# Interdisciplinary Applied Mathematics

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are shown in the lighter shade. Water molecules are also not shown. Chloride occupancy is higher than potassium. A buildup of potassium ions near the COO~ is also observed.

FIGURE 13.15. Positions of individual ions (K+ on the left and Cl_ on the right) inside an end wall functionalized nanotube along the nanotube axis in a (16,16) tube fixed in a slab in a solution of 1.5 M KCl. The axial distance is scaled, with a unit    of    1    being    the    length of the    tube. K+    enters    from the bottom and    Cl_

enters from the top. There is an external electric field of 0.15 V/nm. The rate of chloride ion passage is higher than that of K+ ion passage, indicating a selectivity of anions over cations.

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FIGURE 13.16. K+ trajectory (left) and Cl~ trajectory (right) in a symmetrically functionalized    tube with    COO    groups    on    both    ends    and    protonated    carbon

atoms in the center of the (16,16) tube. The axial distance is scaled such that the length of the tube corresponds to a unit of 1. The chloride current is much higher than that of potassium current wherein there is asymmetric functionalization because K+ ions are electrostatically bound to the COO group at the mouth of the tube. The chloride ion also has to clear a barrier created by protonated carbon atoms in the center to reach the other end.

velocity could be retrieved with reasonable accuracy. External electric field strengths of —0.2 V/nm and 0.2 V/nm are used for positive and negative surface charge densities, respectively.

Figure    13.17    shows    the    water    and    ion    concentration    profile across    the

channel for a positively charged CNT surface. A significant layering of water, as indicated by the strong fluctuation of the water concentration, is observed near the channel wall. The Cl ion concentration is maximum at a position very close to the channel wall (r = 2.34 nm), and decreases toward the channel center. However, such a decrease is not monotonic, and we observe a weak second concentration peak at r = 2.05 nm. It is also observed that the Na+ ion concentration does not decrease monotonically toward the channel wall, but has a peak located at r = 1.93 nm. Both the second peak of    Cl    ion and    the    peak of Na+    ion    are very    close    to    the

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