# Interdisciplinary Applied Mathematics

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The accuracy of this approach depends on how the fine-scale problem is set up and on how significantly the electrochemical potential correction term differs in the two problems (i.e., in the coarse-scale and in the fine-scale problems). In setting up the fine-scale problem, one needs to include the near-wall region, where the electrochemical potential correction is nonzero. By setting up the fine-scale problem using similar operating conditions (e.g., wall charge density, wall structure, and bulk concentration) as in the original system, the electrochemical potential correction will not differ significantly in the two problems. This is because:

1. The wall-ion interaction included in the electrochemical potential correction term is the Lennard-Jones potential, which depends only on the wall structure and the Lennard-Jones parameters, and thus will not change when the channel width is increased.

2. The water-ion interactions depend primarily on the water concentration (i.e., how closely the water molecules are packed). MD simulation results of water concentration profile in channels of different width but with the same surface charge density indicate that the water concentration profile near the channel wall is independent of the channel width.

In summary, the electrochemical potential correction term is primarily due to the wall effects (e.g., ion-wall interactions and wall-induced water layering). Since these interactions are short-ranged, further addition of water layers in the bulk (corresponding to a wider channel) would not affect the electrochemical potential correction term significantly. Hence, the use of the same electrochemical potential correction term for wider channels can produce reasonably accurate results.

The efficiency of this approach depends on whether the length scale of the fine-scale problem can be significantly smaller compared to the original problem. This can be achieved by choosing a small e in Figure 16.6. However, if e is too small in the fine-scale problem, the system behavior in one critical region may be influenced by the system behavior in another critical region (e.g., the ion distribution near the upper channel may be influenced by that near the lower channel wall), which may not exist in the original coarse-scale problem. As an example, for a 6-nm-wide coarse-scale channel, typically a 2-nm-wide fine-scale channel (note that this can depend on a number of parameters such as the surface charge density, and Debye length) is used. MD simulation of a 2-nm-wide channel requires much smaller computational time than the MD simulation of the 6-nm-wide channel. Another good example where the embedding multiscale approach can be efficient is the nanofluidic system studied by (Kemery et al., 1998), where nanochannels are connected by microchannels. In this case, the MD simulation of the nanochannels is possible, but the MD simulation for the entire system is impossible.

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