# Interdisciplinary Applied Mathematics

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(a)

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(b)

FIGURE 12.3. (a) Cl~ ion and water concentration across the channel for case 1 (W = 3.49    nm,    as    = +0.120    C/m2).    The    channel    center    is    located    at г =

0 nm, and the position of water molecules is computed as the center-of-mass position. (b)    Potential    energy    of    Cl_    ions    over    the    channel wall    computed    using

Lennard-Jones potential. It is assumed that the ion can access any position in the xy-plane with equal probability.

results we can infer that the molecular interactions between ion-wall and ion-water play an important role in determining the distribution of ions near the channel wall.

Figure 12.4 shows the concentration profile of Cl_ ions across the channel for case 2, where the channel width is 3.49 nm and the charge density on the channel wall is +0.320 C/m2. Compared to the previous case, the charge density on the channel wall is very high (such a high charge density is realistic in practical systems (Poppe et al., 1996)). The Poisson-Boltzmann equation again underestimates the ion concentration near the channel wall. A clear second peak of Cl_ concentration is observed at a position about 0.45 nm away from the channel wall, whereas such a peak was very weak in case 1. The second peak is primarily caused by the fact that the ions are very densely packed (as indicated by the high concentration) near the channel wall, and the strong electrostatic repulsion between the ions makes it difficult to accommodate more ions in the region within 0.41 nm from the channel wall. As a result, a second peak is observed. Since the ions are more densely packed in the near wall region in this case compared to case 1 (in fact, the average shortest distance between two Cl_ ions within 0.41 nm from the channel wall is found to be 0.54 nm for this case, and 0.69 nm for    case 1),    the    electrostatic    repulsion    between    ions    is    much    stronger

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