# Interdisciplinary Applied Mathematics

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FIGURE 11.8. Water-oxygen atom density profiles along the channel axis direction: (A) Density profile for atomic hydrophobic channels of effective radii 2.6 A (left panel), 4.1 A (central panel), and 5.6 A (right panel). (B) Density profiles for LJ hydrophobic surface described by the LJ 5-3 potential. (C) Density profiles for hydrophilic channels. (Courtesy of S.-H. Chung.)

direction (channel axis direction) for atomic hydrophobic (solid curve), LJ hydrophobic (dotted curve), and hydrophilic (dashed curve) channels. The right panels show the projection onto the radial vector r. In the hydrophobic channels, water molecules have a definite preference to point either up or down the channel axes, and such a trend is especially clear in narrow channels. Most distributions possess symmetry about the origin and result in zero    net    dipole    moment.    However,    in a    very    narrow channel    (e.g.,    r =

2.6 A), the symmetry is broken, and this leads to a rather large net dipole moment along the channel. In contrast, in the hydrophilic channels, the water dipoles prefer to lie in the xy-plane rather than align along the channel axis, and    align    themselves    more    toward    the    channel    wall.    The    net    dipole

moments are zero in all the hydrophilic channels. Clearly, the geometrical confinement    alters    the    behavior    of    water    from    that    in    the    bulk, and    the

effects of different types of surfaces make its behavior further complicated.

This can have a very profound influence on various macroscopic measur-ables. As discussed by (Allen et al., 1999), an ion attempting to pass along a line of dipoles aligned head to tail (as in the narrow hydrophobic channel case) will be impeded less than if it were to pass alongside a line of dipoles oriented perpendicular to the line (as in a hydrophilic channel), and this can lead to very different transport behavior of the ions.

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