Helmholtz model, electrical double-layer

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]. 

Fig. 4.1 Structure of the electric double layer and electric potential distribution at (A) a metal-electrolyte solution interface, (B) a semiconductor-electrolyte solution interface and (C) an interface of two immiscible electrolyte solutions (ITIES) in the absence of specific adsorption. The region between the electrode and the outer Helmholtz plane (OHP, at the distance jc2 from the electrode) contains a layer of oriented solvent molecules while in the Verwey and Niessen model of ITIES (C) this layer is absent...

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

Surface complexation models for the oxide-electrolyte interface are reviewed two models for surface hydrolysis reactions are considered (diprotic surface groups and monoprotic surface groups) and four models for the electric double layer (Helmholtz,... 

For a long time, the electric double layer was compared to a capacitor with two plates, one of which was the charged metal and the other, the ions in the solution. In the absence of specific adsorption, the two plates were viewed as separated only by a layer of solvent. This model was later modified by Stem, who took into account the existence of the diffuse layer. He combined both concepts, postulating that the double layer consists of a rigid part called the inner—or Helmholtz—layer, and a diffuse layer of ions extending from the outer Helmholtz plane into the bulk of the solution. Accordingly, the potential drop between the metal and the bulk consists of two parts ... 

Figure 7.4. Schematic model of the Electrical Double Layer (EDL) at the metal oxide-aqueous solution interface showing elements of the Gouy-Chapman-Stern-Grahame model, including specifically adsorbed cations and non-specifically adsorbed solvated anions. The zero-plane is defined by the location of surface sites, which may be protonated or deprotonated. The inner Helmholtz plane, or [i-planc, is defined by the centers of specifically adsorbed anions and cations. The outer Helmholtz plane, d-plane, or Stern plane corresponds to the beginning of the diffuse layer of counter-ions and co-ions. Cation size has been exaggerated. Estimates of the dielectric constant of water, e, are indicated for the first and second water layers nearest the interface and for bulk water (modified after [6]).

In the absence of specific adsorption of anions, the GCSG model regards the electrical double layer as two plate capacitors in series that correspond respectively, to two regions of the electrolyte adjacent to the electrode, (a) An inner compact layer of solvent molecules (one or two layers) and immobile ions attracted by Coulombic forces (Helmholtz inner plane in Fig. 2). Specific adsorption of anions at the electrode surface may occur in this region by electronic orbital coupling with the metal, (b) An outer diffuse region of coulombically attracted ions in thermal motion that complete the countercharge of the electrode. 

The interphase between an electrolyte solution and an electrode has become known as the electrical double layer. It was recognized early that the interphase behaves like a capacitor in its ability to store charge. Helmholtz therefore proposed a simple electrostatic model of the interphase based on charge separation across a constant distance as illustrated in Figure 2.12. This parallel-plate capacitor model survives principally in the use of the term double layer to describe a situation that is quite obviously far more complex. Helmholtz was unable to account for the experimentally observed potential dependence and ionic strength dependence of the capacitance. For an ideal capacitor, Q = CV, and the capacitance C is not a function of V. 

Figure 4.1 Helmholtz and Gouy-Chapman model of the electric double layer.

At the next level we also take specific adsorption of ions into account (Fig. 4.6). Specifically adsorbed ions bind tightly at a short distance. This distance characterizes the inner Helmholtz plane. In reality all models can only describe certain aspects of the electric double layer. A good model for the structure of many metallic surfaces in an aqueous medium is shown in Fig. 4.6. The metal itself is negatively charged. This can be due to an applied potential or due to the dissolution of metal cations. Often anions bind relatively strongly, and with a certain specificity, to metal surfaces. Water molecules show a distinct preferential orientation and thus a strongly reduced permittivity. They determine the inner Helmholtz plane. 

Fig. 1.10 Schematic view of the electrical double layer in agreement with the Gouy-Chapman-Stem-Grahame models. The metallic electrode has a negative net charge and the solvated cations define the inner limit of the diffuse later at the Helmholtz outer plane (OHP). There are anions adsorbed at the electrode which are located at the inner Helmholtz plane (IHP). The presence of such anions is stabilized by the corresponding images at the electrode in such a way that each adsorbed ion establishes the presence of a surface dipole at the interface...

The static - double-layer effect has been accounted for by assuming an equilibrium ionic distribution up to the positions located close to the interface in phases w and o, respectively, presumably at the corresponding outer Helmholtz plane (-> Frumkin correction) [iii], see also -> Verwey-Niessen model. Significance of the Frumkin correction was discussed critically to show that it applies only at equilibrium, that is, in the absence of faradaic current [vi]. Instead, the dynamic Levich correction should be used if the system is not at equilibrium [vi, vii]. Theoretical description of the ion transfer has remained a matter of continuing discussion. It has not been clear whether ion transfer across ITIES is better described as an activated (Butler-Volmer) process [viii], as a mass transport (Nernst-Planck) phenomenon [ix, x], or as a combination of both [xi]. Evidence has been also provided that the Frumkin correction overestimates the effect of electric double layer [xii]. Molecular dynamics (MD) computer simulations highlighted the dynamic role of the water protrusions (fingers) and friction effects [xiii, xiv], which has been further studied theoretically [xv,xvi]. 

At high ionic strength, the electric double layer is considered to be plane the so-called constant capacity model (Helmholtz model) is applied. 

Since charged particles involve all these processes, including the formation of edge charges (Equations 2.3-2.5), first, the electric properties of interfaces have to be determined. A simple way to do so is the application of a support electrolyte in high concentration. The electric double layer, in this case, behaves as a plane and, as a first approach, the Helmholtz model, that is, the constant capacitance model, can be used (Chapter 1, Section 1.3.2.1.1, Table 1.7). It is important to note that the support electrolyte has to be inert. A suitable support electrolyte (such as sodium perchlorate) does not form complexes (e.g., with chloride ions, Section 2.3) and does not cause the degradation of montmorillonite (e.g., potassium fixation in the crystal cavities). In this case, however, cations of the support electrolyte, usually sodium ions, can also neutralize the layer charges ... 

FIGURE 1.21 Model of the Helmholtz electric double layer. 

FIGURE 2-1 Helmholtz model of the electrical double layer, (a) Distribution of counterions in the vicinity of the charged surface. (b) Variation of electrical potential with distance from the charged surface. 

The spatial charge distribution in the electrical double layer is exactly what causes the electrokinetic phenomena, namely the mutual displacement of the phases in contact in an applied external electric field (electrophoresis and electroosmosis) or the charge transfer that occurs upon the mutual motion of phases (streaming and sedimentation potentials and currents). The following consideration, the simplest consistent with the Helmholtz model, establishes the relationship between the rate of the phase shift, e.g. that of electroosmosis, and the strength of the external electric field, E, directed along the surface3. 

Expression (V.25), referred to as the Helmholtz-Smoluchowski equation, relates the rate of relative phase displacement to some potential difference, Acp, within the electrical double layer. In order to understand the nature of this quantity, let us examine in detail the mutual phase displacement due to the external electric field acting parallel to the surface, taking into account the electrical double layer structure. Let us assume that the solid phase surface is stationary. Figure V-7 shows the distributions of the potential, cp(x ) (line 1), the rate of displacement of the liquid layers relative to the surface in the Helmholtz model, u(x) (line 1/), and the true distribution of the potential in the double layer (curve 2). 

D25.8 (a) There are three models of the structure of the electrical double layer. The Helmholtz model, the... 

Fig. 12. Cyclic voltammogram and model of the electrical double layer at a silver electrode surface. Arrows indicate the direct-ions of molecular dipoles in the water (smallest circles) and pyridine (largest circles, Py) molecules, the arrow head being the positive end. The cations (solvated) could he Na+ or K+, the anions (unsolvated) Cl or SOJ-. IHP and OHP designate the inner and outer Helmholtz planes, respectively, and PZC is the potential of zero charge (see text for further explanations). (Reproduced with permission from ref. 14.)...

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The substantial understanding of the structure of the electric double layer has been summarized in various reviews and books (e.g., Ref. 2, 81, 177-183). The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [3], Gouy and Chapman [5, 6], and Stern [7], and in most of the studies cited in the reviews the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g., as a simple dipolar fluid. Computer simulations can overcome this restriction and describe the solvent in a more realistic fashion. They are thus able to paint a detailed picture of the microscopic structure near a metal electrode. 

To study the effects of electrochemical properties on passive ion transport processes, we developed a model that focuses on ionic processes at membrane and channel surfaces (14). The surface compartment model (SCM) is based on a Helmholtz electrical double layer, where the enhanced concentration of counterions and the depletion of co-ions at charged surfaces is described by straight line gradients. Treatment of the electrical double layer as a compartment greatly simplifies the calculation of ion transport. 

Figure 3.4 The electrical double layer (a) according to the Helmholtz model, (b) the diffuse double layer resulting from thermal motion. Q positive charge, 9 negative charge.

Figure 26. Schematics of the electrical double layer at a solid-liquid interface, (a) the Helmholtz model, (b) the Gouy-Chapman model, and (c) the Stem model.

EDLCs store energy within the variation of potential at the electrode/electrolyte interface. This variation of potential at a surface (or interface) is known as the electric double layer or, more traditionally, the Helmholtz layer. The thickness of the double layer depends on the size of the ions and the concentration of the electrolyte. For concentrated electrolytes, the thickness is on the order of 10 A, while the double layer is 1000 A for dilute electrolytes (5). In essence, this double layer is a nanoscale model of a traditional capacitor where ions of opposite charges are stored by electrostatic attraction between charged ions and the electrode surface. EDLCs use high surface area materials as the electrode and therefore can store much more charge (higher capacitance) compared to traditional capacitors. 

The simplest model of the electrical double layer between a metal and an electrolyte is the simple capacitor visualized by Helmholtz as shown in Figure 14. The diffuse ion distribution in the liquid phase was recognized by Gouy and Chapman- to form a space charge region adjacent to the electrode surface. 

Certain model assumptions are necessary in order to reveal the surface concentration of specifically adsorbed ions in the total surface excess F,-. Usually, the ionic component of the electrical double layer (EDL) is assumed to consist of the dense part and the diffuse layer separated by the so-called outer Helmholtz plane. Only specifically adsorbing ions can penetrate into the dense layer close to the surface (e.g. iodide ions), with their electric centers located on the inner Helmholtz plane. The charge density of these specifically adsorbed ions ai is determined by their surface concentration F Namely, for single-charged anions ... 

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