# Interdisciplinary Applied Mathematics

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second concentration valley (r = 1.94 nm) of the water molecules.

Figure    13.18    shows    the    water    and    ion    concentration    profile across    the

channel for a negatively charged CNT surface. The counterion (Na+ ion) concentration distribution is significantly different from that observed in the positively charged case. Specifically, we observe that:

1.    The location of the counterion adsorption peak is closer to the channel wall than the first water concentration peak, and

2. The peak concentration of the counterion is much higher than the value observed in the positively charged surface.

Both observations are caused by the smaller size of Na+ ion compared to that of the Cl ion, i.e., the bare radius of a Na+ ion is smaller than that

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FIGURE 13.17. Water and ion concentration distribution across the channel for a positively charged CNT surface.

_i_i_._L.

0 1 2

FIGURE 13.18. Water and ion concentration distribution across the channel for a negatively charged CNT surface.

Radial distance (nm)

FIGURE 13.19. Bulk water velocity profile across the channel for a positively charged surface (shown as filled squares) and a negatively charged surface (shown as filled triangles).

of a Cl_ ion and a water molecule (the radii of the Na+ ion, Cl_ ion, and a water molecule are 0.095, 0.181, and 0.14 nm, respectively); hence, the Na+ ion can approach the channel wall more closely compared to a Cl ion or a water molecule.

Figure 13.19 shows the bulk water velocity for positively and negatively charged CNT surfaces. We observe that for a positively charged surface:

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