51. Analytic extensions of the Debye–Hückel approximation to the Poisson–Boltzmann equation
- Author
-
Chien C. Chang, Chih-Yu Kuo, and Chang Yi Wang
- Subjects
General Mathematics ,Analytic model ,Mathematical analysis ,General Engineering ,Charge density ,Perturbation (astronomy) ,Poisson–Boltzmann equation ,Poincaré–Lindstedt method ,symbols.namesake ,Nonlinear system ,Debye–Hückel equation ,symbols ,Perturbation method ,Mathematics - Abstract
The Poisson–Boltzmann equation (P-B) is used as an analytic model in a wide variety of fields in chemistry and physics, because it describes the charge distribution in a solute. Being highly nonlinear, there are only a few known solutions for simple boundary geometries and, beyond, iterative numerical schemes are often employed. This study, on the other hand, presents a systematic perturbation solution of the P-B using a non-dimensional electrokinetic–thermal energy ratio λ which, when it approaches zero, reduces the P-B to the Debye–Huckel approximation. Perturbation-series solutions are obtained for five basic examples, and lead to the surprising result that, even when λ is as large as 3 or larger, the perturbation solution is very accurate with only a few terms included in the series. This is because the perturbation analysis generates very rapidly vanishing coefficients at higher-order approximations. This result has the important implication that the perturbation method presented in this study could be applied quite generally for investigating more complicated problems.
- Published
- 2010