Electrical Helmholtz model

Figure 2.12 Simple capacitor model of electrode-solution interface as a charged double layer (original Helmholtz model). Negatively charged surface. Positively charged ions are attracted to the surface, forming an electrically neutral interphase.

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 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... 

As mention above, the Helmholtz model of an interface enables us to calculate the effective dipole moment p due to the polarisation of the electric field AV/d,... 

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.

Two planes are usually associated with the double layer. The first one, the inner Helmholtz plane (IHP), passes through the centers of specifically adsorbed ions (compact layer in the Helmholtz model), or is simply located just behind the layer of adsorbed water. The second plane is called the outer Helmholtz plane (OHP) and passes through the centers of the hydrated ions that are in contact with the metal surface. The electric potentials linked to the IHP and OHP are usually written as 4 2 and 4f, respectively The diffuse layer develops outside the OHP. The concentration of cations in the diffuse layer decreases exponentially vs. the distance from the electrode surface. The hydrated ions in the solution are most often octahedral complexes however, in Fig. 1.1.2. they are shown as tetrahedral structures for simplification. 

In 1879, von Helmholtz proposed that all of the counterions are lined up parallel to the charged surface at a distance of about one molecular diameter (Figure 10.5). The electrical potential decreases rapidly to zero within a very short distance from the charged surface in this model. Such a model treated the electrical doublelayer as a parallel-plate condenser, and the calculations of potential decay were based on simple capacitor equations. However, thermal motion leads to the ions being diffused in the vicinity of the surface, and this was not taken into account in the Helmholtz model. 

In spite of the above difficulties, a simple theory [26-28] based on Helmholtz model yields a microscopic picture which is useful in understanding the role of pore size and channel length along with the electrical characteristics of this interface in electro-kinetic phenomena. Whereas the macroscopic theory based on irreversible thermodynamics does not depend on any model, the theory discussed below would be valid provided the situation conforms to the model. Both approaches are complementary in understanding the phenomena. 

Electrical double layer EDI). Favorable electron-transfer capabilities make ionic hquids good conductive media and vahd substitutes for conventional electrolytes. Electrolytic properties of ionic hquids were studied to determine the capacitance-layer thickness relationship of the EDL by electrochemical impedance spectroscopy (EIS). EIS data combined with supporting SFG analysis indicate that the EDL formed by ionic hquids at the electrode-ioitic liquid interface follows the Helmholtz model and corresjtonds to a Helmholtz layer of one ion thickness [35,36]. 

Various models have been proposed for the electric double layer at an electrode-electrolyte interface. Briefly explain the structure of the electric double layer starting from the Helmholtz model to the triple-layer model and then identify the key features of each model. 

Presuming the Helmholtz model is valid for the interfacial double-layer region, the interfacial electric field is given by ... 

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. 10 Schematic representation of the proposed surface model (a) the concentration and (b) the electrical potential profiles at the interface of the membrane and aqueous sample solution, x = 0 and 0 are the positions of ions in the planes of closest approach (outer Helmholtz planes) from the aqueous and membrane sides, respectively. (From Ref. 17.)... 

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 electric field or ionic term corresponds to an ideal parallel-plate capacitor, with potential drop g (ion) = qMd/4ire. Itincludes a contribution from the polarizability of the electrolyte, since the dielectric constant is included in the expression. The distance d between the layers of charge is often taken to be from the outer Helmholtz plane (distance of closest approach of ions in solution to the metal in the absence of specific adsorption) to the position of the image charge in the metal a model for the metal is required to define this position properly. The capacitance per unit area of the ideal capacitor is a constant, e/Aird, often written as Klon. The contribution to 1/C is 1 /Klon this term is much less important in the sum (larger capacitance) than the other two contributions.2... 

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 ... 

Fig. 5-8. An interfadal double layer model (triple-layer model) SS = solid surface OHP = outer Helmholtz plane inner potential tt z excess charge <2h = distance from the solid surface to the closest approach of hydrated ions (Helmluritz layer thickness) C = electric capacity.

It is noted that the molecular interaction parameter described by Eq. 52 of the improved model is a function of the surfactant concentration. Surprisingly, the dependence is rather significant (Eig. 9) and has been neglected in the conventional theories that use as a fitting parameter independent of the surfactant concentration. Obviously, the resultant force acting in the inner Helmholtz plane of the double layer is attractive and strongly influences the adsorption of the surfactants and binding of the counterions. Note that surface potential f s is the contribution due to the adsorption only, while the experimentally measured surface potential also includes the surface potential of the solvent (water). The effect of the electrical potential of the solvent on adsorption is included in the adsorption constants Ki and K2. 

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]).

Fig. 6.62. The Helmholtz-Perrin parallel-plate model, (a) A layer of ions on the OHP constitutes the entire excess charge in the solution. (b) The electrical equivalent of such a double layer is a parallel-plate condenser, (c) The corresponding variation of potential is a linear one. (Note The solvation sheaths of the ions and electrode are not shown in this diagram nor in subsequent ones.)...

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