Interdisciplinary Applied Mathematics

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Figure 16.7    (a)    shows    the    MD    concentration    profile    of    Cl_    ion    across


the channel for case 4 (see Table 12.2 for details), where the channel width is 2.22 nm and the surface charge density on the channel wall is 0.120 C/m2. Figure 16.7 (b) shows the electrochemical potential correction term extracted by using equation (12.4). Note that the electrochemical potential correction term is close to zero at about 0.8 nm away from the channel wall. фех reaches a minimum at about 0.34 nm away from the channel wall, and this roughly corresponds to the position of the minimum of the potential


energy due to the Lennard-Jones potential between the wall and the ClU ion    (see    Figure    12.3    (b)).    This indicates    that    the    electrochemical    poten


tial correction term at this position is primarily due to the Lennard-Jones potential between the wall and the ClU ion. Using the electrochemical potential correction    term    shown    in    Figure    16.7    (b),    the    ion    distribution    in


various channels with different widths and similar surface charge densities were calculated. Figure 16.7 (c) shows the comparison of CU concentration in a 3.49-nm channel (case 1, charge density: 0.120 C/m2 (see Table 12.2)) predicted by MD simulation and by the modified Poisson-Boltzmann equation. Figure 16.7 (d) shows the comparison of ClU concentration in a 10.0 nm channel (case 6, charge density: 0.124 C/m2) predicted by MD simulation and by the modified Poisson-Boltzmann equation. The results in Figures 16.7 (c) and 16.7 (d) suggest that the extraction of the electrochemical potential correction using a fine-scale channel and employing it in the modified Poisson-Boltzmann equation to predict the variation of the ion concentration in the coarser channel works very well.

16.3.2 Application to Navier-Stokes Equations


The results    in    Section    12.3    indicate    that    the    continuum    flow theory    is    not

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