# Interdisciplinary Applied Mathematics

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5. Filling and emptying occurs mainly by the sequential addition of wa-

FIGURE 11.29. Water flow through the nanotube. a. Numbering of water molecules leaving the nanotube at time t that entered the nanotube from the other side. Individual water conduction events are smoothed with a 1-ns-wide triangular filter. b. motion (along the tube length) of individual water molecules during the conduction “burst” at 26.5 ns. (Courtesy of G. Hummer.)

ter to or removal from the single-file water chain inside the nanotube.

Figure 11.30 shows the trajectories of individual water molecules entering a silicon dioxide nanopore about 8 A in diameter and 6 nm long as a function of time.    With    longer    tubes,    the    dipoles    at    the    ends    do    not    re

pel each other as much as in a shorter tube, so chains form at both ends. But one of them becomes dominant, and then filling progresses sequentially through that chain till the chains meet. Figure 11.31 shows the filling as distributions of normalized dipole orientations of water molecules with respect to the axis of the tube. Here the behavior of water with respect to time does not    come    into    the    picture,    but    rather the    fact    that    orientation

remains the same in a chain. The orientations are predominantly either —1 or +1, showing that the direction of entry is along the axis of the tube from either end.

The simulations performed on different types of nanotubes show that the water entry is highly sensitive to the interaction of water with the nanotube. A small reduction of the attraction between the carbon nanotube and the water changes the hydration of the nanotube dramatically, and a sharp two-state transition between empty and filled states on a nanosecond time scale occurs in the nanopores (Waghe et al., 2002; Hummer et al., 2001; Beckstein et al., 2001). Table 11.4 summarizes the LJ parameters from various papers, all agreeing the fact that above a particular value of ff0xygenwall, the tube

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