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The same mean-field approximation also underlies a plethora of modified PB equa- tions for concentrated electrolytes [6], which modify the equilibrium distribution ρe(φ) for local [10, 12, 50, 60] or nonlocal [64, 33, 30] volume constraints, dielectric satu- ration ε(∇φ) [36, 29, 6], ion polarizability and dielectric decrement [10, 29, 46], and other effects. But none of these models can predict charge inversion. Charge inver- sion here is defined by the change in sign of the apparent effect of deposited polymer on the surface potential compared to the surface potential of the bare surface. This definition is distinctly different from the definition of charge inversion in terms of the apparent change in sign of the macroscopic zeta potential of the surface as ob- served from beyond the polymer surface, which has already seen some limited work [37, 80, 73, 74] (though this will also be captured in this model) In this work, we provide a simple continuum theory of electrostatic correlations in polyelectrolytes based on the linearized 4th-order BSK model that has been proposed for ionic liquids [8] and concentrated electrolytes [6, 97], which was motivated by approximations of a one-component plasma at intermediate coupling [89, 47]. This model contains only one additional parameter as compared to Equation 5.1, namely a correlation length, which sets the scale for charge oscillations, which we expect to be on the order of the Bjerrum length (the length scale at which electrostatic and thermal energy balance). The model presented here is compared to experimental data obtained using a tran- sistor micro-array detector [31] coated with polyelectrolytes (see figure 5-2), where mean-field theory (without inversion) holds in the absence of the charged polymer, provided by Ulrich Bockelmann’s group at ESPCI in Paris. This data was obtained by measuring the interface potential of transistor arrays that had been spotted with polyelectrolyte solution and then comparing the measured potential with the mea- sured interfacial potential obtained for a bare transistor array. 116PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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