# Interdisciplinary Applied Mathematics

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nm away from the channel wall, and (3) the second Na+ ion concentration peak is much more distinct compared to case 1. The first ion concentration peak is lower in this case because the attractive interaction between the

FIGURE 12.5. Na+ ion and water concentration across the channel for case 3 (W = 3.49 nm, aa = —0.120 C/m2).

Na+ ion and the Si wall atoms—which contributes significantly to the high ion concentration near the channel wall—is much weaker compared to the Cl_ ion concentration with    the    Si    wall    atoms    (see    Table    12.1).    The    second

observation can be    explained    by    the    fact    that    though    a Na+    ion    (bare

radius about 0.095 nm) is smaller compared to the Cl_ ion, its hydrated radius (0.36 nm) is bigger compared to the Cl_ ion (0.33 nm), since a Na+ ion can attract more water molecules around it (the hydration number of a Na+    ion    is    about 4 to 5,    while    the    hydration    number    of Cl    ion    is about

1).

The second Na+ concentration peak is primarily caused by the ion-water interactions. Figure 12.5 indicates that the second ion concentration peak is close to the second concentration valley of water. As explained in case 1, such a water concentration valley is energetically more favorable than the bulk for ions. Since    the    bare    ion    radius of    Na+    is    much    smaller    than that of

the Cl_ ion,    the Na+    ion    can    fit    into    the    water concentration    valley    more

easily than the Cl_ ion. Therefore, the second concentration peak of Na+ ions is more distinct compared to the Cl_ ion case. One consequence of the higher second peak of Na+ ions is that the water concentration valley near the second Na+ peak is shallower compared to that observed in the Cl_ ion case (case 1). This is because the Na+ ion has a higher hydration number. Thus, it can bring more water molecules toward the water concentration valley than the Cl_ ion.

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