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6.3 Analytical Comparison between the PoissonBoltzmann, the Extended PoissonBoltzmann, and the DebyeHückel Model
For better comparison between the PoissonBoltzmann, the extended PoissonBoltzmann, and the DebyeHückel model we study their onedimensional analytical solutions without any charges from macromolecules or due to the sitebinding effect at the oxide surface. The surface potential
will be chosen in a way that all models exhibit the same charge at the surface and that the potential and the electric field
vanish in the limit of infinite distance away from the surface.
In the first step all equations are transformed to dimensionless units.
Reformulating the Laplace term

(6.14) 
and transforming the equations with

(6.15) 
leads to the following differential equations:

(6.16) 
for the PoissonBoltzmann model,

(6.17) 
for the extended PoissonBoltzmann model and

(6.18) 
for the DebyeHückel model.
Assuming vanishing potential and vanishing electric field
for large distances
, integrating these equations twice results in the following solutions:

(6.19) 
for the PoissonBoltzmann model [227],

(6.20) 
or via
as a function of

(6.21) 
for the extended PoissonBoltzmann model [225], and

(6.22) 
for the DebyeHückel model [226].
Unfortunately the analytical expression (6.27) is not as handy as the expressions (6.30) for the DebyeHückel model and (6.25) for the PoissonBoltzmann model. As can be seen in (6.27), only for the position as a function of the potential it is possible to write down a compact analytical expression, while for the inverse function one has to use numerical approaches. However, in the limit
the solution for the PoissonBoltzmann model is recovered [225].
In the next step we assume an equivalent surface charge
for all three models, in order to accomplish a better comparison between them. This is realized by choosing an arbitrary charge at the surface and applying Gauß's law. This way, a surface potential
related to the same surface charge can be found.
The corresponding surface potentials are:

(6.23) 
for the PoissonBoltzmann model [227],
arccosh 
(6.24) 
for the extended PoissonBoltzmann model [225], and

(6.25) 
the DebyeHückel model, respectively.
Figure 6.5:
Illustrating the different screening characteristica for the PoissonBoltzmann, the extended PoissonBoltzmann, and the DebyeHückel model. In the limit of
the extended PoissonBoltzmann model rejoins the PoissonBoltzmann model, while for increasing closest possible ion distance , which corresponds to a decreasing salt concentration, the screening is reduced and resembles for the DebyeHückel model.

Fig. 6.5 shows a comparison between the PoissonBoltzmann, the extended PoissonBoltzmann, and the DebyeHückel model. As already mentioned before, one can see that for
the extended PoissonBoltzmann model and the PoissonBoltzmann model coincide. Increasing the closest possible approach between two ions, leads to a reduction in screening and thus higher surface potential
. Furthermore, for the extended PoissonBoltzmann model equations quite well with the DebyeHückel model. This shows that the extended PoissonBoltzmann model is able to cover a wider range of screening behavior than the PoissonBoltzmann and the DebyeHückel model.
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T. Windbacher: Engineering Gate Stacks for FieldEffect Transistors