Poisson Equation with Trapped Charges

In the presence of traps, the injected charge separates into a distribution with a density Uj of trapped charges and n of free charges. The two distributions together determine the internal field F(x), according to the Poisson equation... 

We summarize recent work showing that condensation can be derived as a natural consequence of the Poisson-Boltzmann equation applied to an infinitely long cylindrical polyelectrolyte in the following sense Nearly all of the condensed population of counter-ions is trapped within a finite distance of the polyelectrolyte even when the system is infinitely diluted. Such behavior is familiar in the case of charged plane surfaces where the trapped ions form the Gouy double layer. The difference between the plane and the cylinder is that with the former all of the charge of the double layer is trapped, while with the latter only the condensed population is trapped. 

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