Double layer electroneutrality

We have to point out that the double-layer electroneutrality condition is here automatically fulfilled, and we have implemented symmetry with respect to the p-axis. 

Since Bis via Gauss s Law of electrodynamics proportional to the local excess free charge it follows that the term fjeV VGj is proportional to the net charge stored in the metal in region G. This net charge, however, was shown above to be zero, due to the electroneutrality of the backspillover-formed effective double layer at the metal/gas interface and thus Dfje w.Gj must also vanish. Consequently Eq. (5.47) takes the same form with Eq. (5.19) where, now, O stands for the average surface work function. The same holds for Eq. (5.18). 

FIG. 9 Schematic illustration of adsorption of poly(styrenesulfonate) on an oppositely charged surface. For an amphiphile surface in pure water or in simple electrolyte solutions, dissociation of charged groups leads to buildup of a classical double layer, (a) In the initial stage of adsorption, the polymer forms stoichiometric ion pairs and the layer becomes electroneutral, (b) At higher polyion concentrations, a process of restructuring of the adsorbed polymer builds a new double layer by additional binding of the polymer. 

When an electrode is in contact with an electrolyte, the interphase as a whole is electroneutral. However, electric double layers (EDLs) with a characteristic potential distribution are formed in the interphase because of a nonuniform distribution of the charged particles. 

Another point of importance about the film structure is the degree to which it can be permeated by various ions and molecules. It is of course essential that supporting electrolyte ions be able to penetrate the film, else the electrical double layer at the electrode/polymer interface could not be charged to potentials that drive electron transfers between the polymer and the electrode. The electroneutrality requirements of porphyrin sites as their electrical charges are changed by oxidation or reduction also could not be satisfied without electrolyte permeation. With the possible exception of the phenolic structure in Fig. 1, this level of permeability seems to be met by the ECP porphyrins. 

The assumption of electroneutrality implies that the diffuse double layer, where there is significant charge separation, is small compared to the volume of the domain, which is normally the case. Because there is no accumulation of charge and electroneutrality has been assumed, the divergence of the total current density is zero... 

This breakdown of electroneutrality in the solution, in the vicinity of the electrode, is a fundamental characteristic of the double-layer region. The next question is how far this double-layer region (interphase) extends out from the electrode into the solution. This question can be answered on the basis of analysis of the potential variation (distribution) in the double layer. 

The second step consists in affirming that there is electroneutrality across the interface, i.e., the charge on the metal is always equal and opposite to the total charge on the solution side of the interface. Before the double layer is formed, the metal is uncharged and in the solution, the charge per unit area of a lamina due to positive and negative ions is zero, i.e.,... 

After the double layer is formed, electroneutrality requires that... 

Summarizing, at equilibrium the entire ED cell is divided into the locally electro-neutral bulk solution at zero potential and the locally electroneutral bulk cat- (an-) ion-exchange membrane at ipm < 0 (> 0) potential. These bulk regions are connected via the interface (double) layers, whose width scales with the Debye length in the linear limit and contracts with the increase of nonlinearity. 

The region extending from the phase boundary out to about 3 nm is quite unlike the solution beyond. Generalizations valid elsewhere in the solution do not necessarily apply here. In this inner zone, the so-called double-layer region [9], we may encounter a violation of the electroneutrality condition (see Sect. 4.1) and large electric fields. Concentrations may be enhanced or depleted compared with the adjacent solution. 

A general relationship that will hold at all times and at all positions in the cell, except within the double layers (see Sect. 1.2), is the electroneutrality condition which reflects the fact that positive and negative charges must occur in equal numbers in any region of macroscopic dimensions. Stated mathematically, this means... 

Grahame derived an equation between a and based on the Gouy-Chapman theory. We can deduce the equation easily from the so-called electroneutrality condition. This condition demands that the total charge, i.e. the surface charge plus the charge of the ions in the whole double layer, must be zero. The total charge in the double layer is /0°° pe dx and we get [59]... 

The arrows above and the symbols below the interfaces indicate the transfer of the charge at each interface when the concentration of NaF in the sample is abruptly increased. It is possible to estimate the actual number of ions that are required to establish the potential difference at the interfaces. A typical value for the doublelayer capacitor is 10 5 F cm 2. If a potential difference of n = 100 mV is established at this interface, the double-layer capacitor must be charged by the charge Q = nCdi = 10 6 coulombs. From Faraday s law (6.3), we see that it corresponds to approximately 10 11 mol cm 2 or 1012 ions cm 2 of the electrode surface area. Thus, a finite amount of the potential determining ions is removed from the sample but this charge is replenished through the liquid junction, in order to maintain electroneutrality. 

What happens when the dimensions are furthermore reduced Initially, an enhanced diffusive mass transport would be expected. That is true, until the critical dimension is comparable to the thickness of the electrical double layer or the molecular size (a few nanometers) [7,8]. In this case, diffusive mass transport occurs mainly across the electrical double layer where the characteristics (electrical field, ion solvent interaction, viscosity, density, etc.) are different from those of the bulk solution. An important change is that the assumption of electroneutrality and lack of electromigration mass transport is not appropriate, regardless of the electrolyte concentration [9]. Therefore, there are subtle differences between the microelectrodic and nanoelectrodic behaviour. 

When an open tube with fixed charges at the tube wall is filled with an electrolyte solution, the ionic atmosphere forms an electrical double layer [31-33]. Since the double layer has a higher concentration of counterions than the bulk solution, electroneutrality requires that the bulk electrolyte outside the double layer has the same amount of excessive coions. 

Here C is the specific differential double layer capacitance. The two terms on the left side of Eq. (4) describe the capacitive and faradaic current densities at a position r at the electrode electrolyte interface. The sum of these two terms is equal to the current density due to all fluxes of charged species that flow into the double layer from the electrolyte side, z ei,z (r, z = WE), where z is the direction perpendicular to the electrode, and z = WE is at the working electrode, more precisely, at the transition from the charged double layer region to the electroneutral electrolyte. 4i,z is composed of diffusion and migration fluxes, which, in the Nernst-Planck approximation, are given by... 

Let us consider an electrode immersed in an electrolyte. A potential difference arises at the interface between the electrode and the surrounding electrolyte solution. The potential difference arises due to charge separation. If electrons leave the electrode and reduce the cations in solution, the electrode acquires a positive charge, the solution loses electroneutrality and anions would move closer to the positively charged electrode. The situation is depicted in Figure 1.11. Thus, we have a pair of positive and negative sheets, which is known as the electrical double layer. 

The number of surface sites, surface area, and structure of the electric double layer, and its surface charge and potential have to be known in order to use these programs. The law of mass action, electroneutrality, and mass balances have to be taken into consideration. 

When ions specifically adsorb at the interface, the excess surface charge density is divided into three parts, the charges in the two diffuse parts of the double layer, q and q, and the charge due to the specifically adsorbed ions, q° [25]. The electroneutrality condition of the entire interfacial region is... 

All non-stoichiometric models adopt the electrical double layer concept and disagree with the stoichiometric hypothesis of an electroneutral stationary phase they emphasize the higher adsorbophilicity of the IPR compared to that of its counter ion a surface excess of IPR ions generates a primary charged layer and a charged interface. Like-sign co-ions are repelled from the surface while IPR counter ions are attracted by the charged surface. 

This imbalance of electroneutrality and creation of field should not be confused with that arising from the presence of the electrode, which causes an anisotropy in the forces on the particles in the electrode-electrolyte interphase region. That anisotropy also produces an unbalance of electroneutrahty and an electric double layer (Chapter 6) with a field across the interface, but it occurs only within the first few tens of nanometers of the surface. 

As a whole, electric double layers are always electroneutral. Hence, if the countercharge is purely diffuse... 

Because double layers are electroneutral, effectively, only adsorption of electroneutral entitles takes place hence operationally the definition Is... 

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