The Electrical Double Layer

The integral of p over all space gives the total excess charge in the solution, per unit area, and is equal in magnitude but opposite in sign to the surface charge density a  

This produces a double layer of charge, one localized on the surface of the plane and the other developed in a diffuse region extending into solution. 

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation 

Debye-Hiickel equation Debye-length Condenser capacity 

The treatment in the case of a plane charged surface and the resulting diffuse double layer is due mainly to Gouy and Qiapman. Here may be replaced by d /dx since is now only a function of distance normal to the surface. It is convenient to define the quantities y and yo as 

Any reactions that occur must occur at the electrode-solution interface and the reacting species must be brought to the electrode surface by diffusion or mass transport through stirring of the solution (convection). The ions in the bulk of the solution are not attracted to the electrodes by potential difference there is no potential gradient in the bulk of the solution. Even when we are interested in the bulk properties of the solution, we are only analyzing an extremely small amount of the solution that is no longer 

The electrical double layer is tire aiTay of eharged particles and/or oriented dipoles that exists at every material interfaee. In eleetrochernistry, such a layer reflects the ionic zones formed m tire solution to eompensate for the excess of eharge on the elecfiode ( g). A positively ehar ged eleetrode thus attr-acts a layer of negative ions (and vice versa). Smee tire interfaee must be neirfral, = Q (where q is the 

FIGURE 1-11 Schematic representation of the electrical donhle layer. IHP = inner Helmholtz plane OHP = onter Hehnoltz plane. 

The total charge of the compact and diffiise layers equals (and is opposite in sign to) the net charge on the elecfiode side. The potential-distance profile aeross tlie double- 

The eleehieal double layer resembles an ordmai-y (parallel-plate) eapaeitor. For an ideal eapaeitor, the eharge (q) is direetly proportional to the potential difference  

The capacitance of the double layer consists of combination of the capacitance of die compact layer in series with that of the difflise layer. For two capacitors in series, the total capacitance is given by 

Gouy and Chapman realised that in an electrolyte solution, the charges are free to move and are subject to thermal motion. They retained the concepts of the electrostatic theory to describe the coulombic metal-counter ion interaction but, in addition, they allowed for the random motion of the ions. The result is a diffuse layer of charge in which the concentration of counter ions is greatest next to the electrode surface and decreases progressively towards a homogeneous distribution of ions within the bulk electrolyte (Fig. 5.1b). 

However, the theory neglects the finite size of the ions, and it was Stern who postulated that ions could not approach the electrode beyond a plane of closest approach, thereby introducing in a crude way the ion size (Fig. 5.1c). Although formulated in a complex manner [1], the basis of Stern s model is a combination of the Helmholtz and Gouy-Chapman approaches. It may be noted that Fig. 5.1 also shows the potential and charge distributions resulting from the models. These will be discussed later. 

The thermodynamic theory of the ideally polarised electrode has been extensively reviewed in the past few decades [1-5], and the relationship with the ideally non-polarisable interface has been derived in an elegant treatment by Parsons [6]. The starting point in all derivations is the Gibbs-Duhem equation which defines the relationship between the extensive thermodynamic variables. For a bulk phase this has the form  

In order to use these equations it is necessary to devise a model which enables the surface excess concentration of species i in the interphase to be defined. The interphase itself is defined by enclosing it in two arbitrary planes positioned in such a way that the bulk phases extend homogeneously up to these planes. Any changes in the thermodynamic properties must occur between these planes (Fig. 5.3). If the thickness of the phase is known, then the concentration of species i can be mathematically expressed this is the Guggenheim model. An alternative approach due to Gibbs introduces a third arbitrary plane called the dividing surface which acts as a reference surface. The concentration of species i can then be expressed in terms of an excess or deficiency of component i at the reference surface with respect to its concentration in the bulk phase. 

Let us adopt the Guggenheim description for the interphase, and apply these concepts to a specific experimental cell consisting of a dropping mercury electrode (Hg ) in contact with an aqueous electrolyte (KCl) and calomel reference electrode  

As mentioned above, the existence of a charged surface layer induces an equal and opposite charge in the adjacent solution. This situation is described as the electrical double layer (EDL). The reactions controlling the charges on the surface may be modeled as described above, but the existence of the electrical field adds a complication not present in the usual aqueous reactions. 

In such charged fields, chemical potentials, activities, and equilibrium constants are different from the values they would have in the absence of the charged field. By convention, equilibrium constants are given their values at the surface, before any 

The correction factor for positive and negative ions will have opposite signs and will cancel. However, although individual surface complexation reactions (e.g., Equations (7.12) - (7.14)) are charge balanced, there is often a net change in charge of surface species (sites). For example, in Equation (7.12), Az = — 1. This is the number of charges that must be transferred from the solution to the surface, or vice versa. The ArG° of the surface complexation reaction will therefore have a correction factor of A or 

Modeling programs deal exclusively with equilibrium constants defined for the aqueous solution, so the intrinsic or surface equilibrium constants. surface obtained from the literature (e.g., Dzombak and Morel, 1990) are corrected using Equations (7.17) or (7.18). 

There remains one problem - what is the value of for a charged surface  

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IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Properties of the Electrical Double Layer at the Electrocapillary Maximum... 

The treatment may be made more detailed by supposing that the rate-determining step is actually from species O in the OHP (at potential relative to the solution) to species R similarly located. The effect is to make fi dependent on the value of 2 and hence on any changes in the electrical double layer. This type of analysis has permitted some detailed interpretations to be made of kinetic schemes for electrode reactions and also connects that subject to the general one of this chapter. 

M. J. Spamaay, The Electrical Double Layer, Pergamon, New York, 1972. 

A. L. Loeb, J. Th. G. Overbeek, and P. H. Wiersema, The Electrical Double Layer Around a Spherical Particle, MIT Press, Cambridge, MA, 1961. 

Most studies of the Kelvin effect have been made with salts—see Refs. 2-4. A complicating factor is that of the electrical double layer presumably present Knapp [3] (see also Ref. 6) gives the equation... 

Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

This interface is critically important in many applications, as well as in biological systems. For example, the movement of pollutants tln-ough the enviromnent involves a series of chemical reactions of aqueous groundwater solutions with mineral surfaces. Although the liquid-solid interface has been studied for many years, it is only recently that the tools have been developed for interrogating this interface at the atomic level. This interface is particularly complex, as the interactions of ions dissolved in solution with a surface are affected not only by the surface structure, but also by the solution chemistry and by the effects of the electrical double layer [31]. It has been found, for example, that some surface reconstructions present in UHV persist under solution, while others do not. 

Manne S, Cleveland J P, Gaub FI E, Stucky G D and Flansma P K 1994 Direct visualization of surfactant hemimicelles by force microscopy of the electrical double layer Langmuir 10 4409-13... 

Here we consider the total interaction between two charged particles in suspension, surrounded by tlieir counterions and added electrolyte. This is tire celebrated DLVO tlieory, derived independently by Derjaguin and Landau and by Verwey and Overbeek [44]. By combining tlie van der Waals interaction (equation (02.6.4)) witli tlie repulsion due to the electric double layers (equation (C2.6.lOI), we obtain... 

Attard P 1996 Electrolytes and the electric double layer Adv. Chem. Phys. 92 1-159... 

Splelman L A and Friedlander S K 1974 Role of the electrical double layer In particle deposition by convective diffusion J. Colloid. Interfaoe. Sol. 46 22-31... 

A current in an electrochemical cell due to the electrical double layer s formation. 

The 2eta potential (Fig. 8) is essentially the potential that can be measured at the surface of shear that forms if the sohd was to be moved relative to the surrounding ionic medium. Techniques for the measurement of the 2eta potentials of particles of various si2es are collectively known as electrokinetic potential measurement methods and include microelectrophoresis, streaming potential, sedimentation potential, and electro osmosis (19). A numerical value for 2eta potential from microelectrophoresis can be obtained to a first approximation from equation 2, where Tf = viscosity of the liquid, e = dielectric constant of the medium within the electrical double layer, = electrophoretic velocity, and E = electric field. 

On the electrode side of the double layer the excess charges are concentrated in the plane of the surface of the electronic conductor. On the electrolyte side of the double layer the charge distribution is quite complex. The potential drop occurs over several atomic dimensions and depends on the specific reactivity and atomic stmcture of the electrode surface and the electrolyte composition. The electrical double layer strongly influences the rate and pathway of electrode reactions. The reader is referred to several excellent discussions of the electrical double layer at the electrode—solution interface (26-28). 

Pig. 3. Representation of the electrical double layer at a metal electrode—solution interface for the case where anions occupy the inner Helmholtz plane... 

Electrically, the electrical double layer may be viewed as a capacitor with the charges separated by a distance of the order of molecular dimensions. The measured capacitance ranges from about two to several hundred microfarads per square centimeter depending on the stmcture of the double layer, the potential, and the composition of the electrode materials. Figure 4 illustrates the behavior of the capacitance and potential for a mercury electrode where the double layer capacitance is about 16 p.F/cm when cations occupy the OHP and about 38 p.F/cm when anions occupy the IHP. The behavior of other electrode materials is judged to be similar. 

Activation Processes. To be useful ia battery appHcations reactions must occur at a reasonable rate. The rate or abiUty of battery electrodes to produce current is determiaed by the kinetic processes of electrode operations, not by thermodynamics, which describes the characteristics of reactions at equihbrium when the forward and reverse reaction rates are equal. Electrochemical reaction kinetics (31—35) foUow the same general considerations as those of bulk chemical reactions. Two differences are a potential drop that exists between the electrode and the solution because of the electrical double layer at the electrode iaterface and the reaction that occurs at iaterfaces that are two-dimensional rather than ia the three-dimensional bulk. 

Fig. 7. (a) Simple battery circuit diagram where represents the capacitance of the electrical double layer at the electrode—solution interface, W depicts the Warburg impedance for diffusion processes, and R is internal resistance and (b) the corresponding Argand diagram of the behavior of impedance with frequency, for an idealized battery system, where the characteristic behavior of A, ohmic B, activation and C, diffusion or concentration (Warburg... 

Fig. 1. The structure of the electrical double layer where Q represents the solvent CD, specifically adsorbed anions 0, anions and (D, cations. The inner Helmholtz plane (IHP) is the center of specifically adsorbed ions. The outer Helmholtz plane (OHP) is the closest point of approach for solvated cations or molecules. O, the corresponding electric potential across the double layer, is also shown.

The physical separation of charge represented allows externally apphed electric field forces to act on the solution in the diffuse layer. There are two phenomena associated with the electric double layer that are relevant electrophoresis when a particle is moved by an electric field relative to the bulk and electroosmosis, sometimes called electroendosmosis, when bulk fluid migrates with respect to an immobilized charged surface. 

Two kinds of barriers are important for two-phase emulsions the electric double layer and steric repulsion from adsorbed polymers. An ionic surfactant adsorbed at the interface of an oil droplet in water orients the polar group toward the water. The counterions of the surfactant form a diffuse cloud reaching out into the continuous phase, the electric double layer. When the counterions start overlapping at the approach of two droplets, a repulsion force is experienced. The repulsion from the electric double layer is famous because it played a decisive role in the theory for colloidal stabiUty that is called DLVO, after its originators Derjaguin, Landau, Vervey, and Overbeek (14,15). The theory provided substantial progress in the understanding of colloidal stabihty, and its treatment dominated the colloid science Hterature for several decades. 

Fig. 7. The van der Waals potential between droplets is increasingly negative with reduced interdroplet distance, whereas the electric double-layer potential...

The relative value of the two potentials reveals the destabdization action of salts added to the emulsion. Addition of an electrolyte to the continuous phase causes a reduction of the electric double-layer repulsion potential, whereas the van der Waals potential remains essentially unchanged. Hence, the reduced electric double-layer potential causes a corresponding reduction of the maximum in the total potential, and at a certain concentration of electrolyte the maximum barrier height is reduced to a level at which the stabdity is lost. 

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