The Debye length

We can easily get some idea of what this theory predicts by looking at the limit of low potentials, that is where Yq 1 (which for 1 1 electrolytes corresponds to /o 25mV). In this case (6.13) can be shown to rednce to the simple resnlt 

The other important parameter entering in electric forces, the Debye length, is defined as  

Note that the ionic valency, z includes the sign of the ion charge. For example, S04 has z = -2 and Ca has z = +2. The atomic valency, on the other hand, refers to the number of possible bonds an atom can form with other atoms and is always positive. The term K (as opposed to is called the Debye-Hiickel parameter. The Debye length, is often referred to as the thickness of the double layef even though the region of varying potential is of the order of 3/k to 4/k (Hunter, 1993). The Stem layer is, in most cases, much smaller than the diffuse layer and is of the order of the counter-ion diameter. 

The Debye length decreases with increasing salt (electrolyte) concentration and type (as expressed by the ionic valency, z). 

The summation in Equation 10.11 requires knowledge of the type of electrolyte. The equation can be simplified in specific cases, e.g. for an aqueous solu- 

The summation requires that we know what type of electrolyte we have, for example  

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Assuming that the gas is electricaUy neutral over regions having dimensions larger than the Debye length, typicaUy of the order 10 m in an MHD generator, the electron and ion densities in the bulk of the gas are equal. 

An important characteristic of plasma is that the free charges move in response to an electric field or charge, so as to neutralize or decrease its effect. Reduced to its smaUest components, the plasma electrons shield positive ionic charges from the rest of the plasma. The Debye length, given by the foUowing ... 

The region of the gradual potential drop from the Helmholtz layer into the bulk of the solution is called the Gouy or diffuse layer (29,30). The Gouy layer has similar characteristics to the ion atmosphere from electrolyte theory. This layer has an almost exponential decay of potential with increasing distance. The thickness of the diffuse layer may be approximated by the Debye length of the electrolyte. 

In most electrochemical systems, the double layer is very thin (1—10 nm). The thickness is characterized by the debye length, X,... 

Selecting the values of the parameters for the calculations we have in mind a 1 1 aqueous 1 m solution at a room temperature for which the Debye length is 0.3 nm. We assume that the non-local term has the same characteristic length, leading to b=. For the adsorption potential parameter h we select its value so that it has a similar value to the other contributions to the Hamiltonian. To illustrate, a wall potential with h = 1 corresponds to a square well 0.1 nm wide and 3.0 kT high or, conversely, a 3.0 nm wide square well of height 1.0 kT. 

In concentrated NaOH solutions, however, the deviations of the experimental data from the Parsons-Zobel plot are quite noticeable.72 These deviations can be used290 to find the derivative of the chemical potential of a single ion with respect to both the concentration of the given ion and the concentration of the ion of opposite sign. However, in concentrated electrolyte solutions, the deviations of the Parsons-Zobel plot can be caused by other effects,126 279"284 e.g., interferences between the solvent structure and the Debye length. Thus various effects may compensate each other for distances of molecular dimensions, and the Parsons-Zobel plot can appear more straight than it could be for an ideally flat interface. 

The Debye length of the electrode material can be determined from the constant B, and the sensitivity factor S from C, provided the diffusion length and the diffusion constant for minority carriers are known. 

With electrochemically studied semiconductor samples, the evaluation of t [relation (39)] would be more straightforward. AU could be increased in a well-defined way, so that the suppression of surface recombination could be expected. Provided the Debye length of the material is known, the interfacial charge-transfer rate and the surface recombination... 

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

As an example Fig. 6 shows the distribution of the ions for a potential difference of A(j) = 0(00) — 0(—00) = kT/cq between the two bulk phases. In these calculations the dielectric constant was taken as e = 80 for both phases, and the bulk concentrations of all ions were assumed to be equal. This simplifies the calculations, and the Debye length Lj), which is the same for both solutions, can be used to scale the v axis. The most important feature of these distributions is the overlap of the space-charge regions at the interface, which is clearly visible in the figure. 

The ionic atmosphere can thus be replaced by the charge at a distance of Lu = k 1 from the central ion. The quantity LD is usually termed the effective radius of the ionic atmosphere or the Debye length. The parameter k is directly related to the ionic strength I... 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

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