Debye screening lengths inverse

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. 

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...

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. 

Free ions are usually not included explicitly in the simulations, but their overall effects on monomer-monomer and monomer-particle interactions are described via the dependence of the inverse Debye screening length (m" ) on the electrolyte concentration according to... 

Here, B is an adjustable amplitude of the electrostatic repulsion in thermal energy units, k the inverse Debye screening length, and m the buoyancy-corrected mass of the probe. The depletion contribution is twice the potential between two equally sized spheres according to the Derjaguin approximation. 

Figure 7 shows the dimensionless bubble translation speed Ca as a function of the surfactant (anionic surfactant since the glass capillary used has positive surface charge) ionic concentration for various electric field strengths [17]. At low concentrations, given that the Debye screening length scales as the inverse of the square root of the concentration, the electric double layer thickness becomes compa-... 

Fig. 3 Illustration of the adsorption behavior of a polyelectrolyte on planar, cylindrical, and spherical surfaces with the corresponding dependencies of the critical surface charge densities on the inverse Debye screening length k [48]...

The exact solution for the Debye-Hiickel potential predicts a linear dependence of the critical colloid charge density on the inverse Debye screening length for Kfl 1 (Fig. 5). This is different from the predicted dependence IcTcI based on... 

Fig. 5 Critical charge density luj as function of the inverse Debye screening length. The black lines are the analytical approximations (25) for Ka 1, respectively. No adsorption is obtained in the area on the right of the curve [58]...

The Debye screening length, k, is inversely proportional to the square root of I ... 

For 1 1 electrolytes, this assumption implies that the surface potential is less than 25 mV. The x-parameter inside the exponent is the inverse of the Debye screening length it is a measure of the thickness of the double layer. When x = k, then the potential has been decreased to 1/e of its original value. The Debye length is, in Figure 10.22, between 1 and 300 nm. Notice that the Debye length and the potential depend a lot (actually decrease) with increasing electrolyte concentration. Thus, the... 

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