Interdisciplinary Applied Mathematics

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Residence Time

The mean    time spent    by    a    water    molecule    in    its    first    coordination    shell

(see Section 11.2.2 for the definition) can be characterized by a residence time function Cres(t). The spatial distribution of various molecules and ions closely surrounding a water molecule forms the coordinate shells of the water molecule. The first coordination shell gives rise to a noticeable

FIGURE 11.21. Residence time rres of a water molecule in its first coordination shell as a function of temperature in different-sized nanotubes at a density of 0.83 g/cm3. (Courtesy of J. Marti.)

cluster, whereas the second coordination shell is more diffuse, and it becomes difficult to discern a pattern for subsequent coordination shells. The residence time for a given water molecule is defined as the number of water molecules in the first coordination shell of the tagged molecule during the time interval of length t. Usually, an exponential-like behavior of Cres(t) is observed, and a residence time Tres can be obtained from the fitting of Cres(t) to a single exponential:

Cres(t) « ce tjT

A longer residence time suggests that water molecules form a more stable structure. Figure 11.21 (Marti et al., 2002) shows the residence time Tres of a water molecule in its first coordination shell as a function of the temperature in different-sized nanotubes. We observe that at room temperature, the residence time Tres increases as the nanotube diameter decreases, indicating that a water molecule inside a narrow pore will spend longer time in its neighborhood than in larger pores. As the temperature increases, the residence time decreases sharply, and the deviation of Tres in different-sized tubes also decreases.

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