Interdisciplinary Applied Mathematics

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In principle, one can calculate фех provided the wall-ion, water-ion, and ion-ion interactions can be computed explicitly. However, such a calculation, if possible, is very difficult. For example, to account for the molecular nature of water, the charge-dipole interaction between water and the ion as well as other molecular interactions (e.g., the van der Waals interaction as included in the Lennard-Jones potential) will need to be considered explicitly. In addition, since the water concentration profile in the channel is related to the ion concentration profile (as demonstrated by the correlation between the second peak of the ion concentration with the water concentration valley in Figures 12.3 (a), 12.4, and 12.5), the concentration profiles of water and the ion must be computed self-consistently. In Section 16.3, we describe a multiscale approach to computing the electrochemical potential correction and to computing the ion concentrations using the modified Poisson-Boltzmann equation.

12.3 Velocity Profiles

The comparison of velocity profiles obtained from continuum and MD simulations is presented in this section. Figure 12.6 shows the velocity profile across the channel for case 1, where the channel width is 3.49 nm. The velocity profile based on the continuum flow theory, calculated by substituting the ion concentration obtained from the MD simulation into equation (12.1b) and using the boundary condition specified by equation (12.2b), is also shown for comparison. Figure 12.6 shows that the continuum flow theory prediction using a constant viscosity of 0.743 mPa-s overestimates the velocity in the entire channel. This is because the continuum calculation fails to take into account the fact that the viscosity near the channel wall is much higher than its bulk value. It is reasonable to assume (Freund,

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