Electric double-layer Debye screening length

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

In simulations of the full electric double layer, ionic density profiles are oscillatory in the concentration range between 1 and 3 mol/1. Surface charges are screened by free ions over a distance of several Debye lengths. 

The charge density accumulation cannot be neglected for interphase layers such as the electrical double layer. Using a nondimensionalized form of Equation 2.2, it was shown that the electroneutrality condition is a direct result of the Debye screening length, where the ratio of the Debye length Xx, to the field length L is described as follows (Eq. 2.3) (Chu, 2005) ... 

When solid sm faces are in contact with a liquid, they may become charged either because of (i) the dissociation of surface chemical groups into the liquid, or (ii) the selective adsorption of ions from the liquid onto the surface [42]. In both cases, a surface electric field is developed due to the charged surfaces, and the electric field extends over a Debye screening length into the bulk liquid. This is due to the presence of charged ions in the solution that redistribute themselves due to the electric field and form a double layer, which screens the electric field, emerging from the surfaces. 

Figure 7 shows the dimensionless bubble translation speed Ca as a function of the surfactant (anionic surfactant since the glass capillary used has positive surface charge) ionic concentration for various electric field strengths [17]. At low concentrations, given that the Debye screening length scales as the inverse of the square root of the concentration, the electric double layer thickness becomes compa-... 

The concentration and nature of the electrolyte also has a significant impact on the stability of charged colloid dispersions. This was discussed in Section 3.3.2, where the concept of electric double layers was introduced. The electric double layer results from the atmosphere of counterions around a charged colloid particle. The decay of the potential in an electric double layer is governed by the Debye screening length, which is dependent on electrolyte concentration (Eq. 3.8). In the section that follows, the stability of charged colloids is analysed in terms of the balance between the electrostatic (repulsive) forces between double layers and the (predominantly attractive) van der Waals forces. 

An electric field due to dipoles emanating from a material in a salt solution will be screened according to the Gouy-Chapman theory in an exponential decay fashion typical for the electric double layer (see Chapter 4). The decay length will be given by the Debye length ko (Eq. (4.8)). For the interaction between two parallel surfaces over a gap of width x filled with electrolyte solution, the field from surface 1 will have decayed to exp —x/ k-c) when it has reached the other surface. The dielectric response of surface 2 will be shielded the same way. This will lead to damping of the interaction that we can describe by a simple correction factor... 

Tlie lengtli is called screening due to the sufficient analogy with Debye-IIiickel s screening length (interaction between a charged surface and ions in solution in the problem of a double electrical layer). 

The final charge on the surface is balanced by counterions from the solution, establishing the so-called diffuse double layer. The solution of the non-linear Poisson-Boltzmann (PB) equation yields the density of counterions, the potential and the electric field at any point [19]. The characteristic length or thickness of the diffuse double layer is the so-called Debye length, which determines the exponential decay of the counterion density away from the surface. It can also be viewed as a screening length and depends solely on the properties of the liquid such as the concentration of the electrolyte, and not on any property of the surface. For a 1 mM solution of a 1 1 electrolyte it is approximately 10 nm [19]. 

The electrical forces of particles are due to the overlap of their (diffuse) double layers (Figure 10.4). The inner Stem layer is essentially the layer of counterions with a concentration that gradually decreases in the difruse layer. The Debye length (or screening length) is a measiue of the length (thickness) of the double layer and depends on the solvent, temperature and the ionic strength of the solution. It can be... 

---