# Interdisciplinary Applied Mathematics

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The boundary conditions for equation (12.1b) and equation (12.1b) are

d^(z)

dz    z=±h/2

u(z)z=±h/2

(12.2a)

(12.2b)

where z = ±h/2 corresponds to the location of the lower and the upper channel walls and as is the charge density on the channel walls. Analytical solutions for    equation    (12.1b)    and    equation    (12.1b)    are    available    for    the

boundary conditions given in equations (12.2b) and (12.2b) (Israelachvili, 1992a; Eikerling and Kornyshev, 2001). However, to use the analytical solution, one needs to first solve a transcendental equation numerically, so equations    (12.1b)    and    (12.1b)    are    typically    solved    numerically.    The    rela

tive permittivity of water is typically taken as 81, since this is the reported value for SPC/E    water    at    300 K (van    der    Spoel    et    al.,    1998).    The    dynamic

viscosity of water is taken as 0.743 mPa-s in the continuum simulations, since this gives the best match to the velocity profile in the central portion of the channel.

Nonequilibrium molecular dynamics (NEMD) simulations, as described in Chapter 16,    were performed    for    systems    consisting    of    a    slab    of    water

molecules    and    ions    sandwiched    by    two channel    walls.    Figure    12.2    shows

a schematic diagram of the system under investigation. The two channel walls are symmetrical with respect to the channel center line. Each wall is made up of four layers of silicon atoms oriented in the (111) direction. Typical lateral dimensions of the channel wall are 4.66 nm x 4.43 nm, which corresponds to 161 silicon atoms for each layer of the channel wall. The channel width is varied from 0.95 nm to 10.0 nm in the simulations. For the simulation of electroosmotic flow, the outermost wall layers (i.e., layer I of the lower channel wall and its counterpart in the upper channel wall) are partially charged. We assume that the charges are uniformly distributed among the wall atoms; i.e., wall atoms are partially charged. The wall atoms are fixed to their original positions during the simulation. The water is modeled by using the SPC/E model (Berendsen et al., 1987) (see Chapter 16 for details). We consider two types of interaction potentials, i.e., Lennard-Jones and Coulomb potentials. The Lennard-Jones potential is considered for every atom pair (except the atom pairs that have a hydrogen atom and the Si-Si pair). The parameters for the Lennard-Jones potential are taken from the Gromacs forcefield (Spoel et al., 2001) and are summarized in Table 12.1. The Coulomb potential is considered for every charged atom pair.

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