# Interdisciplinary Applied Mathematics

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For this approach to be accurate, the size of the fine-scale problem must be large    enough    such    that    all    the    critical    regions in    the    coarser    length

scale problem, where atomistic details are important, are included in the auxiliary problem. For this approach to be efficient, the size of the fine-scale problem    must    be    much    smaller    than    the    size    of    the    original    system,

i.e., W ^ W0. In practice, the size of the auxiliary problem is chosen as a compromise between these two objectives.

In the rest of this section, we describe two embedding multiscale examples to compute the ion concentration and the velocity distribution in a nanochannel electroosmotic flow.

W

0

i

W1

(a)    (b)

FIGURE 16.6. Schematic of the embedding technique for multiscale simulation. (a) represents the original coarser length scale problem and (b) represents auxiliary fine    scale    problem    set    up to solve    the    coarser    length    scale    problem.    W0

and Wi are the characteristic length scales of the two systems. The shaded areas of width    S    denote    the critical    regions    where    atomistic    details    are    important in

determining the system behavior. The region of width e in panel (b) is a buffer region.

###### 16.3.1 Application to the Poisson-Boltzmann Equation

In Section 12.2.1, we presented a modified Poisson-Boltzmann equation to account for the wall-ion, water-ion, and ion-ion interactions in a more accurate manner. The key issue in the implementation of the modified Poisson-Boltzmann equation is to compute the excess chemical potential for an ion, i, denoted by феху (see Section 12.2.1 for details). In the embedding multiscale approach, we extract the electrochemical potential correction term from the ion concentration profile obtained from MD simulation of a smaller width channel using equation (12.4). Once the electrochemical potential correction term is obtained, one can use it in the modified Poisson-Boltzmann equation (12.5b) to simulate the ion distribution in a bigger channel with the same wall structure and similar surface charge density. In such an approach, one circumvents the difficulty of obtaining a closed form expression for the electrochemical potential correction term by utilizing the MD simulation results.

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