Effective Debye length

Figure 5-47 shows the Mott-Schottky plot of n-type and p-type semiconductor electrodes of gallium phosphide in an acidic solution. The Mott-Schottl plot can be used to estimate the flat band potential and the effective Debye length I D. . The flat band potential of p-type electrode is more anodic (positive) than that of n-type electrode this difference in the flat band potential between the two types of the same semiconductor electrode is nearly equivalent to the band gap (2.3 eV) of the semiconductor (gallium phosphide). 

The thickness of depletion and deep depletion layers may be approximated by the effective Debye length, Lo, ff, given in Eqn. 5-70 Ld, is inversely proportional to the square root of the impiuity concentration, In ordinary semiconductors... 

At constant surface charge, an increase in the Debye length implies an increase in the repulsion at any distance. However, this is not true for constant surface potential, since the function (Z/22)2 cosh2(Z/22) (for fixed Z) has a maximum at 1/2X 1.2. Consequently, an increase in the effective Debye length corresponds to an increase in repulsion only at large separations (Z > 2.47.) but to a decrease in repulsion at smaller separations. 

The situation gets more complicated when the free polymers are (like-)charged as well. Work of Israelachvili et al. [89] revealed that the addition of free polyelectrolyte mainly decreases the effective Debye length in the aqueous salt solutions, leading to a decrease in the double layer repulsion. 

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 ... 

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 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 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). 

Note that when the concentration of added salt is very low, Debye length needs to be modified by including the charge contribution of the dissociating counterions from the polyelectrolytes. Because the equilibrium interaction is used, their theory predicts that the intrinsic viscosity is independent of ion species at constant ionic strength. At very high ionic strength, the intrachain electrostatic interaction is nearly screened out, and the chains behave as neutral polymers. Aside from the tertiary effect, the intrinsic viscosity will indeed be affected by the ionic cloud distortion and thus cannot be accurately predicted by their theory. 

K = is the reciprocal Debye length and e, is the dielectric constant of the solvent. Through most of what follows we use the ideal conductor approximation, Eq. (66). Ionic effects will be considered in Section IV.D. 

In addition, the effect of ionic screening was analyzed [89] using Eqs. (67). The ionic concentrations considered (1.0 M, 0.1 M, 0.01 M) correspond to Debye lengths Ijj = 0.3,1 and 3 nm. The results for /o = 3 nm, the most dilute electrolyte, where ionic screening is least effective, are presented in Table 3. 

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]. 

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. 

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)) ... 

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