# Interdisciplinary Applied Mathematics

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Figure 12.11 shows the variation of water and ion concentration as well as the screening factor across the channel. Apart from the well-known layering of water near the channel wall, we also observe that:

1. a significant number of Na+ ions are accumulated within only 0.5 nm of the channel wall,

2. the concentration of Cl_ ion is very low in the region z < 0.5 nm even though its access to the region 0.35 nm < z < 0.5 nm is not limited by its finite size, and

 (b) : Г» z = 0.37 nm

0    0.5    1    1.5

Distance from the channel wall (nm)

FIGURE 12.11. Water/ion concentration profiles and the screening factor Sf (z) across the channel (only the lower portion of the channel is shown because of the symmetry). Note that z = 0 denotes the position of the lower channel wall.

0

2

100

0

1.5

3. the surface charge is overscreened at a position 0.37 nm away from the channel wall (i.e., Sf (z) ^ 1 for z > 0.37 nm), and charge inversion occurs in the region z > 0.53 nm.

The overscreening and charge inversion cannot be predicted by the classical EDL theories, e.g., the Poisson-Boltzmann equation. Observation (1) can be understood by analyzing the mean force acting on a Na+ ion as shown in Figure 12.12. From the PMF definition and the Boltzmann distribution, it follows that the accumulation of the Na+ ions in the region z < 0.5 nm is mainly caused by the negative total mean force in regions II and IV of Figure 12.12 (a). Figure 12.12 (a) also shows that the nonelectrostatic Na+—water interactions contribute significantly to the negative total mean force in these regions (the mean force due to the nonelectrostatic interactions between Na+—Na+, Na+— Cl_, and Na+—wall is not shown, since its contribution to the total mean force is less significant). The magnitude and sign of the nonelectrostatic Na+—water mean force depends mainly on the repulsive nonelectrostatic interactions between the Na+ ion and its nearby water molecules and on the asymmetrical distribution of the water molecules around the ion (a symmetrical distribution will result in a zero mean force). For    an    ion    located    near    the    channel    wall,    the    asymmetrical

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