Debye solid electrolytes

This chapter is concerned with the mechanisms of formation of the electrical double layer at a dielectric solid/electrolytic solution interface. When such a contact occurs, the solid s surface acquires a certain charge due to the dissociation of surface ionizable groups and adsorption of ions from solution. Since the whole system is assumed to be electroneutral, the solution has to bear an equal charge of opposite sign. This charge is effectively confined to a thin layer near the dividing surface, termed the diffuse double layer. Its thickness is characterized by the reciprocal Debye length. 

Solid electrolytes are not usually solutions of a conducting solute in a solvent matrix. Liquid electrolyte solutions are often sufficiently dilute (1-10 millimolar) to be described by the textbook theories of Debye-Hiickel or Onsager and oppositely charged ions are sufficiently dispersed for interaction between anions and cations to be minimized. By contrast, molten salts are very concentrated (typically 2-20 molar), ion-ion interactions are pronounced, and alternative theories such as that of Fuoss [105] are required. Polymer electrolytes typically have [repeat unit] [cation] ratios, n, in the range 8 to 30, corresponding to 0.7 to 2.5 molar for PEOn LiC104 [106], and ion clustering is an important feature of their behaviour. To account for both the ion-polymer and ion-cluster interactions, Ratner and Nitzan have developed dynamic percolation theory [107]. 

VOINOV You said the Debye length you calculate in the case of solid electrolyte is smaller than atomic dimension and consequently diffuse layer effects are irrelevant- Presumably, you use for this calculation the Debye length formula developed for the case of aqueous electrolytes. As you have stressed, in many solid electrolytes, only one species can migrate and in that case the solution of Poisson s equation is not the same as in the case of liquid electrolytes. As long as you do not have this solution and have shown it can be approximated by an exponential, and that this exponential is the same as in the case of liquid electrolytes, it seems to me difficult to calculate a Debye length in solid electrolytes. 

Debye length in solid electrolytes. The Debye length Id describes the shielding of an electric field by the charge carriers. The electrostatic potential drops to 1/e of its value within Id- In contrast to diluted media with small concentrations of ions, such as liquid electrolytes, solid electrolytes have very small Debye lengths. 

Figure 7.8 Comparison of experimental ln7 for 1 1, 2 1, and 2 2 electrolytes. The symbols indicate the experimental results, with representing HC1 (z+ = 1, z = — 1) representing SrC ( + = 2, r = — 1) and A representing ZnS04 (z+ = 2, z = -2). The lines are the Debye-Huckel predictions, with the solid line giving the prediction for (z+ = 1, z = -1) the dashed line for (z+ = 2, r = -1) and the dashed-dotted line for (z+= 2, z =-2). In (a), In 7- calculated from the limiting law [equation (7.45)] is shown graphed against I 2. In (b). In 7- calculated from the extended form [equation (7.43)] is shown graphed against 7m2.

The basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration of charge carriers is very low in semiconductors (see Section 2.4.1). For this reason and also because the permittivity of a semiconductor is limited, the semiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths of the order of 10 4-10 6cm. This layer is termed the space charge region in solid-state physics. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

When one first thinks of the electrical double layer (edl) one imagines the description conceived by the originators, Debye and Huckel [2], Gouy and Chapman [3], Verwey and Overbeek [1], of a sharp and well-defined boundary between two phases. One of the phases usually being an aqueous medium in which a strong electrolyte is dissolved to a molar concentration of cs. The other phase is usually a solid, impermeable to either the electrolyte... 

FIGURE 1.10 Scaled surface charge density scaled surface potential ya = ze>pJkT for a positively charged sphere in a symmetrical electrolyte solution of valence z for various values of ku. Solid line, exact solution (Eq. (1.86)) dashed line, Debye-Hiickel linearized solution (Eq. (1.76)). 

Figure 1.11 gives the scaled potential distribution y(r) around a positively charged spherical particle of radius a with yo = 2 in a symmetrical electrolyte solution of valence z for several values of xa. Solid lines are the exact solutions to Eq. (1.110) and dashed lines are the Debye-Hiickel linearized results (Eq. (1.72)). Note that Eq. (1.122) is in excellent agreement with the exact results. Figure 1.12 shows the plot of the equipotential lines around a sphere with jo = 2 at ka = 1 calculated from Eq. (1.121). Figures 1.13 and 1.14, respectively, are the density plots of counterions (anions) (n (r) = exp(+y(r))) and coions (cations) ( (r) = MCxp(—y(r))) around the sphere calculated from Eq. (1.121). 

FIGURE 1.16 Potential distribution y(x) around a positively charged plate with scaled surface potential >>o = ze JkT in a symmetrical electrolyte solution fory o =1,2, and 5. Solid lines are exact results (Eq. (1.37)), dashed lines are asymptotic results (Eq. (1.178)), and dotted lines are the Debye-Hiickel approximation (Eq. (1.25)). 

A plane (m) is associated with an excess concentration of elections near the physical surface of the electrode, represented by a solid line. The inner Helmholtz plane (ihp) is associated with ions that are specifically adsorbed onto the metal surface. The outer Helmholtz plane (ohp) is the plane of closest approach for solvated ions that are free to move within the electrolyte. The ions within the electrolyte near the electrode surface contribute to a diffuse region of charge. The diffuse region of charge has a characteristic Debye length. 

In the layer with decreased relative permittivity surrounding the ions, the free energy of the solvent is lower than in the absence of the electric field of the ions. The approach of the ions towards one another requires the mutual inter-penetration of the solvate spheres, i.e., the release of a certain amount of solvent from ihc solvate sphere of the ions. This process needs work, and this work appears as a repulsive force between the ions, (This effect lends stability to electrolyte solutions, for in the absence of such repulsive forces, attraction between the charges would favour the precipitation of the solid salts.) By taking into account such repulsive forces, it was possible to interpret the positive deviation of the average activity coefficients of the ions from the Debye-Hiickel limiting law (hypernetted chain equations, HNC, calculation by the Monte Carlo method [Ra 68, Ra 70],... 

The separation of Tp into ratios for the solution and solid phases is useful when solution phase activity coefficients are available (5) because it allows evaluation of variations from ideality of the clay phase alone over a range of experimental conditions. When measurements for the aqueous mixed electrolyte systems in question are not available, adequate estimates can frequently be made by Debye-Huckel equations or by various methods from measurements on two-component systems (6). 

An important quantity in solid and liquid electrolytes is the Debye length Ld, given by... 

In the twentieth century. X-ray diffraction experiments have shown that salts exist entirely in the form of ions even in the solid state. The variation in conductivity of solutions of these electrolytes with change in concentration could no longer be explained in terms of increasing degree of dissociation as the dilution was increased. The explanation was provided in 1923 by Peter Debye (1884-1966)... 

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