Debye-Huckel charge densities

In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

Tanford examined the application of Debye-Huckel theory and found the theory not to be valid because the high charge density generatedby the closely spaced head groups leads to substantial charge neutralization by counter ions Alternatively, he equated the work of... 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

Begin with the Poisson equation but keep the e matrix inside the divergence operation V (eV0) = -4jrpext (see Fig. L3.24). The net electric-charge density pext at a given point depends on the magnitude of potential as in Debye-Huckel theory. As before in relation (L3.175),... 

Fig. 3.7. The Debye-Huckel model is based upon selecting one ion as a reference ion, replacing the solvent molecules by a continuous medium of dielectric constant e and the remaining ions by an excess charge density (the shading usually used in this book to represent the charge density is not indicated in this figure).

Here we treat a planar plate surface immersed in an electrolyte solution of relative permittivity e,. and Debye-Huckel parameter k. We take x- and y-axes parallel to the plate surface and the z-axis perpendicular to the plate surface with its origin at the plate surface so that the region z>0 corresponds to the solution phase (Fig. 2.1). First we assume that the surface charge density a varies in the x-direction so that a is a function of x, that is, cr = cr(x). The electric potential ij/ is thus a function of x and z. We assume that the potential i/ (x, z) satisfies the following two-dimensional linearized Poison-Boltzmann equation, namely,... 

As an illustrative example taken from Russel et al. (1989), let us consider a 0.01 molar solution of sodium chloride in contact with a surface charged at a density of 5 x 10 negative charges per square meter at room temperature, 298°K. Equation (2-52) gives /c = 3 nm. The dimensionless surface potential exfJkrtT. obtained from Eq. (2-45), is —5.21, and Eqs. (2-46) and (2-49) give respectively the exact and the Debye-Huckel approximations for the potential as a function of distance from the surface. The results are plotted in Fig. 2-13. Note that since — s/ ks T > 1, the Debye-Huckel approximation is... 

Metals, semiconductors, electrolyte solutions, and molten salts have in common the fact that they contain given or variable densities of mobile charge carriers. These carriers move to screen externally imposed or internal electrostatic fields, thus substantially affecting the physics and chemistry of such systems. The Debye-Huckel theory of screening of an ionic charge in an electrolyte solution is an example familiar to many readers. 

FIG. 5 (a, c) Surface charge density and (b, d) surface potential as functions of pH and pore size. (a. b) Linear potential-to-charge relation (Debye-Huckel) (c, d) nonlinear potential-to-charge relation (Poisson-Boltzmann). The charge regulation model used in the calculation was based on reaction set (34) with the following model parameters ... 

Fig. 8 Radial monomer density distribution Iot the charge density al( 2neP ap l(ek Tr)) = 3 and the Debye-Huckel screening parameters Kfl = 0.1,0.5,0.8,1.0, and 1.1 (from left to right) [60]...

Consider as a model for a lyophobic colloid an equilibrium system of charged particles of finite volume and suspended in an electrolyte solution. Denote by D the space taken up by the solution. This means that D is a multiply-connected domain bounded externally by the walls of the container and internally by the surfaces of the various charged particles. D is the stage of action of an electric field whose behavior is governed by the basic laws of the Debye-HUckel theory (not merely its linear approximation). The feature of that theory which is essential for our purposes is the idea that the space-charge density p of the solution is a given function of the electrostatic potential ijj so that Poisson s law reads... 

Fig. 2 Electrical part of the modulus of Gaussian curvature versus surface charge density. The dashed and dashed-dotted lines represent the results of Debye-Huckel and Poisson-Boltzmann theory, respectively. The Debye lengths and other parameters are the same as in Fig. 1. The bilayer thickness was taken to be 4 nm. The results apply if the compensating mechanical tension resides in the interlace (and has no effect on the mechanical part of the beading rigidity). (From ref. [17].)...

The charge density on the alkali metal cations follows the sequence Li > Na > K . As a consequence Li cations are the most strongly hydrated in aqueous solution (as evidenced in Table 2.1) and K" " cations the least as shown in Fig. 2.9. This observation suggests, as is the case, that the deviation from the Debye-Huckel limiting law and the ext ykd Debye-Hiickel law at higher concentrations is due to ion-solvent t. is. 

One other relevant theoretical set of work concerns the counterion distribution particularly in the dilute limit. Manning solved the Debye-HUckel equation for a single infinitely thin polyelectrolyte. He found that when a < Xb the counterions condense onto the hne polymer reducing the charge density until the charge separation becomes equal to the Bjerrum length. The details are altered when the Poisson-Boltzmann approximation is used for a cylindrical polyelectrolyte, " but the basic point of condensation occuring for A > 1 remains. In a similar vein, Oosawa proposed a two-phase model of bound and free counterions. These results are especially relevant, since many prototypical polyelectrolytes, such as DNA and NaPSS, have A 3,... 

Figure 3 The Gouy-Chapman and Dehye-Huckel surface potentials <() (Eqs. [25] and [32], respectively) for a positively charged plane with a surface charge density Oa (in eo/A ) in symmetric mono- and divalent electrolytes at concentrations of 0.01 M (solid lines) and 0.1 M (dashed lines) Debye-Hiickel potentials are given by straight lines.

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