Debye length, calculation

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. 

As an example Fig. 6 shows the distribution of the ions for a potential difference of A(j) = 0(00) — 0(—00) = kT/cq between the two bulk phases. In these calculations the dielectric constant was taken as e = 80 for both phases, and the bulk concentrations of all ions were assumed to be equal. This simplifies the calculations, and the Debye length Lj), which is the same for both solutions, can be used to scale the v axis. The most important feature of these distributions is the overlap of the space-charge regions at the interface, which is clearly visible in the figure. 

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. 

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...

We shall now consider what happens when the film thickness is of the order of the Debye length. In such a situation, no analytical expressions can be derived and numerical calculations should be used [125]. The real situation could be even more complicated, since an ill-defined film thickness can exist, like the example in Figure 2.6. We can use the molecular theory to obtain a self-consistently determined electrostatic potential profile across the interface as was shown in Figure 2.7 (see... 

Because the inverse Debye length is calculated from the ionic surfactant concentration of the continuous phase, the only unknown parameter is the surface potential i/io this can be obtained from a fit of these expressions to the experimental data. The theoretical values of FeQx) are shown by the continuous curves in Eig. 2.5, for the three surfactant concentrations. The agreement between theory and experiment is spectacular, and as expected, the surface potential increases with the bulk surfactant concentration as a result of the adsorption equilibrium. Consequently, a higher surfactant concentration induces a larger repulsion, but is also characterized by a shorter range due to the decrease of the Debye screening length. 

On the basis of this description, a relationship between the two lengths 8 and K can be established. Different 5 values are obtained by gradually increasing the amount of micelles and fitting the force profiles. The evolution of 5 as a function of the calculated Debye length is plotted in Fig. 2.8. The thickness 5 increases linearly with The inherent coupling between depletion and doublelayer forces is reflected by this empirical linear relationship which is a consequence of the electrostatic repulsion between droplets and micelles. The thickness 5 may be conceptually defined as a distance of closer approach between droplets and micelles and thus may be empirically obtained by writing ... 

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). 

Calculate the Debye lengths for 0.1 mM, lOmM and lOOmM aqueous solutions of NaCl and MgS04, assuming that the salts are completely ionized. 

The thickness of the Donnan phase layer rD was equated to the Debye length so that wD/w = rD/r = l/r /pCs. However, experimental values of wD/w calculated from measurements of SP and SN in cellulose-KCl by means of a combination of Eq. (42), the corresponding equation for the counterion and the proper version of Eq. (37), namely SpS = w2D, did not agree very well with the aforesaid theoretical rD values 116). Recently, this difficulty has been resolved by showing 110) that Eqs. (41) and (42) become identical under the proper conditions (namely at high Cs), upon setting... 

Rh is the hydrodynamic radius of the analyte, k is the inverse of the Debye length, r is the viscosity of the separation buffer, e is the fundamental unit of charge, and ft is a function that describes the effect of the molecule (or particle) on the electric field and is defined between two limits (i) the Htickel limit,/ = 1 when k,Rh < 1 (when the hydrodynamic radius is lower than the Debye length) and (ii) the Helmholtz-Smoluchovski limit, fi= /2 when k,Rh > 10 (when the hydrodynamic radius is higher than the Debye length). Between the limits / is calculated from the following equation ... 

Figure 40. Numerically calculated impedance for ion blockage. The variation of the defect concentration leads to the transition from Warburg to a pure semicircular behavior. The impedances are normalized such that the points of highest frequency coincide. (For the mobility ratio, ratio of thickness to Debye length and charge numbers the following values are assumed = lOultJi, UA = 101, = l = -z )m...

AH rate calculations presented above have used the double-layer interaction given by Eq. [p3 as applied to approach at constant surface potential. Suppose the charge density had been held constant instead. How would this change affect the rate of deposition The answer is illustrated in Fig. 5. When the sphere and plate are many Debye lengths apart, the... 

The calculations presented in this section show that the behavior of the black films can be understood in terms of the interaction energy between planar films. However, they cannot explain why, for the same electrolyte concentration, the transition from the common to the Newton black film occurs at various pressures (for example, for Ce = 10 3 mol/dm3,p = (2.5 4- 9.8) x 104 N/m2).2 In addition, the thickness at the transition apparently does not depend on the electrolyte concentration (while the Debye length A2 does) and is larger than the upper bound 3A2 (which is obtained, when only the double layer and van der Waals interactions are present, using the approach employed to derive eq 21). In the next section, it will be shown that by accounting for the thermal fluctuations of the interfeces one can provide answers to these questions. 

The quantity k, also known as the Debye length, has the dimensions of distance and is an approximate measure of the thickness of the ionic atmosphere over which the electrostatic field of the ion extends with an appreciable strength. The k term can be calculated from the following relationship... 

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