# Interdisciplinary Applied Mathematics

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distribution of water molecules around the ion can be significantly influenced by the fluctuation of water concentration near the channel wall (see Figure 12.11), which then leads to a fluctuating mean force in the region 0 < z < 0.75 nm of Figure 12.12(a).

FIGURE 12.12.    (a)    Total    mean    force    acting    on    a    Na+    ion    and    its    component

due to the nonelectrostatic Na+—water interactions, (b) the electrostatic mean force acting on a Na+ ion with and without the surface charge density. The zero surface charge case has a bulk ion concentration of 1.0 M.

The contribution of the electrostatic interactions is evaluated by studying the electrostatic mean force acting on the Na+ ion, which is computed by subtracting the nonelectrostatic component from the total mean force. As shown in Figure    12.12    (b),    the mean    force acting    on    a Na+    ion    due    to    its

electrostatic interaction with all other molecules in the system is positive for z < 0.35 nm. Such a positive mean force arises mainly from the fact that when the Na+ ion is very close to the lower channel wall, the electrostatic interactions between the ion and its hydration water molecules, which are distributed mainly above the ion, tend to “pull” the ion from the surface. To understand how the surface charge contributes to the accumulation of Na+ ions near the charged channel wall, we computed the electrostatic mean force experienced by a Na+ ion in a reference case, where the surface charge density is zero and the bulk ion concentration is the same as in the present case. Figure 12.12(b) shows that when the surface is charged, though the    electrostatic    mean    force    acting    on    the Na+    ion    near    the    wall    is

positive, it is significantly lower compared to the case in which the surface is not charged. From this we see that when the molecular nature of water is considered, the electrostatic interactions contribute to the accumulation of Na+ ions near the charged channel wall in a different way from what one would expect in a classical EDL theory.

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