearing it at the solid-liquid border. Rather a thin adhering layer remains stagnant. Freundlich also real- ized that the very fact that electrokinetic phenomena could be observed at all must mean that the double layer must have a certain thickness, supporting the existence of a diffuse part. He quoted Smoluchovski who already in 1914, two years before the publication of Gouy theory, arrived at the same conclusion.18 Figure 1 sketches these ideas. Qualitatively, the shapes of the two curves do not differ that much from our present insights. Curve 1 applies to simple double layers, curve 2 for systems with overcharging (as it is nowadays called). The most strik- ing difference with (good) recent pictures is the absence of anything quantitative. There are no ideas about the potential inside the solid, it is not clear why there is a bending point in curve 1 and whether the position of that point coincides with the minimum in curve 2. That minimum (x = d) is apparently the distance from the surface where specific adsorption takes place. However, Freundlich identifies the potentials at that distance as ζ-potentials and this suggests that the plane of specific counterion adsorption is assumed to coincide with the boundary between tangentially moving and tangentially stagnant water, nowadays called the slip plane. No argu- ment for this is given; at any rate the identity between the two is not obvious: the distance from the solid sur- face where bound counterions are situated is determined by short range forces and water structure, whereas the position of the slip plane is determined by the tangen- tial motion of many water molecules. There is no a priori reason for their identity. We shall return to that in sec. 6. 3.1 Application of Gauss’s Law One feature underlying the sketches of figure 1 that Freundlich did not further elaborate, although its basic physics were perfectly well known to him, was the infer- ence on the various surface charges that can be drawn, using Gauss’s vector theorem (also known as Ostro- gadski’s or Green’s theorem). It is a general theorem for a vector field E in a volume V, bound by a closed area A of arbitrary shape. The space (triple) integral over the divergence of E equals the surface (double) integral over the inner product E·n where n is the unit vector normal to the surface and directed outward. The math- ematics becomes easy if applied to a flat interface, say for a charged sphere with radius r→∞ Then E·n simply becomes the field strength E = -dψ/dx normal to the sur- face. E corresponds to the slope of the curves drawn in Figure 1, integrated over the entire surface A. The triple integral over div·E can be written as div·(gradψ), mostly written as ∇2ψ which, according to Poisson’s law equals -ρ/(εε0) where ρ is the local space charge density, ε is the local relative dielectric permittivity and εo is the dielec- tric permittivity of free space. This triple integral leads to the total charge in the volume V. In our specific case with no space charges in the solid interior, this charge is the surface charge σ multiplied by the area A. Combin- ing all of this leads for one-dimensional situations to the simple expression ψ σ ε ε =− d dx 0 (1) Eq. (1) is generally valid: at any position in the dou- ble layer can we make a cross-section where the part on the left and the part on the right have equal, but oppo- site charges, given by this equation. For instance, just to the right of the x = 0 plane the slope for curve 2 is steep- er than that for curve 1, which means that in the second case the surface charge is higher. Phenomenologically this trend is correct: stronger screening leads to higher surface charge at given surface potential. However, curve 2 cannot be correct, because it predicts a maximum in Figure 1. Sketch of the potential distribution ψ(x) in a flat double layer after Smoluchovski (1914) and Freundlich (1922). The border between the solid (hatched) and the liquid is situated at x = 0. Dis- cussion in the text. 79Interfacial Potentials: Measuring the Immeasurable? the countercharge around the slip plane, which is physi- cally unrealistic. Moreover, it would also predict a zero charge coinciding with the slip plane for which also no physical reason can be given. Far from the surface all charges are zero, which is obviously correct. Although all of this must have been known around 1920, little of such argumentation is encountered in Fre- undlich’s book. In his attempts to get hold of the charg- ing principles he collected many data on the electroki- netic properties of a great variety of systems. Regretta- bly, in that respect he was not very successful, mainly because he did not have sufficiently well-defined systems at his disposal. 4. LANGE AND KOENIG About ten years after Freundlich, Lange and Koenig wrote a completely different approach, in which they considered electrical potentials thermodynamically.1 Not surprisingly, this approach also involved the ques- tion of the measurability of surface potentials. The paper is “deutschgründlich” written, with the excuse that they could not present it in even more detail because of lack of space. The Lange–Koenig paper helped the under- standing of interfacial potentials in at least two respects: distinguishing between measurable and immeasurable potentials and the introduction of the notion “potential- determining ions”. Their most significant contribution is the insight that there is in principle no way for measuring the (elec- tric) potential difference between two adjacent, chemi- cally different phases. Let us by way of paradigm con- sider the potential of a solid (particle or electrode) with respect to the solution in which it is imbedded. Measur- ing electric potential differences, say between locations A (the solution) and B (the solid), requires bringing a unit charge from A to B and determining the electric work per unit charge involved. However, when points A and B are located in chemically different condensed phases, the transport of the charge carrier also involves chemical work. For colloids the charge carriers are ions, which have finite sizes and hence they also inter- act non-electrostatically with their surroundings. The Gibbs energy of that interaction is non-zero even if the phases are uncharged. This chemical contribution can- not unambiguously be separated from the electrical one. Hence, absolute values for the potentials in condensed phases are in principle immeasurable. It is historically interesting that, in the first quote of the present paper Lange and Koenig take a less absolute stance; they just state that separation is as yet not pos- sible, apparently they believed in technological solutions for all scientific problems. Nowadays the insight is that the impossibility of measuring absolute potentials in condensed phases is intrinsically coupled to the impos- sibility of measuring thermodynamic potentials of sin- gle ionic species. In no way can the energy and entro- py of, say a mole of protons be determined because no process can be envisaged to prepare and transport that mole without simultaneously also transporting the same amount of anions. Otherwise stated, potential differ- ences between different condensed phases are inoperable. Lange and Koenig realized that. Given this impossibility, the problem was, and is, how to cope with that in practice. Generally, for such issues two ways are open: avoiding it or subjecting it to penetrating theoretical analyses, two options that are not very handy in daily practice. Lange and Koenig came with two suggestions, the introduction of the notion of potential-determining (pd) ions and distinguishing various types of potentials, some measurable, others immeasurable. 4.1 Charge-Determining Ions As to the former, although absolute potential dif- ferences between adjacent condensed phases of different compositions are inoperable, changes therein are often accessible, in particular when the material under study can function as an electrode in a Galvanic cell. The sim- plest example is a cell containing the electrode under study in an electrolyte solution and a reference elec- trode. The cell potential Ecell is in principle measurable and consists of the sum of the sought electrode-solution potential difference Δψ and the reference electrode- solution potential jump Δψref. With such a cell one can measure changes in the cell potential as a function of changes in the composition of the solution, like chang- es of the electrolyte concentration or changes in pH. If the reference electrode is insensitive to those changes, as is the case for a suitable calomel electrode, the change in cell potential comes only on the account of Δψ. So, although Δψ is inoperable, dΔψ is relatively well acces- sible. It was already known to Freundlich that in many systems dΔψ shows Nernst behavior, according to which dΔψ is proportional to the log of certain electrolyte con- centrations to the amount of 58 mV per decade of the concentration of that ion. In many cases that electro- lyte contains the proton, H+ and this rule is the basis for applying glass electrodes for the pH measurements. The thermodynamic background of Nernst’s law led to the introduction of the notion of potential determining ions by Lange and Koenig. 80 Johannes (Hans) Lyklema According to Lange and Koenig, potential-determin- ing ions are thought as being present in both of the adja- cent phases in sufficient amount as to be thermodynami- cally significant. Then, an electrochemical potential can be assigned to the activity of the proton in the solution and the solid and their equalization leads to the estab- lishment of the dΔψ(dpH) relation. Mathematically, elec- trochemical potentials consist of three terms, for pd ion i generally written as μoi + zi Fψ + RT ln ai.We are inter- ested in the variation upon changes in ai. In the solution this leads to ziFdψ(solution) + RTdln ai and in the solid to ziFdψ(solid). Equating these two changes in chemical potentials, writing ψo for ψ(solid) – ψ(solution), leads to ziFdψ0 = RTdlnai (2) which predicts dψo/(RT dlnai) to be constant and equal to RT/(zF) per decade of the activity of z-valent pd ions. At room temperature this amounts to 58 mV/z per factor 10 in the activity of i, i.e. the observed Nernst behavior. The above derivation gives some insight into the assumptions that had to be made in order to arrive at Nernst’s law. The observed applicability of this law for a given solid-liquid interface is a justification in retrospect for the applicability of the model assumptions for that particular system. As to the measurement it has to be added that measuring relative activity changes of single ionic species (dlnai) is not possible; only activity changes of neutral electrolytes or electrolyte mixtures are meas- urable. To overcome this problem in practice, the pro- cedure is working in swamping electrolyte that has the anion in common with the acid that produces the pro- tons to be measured. For example, if the activity of pro- tons must be measured in HCl, the measurement is car- ried out in swamping KCl, of which the concentration is fixed so that in dlna(H+) + dlna(Cl-) the second term vanishes. Calibration is necessary to ensure that this sit- uation is attained. If it is not sufficiently controlled the trend is that lower than 58 mV is found for the Nernst slope. It may be added that “swamping” means “large as compared to the concentration of i”, implying that not necessarily high salt concentrations are needed. The notion of potential-determining ions has been generally accepted and used for several decades. Nowa- days these ions are rather dubbed “charge-determining (cd) ions”, just because of the reason that the absolute value of the potential in a chemically different phase cannot be determined, whereas charges can be very well measured. Below we shall adhere to the term “charge- determining”. The conclusion of this part is intuitively sound: Nernst behavior means that upon transport of one ion from one condensed phase to the adjacent requires an energy of kT per unit charge zie and as energy = charge x potential its quotient is a unit potential kT/(zie) = 58/z mV. Ions for which this is in practice measurable are charge-determining ions. 4.2 The χ-potential As to the fundamental impossibility of determin- ing absolute values of the potential difference Lange and Koenig proposed to distinguish between different types of potentials, each with its own symbol: the Galvani (φ), Volta (ψ), real (α) and χ-potentials. Ref. 1 is replete with relationships between these potentials, and with the thermodynamic and electrochemical potentials in solu- tions. Galvani potentials are the (immeasurable) inner potentials, in our example the potentials in the heart of a colloid particle or electrode with respect to the bulk of the solution. Volta potentials are measured in the solu- tion so close to the outside of the condensed phase that the effect of the nature of that phase is not felt, but close enough to pick up any longer range effect of the elec- tric charge on the phase. By this definition Volta poten- tials are in principle measurable, but the information they carry is much less interesting than that of Galvani potentials. Real potentials refer to a specific ionic spe- cies, say i, and they are combinations of μi and the Volta potential. We shall not use this quantity but note that it is identical to the electronic work function for extract- ing an electron from a metal (or ion from a non-metal) and take it to infinity. Lange and Koenig distinguished between Galvani and Volta potentials through the differ- ent symbols φ and ψ, respectively. Hence the distinction between dφ and dψ in the first quote of this paper.1 We shall not use this distinction and only use ψ for the elec- tric potential. Where needed, we shall provide enough information on the way in which the non-electric contri- butions are sequestered. The last suggestion by Lange and Koenig, was the introduction of the notion of χ-potential, also recom- mended by IUPAC. We shall call it the interfacial poten- tial jump. The potential jump in going from A to B we shall denote as χAB. In the Lange-Koenig nomenclature χ = φ – ψ. The quantity χ is immeasurable but has a clear physical meaning: it accounts for the potential jump at the interface between two phases caused by the local polarization of the molecules at the interface, orientation of water dipoles at the interface, polarization of the sur- face layer of the solid; in short, all contributions to the potential difference between uncharged adjacent phases, that is: at the point of zero charge. The reason why the introduction of χ is so handy, even though it is immeas- 81Interfacial Potentials: Measuring the Immeasurable? urable, is that all contributions grouped into it are short- range, a few molecular diameters at most, whereas elec- tric double layers have thicknesses of the order of κ-1 which can be orders of magnitude larger. For colloids an additional argument is that surfaces are rarely per- fectly flat. Roughness of the order of a few molecular layers can rarely be avoided. An additional considera- tion for using χ is that the properties of diffuse double layers that are relevant for the daily practice of colloid science (say for the interpretation of colloid stability and electrokinetics) can to a large extent be understood by considering the diffuse part only. However, assum- ing that the surface charge σo has been measured, the very relevant and interesting issue is which fraction of the countercharge (-σo) finds itself in the region very close to the surface, that is: in the Stern layer, and which fraction remains for the diffuse part. This is one of the most recurrent question of double layer science anyway, because the Stern layer coincides with the layer deter- mining χ. One typical illustration of employing this insight is in establishing the point of zero charge (p.z.c.) for amphoteric colloids, that is the point where the total amount of positive charge on the surface is the same as that for negative charge. This is a measurable quantity. For oxides it is a specific value of the pH, to be indicated as pHo. Tables of pHo values can be found in the litera- ture.19 The point is that these points are not identical to the corresponding points of zero potential. The differ- ence between the two zero points is just χ, the elusive unknown. “Elusive” is a disputable notion. Nowadays χ is a popular topic of investigation because many modern techniques do shed some light on it, if not in the liter- ary sense of optical measurements but also by molecu- lar simulation, statistical mechanics, colloid chemical, and even thermodynamic tools. As to the last one, ther- modynamics are phenomenological, hence cannot tell something about molecular organization but do teach us how to obtain surface excess entropies, which in them- selves are challenging quantities. For example, for the surface of pure water the surface excess entropy, (that is: the entropy of the layer that is responsible for the χ-potential) has been obtained from the temperature dependence of the surface tension.19 In this way the dif- ference between the surface energy and the surface ten- sion could be quantified. On the other hand, Guggen- heim refused to discuss χ at all because of its immeasur- ability.20,21 However, if one is not from the very onset at war with model assumptions or approximating experi- ments, some information about the χ potential of water (χw) can, and has been, obtained. It stems from different sides, like making assumptions for single ionic activities for some untypical electrolytes, comparisons between different electrolytes or of ionic Gibbs energies in dif- ferent media and simulations, jokingly called “experi- ments”. All these “partial” or approximate analyses led to roughly the same order of magnitude of a few tens of mV positive.19 The positive sign means that water is positive with respect to water vapor: the negative sides of the superficial water molecules (the oxygens) are on the average pointing outward. Recent studies of the pre- sent author seem to indicate that that is also the case for water in contact with condensed apolar media like mer- cury or silver iodide. The conclusion is that the immeas- urability of potential jumps between condensed phases is basically upheld, but that there are interesting attempts of measuring the immeasurable. See also briefly sec. 8.2. Figure 2 illustrates how Lange and Koenig visual- ized the χ-potential at the boundary between a con- densed phase (left) and an aqueous solution (right). The picture is extremely simplified. As to χw the authors are thinking of a monolayer of oriented parallel water dipoles, This is a very unrealistic model because forcing all those molecules into parallel positions is entropically very improbable. In fact, the numbers for χ and ψo that the authors substitute by way of illustration are about a factor of 10 higher than we now know and the peak at x = d cannot be that sharp. The figure also contains a dif- fuse double layer part that in their text is not discussed; apparently they envisage that the creation of a double layer by adsorption of charge-determining ions simply comes additively, a view that is mostly adhered to till today. Figure 2. Sketch of the potential jump at a solid-liquid phase boundary as caused by the χ-potential. Modified after ref. 1, their figure 1. Discussion in the text. 82 Johannes (Hans) Lyklema In the last pages of their longish (190 pages) paper Lange and Koenig discuss electrokinetic phenom- ena, emphasizing the complicated inf luences on the ζ-potential by various electrolytes. In their figure 75 they place the slip plane at the boundary between the Stern layer and the diffuse part, subsuming the χ-potential in the potential of the solid, which is not further discussed, leaving a variety of issues about double layer potentials and charges for their successors., as we shall do in sec.6). The conclusion of this section is that essentially the impossibility of measuring the absolute value of the potential difference between adjacent condensed phases of different chemical composition is translated into the problem of evaluating the corresponding χ-potential. For many purposes working with charges rather than with potentials is preferable. 5. SILVER IODIDE, THE LUCKY STRIKE OF THE DUTCH SCHOOL In the years before World War II one of the major challenges in the domain of interfacial potentials and charges was finding systems for which potentials and/ or charges could be assessed, if not measured, both for macroscopic surfaces and for colloids. For macroscopic surfaces, in electrochemical cells double layer capaci- tances could at that time be measured with great preci- sion for mercury electrodes, using Wheatstone bridge techniques. For colloids, stability against coagulation and electrokinetic potentials could be measured for sev- eral systems with reasonable confidence. However, from mercury it is virtually impossible to make colloidal sols and for colloids it was impossible to determine the sur- face potential or surface charge. Otherwise stated, sys- tems for which surface potentials/charges and electro- kinetic potentials/charges were both measurable did not (yet) exist. It appeared that silver iodide was an excellent candidate to fill that gap. Colloidal studies with the AgI system have been carried out over more than a century with a major con- tribution of the Dutch School of Kruyt, Overbeek and offspring. As a model substance for both interfacial elec- trochemistry and colloidal studies, AgI has many attrac- tive features. The material is chemically nearly inert, its solubility in water is very low (solubility product about 10-16 at room temperature), stable sols (many years with- out perceptible coagulation) can be made of it and the material is very hydrophobic (contact angles for water droplets around 120°). AgI sols are really hydrophobic sols; their stability is exclusively determined by the inter- action of overlapping electric double layers. Changes in the surface charge of dispersed AgI can be measured by titration with KI or AgNO3: changes in ψ° obey Nernst’s law, with silver and iodide ions charge-determining, i.e. changes in the surface charge σ° can be measured as a function of changes in pAg or pI. This last fact means that dσ° as a function of dψ° is measurable, so that the differential capacitance C = dσ°/dψ° can be established. For mercury, the primary measurable is also the dif- ferential capacitance, hence properties of very disparate materials in different states (macroscopic surfaces ver- sus finely dispersed) can be compared with each other, a most rewarding exercise! Studying all of that does not require information on the absolute values of ψ°. Hence, they can be carried out without being thwarted by its principally immeasurability. Given the fact that the point of zero potential cannot be established, one usual- ly refers the point of zero charge for which good experi- mental methods are available (the common intersection point in Figure 5, see below). The development of all that insight took several dec- ades and in the following sections some typical achieve- ments in line with the present theme will be considered. For historical reasons it be noted that Kruyt et al. were not the only investigators of the AgI system. For exam- ple, older, but more primitive titrations have been car- ried out by Lange and Berger22 and for AgBr by Lotter- moser and Petersen.23 One of the first, but still rather primitive, titration of AgI with different Ag salts was conducted by De Bruyn.24 5.1 Interaction at Constant Potential or at Constant Charge? During the Second World War colloid science got a tremendous boost by the publication of the DLVO theory for the stability of hydrophobic colloids.25,26 The abbreviation stands for Deryagin, Landau, Verwey and Overbeek. The history is well known: the Russians wrote their paper in 1941 in a Russian journal,25 which because of the war conditions did not make it to the West, whereas the Dutch couple elaborated the theory dur- ing the War and then wrote an extensive monography on the matter.26 It was only after the War that it became clear that the two theories were identical as to the main principles stability is determined by a balance of van der Waals attraction and double layer repulsion. The qualita- tive insight that the stability of hydrophobic colloids had its origin in the interaction between electric double lay- ers was more or less generally accepted in 1940 but the quantifications, some of them by respectable scientists like Levine and Langmuir, were not successful: after all, it is not so obvious to prove that interaction between 83Interfacial Potentials: Measuring the Immeasurable? two double layers that as a whole are electroneutral are repulsive at any distance between the surfaces. In addi- tion, it must be realized that, say around 1935, the role of attractive van de Waals forces was not yet quantita- tively established, so that some investigators even con- sidered electric double layers, that, although electroneu- tral as a whole, could nevertheless be electrically repul- sive over some part of the interaction, but attractive over another range. Now we know that attractive van der Waals forces are quantitatively comparable to electro- static double layer repulsion forces. However, this part of the story, interesting for its own sake, does not belong to the present theme. Let us state that at the start of the Second World War the main problem was to develop a quantitative theory for the electric double layer repulsion that was strong enough to protect the particles against coagulation by van der Waals attraction. Needless to state that this repulsion is sensitive to the potential - and charge distribution during overlap, which brings us back to the theme of this paper. A crucial proof of the pudding was to explain the observed extreme sensitivity of the stability of hydro- phobic sols to the valence of the counterion (the so- called Schulze-Hardy rule). Both DL and VO were able to do so quantitatively and the two couples arrived at the same law. As to the way of presentation, in particular the motivation of some decisions, the VO book is far more useful than the DL paper: the former gives a careful consideration of all the decisions taken, whereas in DL theory sometimes hand-waving arguments are used. For example, when V and O make a conscientious distinc- tion between double layers at high potential and those at high charge, DL only speak of “strong double layers”. In connection with the present theme, we are now deal- ing with double layers resulting from free ionic charges, which may reside in the diffuse or Stern part; the fact that there exists something like a χ-potential does not (yet) play a role, but may show up later. As to the elaboration of the electric interaction, DL and VO have in common that both compute the charge and potential distribution between two approaching par- ticles at any distance d using a charging process. In this way, not the energy but the Gibbs energy of interaction is obtained, which also contains the entropic contribution. The interaction Gibbs energy is then found as the iso- thermal-reversible work to adjust the charge and poten- tial distribution upon increasing the overlap between the two surfaces. In passing, it is a pity that this procedure seems to be forgotten when interaction between more complicated colloids has to be considered, for example upon the interaction of oppositely charged polyelectro- lytes, as in the formation of complex coacervates. Overlap of (diffuse) double layers implies that the two original ψ(d) curves will change. The sum of the two contributions at given distance d is not necessar- ily additive. For the elaboration of the differential equa- tions describing the potential and charge distributions it is mandatory to know the boundary conditions. Will ψ° remain constant or will it increase? And what will happen with the concomitant charge σ °? Pondering this question, it becomes clear that the problem exceeds that of something purely mathematical: basic physi- cal processes are involved. This becomes immediately clear when eq. (1) is considered. See Figure 3B. Keeping the potential fixed upon the approach of a second dou- ble layer of the same charge sign can only be realized by reducing the charge, that is: by desorption of charge- determining ions. This is a chemical process. When the double layers are far apart the double layer forms spon- taneously; its Gibbs energy of formation is negative. Thus, the origin of the stability has a chemical nature. Case A is the other extreme. Then no ion desorption takes place; upon approach the potential must shoot up and now electric work must be done to let the surfaces approach. In reality always repulsion is observed, at any d, but the mechanism can be different. Otherwise stated, equal double layers always repel but not necessarily for Figure 3. Sketch of the difference between interaction between two diffuse double layers at constant charge (A) and at constant potential (B). In case A, ψo increases; the nature of the interaction is electric, whereas in case B, σo decreases and the interaction is of a chemical nature. Dashed lines: the slopes, proportional to the charges because of eq. (1). Reprinted from J. Lyklema, Fundamen- tals of Interface and Colloid Science. Chapter 3: Pair Interactions. 2005. Elsevier, Amsterdam. Copyright © 2005 Elsevier 84 Johannes (Hans) Lyklema electrostatic reasons. Case B could well apply to the AgI system, and most of the oxides. Adsorbed Ag+ ions have to desorb upon approach. Case A is representative for clay minerals for which the charge is more or less fixed because σo is determined by isomorphous substitution of ions inside the solid. On closer inspection, the distinction between con- stant potential and constant charge has a dynamic ori- gin. The critical question is whether or not the adsorbed charge-determining ions have enough time to desorb during particle encounter. Supposed we would be able to shoot the particles very rapidly onto each other, so fast that the adsorbed ions would not have enough time to escape, repulsion would still prevail, but the nature of the repulsion would be different. Returning to Verwey and Overbeek, they realized all of this but for practical reasons had to make a choice which system to elaborate in detail. Not surprisingly, they had the AgI system in mind. Their decision was to treat purely diffuse double layers interacting at constant potential. Theory for such double layers at large separa- tion was available.11,12 Going for diffuse double layers implied ignoring ion specificity and was motivated by the fact that the challenge of explaining the very dra- matic ion valence effect on sol stability was generic (as diffuse layers are) whereas the less pronounced ion spe- cific effect (typically in the Stern layer) depends on the nature of the colloidal material. (In passing, Ref. 26 also contains a section on Stern layers). As a consequence, the DLVO model is that of a diffuse double layer with surface potential ψ° and surface charge σ°. The choice for interaction at constant potential was also made with the AgI system in mind: because of the Nernst behavior: ψ° is determined by pAg and as the latter remains con- stant upon particle interaction, so does the former. Now- adays dynamic arguments support this choice: the time for particle interaction is large compared to the relaxa- tion time of a diffuse double layer. In summary, the the- ory considers interaction between two (flat) double lay- ers at fixed ψ°. In fact, the Russians arrived at the same model, although they did not justify their arguments in so much detail. In order to apply the theory quantitatively a meas- ure for the stability had to be developed. Nowadays that would go in terms of turbidities, but V and O did not yet have such data at their disposal. The alternative of that time were critical coagulation concentrations (cc), that are the salt concentrations above which the sol is unstable. At that time, by trial and error a proce- dure was developed to determine such data with a sur- prisingly good reproducibility. The procedure involved making series of test tubes containing the sol to which increasing amounts of the electrolytes were added. Using a smart empirical mixture of what is now called periki- netic and orthokinetic coagulation, sharp boundaries between fully coagulated and fully stable sols could be created. The virtue of this time-consuming method was that the relatively complex phenomenon of aggregation could be translated into only one single number, the critical coagulation concentration cc, a parameter want- ing theoretical interpretation. 5.2 DLVO Stability and the Schulze-Hardy Rule The DLVO relation between the critical coagulation concentration cc, and the valence of the (z-z) electrolyte can for the present purpose be written as ( ) =c const tanhzy A z ·c 0 4 2 6 (3) where the constant contains known quantities (like nat- ural constants, temperature, dielectric constants, etc.) and A is the Hamaker constant, a measure of the van der Waals attraction between the particles, not surpris- ingly in the denominator. The most crucial parameter is the surface potential, for which the dimensionless abbre- viation ψ ψ = =y F RT y F RT etc,  ,  .0 0 (4) is used. For the interpretation of the role of the surface potential the dependency of tanhy on y is critical. At suf- ficiently high potentials the hyperbolic tangent is inde- pendent of y and equal to unity. Verwey and Overbeek pondered the issue. They wondered whether they should substitute y° or the dimensionless electrokinetic poten- tial yek = Fζ/RT. One argument for the latter was that for hydrophobic sols often a good correlation between zeta potentials and stability was found. However, the desired data available at that time were not yet sufficiently repro- ducible and it was not clear where the slip plane was located. Hence they fell back on the well-known surface potential, which at the pAg where the stability measure- ments had been carried out was several hundreds of mV negative, that is, well into the domain where the tanh function equals unity. In that case the critical coagula- tion concentration is independent of the surface potential and inversely proportional to the 6th power of the (coun- ter)ion valency. In passing, as the Russians considered “strong” double layers, they also arrived at the z-6 power. 85Interfacial Potentials: Measuring the Immeasurable? Establishing this law was the beautiful vindication and quantification of the qualitative Schulze-Hardy rule and it became the justification of DLVO theory, even to the extent that the 6th power was sometimes identified as the Schulze–Hardy rule. Verwey and Overbeek tested the inverse 6th power law for critical coagulation concentra- tions of AgI and other hydrophobic sols. They confirmed the quantitative applicability of the law for the difference between z =1 and z = 2, with as a trend a lower power for the difference between z = 2 and z = 3. This solved one of the main problems of colloid stability at that time. However, it was a pity that for the present issue the result is not useful because the potential does not play a role when it is high enough. 5.3. Hofmeister Series and Stern Layers Superimposed on the 6th power law of DLVO there is a less dramatic, but certainly quite interesting and not negligible ionic specificity effect. At that time, series of data for different ions of the same valency were called lyotropic series. Nowadays, the term Hofmeister series is more common. For the critical coagulation concen- trations of AgI sols these data were well known. Verw- ey and Overbeek were familiar with the data by Kruyt and Klompé,27 which were also obtained during the war time. (Marga Klompé was a PhD student of Kruyt; later she became the first female Dutch Minister of educa- tion). In the table her results are given for monovalent nitrates. Table 1. Coagulation concentrations of some (1-1) electrolytes for AgI sols. Data by Kruyt and Klompé (1942). LiNO3 165 mM NaNO3 140 mM KNO3 136 mM RbNO3 126 mM For alkali ions, the differences in the ion specifi- cities are relatively large, about 30% between Li+ and Rb+ as the counterion. Other investigators of AgI find somewhat different concentrations but always the same sequence. As a digression, it depends on the nature of the colloid whether the sequence is in the direction of increasing or decreasing ion radius. Otherwise stated, the Hofmeister sequence depends on the surface proper- ties of the colloid. In fact, recent studies have confirmed the fact that generally such series are not properties of isolated ions but of pairs. In passing, it is an idiosyncra- sy of the AgI system that for HNO3 cc = 137 mM, similar to that for the cations mentioned in the table: the pro- ton behaves as an ordinary counterion, in contradistinc- tion to most other systems. For oxides protons are even charge-determining. Not knowing how to deal with ion specificity, Ver- wey and Overbeek just took the average value, 142 mM for the monovalent alkali ions. By the same token, for seven bivalent cations they obtained the average of 2.43 mM. The ratio 142/2.43 = 58.4, is close to 26 = 64. They also tested other colloidal systems, like As2S3 and Au sols, and positive sols, like Fe2O3 and Al2O3 which gave rise to Hofmeister series of anions, and grossly confirmed the z-6 dependence. So, the interpretation of stability by overlap of the diffuse layers is almost quan- titatively explained. This result is still considered as the most important quantitative success of DLVO theory. Around 1955, when I started my PhD work with Overbeek I focused on the, then still open Hofmeister trend, trying to exploit the other characteristic pro- pensity of the AgI system, namely that it is possible to measure double layer charges by titration. Do the titra- tion charges also depend on the nature of the (counter) ion? In figure 4 results are presented for three of the four alkali nitrates considered in the table. The p.z.c. is located at pAg = 5.6. To the left of it the surface is positively charged; there the nitrate is the (common) counterion so that no cationic specificity is observable; to the right, at sufficiently negative charge, Hofmeister effects do show up upon increase of the sur- face charge in the direction Li+ < K+ < Rb+. It was a satis- fying result of this research that the ion specificity found in the stability recurred in that of the double layer. At Figure 4. The surface charge on AgI as a function of the pAg in decimolar solutions of the indicated electrolyte, that is the order of magnitude of the cc. Reproduced from B.H. Bijsterbosch, J. Lykle- ma, Adv. Coll. Interface Sci. 1978, 9, 147, Copyright 1978, with per- mission from Elsevier. 86 Johannes (Hans) Lyklema the same time it posed a problem in that the direction was unexpected: along the lines of DLVO one would not expect that sols for which the surface charge at given pAg is higher would correspond to less stable sols. On mercury the same sequence was found in the surface charge but the alkali ion specificity was lower by about a factor of 10.28 In fact, had the specificity been so low on AgI, it would have been impossible to meas- ure it at all. Had it been possible to measure the stability of mercury sols, the result as to the Hofmeister effect for alkali ions, would be very dull. This comparison indi- cates that not only the sequence but also the extent of the specificity depend on the nature of the adsorbent. Returning to the AgI case, the seemingly contrast- ing Hofmeister sequences between cc and σ° proves that one is dealing with the non-diffuse part of the double layer, that is: with Stern layers. For AgI the non-electro- static affinity increases in the sequence Li+ < K+ < Rb+. Rb+ ions screen the negative potentials near the surface better than Li+ ions do, apparently because they adsorb stronger. Consequently, at given ψ°, σ° will become more negative but at the same time more of the countercharge is compensated in the Stern layer so that less charge is left for the diffuse part, which controls interaction, hence the lower cc values. The combined action of electric and specific interac- tions in determining double layer properties is also very well reflected in by overall σ°(pAg) curves, given in fig- ure 5. The 10-1 curve corresponds to the KNO3 curve of figure 4. On the r.h.s the surface becomes more negative with increasing salt concentration because of screen- ing of the surface charge by electrolyte. The counterion plays the more important role. A large part has an elec- trical origin, where, typically for PB theory, the vertical distances between the curves scale as √c. The chemi- cal contributions become visible at higher charges and higher salt concentrations, see also Figure 4. On the left hand side of the point of zero charge the surface is positively charged, this charge becomes even more posi- tive by the addition of electrolyte for the same reason as what happens at the negative side. Only at the point of zero charge there is no charge to screen, hence there is no effect of electrolyte, unless one of the ions adsorbs specifically. In fact, this is the common way of establish- ing the (pristine) p.z.c., that is the p.z.c. in the absence of specific adsorption. For AgI the common intersection point is located at pAg = 5.6 till concentrations of 1 M, where the surface is no longer pristine: a slight shift to the right is observed, indicating specific adsorption of the anion, NO3- in this case. So, the double layer properties of colloidal AgI are well understood, without the need of measuring ψ°. It be repeated that the point of zero charge is not identical to the point of zero potential, which remains immeasurable. The difference between the pAg of zero potential and that at zero charge is just the χ-potential at the AgI-water interface. When comparing with mercury, the model surface for the electrochemist, it appears that the AgI did become a grown-up counterpart mod- el for which the lesser reproducibility and precision is outweighed by the propensities that double layer charges can be measured because the areas are so large and because the charge-determining ions have been identified. At this place it is interesting to note that curves like those of figures 4 and 5 have also been measured by the “mercury“ method, that is by direct measurement of the differential capacitance of AgI electrodes. The trick is that the adsorption of charge-determining ions is sup- pressed by working at very high frequencies. The results are very close to those obtained by titration.19,29 This does not yet exhaust the analysis. It is obvious from the above that the DLVO idea of a purely diffuse double layer is untenable. At least a Stern layer has to be added, to account for the fact that even the closest coun- terions cannot approach the layer where the surface ions are located because of their non-zero volume. That gives rise to a thin charge-free layer close to the surface, the so-called zeroth order Stern layer. It has a (differential) capacitance Ci that is in series with the capacitance Cd of the outer, diffuse part of the double layer. The sum capacitance C is given by = + C C C 1 1 1 i d (5) Figure 5. Surface charge as a function of pAg for silver iodide. T = 25o. Electrolyte, KNO3. The pAg axis may also be read as a ψ° axis with 58 mV per unit of pAg. 87Interfacial Potentials: Measuring the Immeasurable? The total capacitance is dominated by the lower one of the two. When, in addition to the ion size exclusion, also specific ion adsorption has to be considered, eq. (5) has to be modified The point is that Figure 5 shows that the double layer is completely diffuse only over a narrow range of very low potentials. The differential capacitance of a purely diffuse double layer depends on ψ° according to a cosh function, which has a minimum at the p.z.c. For AgI these capacitances are, except for a constant, the slopes of the curves in this figure. Obviously such a minimum is clearly visible only in very dilute solutions around the p.z.c. This conclusion is at variance with the popular statement that one has worked with a “diffuse double layer with Stern corrections”. Nevertheless, as long as no specific adsorption is detectable, the relation between charge and potential for diffuse layers remains valid, except that the diffuse part does not start at the surface but at a short distance from it, corresponding to the thickness of the Stern layer. Figure 6 illustrates how we look nowadays at the structure of double layers. The distinct subdivision between a diffuse and a non-diffuse part is well-motivat- ed and has many practical advantages, even if we neither know exactly how thick the Stern layers are, nor how much charge they contain. All short-range effects are sub- sumed in it, so that beyond this layer the layer is purely diffuse, where Gouy-Chapman theory does apply with impunity. Depending on conditions the diffuse part may contain all or only a minor part of the countercharge but even if this is only a low fraction, it nevertheless plays a dominant role in colloid stability and rheology. Figure 6a is the most simple double layer with a charge-free Stern layer. In this case the diffuse charge σd equals minus the surface charge, which is measurable. For a flat double layer (in practice, for κa>>1) Gouy the- ory has derived the following useful equation,19 relating the surface charge to the diffuse potential ψd (yd = Fψd/ (RT)): σ εε=− RT sinh zy 8 2 d d 0 (6) In the inverse situation, when the charge has been measured, the potential follows from its inverse: σ εε = −−y z sinh RT 2 8 d d 1 0 (7) This equation is often used in electrokinetics, where the charges on moving particles are measured but where these charges are by force of habit converted into ζ-potentials, (sec. 6). Figure 6b is characteristic for situations where ion specificity is observed, as in the AgI case of figure 4. The specifically adsorbed charge is called σi and the potential at their locus of adsorption is denoted ψi. Sometimes the planes where the specifically adsorbed ions reside and the plane from whereon the layer is diffuse are called the “inner and outer Helmholtz layer” (iHp and oHp, respectively). As to our present theme, ψi is immeasur- able. Eq. (6) remains valid, of course, but only for the diffuse part. Beyond that, its application becomes prob- lematic because from the charge balance σ° + σi +σd = 0 the diffuse charge σd cannot be computed (unless addi- tional data are available, see sec 5.4). In this connection, eq. (6) is often used under conditions where it is not (a) Gouy-Stern double layer without specific adsorption. Finite counterion size. (b) Gouy-Stern double layer with specific adsorption. (c) Gouy-Stern double layer with superequivalent adsorption. Figure 6. Actual view of Gouy-Stern double layers. Taken from Ref. 19 with permission. Copyright © 1995 Academic Press. Published by Elsevier. Discussion in the text. 88 Johannes (Hans) Lyklema allowed. Perpetrators are psychologically lured by the scientific magic and apparent rigor, and by the sugges- tion that unknown potentials can seemingly, but errone- ously be identified. With three unknown potentials and two unknown charges there are six ways of going wrong and only one of them is correct. Turning to Figure 6c, this is a sketch of the poten- tial distribution in cases where the double layer con- tains more countercharge in the Stern layer than needed to compensate the surface charge. This phenomenon is commonly called overcharging. Several theories have been proposed to explain it. For the AgI case, and prob- ably much more generally, overcharging occurs for counterions of higher valency if they hydrolyse in the adsorbed state. The hydrolyzed complexes have a strong chemical affinity to the surface. That this is the right mechanism can be proven by systematically investigat- ing the pH dependence of the charge inversion.30 So, it is a chemical, rather than electrical phenomenon. 5.4 DLVO Theory Revisited It is obvious from the previous sections that the prem- ises of DLVO theory, notwithstanding its successes, have to be reconsidered. The main correction is that only part of the counterlayer is diffuse and that the potential of the diffuse layer is not ψ° but ψd. The latter potential is quite a bit lower than the former and depends on the nature of the counterion. Let us consider these points systematically. Is ψd measurable? There is no argument against measurability of the nature as we invoked when explain- ing why ψ° is immeasurable, because the location of the potential ψd is in the same solution as the reference. On the other hand, ψd is a model parameter; it is based on a number of assumptions. The most relevant is that of having replaced some gradual functions by step distri- butions, say the extent to which the dielectric permit- tivity changes with distance. So, in practice quantifica- tion is rather a matter of doing independent experiments that give information on the charge of the diffuse part of the double layer only and then subtracting that from the total (titration) charge to find how much is left for the inner layer. Electrophoresis is one of these (a double layer with only a Stern layer as the counterlayer will not move in an electric field), to which we shall come back in sec. 6. Electrokinetic potentials are difficult to meas- ure for systems on the verge of instability, but the order of magnitude is at least a factor of ten lower than the ca. 400 mV assumed for ψ° in DLVO theory. This means that in (3) it is no longer allowed to replace the hyper- bolic tangent by its high potential limit (unity), rather tanhy must rather be replaced by the first term of its series expansion, tanhy ≈ y. Consequently for the coagu- lation concentration we obtain instead of (3), ψ( ) =c const A z 'c d 4 2 2 (8) So, for low diffuse double layer potentials ψd can be obtained from stability measurements. The inference is that the interpretation of the Schulze-Hardy rule is more complicated than expected by DLVO. An exact z-6 power would require ψd to scale as z-4. More pertinent to the present theme is the finding that ψd depends on the nature of the counterion in the Stern layer, hence the Hofmeister effect is automatically incorporated. It is a small step to consider again charges instead of poten- tials, by moving further and use (6) to compute σd, subtract that from σ° (from titration) to find the Stern charge σi. In many situations the charge in the dif- fuse part is only a minor fraction of the total. In those cases one does not have to estimate σd very accurately but still find for σi an acceptably correct value. Apply- ing the Frumkin-Fowler-Guggenheim adsorption iso- therm equation (a modified Langmuir isotherm) it is even possible to estimate the specific ionic Gibbs energy of adsorption of the alkali cations on silver iodide.31 The ranges are 2.3 - 3.6 kT, 2.5 - 4 kT and 2.9 - 4.2 kT for Li+, K+ and Rb+, respectively. (The uncertainty margins are mainly caused by the uncertainties in the assessment of ψi). That is about as a far as we can get with classical means for a well-studied model system. Similar exercises have also been carried out for other systems, like oxides. The general message is that for a proper handling of interfacial potentials measuring absolute values for the surface potentials are not needed and that there are advantages in considering charges instead. Introduction of a Stern layer has also an important advantage when considering the interaction dynamics. Recall that, according to DLVO theory, interaction at constant (surface) potential required discharging of the particles upon encounter. For negatively charged AgI sols that would mean desorption of adsorbed I- ions and transporting them through the narrowing gap between the approaching surfaces toward the open space further away. This does not look like a simple and fast process. On the other hand, if there are two Stern layers between the surfaces, discharging can be rapidly realized by the formation of ion pairs between surface charges and the close-by counterions. This, somewhat extended, review of the AgI work illustrates how far one can get nowadays with the simple Gouy-Stern picture without worrying about the immeas- 89Interfacial Potentials: Measuring the Immeasurable? urability of ψ°. It is questionable whether more advanced models would work better because of the inherent imperfections of solids such as the reproducibility of the interface, and the homogeneity of the surface charge dis- tribution. 6. ELECTROKINETIC POTENTIALS. Having dealt in some detail with the (im)measur- ability of various static potentials it is appropriate to also consider the domain of electrokinetic, or zeta (ζ) poten- tials. The main reason for that decision is their outstand- ing practical relevance for colloid stability and hence for much industrial process control. As over the years the instrumentation has become quite sophisticated, measuring them is nowadays more or less routine, but instead now the interpretation is the issue. In particular the question is how electrokinetic potentials match into the static pictures of figure 6? Anticipating the discus- sion, we shall show that in many practical cases identi- fying the slip plane with the (oHp) plane where the dif- fuse part of the double layer starts appears to be a very acceptable approximation. This identification has no sci- entific background, but it adds to the accessibility of ψd. Electrokinetic potentials are very different from stat- ic potentials. They do not relate to equilibrium situations but to stationary state processes, involving tangential shear parallel to the surface. It leads to slip between a thin liquid layer, close to the surface, that remains stag- nant and a mobile layer in which charge transport takes place. The shearing motion creates a potential difference between the tangentially mobile and the tangentially stagnant parts of the liquid. Phenomenologically it looks as if the fluidity of the liquid decreases rapidly with distance from the (solid) surface, but for lack of simple theory about this, the fluidity profile is usually replaced by a step function with a slip plane, where the fluid- ity increases jump wise from zero to its bulk value. The double layer potential at the slip plane is identified as ζ. Of the many electrokinetic phenomena we shall emphasize electrophoresis, perhaps the most famil- iar illustration, and indeed suitable to discuss how ζ-potentials fit into the picture. For a more extended review see Ref. 32. Many equations have been derived for converting electrophoretic mobilities into ζ-potentials, depending on the nature, size and shapes of the parti- cles and their conductivities. Here, we shall consider two limiting cases: spherical, non-conducting particles for high and low κa. The equation for converting the elec- trophoretic mobility u into a zeta potential depends on κa where κ is the reciprocal Debye length and a the radi- us of the (spherical) particle. For the κa >> 1 limit the Helmholtz-Smoluchovski relation is ε εζ η =u 0 (κa >> 1) (9) whereas for κa << 1 the Hückel-Onsager equation applies, which reads ε εζ η =u 2 3 0 (κa << 1) (10) Equation (9) is characteristic for dilute sols of small particles whereas (10) rather represents macroscopically flat surfaces, where the flow takes mainly place parallel to the surface. Equations also exist for intermediate κa cases. We shall not worry here about these and neither about surface roughness problems, although these are very relevant for daily practice. Equations (9) and (10) are straightforward: a mobil- ity is measured and the ζ-potential follows directly, by force of habit. This procedure looks as a direct measure- ment of a potential. However, as stated in sec. 1, things are not as they like. Rather a charge is measured, which is converted into a potential. For example, in the deri- vation of (10) this step involves the writing of the total charge Q as 4πaεoεζ. This is a decent equation of electro- statics where ζ stands for the potential at the surface of a charged sphere with radius a. However, electrostatics does not consider something like a slip plane or stag- nant layer; hence the border between the moving par- ticle and the stationary fluid is taken to be identical to the borderline between the solid and liquid, demanding that ζ and ψ° coincide. A colloid scientist would there- fore rather say that a is the particle radius, inclusive the stagnant layer. A similar argument can be presented for the derivation of (9). In this case the Poisson equation is used, which relates the space charge density to the sec- ond derivative of the potential, similar as in the deriva- tion of (1). For all electrokinetic phenomena the question is what determines the existence of a slip plane. By what factors is it determined? It is imaginable that some of the fluid molecules, in close or immediate contact with the surface could remain somewhat adhered to that sur- face, and it could also be imagined that upon increasing distance from the surface the fluidity would gradually increase with increasing distance. As stated, the basic electrokinetic assumption is that for practical purposes the gradual transition from stagnant to mobile may be 90 Johannes (Hans) Lyklema replaced by a step function. Such a step function was also assumed for the double layer potential to introduce ψd. Given the very different origins for assuming a step function in the potential and in the fluidity there is no a priori reason for assuming the outer Helmholtz plane and the slip plane to be identical. Or, for that matter, assuming ψd and ζ to be identical. Nevertheless, experi- ence has shown that often the two are so close that they are experimentally indistinguishable. Perhaps the most cogent argument for identifying the oHp and the slip plane is in the very meaning of the slip plane: it separates the electrokinetically active charg- es from those inactive. The former category coincides with the diffuse parts of the double layer, the latter with the Stern part, just as is the case for the oHp. Ions in the Stern layer are not electrokinetically active (a particle with a Stern layer around it will not move in an electric field), although the ions there are not immobile (they do contribute to the surface conductivity). So, the oHp and the slip plane have in common that for both the outer side is the diffuse part of the double layer. Hence, for both can we apply equations (6) and (7) to relate charge and potential of the diffuse part. In order to better understand the slip process, in Figure 7 for some amphoteric model systems the elec- trokinetic and surface charge are plotted. These are all experimental data, the only difference with more classi- cal measurements is that charges are plotted instead of potentials. This makes the role of the position of the slip plane better visible. The skewed straight line (σek = σ°) would apply if the thickness of the stagnant layer would be zero (slip plane coincides with solid-liquid boundary), or if the diffuse layer is so extended that no measurable amounts of charge are detectable between the surface and the slip plane. This limit is always reached for very low sur- face charge, say about 1 μC cm-2. But when the surface charge becomes a bit higher, the electrokinetic charge always falls far below that of the surface charge, mean- ing that a relatively large fraction of the countercharge resides within the stagnant layer. Electrokinetic charges are not very high. About 2.5 μC cm-2 appears the maxi- mum that can be reached, independent of the nature of the surface. This charge can be compared with the sur- face charge which can become 10 times as high, espe- cially for oxide surfaces. So, only a minor fraction of the counterlayer is electrokinetically active. The same may be stated about the fraction of the countercharge that is accounted for by the diffuse parts of the double layer. Also in this respect there is a definite analogy between the diffuse part and the electrokinetically active double layer part. A second striking point is that the inf luence of indifferent electrolyte is not as strong as when the data would have been presented in terms of potentials. For iron oxide the 10-2 and 10-3 curves even coincide. It would be interesting if more data of this kind would become available to generalize this observation. Given the present query about the nature of the slip process this figure contains some interesting pieces of information. Consider first the situation with the iron oxide (FeOOH), which is typical in that it is rather sym- metrical with respect to the point of zero charge. This important observation means that the stagnant layer is the same to the right (surface negative) as to the left (surface positive) of the point of zero charge; it does not depend on the sign of the charge. Stagnant layers must also be present around the zero point of charge, even though there we cannot electrokinetically measure them. We conclude that the formation of macroscopically stag- nant layers is a hydrodynamic phenomenon; it does not have an electrical origin and the double layer is there only to measure and quantify its existence. For rutile (TiO2) the situation is less symmetrical but here the presence of a stagnant layer on the uncharged surface is beyond discussion. Stagnant layers are not very different between hydrophilic and hydrophobic surfaces: compare the hydrophobic AgI with the hydrophilic oxides. This observation means that attraction between water in the first adjacent rows and the surface does not lead to thicker stagnant layers. Neither is there a thin empty layer between the AgI surface and the first adjacent bulk water along which the liquid could slide away (of course, sliding might occur if the interfacial layer would contain microscopic air bubbles). Figure 7. Comparison between electrokinetic (σd) and surface charges (σ°) for a number of amphoteric model colloids. Taken from Ref. 32 with permission. Copyright © 1995 Academic Press. Published by Elsevier. 91Interfacial Potentials: Measuring the Immeasurable? The collected inference on the slip process is that stagnancy results from the mutual repulsion between adjacent fluid molecules. This repulsion gives rise to a layer-like ordering in the fluid: the density distribution exhibits a few maxima and minima, petering out rapidly with distance. As this mutual stacking of the water mol- ecules holds for all liquids in contact with solid surfaces the phenomenon is general. One must therefore expect that also for non-aqueous systems such a process with slip occurs. The general conclusion is that the observations made are sufficiently relevant to look for additional informa- tion on well-defined systems 7. POTENTIAL DANGER? In May 2014 a conference on electrokinetic phe- nomena took place in Gent (Belgium). At that occasion a problem surfaced with the measurement of electro- kinetic potentials. As this issue has its ramifications it deserves paying attention. The problem had to do with the sophistication of instrumentation that went so far that the very truth was huddled in uncertainties about the software. At the beginning of electrokinetic investigations, say about a century ago, measuring electrophoretic mobili- ties was very laborious. Stable sols had to be made, thin homogeneous glass tubes blown, arrangements made to view the motion of individual particles ultramicroscopi- cally, and exhausting velocity measurements carried out on different levels inside the capillary. Inconvenient as such measurements were, they had at least the advan- tage that the individual behaviour of the particles could be directly observed. With time novel techniques were developed and marketed; in the first place those based on laser-Doppler techniques. More recently very power- ful instrumentation was developed on the basis of mod- ern techniques that could, hand-waivingly summarized as electroacoustics.32 Very sophisticated instruments based on these principles are now commercially availa- ble. They allow fast electrokinetic and rheological studies of concentrated systems in a variety of media. Industrial labs are willing to pay prices of the order of 105 € apiece. The challenge is that the road from primary data to the desired end product is long and replete with problems. The ways in which these problems are solved depend on the different providers and is so deeply hidden in the software that the average user will not notice them. During the conference mentioned above, a discus- sion developed when measurements of the electrolyte concentration dependency of silica sols in alcohol, meas- ured by two electroacoustic apparatus of differing prov- enance, did give different results for the zeta potentials.33 (The difference was smaller if electrokinetic charges were considered instead of potentials). The roots of this dif- ference must be attributed to differences in the software and in the assumptions made in developing it. At the time of the meeting representatives of both industries were present but they were not available to explain how their software worked on grounds of propriety argu- ments. Although the dispute did not assume Faustian dimensions it is mandatory to remain alert on the possi- bility that such issues may develop more easily than with less sophisticated apparatus. Anticipating and dealing with them involves also an ethical aspect. 8. STILL OTHER POTENTIALS Or should we say “still other ways to measure (inter- facial) potentials”? More recently, increasingly sophisti- cated techniques have been developed for “measuring” surface potentials and more will become available. Do these attribute really new insights or do they just reflect a repetition of techniques already considered? We shall briefly discuss two of those developments, anticipated in two of the quotes with which we started this discus- sion.4,5 8.1 Surface Potentials from AFM and Related Techniques To this category belong the several techniques by which properties of a surface are measured by probing it with a probe, measuring the force of interaction as a function of distance and position. When the force that the probe experiences as a function of distance obeys classical laws of colloid interaction (say, an exponen- tial decay with distance) extrapolation to zero distance yields a “surface potential”. However, this is not our (immeasurable) ψ° but ψd because the exponential decay only applies to the interaction of diffuse double layers. So, the immeasurability is not affected although measur- ing ψd has its virtues. 8.2 Surface Potentials from Non-Linear Optics This is a new development with a physical basis, which offers promised additional information on meas- uring interfacial potentials. The techniques involve mod- ern developments of surface spectroscopy and go under acronyms like SHG (second harmonic generation), and 92 Johannes (Hans) Lyklema SFG (sum frequency generation).34-36 For our present purpose the challenging new element is that molecules can be studied that are located under non-centrosym- metry conditions, implying that in a two-phase system molecules at the interface are selectively measurable, in fact through their molecular susceptibilities. Charges at the interface lead to an additional dipole contribution, which is measurable and can be analyzed to produce an interfacial potential. Which potential? It is obvious that, as far as our present theme con- cerns, the added value refers to the Stern part of the double layer, not to the diffuse part. This is immediately seen in figure 6, where the high field strengths (slopes of the curves) are almost exclusively in the Stern part. In fact, the habit of dividing the counterlayer in a dif- fuse and a non-diffuse part tacitly presumed ideal- ity, and hence centrosymmetry, of all ions in the diffuse part. More arguments can be given, such as the fact that the charges in the Stern part (figure 7) are too low for dielectric saturation. All of that is a happy coincidence because the Stern part is much less understood than the, relatively dull, diffuse part. In practice, so far only very few elaborations have been given leading to values of a potential, be it ψ° or ψi or still something else. These elaborations are some- what suspicious because they contain hyperbolic func- tions, typical for the diffuse part only. The conclusion is that here is a field awaiting systematic elaboration from a team containing physicists and colloid scientists. 9. CONCLUSIONS The issue of surface potential measurement is con- sidered in a historical context. The absolute value of the potential difference between two condensed phases is and remains principally immeasurable. Changes in it as a function of the composition of one of the phases are sometimes accessible, with Nernst’s law as a typical illus- tration. Immeasurability does not mean that no theories can be made for them. Many of these attempts belong to the domain of model approaches to assess interfacial potential jumps χ. Avoiding these immeasurabilities, a variety of other interfacial potentials have been intro- duced. These are discussed with a critical evaluation of the steps and assumptions to be made in order to assess them. Several interfacial systems are more transparently analyzed if interpreted in terms of charges rather than of potential. It is recommended to change the notion of “potential-determining ions” into “charge-determining ions”. REFERENCES 1. E. Lange, F.O. Koenig, Elektrochemie der Phasen- grenzen, in Handbuchder Experimentalphysik, (Eds. W. Wien and F. Harms), Akademische Verlagsgesells- chaft mbH, Dresden, 1933, part VII, 265. 2. S. Trasatti, Pure Appl.Chem. 1986, 58, 955. (IUPAC recommendation on the absolute electrode potential). 3. Undisclosed author of a paper that the present author was asked to review. 4. H. Yamada, T. Fukuma, K. Umeda, K. Kobayashi, K. Matsushige, Appl. Surf. Sci. 2002, 188, 391. 5. One of the main themes of a workshop “Surface Potentials - Facts, Findings and Fantasies”, organized in Lausanne (Switzerland) by S. Roke, Sept. 2015. 6. D. H. Everett, Pure Appl. Chem. 1972, 31, 579 Defini- tions, Terminology and Symbols in Colloid and Sur- face Chemistry. Part I. (IUPAC report). 7. R. Parsons, Pure Appl. Chem. 1974, 37, 501. Manual of Symbols and Terminology for Physico-Chemi- cal Quantities and Units. App III Electrochemical Nomenclature. (IUPAC report). 8. I. Mills, T. Cvits, K. Homann, N. Kallay, K. Kuchitsu, Quantities, Units and Symbols in Physical Chemistry, (IUPAC report), Blackwell, Oxford, 1988. 9. J. Lyklema, Pure Appl. Chem. 1991, 63, 895. (IUPAC recommendation on the measurement and interpre- tation of Interfacial Electrochemical Data in Aqueous Disperse Systems). 10. A.V. Delgado, F. González-Caballero, R.J. Hunter, L.K. Koopal, J. Lyklema, J. Coll. Interface Sci. 2007, 309, 194. (IUPAC report on the Measurement and Interpretation of electrokinetic phenomena). 11. G. Gouy, Comptes Rendus 1909, 149, 654; J. Phys.(4) 1910, 9, 457; Ann.Phys. (9) 1917, 7, 129 12. D.L. Chapman, Phil. Mag. 1913, 25, 475. 13. H.B.G. Casimir, Tweede Symposium over sterke Ele- ktrolyten en de ElektrischeDubbellaag, Dutch Chem. Soc., 1944, 6. 14. F.G. Donnan, A.B. Harris, J. Chem. Soc. 1911, 99, 1554; F.G. Donnan, Z. Elektrochem. 1911, 17, 572; F.G. Donnan, Chem. Rev. 1924, 1, 73. 15. O. Stern, Z. Elektrochem. 1924, 30, 508. 16. P. Debye. E. Hückel, Physik. Z. 1923, 24, 49, 185 and 305, 1924, 25, 97. 17. H. Freundlich, Kapillarchemie: eine Darstellung der Chemie der Kolloide und verwandter Gebiete. Akad. Verlagsgesellschaft, Leipzig, 1922. 18. M. von Smoluchovski in Graetz, Handbuch der Elek- trizität und des Magnetismus. Vol II, 1914, p. 393. 19. J. Lyklema, Fundamentals of Interface and Colloid Sci- ence. Vol. II Solid-Liquid Interfaces; chapter 3, Electric 93Interfacial Potentials: Measuring the Immeasurable? double layers. Academic Press. London, 1995. 20. E.A. Guggenheim, J. Phys. Chem. 1929, 33, 847. 21. E.A. Guggenheim, Thermodynamics. 4th ed, North Holland Publ. Com. Amsterdam, 1959, ch. 9. 22. E. Lange, R. Berger, Z. Elektrochem. 1930, 36, 171. 23. A. Lottermoser, W. Petersen, Z. Phys. Chem. 1928, 133, 69. 24. H. de Bruyn, Rec. Trav. Chim. 1942, 61, 21. 25. B.V. Derjaguin, L.V. Landau, Acta Physicochim. URSS, 1941, 14, 637. 26. E.J.W. Verwey, J. Th. G. Overbeek, Theory of the Sta- bility of Lyophobic Colloids. The Interaction of Sol Particles having an Electric Double Layer, Elsevier, Amsterdam, (Also available as a Dover reprint), 1948. 27. H.R. Kruyt, M.A.M. Klompé, Kolloid Beihefte 1942, 54, 484. 28. D.C. Grahame, J. Electrochem. Soc. 1951, 98, 343. 29. J.H.A. Pieper, D.A. de Vooys, J. Electroanal. Chem. 1974, 53, 243. 30. J. Lyklema, T. Golub, Croat. Chem. Acta 2007, 80, 303. 31. J. Lyklema, Adv. Coll. Interface Sci. 2013, 100-102, 1. 32. As ref. 19, but now chapter 4, Electrokinetics. 33. R.J. Kortschot, J. Lyklema, A.P. Philipse, B.H. Erné, J. Coll. Interface Sci. 2014, 423, 65. 34. K.B. Eisenthal, Chem. Rev. 2006, 106, 1462. 35. E.C.Y. Yan, Y. Liu, K.B. Eisenthal, J. Phys. Chem.B 1998, 102, 6331. 36. S. Roke, Chem. Phys. Chem. 2009, 10, 1380.