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fits did not visibly change and that the trends are still the same. Furthermore, the product of the thickness and the background charge density is still roughly the same, which should be the case, since the same amount of polymer should still be adsorbed. 5.3.5 Incorporation of Born Energy In addition to taking in account the varying thickness of the polymer layer, we need to also think about taking in account solvent effects, since the dielectric constant and solvent environment for the electrolyte in domain I can differ quite significantly from that in the bulk electrolyte in domain II. One simple approach for solvent effects is to add the Born energy into the Poisson-Boltzmann equation [11, 111]. Since we will assume that the Born energy is only a function of the bulk electrolyte concentration, it is independent of the potential profile and hence is simply a constant in the differential equation. As such, we can simply replace the constant ρ ̃b in equation 5.8 with a constant (γ ̃) that not only takes in account the constant background charge density, but also the Born energy. γ ̃=ρ ̃b−E ̃Born=λ2peρb − λ2pe4c0 1−1zi3si (5.45) εpkBT 8πεp (kBT)2 εp εo i ai where si is each ion’s stoichiometric coefficient and ai is the ion’s Born radius. The Born radii used in this work are 1.95 Å for K+, 1.42 Å for Mg2+, 2.26 Å for Cl−, and 2.47 Å for NO−3 [104]. The resulting curves and their corresponding parameters, when incorporating the Born energy into the model are shown in Figures 5-10 – 5-12 and Table 5.4. As would be expected, incorporating the Born energy did not have a very large effect on the systems that used KCl, since the dielectric constant in these systems did not vary much between the bulk electrolyte in domain II and domain I. Furthermore, it is interesting to note that the effect of the Born energy appears to be larger for MgCl2 than for KNO3, despite the fact that the dielectric constant in domain I is larger for systems that used KNO3. This effect is probably due to the larger charge and larger number of ions that for MgCl2. Ions with larger charges are associated with a higher Born energy and the larger number of ions for MgCl2 133PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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