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where lc is the correlation length, φp is the potential, and εp is the permittivity. In this case, the charge density in the polymer region consists of both, the charge density due to the presence of ions (ρion) and the charge density due to the polymer (ρb). The ion charge density can be written as follows: 0 zieφp ρion = zieni = zieni exp − kBT (5.4) ii where zi is the charge number, e is the elementary charge, n0i is the bulk number den- sity (concentration), kB is the Boltzmann constant, and T is the temperature. After linearizing this charge density and plugging it into the modified Poisson-Boltzmann equation, we obtain a modified Debye-Huckel equation: λ2p 1 − lc2∇2 ∇2φp = φp − λ2pρb (5.5) εp where λp is the Debye length inside the polymer region. We can then non-dimensionalize this equation using ∇ ̃ = λ ∇ and φ ̃ = eφp and obtain: Debye-Huckel equation: p pkBT 1−δ2∇ ̃2∇ ̃2φ ̃ =φ ̃ −ρ ̃ (5.6) c ppb where δc = lc and ρ ̃b = λ2peρb . Equation (5.6) will be our model for inside the polymer λp εpkBT layer of thickness L (L ̃ = L ). Outside of the polymer layer, we will use the regular λp ∇ ̃2φ ̃ =r2φ ̃ oλo (5.7) (5.8) (5.9) Both, equations (5.6) and (5.7) have analytical solutions in one dimension: φ ̃(x ̃)=c eb1x ̃ +c e−b1x ̃ +c eb2x ̃ +c e−b2x ̃ +ρ ̃ p1234b φ ̃ (x ̃) = c erλx ̃ + c e−rλx ̃ o56 √ √ 1− 1−4δc2 1+ 1−4δc2 λp 2δc2 ,b2 = 2δc2 ,andrλ = λo. Wenowneedtofindthe coefficients. We have several conditions on the displacement field and the potential whereb1 = 119PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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