Debye length electrolytes

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 the limit of large Debye lengths (low electrolyte concentrations) the roughness would not bear on capacitance, which would thus obey Eq. (59). 

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. 

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. 

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 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. 

With the proper definitions of ex and k0, this equation is applicable to the metal as well as to the electrolyte in the electrochemical interface.24 Kornyshev et al109 used this approach to calculate the capacitance of the metal-electrolyte interface. In applying Eq. (45) to the electrolyte phase, ex is the dielectric function of the solvent, x extends from 0 to oo, and x extends from L, the distance of closest approach of an ion to the metal (whose surface is at x = 0), to oo, so that kq is replaced by kIo(x — L). Here k0 is the inverse Debye length for an electrolyte with dielectric constant of unity, since the dielectric constant is being taken into account on the left side of Eq. (45). For the metal phase (x < 0) one takes ex as the dielectric function of the metal and limits the integration over x ... 

Table 2.1 Debye lengths Tor various electrolytes calculated from equation (2.28) in water...

FIGURE 7.5 FED with 10-bases DNA molecule oriented normal to the sensor surface and Debye length XD in the electrolyte with different ionic strengths (schematically). Increasing ionic strength of the electrolyte solution decreases the fraction of DNA charge that can be detected by the FED. 

Table 3.1 Debye length for an aqueous solution of a completely dissociated 1-1 electrolyte at room temperature.

Ld = 1/A is the Debye length Table 3.1 shows values for several concentrations of a 1-1 electrolyte in an aqueous solution at room temperature. The solution compatible with the boundary condition oo) = 0 has the form 4>(x) = Aexp(—kx), where the constant A is fixed by the charge balance condition ... 

Thus, the time constant td is directly related to the time necessary for ions to migrate over a distance equal to the Debye length i< 1. For example, for a 10 3 mol dm aqueous electrolyte solution with 1) 10 9 m2 s, and... 

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. 

The disjoining pressure vs. thickness isotherms of thin liquid films (TFB) were measured between hexadecane droplets stabilized by 0.1 wt% of -casein. The profiles obey classical electrostatic behavior. Figure 2.20a shows the experimentally obtained rt(/i) isotherm (dots) and the best fit using electrostatic standard equations. The Debye length was calculated from the electrolyte concentration using Eq. (2.11). The only free parameter was the surface potential, which was found to be —30 mV. It agrees fairly well with the surface potential deduced from electrophoretic measurements for jS-casein-covered particles (—30 to —36 mV). 

Let us now examine how we can obtain an estimate of /q from the measured electromobility of a colloidal particle. It turns out that we can obtain simple, analytic equations only for the cases of very large and very small particles. Thus, if a is the radius of an assumed spherical colloidal particle, then we can obtain direct relationships between electromobility and the surface potential, if either Kit > 100 or Kd < 0.1, where K" is the Debye length of the electrolyte solution. Let us first look at the case of small spheres (where Kd < 0.1), which leads to the Hiickel equation. 

---