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Furthermore, the equations for the other constants also had to be modified in a similar manner for KNO3 and MgCl2. b2 q ̃ exp (−2b L ̃) − cˆ exp (−b L ̃) 2s161 c1 = b1 (b2 − b21) 1 − exp (−2b1L ̃) b2 q ̃ − cˆ exp (−b L ̃) 2s61 c2 = b1 (b2 − b21) 1 − exp (−2b1L ̃) b2 q ̃ exp (−2b L ̃) − cˆ exp (−b L ̃) 1s262 c3 = 2b2 (b21 − b2) 1 − exp (−2b2L ̃) b2 q ̃ − cˆ exp (−b L ̃) 1s62 c4 = 2b2 (b21 − b2) 1 − exp (−2b2L ̃) (5.39) (5.40) (5.41) (5.42) Using this form of the equations proved to be useful for evaluating the constants even at relatively high L ̃ (though it was still generally limited to about L < 1 μm). Another interesting aspect of this model is that at large L (generally L > 20λp), the model actually becomes very insensitive to L, since L only appears in exponents, which will significantly impact the optimization. In fact, in the regime of large L, one can use an approximation that assumes L to be infinite without much error. Taking this limit yields ( b 3 − b 3 ) q ̃ q ̃ lim∆φ ̃s= 2 1 s +ρ ̃s−s (5.43) L ̃→∞ b1b2 (b2 − b21) ξ 5.3 Results and Discussion 5.3.1 Traditional Debye-Huckel Model As mentioned in the last section, traditional Debye-Huckel and Gouy-Chapman mod- els are unable to effectively model the "charge inversion" observed upon deposition of a PLL polymer on a silica surface. For the purposes of this discussion, we will look at the Debye-Huckel model. If we don’t take differences in dielectric constants 124PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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