Interdisciplinary Applied Mathematics

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1. There is no Cl_ ion within 0.24 nm from the channel wall,


2. The peak concentration of Cl_ ions occurs at a position about 0.35 nm away from the channel wall, and the peak value is about 88% higher compared to the Poisson-Boltzmann equation prediction, and


3. The ion concentration from MD does not decrease monotonically to its value in the channel center. In particular, there is a plateau located at about 0.49 nm to 0.60 nm away from the channel wall.


These deviations can be understood by looking at the molecular aspects of the ions, wall atoms, and the water molecules. First, since a bare Clion has an effective radius of about 0.18 nm (Israelachvili, 1992a), a Clion cannot    approach too    close    to    the    channel    wall.    Second, the    ion    and


the wall    interact    with each    other    via    the    Lennard-Jones    potential    in    the


MD simulation. Such an interaction can contribute to the attraction of ions toward the    wall.    Figure    12.3    (b)    shows the potential    energy    of a    Cl_    ion


due to the Lennard-Jones potential between the Cl_ ion and the channel wall.    In    this    calculation,    we    have    assumed that    the    ion    can access    any


position in the xy-plane with equal probability. Figure 12.3 (b) indicates that the potential energy due to the ion-wall Lennard-Jones interaction is about —1.8    kB    T    (T    = 300 K)    at    a    position    about 0.39 nm    away from the


channel wall. Since a location with lower potential energy is more favorable for the    ions,    the    potential    energy    valley    can    attract    more    ions    toward    it.


The second deviation is primarily caused by this effect. Third, the molecular interaction between the ion and the water molecules also plays an important role in determining the ion concentration. Figure 12.3 (a) shows that the water concentration is not uniform across the channel, and a significant layering of water is observed near the channel wall. Such a layering effect is well known and has been already discussed in Chapter 10. Since the water molecules are less closely packed near the density “valley” than in the bulk, the energy required to insert a finite-sized ion into the density “valley” is lower compared to inserting an ion in the bulk. Hence, more ions are attracted toward the density “valley” of water. In fact, a very weak peak of Cl_ ions is observed near the second density valley of water. From these

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