# Interdisciplinary Applied Mathematics

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Marry and colleagues have estimated the slip length to be 8 = 6 A. Using this slip length in equation (12.13) and using the Poisson-Boltzmann equation for the concentration profile results in the electroosmotic profile shown by the solid curve in Figure 12.9. The agreement with the macroscopic model is excellent for the three simulated channel widths shown in Figure 12.9. The influence of the hydrodynamic boundary condition can be significant even for large channel widths. As shown in Figure 12.10, even for a channel width of 100 A, the slip boundary condition correction is not negligible.

FIGURE 12.9. Comparison of electroosmotic velocity profiles ((u(z)/Eext)) obtained for different channel widths; MD results are shown by solid circles (number of solvent molecules is 200), open circles (number of solvent molecules is 400), and filled triangles (number of solvent molecules is 600); PB/NS results with no-slip boundary conditions are shown by the dotted lines; PB/NS results with slip boundary conditions (S = 6 A, zo = -W/2) are shown by the solid lines; the vertical dashed lines denote the position of the channel walls. (PB/NS refers to Poisson-Boltzmann/Navier-Stokes equations.) (Courtesy of P. Turq.)

FIGURE 12.10. Mean electroosmotic velocity (u/Eext) as a function of the channel width. MD data are shown as diamonds; PB/NS equation with no-slip boundary conditions is shown as a dotted line; PB/NS with slip boundary conditions are shown by the solid    line;    L in    this    plot    is    the    channel    width,    and    the    error

bars are for the MD data. (PB/NS refers to Poisson-Boltzmann/Navier-Stokes equations.) (Courtesy of P. Turq.)

##### 12.5 Charge Inversion and Flow Reversal

Charge inversion refers to the phenomenon that the coion charge density exceeds the counterion charge density in a certain region of the electric double layer (EDL) (Qiao and Aluru, 2004). Consider again the channel system shown in Figure 12.2 with NaCl solution sandwiched between the two channel walls. A total charge of —70e is evenly distributed among the atoms of the innermost wall layers, giving an average surface charge density (as) of —0.285 C/m2. Such a charge density can be considered high, but it is not impractical, since the typical charge density of a fully ionized surface can exceed 0.3 C/m2 in magnitude. The system contains 108 Na+ ions, 38 Cl_ ions, and 2144 water molecules. MD simulations were performed using the parameters and models discussed in Section 12.1. Starting from a random configuration, the system was simulated for 2.0 ns to reach steady state, followed by a 15-ns production run. The flow was driven by an electric field, Eext, applied in the ж-direction. Because of the extremely high thermal noise, a strong electric field (Eext = 0.55 V/nm) is necessary so that the fluid velocity can be retrieved with reasonable accuracy. A strong external electric field can induce noticeable water alignment along the field direction, which can influence the ion distribution. To understand how this influences the charge inversion results presented here, simulations were also performed with zero external field. The ion distribution was found to be only slightly different from what is reported here, and the charge inversion is still observed.

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