Debye-Hiickel solution

Using the Debye-Hiickel solution for the potential distribution in the double layer enables Eq. (7.2.13) to be integrated to give a result that may be written in the form... 

This equation is known as the Poisson-Boltzmann equation. By solving this equation, we may know the potential distribution and the ion distribution around the ion concerned. The Debye-Hiickel solution is based on the approximation of the exponential in Eq..(7) by taking the leading terms (the first two terms) in the series expansion and applying the electroneutrality condition for bulk ionic solution (Zy=i = 0) ... 

The essential criticism of Eq. 18 is that it provides us with absolutely no information regarding the flow field within the double layer. We can capture this information by incorporating the Debye-Hiickel solution, Eq. 11, into Eq. 17 above which yields... 

By comparing the actual charge density to the apparent charge density in Fq. [94], we can define the fraction of surface charge neutralized by counterions, /iieutj according to the apparent Debye-Hiickel solution, as... 

Bulk Model Nonlinearizing the Debye—Hiickel Solution... 

Comparison with the standard Debye-Hiickel solution obtained by neglecting the denominator on the left-hand side of Eq. [168] ... 

Far from the surface the ADH potential is small, and only the first term in the series expansion of tanh need be retained. The NLDH potential then reduces to the Debye-Hiickel solution but with a reduced charge density given by Eq. [172]. It is convenient to write Eq. [174] in an alternative form equivalent to Eq. [22] for a charged plane ... 

The Debye-Hiickel solution for a cylindrical capillary follows along lines similar to those of the previous section except that we again extract the solution valid at r = 0. For this system, Eq. [258], which gives the Debye constant Ko at the position of the potential gauge (r = 0), simplifies to... 

While no exact analytical solution to Eq. [292] is available, approximate nonlinear expressions corresponding to the PGC and NLDH solutions as well as the weak-field Debye-Hiickel solution are given below. 

For low-polyelectrolyte concentrations, R 00 and Eqs. [304] and [305] give the standard bulk Debye-Hiickel solution for the potential near an ion of radius a... 

Figure 21 represents the interaction of two charged particles. The appropriate equations for the NLDH potential and pressure were derived previously and are given by Eqs. [196], [197], [201], [202], and [209]. The development here is similar to that used previously for two interacting cylinders. To implement the general expressions to treat two interacting spheres, we use the Debye-Hiickel solution [307] to put... 

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