Debye-Hiickel linearization

FIGURE 1.4 Potential distribution y x) = ze>J/ x)/kT around a positively charged plate with scaled surface potential yo = ze[j/JkT. Calculated foryo= U 2, and 4. Solid lines, exact solution (Eq. (1.37)) dashed lines, the Debye-Hiickel linearized solution (Eq. (1.25)). 

FIGURE 1.10 Scaled surface charge density scaled surface potential ya = ze>pJkT for a positively charged sphere in a symmetrical electrolyte solution of valence z for various values of ku. Solid line, exact solution (Eq. (1.86)) dashed line, Debye-Hiickel linearized solution (Eq. (1.76)). 

Figure 1.11 gives the scaled potential distribution y(r) around a positively charged spherical particle of radius a with yo = 2 in a symmetrical electrolyte solution of valence z for several values of xa. Solid lines are the exact solutions to Eq. (1.110) and dashed lines are the Debye-Hiickel linearized results (Eq. (1.72)). Note that Eq. (1.122) is in excellent agreement with the exact results. Figure 1.12 shows the plot of the equipotential lines around a sphere with jo = 2 at ka = 1 calculated from Eq. (1.121). Figures 1.13 and 1.14, respectively, are the density plots of counterions (anions) (n (r) = exp(+y(r))) and coions (cations) ( (r) = MCxp(—y(r))) around the sphere calculated from Eq. (1.121). 

In this section, we present a novel linearization method for simplifying the nonlinear Poisson-Boltzmann equation to derive an accurate analytic expression for the interaction energy between two parallel similar plates in a symmetrical electrolyte solution [13, 14]. This method is different from the usual linearization method (i.e., the Debye-Hiickel linearization approximation) in that the Poisson-Boltzmann equation in this method is linearized with respect to the deviation of the electric potential from the surface potential so that this approximation is good for small particle separations, while in the usual method, linearization is made with respect to the potential itself so that this approximation is good for low potentials. 

In the usual Debye-Hiickel linearization approximation, Eq. (9.162) is linearized with respect to y itself, namely. 

The next-order correction terms to Derjaguin s formula and HHF formula can be derived as follows [13] Consider two spherical particles 1 and 2 in an electrolyte solution, having radii oi and 02 and surface potentials i/ oi and 1/ 02, respectively, at a closest distance, H, between their surfaces (Fig. 12.2). We assume that i/ oi and i//q2 are constant, independent of H, and are small enough to apply the linear Debye-Hiickel linearization approximation. The electrostatic interaction free energy (H) of two spheres at constant surface potential in the Debye-Hlickel approximation is given by... 

Table 3 Results of Debye-Hiickel linearization of Eq. (13) and Derjaguin approximation...

The third group of the approximate models includes various improvements of the Derjaguin approximation, linearization, and approximate solutions of PB Eq. (13) for spherical particles. The first improvement on the Derjaguin approximation for the interaction energy between identical spheres was probably obtained by the Debye-Hiickel linearization and the superposition approximation,given by ... 

Derjaguin approximation in the framework of the Debye-Hiickel linearization can be expressed... 

With the Debye-Hiickel linearization the solution is simply... 

For further simplifications, one may note that for small values of the argument, it is possible to neglect higher-order terms in an exponential series and approximate exp(x) as 1 -1- x. This consideration, when appHed to the potential distributions depicted by Eqs. 12a and 12b, forms the basis of the Debye-Hiickel linearization principle [2], which effectively linearizes the pertinent exponential variation of ionic charge distribution for small values of e4>/k T. Under such approximations, Eq. 13 can be simplified as... 

The mathematical descriptions outlined so far for analyzing cases 1 and 2 implicitly assume the validity of the celebrated Debye-Hiickel linearization principle, as described earlier. However, for higher pH values (such as pH > 8), the surface potential may be such that the value of eij/lksT cannot be taken to be small at all locations. A limiting condition that constraints the applicability of the Debye-Hiickel approximation occurs for e j/lk T 1, which, for standard temperatures. 

The potential near a protein in salt solution. Consider a protein sphere with a radius of 18 A, tind charge Q = -lOe, in an aqueous solution of 0.05 M NaCl at 25 °C. Consider the small ions as point charges and use the Debye-Hiickel linear approximation of the Poisson-Boltzmann equation. 

The electric double layers formed at the microchannel walls do not overlap, and the Debye—Hiickel linearization principle remains as vahd. 

M. A. Lampert and R. S. Crandall, Chem. Phys. Lett., 68,473 (1979). Spherical, Nonlinear Poisson-Boltzmann Theory and Its Debye-Hiickel Linearization. 

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