Debye potential

In the previous section, we verified that the screened Debye potential (314) does indeed give a satisfactory approximation for the collective interaction in the conductance of electrolytes. However, we have not yet considered the effect of the interactions with the solvent. 

Vectorial fields M = VA ( ) and N = VAVA ( ) are respectively known as toroidal and poloidal fields. The scalar potentials and are called Debye potentials, while the vector potentials F = and G = are known as Whittaker potentials. 

The van der Waals forces between atoms or molecules are the sum of three different forces, all proportional to 1/r , where r is the distance between the atoms or molecules. The corresponding potentials are the orientation or Keesom potential, the induction or Debye potential, and the dispersion or London potential, respectively. For two dissimilar polar molecules, the van der Waals potential is... 

The SPC/E model approximates many-body effects m liquid water and corresponds to a molecular dipole moment of 2.35 Debye (D) compared to the actual dipole moment of 1.85 D for an isolated water molecule. The model reproduces the diflfiision coefficient and themiodynamics properties at ambient temperatures to within a few per cent, and the critical parameters (see below) are predicted to within 15%. The same model potential has been extended to include the interactions between ions and water by fitting the parameters to the hydration energies of small ion-water clusters. The parameters for the ion-water and water-water interactions in the SPC/E model are given in table A2.3.2. 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

Explicit calculation of this sum shows that it is the Debye screened potential... 

This has the fomi of a second virial coefficient in which the Debye screened potential has replaced the Coulomb potential. Expressions for the other excess themiodynamic properties are easily derived. 

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. 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

The total van der Waals interaction potential is obtained by simply adding the individual contributions arising from the Keesom, Debye, and London interactions. Because the radial power-law dependencies of all these interactions vary as 1 /r, the total van der Waals interaction can be expressed simply as... 

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. 

Let us now consider a pair of ions in aqueous solution from such a crystal. In the Debye-Hilckel theory it is assumed that in pure solvent, the mutual potential energy is — e2/ r, where e is the macroscopic dielectric constant of the solvent,2 until the ions come into contact with each... 

Forces Superimposed on the Coulomb Forces. The discussion has been based on the idea that, superimposed on the electrostatic forces between a pair of ions, there are rather short-range forces of other origin, which may be attractive or repulsive. Consider now what the situation will be if these forces cause the mutual potential energy to fall at short distances, below the value — e2/er that is assumed in the Debye-Hlickel theory. In Fig. 74 let the broken curve be a plot of — e2/er, while the full curve gives the actual potential energy between a certain pair of... 

Here, x and y are the dimensionless distance and potential defined by x = xr and y = e

elementary charge, T the absolute temperature, k the Boltzmann constant, and x the Debye screening parameter defined by x = (8jtne2/skT)1/2. 

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. 

For low wall potentials the Debye-Huckel linearization holds and the excess charge distribution is... 

Thermal diffusivity Temperature sensitivity Temperature difference Thickness of tube Aspect ratio, relation of Cp/Cy Fluid dielectric constant Wall zeta potential Dimensionless temperature Friction factor, Debye length Mean free path Dynamic viscosity Kinematic viscosity Bejan number Density... 

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). 

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