# Interdisciplinary Applied Mathematics

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Ditsance from the channel wall (nm)

FIGURE 12.13.    (a)    Total    mean    force    acting    on a    Cl~    ion and its    nonelectrostatic

component, (b) and (c) are the mean forces experienced by a Cl ion due to its nonelectrostatic interactions with Na+ and Cl ions, wall atoms and water molecules.

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Observation (2), which deals mainly with the low Cl ion concentration in the region z < 0.5 nm, can be explained by performing an analogous mean force analysis for the Cl ion. Figure 12.13 shows that the total mean force acting on a Cl ion is strongly positive in the region 0.4 nm < z < 0.6 nm (see panel (a)), and it is dominated by the nonelectrostatic interactions between the Cl — Na+ ions (see panel (b)) and Cl —water molecules (see panel (c)). This indicates that the low concentration of Cl ion in the region

0.35 nm < z < 0.5 nm is mainly caused by the nonelectrostatic interactions between Cl — Na+ and Cl—water. The depletion of Cl ion in the region z < 0.35 nm is caused by the finite size of the Cl ion.

Observation (3), which deals with overscreening and charge inversion, can be explained by the following two mechanisms. First, as discussed above, in addition to the electrostatic interactions, the nonelectrostatic Na+—water interactions also contribute significantly to the accumulation of Na+ ions near the channel wall. Together, these interactions bring many more Na+ ions toward the channel wall than is predicted by the classical EDL theory, where the molecular nature of water is not considered. Second, because of the nonelectrostatic interactions between the Cl ions and the wall atoms and the water molecules, and because of the accumulation of Na+ ions near the channel wall,    the    Cl    ions    tend to    stay    away    from    the    channel    wall

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