Length, inverse Debye

The solution is implicit and consists of a pair of nonlinear algebraic equations, which can be solved numerically. In the limit of low concentrations, where the Debye inverse length k = 1/Td tends to zero, explicit results can be derived. The distribution of the ions gi(x) and of the dipoles gs(x) as a function of the separation x from the electrode show oscillations, which are caused by the packing of the particles (see Fig. 3) they die out far from the surface. 

In the non-linear differential equation Eq. (43), k is related to the inverse Debye-Hiickel length. The method briefly outlined above is implemented, e.g., in the pro-... 

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. 

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). 

Thus, 3>n is only a function of the inverse Debye screening length To, defined by... 

Here, T is the absolute temperature, e is the bulk dielectric constant of the solvent, P is the number of phosphate charges, k is the inverse of the Debye screening length, kB is the Boltzmann constant, qna is the renormalized charge, the interaction between the charges is screened Debye-Hiickel potential, and — j b is the distance between a pair of charges labeled i and /... 

Ionic screening constant for solution in region i, inverse to the Debye screening length. 

Firstly, the physical origin of the empirical parameter x is not known. As suggested by Petrache et. al [13], it might simply represent a correction due to the interlamelar salt deficit (compared to the salt concentration in reservoir). However, since the Debye-Hiickel length is inverse proportional to the square root of the ionic strength, to account for the value x = 0.2 the... 

As elsewhere, we will confine our attention to a uni-univalent electrolyte solution for the sake of simplicity. The inverse Debye screening length k is then given by... 

Table 7 Fraction of free counterions, fc (normalized to the number of chemically quater-nized monomers), the effective charge density, / (normalized to the total number of monomers), the effective charge density per main chain monomer fma the cross sectional radius of gyration RgjC) the mean concentration of counterions cc and the mean inverse Debye screening length Xb l within the volume of a cylindrical brush molecule due to condensed counterions...

In these expressions eQ and are the static and high-frequency dielectric constants of the host, vs is a (mean) velocity of sound therein, Cpe is the piezoelectric coupling constant, CQ a numerical factor that depends on the crystal structure and kD the Debye inverse screening length. 

A typical disjoining pressure variation with film thickness is shown in Figure 7a. At small thicknesses, a repulsive force is observed which can be fitted with an exponential form exp-(jch), as expected for screened electrostatic repulsion k is close to the calculated inverse Debye Huckel length in the solution ... 

This is an inverse length k is known as the Debye screening length (or double layer thickness). As demonstrated below, it gives the length scale on which the ion distribution near a surface decays to the bulk value. Table C2.6.4 gives a few numerical examples. 

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