Debye charge density

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 order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

In the second approximation, Debye and Hiickel introduced the idea that the centers of the ions cannot come closer than a certain minimum distance a, which depends on ion size the ions were now treated as entities with a finite radius. The mathematical result of this assumption are charge densities Qy, which are zero for r[Pg.120]

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

So far in our revision of the Debye-Hiickel theory we have focused our attention on the truncation of Coulomb integrals due to hard sphere holes formed around the ions. The corresponding corrections have redefined the inverse Debye length k but not altered the exponential form of the charge density. Now we shall take note of the fact that the exponential form of the charge density cannot be maintained at high /c-values, since this would imply a negative coion density for small separations. Recall that in the linear theory for symmetrical primitive electrolyte models we have... 

The theory of Debye and Hiickel started from the assumption that strong electrolytes are completely dissociated into ions, which results, however, in electrical interactions between the ions in such a manner that a given ion is surrounded by a spherically symmetrical distribution of other ions mainly of opposite charges, the ionic atmosphere. The nearer to the central ions the higher will be the potential U and the charge density the limit of approach to the central ion is its radius r = a. 

At the final concentration c, the potential j)k at distance r is given not only by the potential of the ion, )°ky but also by the potentials of the surrounding ions. Debye and Hiickel assumed that a spherical ionic atmosphere of statistically prevailing ions with opposite charge forms around each ion, giving rise to the potential ipa. Thus xpk = ipk + The potential of the space charge density p is given by the Poisson equation... 

The charge distribution in the immediate vicinity of the interface will play a critical role in transferring the hybridization-induced signal to the FED. Only effects of charge-density changes that occur directly at the surface of the FED or within the order of the Debye length D from the surface can be detected as a measurable biosensor signal (see also Eq. (3)) ... 

Here z is the ion valency, nQ is the number density of added electrolyte and q is the surface charge density of the particles. Figure 3.21 clearly illustrates the sensitivity at the lower added electrolyte concentrations where the diffuse layer Debye length is equivalent to that which would be estimated for an added electrolyte concentration two orders of magnitude higher, at a volume fraction of 0.5. Clearly higher concentrations of added electrolyte are not as sensitive but the variation is still significant. 

Tanford examined the application of Debye-Huckel theory and found the theory not to be valid because the high charge density generatedby the closely spaced head groups leads to substantial charge neutralization by counter ions Alternatively, he equated the work of... 

In this expression, the dipole dipole interactions are included in the electrostatic term rather than in the van der Waals interactions as in Eq. (9.43). Of the four contributions, the electrostatic energy can be derived directly from the charge distribution. As discussed in section 9.2, information on the nonelectrostatic terms can be deduced indirectly from the charge density. The polarizability a, which occurs in the expressions for the Debye and dispersion terms of Eqs. (9.41) and (9.42), can be expressed as a functional of the density (Matsuzawa and Dixon 1994), and also obtained from the quadrupole moments of the experimental charge density distribution (see section 12.3.2). However, most frequently, empirical atom-atom pair potential functions like Eqs. (9.45) and (9.46) are used in the calculation of the nonelectrostatic contributions to the intermolecular interactions. 

The function on the right hand side of Eq. (34) consists of a series of elliptic integrals, which depend not only on the unknown electrostatic force but also on the surface charge densities, q and on the interface and protein surface, respectively, and on the inverse Debye screening length (1/K). 

Show graphically how the surface charge density varies with the surface potential for a planar surface in different Debye-length solutions. 

Note The Debye Length is the distance over which the thermal energy of the particles causes major differences in the pos and neg charge densities, hence the max distance within a plasma that a particular charged (either pos or neg) particle can be "seen . The Debye length is not constant. 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

Equation (45) provides a relationship between the surface charge density and the slope of the potential at the surface. Next, we turn to Equation (37) —the Debye-Hiickel approximation for p — to evaluate (dip/dx)0. Differentiation leads to the value... 

The Poisson equation (see Equation (11.18)) gives the fundamental differential equation for potential as a function of charge density. The Debye-Hiickel approximation may be used to express the charge density as a function of potential as in Equation (11.28) if the potential is low. Combining Equations (11.24) and (11.32) gives... 

Figure 2 shows that, for an infinite flat plate for which u0 is 4.0 and Cb is 0.01M, the surface excess (calculated) is independent of the choice of depth of the surface region provided that the latter is equal to or greater than the Debye double layer thickness (33 A. for this concentration). Note that the corresponding surface charge density is larger than the surface excess by a factor of 1.44. That expulsion of similions from the immediate vicinity of the surface is the basic source of the... 

The space charge density is g = (F/Vm)-(Nd+ -Nd-), and their characteristic width (Le., the Debye-HGckel length, which is an equilibrium property and independent of the d-mobilities) is obtained as... 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

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