Beyond Poisson-Boltzmann theory

Vlachy V 1999 Ionic effects beyond Poisson-Boltzmann theory Ann. Rev. Phys. Chem. 50 145... 

Vlachy, V. Ionic effects beyond Poisson-Boltzmann theory. Annual Review of Physical Chemistry, 1999, 50, p. 145-165. 

It follows from the preceding section that the limiting case xa 1 (double layer thin as compared with radius of curvature) is simple then we can simply apply the flat layer theory, discussed extensively in secs 3.5a-d. Beyond this limit, the appropriate Poisson-Boltzmann equation (with p in (3.5.631 depending on the geometry) has to be solved with the appropriate boundary condition, l.e. dy/dr for r = 0, so in the centre of a sphere or infinitely long cylinder, the field strength is zero because of symmetry. However, at that location y is not necessarily zero, because double layers from the opposite sides may overlap. This Is a new feature as compared with convex double layers around non-interacting particles. 

There have been considerable efforts to move beyond the simplified Gouy-Chapman description of double layers at the electrode-electrolyte interface, which are based on the solution of the Poisson-Boltzmann equation for point charges. So-called modified Poisson-Boltzmann (MPB) models have been developed to incorporate finite ion size effects into double layer theory [61]. An early attempt to apply such restricted primitive models of the double layer to the ITIES was made by Cui et al. [62], who treated the problem via the MPB4 approach and compared their results with experimental data for the more problematic water-DCE interface. This work allowed for the presence of the compact layer, although the potential drop across this layer was imposed, rather than emerging as a self-consistent result of the theory. The expression used to describe the potential distribution across this layer was... 

Manning proposed a linear counterion condensation theory to account for the low activity of counterions in polyelectrolyte solutions 14). The basic idea of the theo is that there is a critical charge density on a polymer chain beyond which some counterions will condense to the polymer chain to lower the charge density, otherwise the ener of the system would approach infinite. The concept of this theory has been widely accepted. The shortcoming of the linear countmon condensation is that it predicts that counterion condensation is independent of ionic strength in the solution, which is not in agreement with experimental observations. Counterion condensation can be obtained duectly by solving the nonlinear Poisson-Boltzmann equation. 

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