# Interdisciplinary Applied Mathematics

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and accumulate in the region z > 0.5 nm. The combination of these two mechanisms then makes it possible for the surface charge to be overscreened at    a    short    distance    from    the    channel    wall    (z    = 0.37    nm),    and    for    the

concentration of    Cl    ions to exceed    that    of    the    Na+    ions    at a distance    of

0.53 nm away from the channel wall.

FIGURE 12.14. (a) Water velocity profile across the channel obtained from MD and continuum simulations. (b) driving force for the flow across the channel using the ion concentrations obtained from MD simulation.

Flow reversal refers to the phenomenon that the electroosmotic flow is in the    opposite    direction    to    the    one    typically    expected    (Qiao    and Aluru,

2004). For example, when the surface is negatively charged and the external electrical field is in the positive x-direction, one typically expects the electroosmotic flow to be in the positive x-direction. However, an electroosmotic flow in the negative x-direction will be observed when flow reversal occurs.

For the channel system discussed above, Figure 12.14 (a) shows the water velocity profile across the channel obtained by using MD and continuum calculations. In the continuum calculation, the Poisson-Boltzmann (PB) equation is solved to obtain the ion concentration, which is then used to calculate the driving force for the flow given by the expression

Fe(z) = q[cNa+(z)cCi- (z)]Eext,

where q is the electron charge. The flow Fe(z) is used as the driving force in the    Stokes equation to    compute    the    water    velocity.    In    solving    the    PB

equation, the bulk concentration of the NaCl solution is taken to be 1.0 M, which is consistent with the result shown in Figure 12.11. For the Stokes equation, a no-slip boundary condition is applied at positions z = 0.13 and 3.36 nm, which is consistent with the MD observation. The dielectric constant and the viscosity of the water are taken as 81 (this is the relative dielectric constant) and 0.743 mPa-s, respectively. From Figure 12.14 (a), we observe that while the continuum theory based on the PB and the Stokes equations predicts a positive velocity in the entire channel, the MD simulation shows a velocity that is slightly positive near the channel wall, but is negative in the region 0.42 nm < z < 3.01 nm. Clearly, a flow reversal is observed.

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