Interdisciplinary Applied Mathematics

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Cl,a(t) = (cos ea(t)) ,


C-2,„(!) = ^ (cos2Oa{t) — l) ,


where cosва = ua(t)ua(0). The characteristic reorientational times (т“,та) are computed by



Ci,a(t) dt,    (i = 1, 2)


The reorientational motion of water molecules is characterized by the dipole moment reorientation time т^2 (Marti et al., 2002). A smaller т corresponds to a faster reorientation motion of the water molecule. Marti


2.2


A


■w 1.2 a



0


— 0


a


0.2


5


5


О о


5.5


*


л3.5


CO


Cl


a


H*


о О


A Bulk 0 (8.8) tube О (10,10)


□ (12,12)

1.5


FIGURE 11.20. Molecular dipole moment reorientation time t^2 as a function of temperature at a density of 0.83 g/cm3. Computation with the first (bottom) and second (top) Legendre polynomials. (Courtesy of J. Marti.) and coworkers (Marti et al., 2002) studied the reorientation time of water molecules confined in carbon nanotubes of length 7.45 nm and internal radii of 2.65 A((8, 8) nanotube), 4 A ((10,10) nanotube) and 5.33 A ((12,12) nanotube); see Section 13.2.1 for details on carbon nanotubes. Figure 11.20 shows the reorientation time of the water dipole moment at different temperatures and in different-sized nanotubes. At room temperature, the smaller the nanotube diameter, the faster the reorientational motion, and in the largest nanotube, the reorientation time approaches that of the bulk water. From this result we can conclude that confinement tends to speed up molecular reorientations. The faster reorientational motion in small diameter nanotubes can be attributed to the partial breakdown of the tetrahedral hydrogen-bond network, which is typical when water is confined.

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