Interdisciplinary Applied Mathematics

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s2 = 2Ctn.


Figure 10.12 shows the log-log plots of the mean-square displacement for methane, ethane, and ethylene in a (10,0) carbon nanotube (diameter: 0.8 nm). Clearly, the diffusion of methane is a normal-mode diffusion and the diffusions of ethane and ethylene are transition-mode diffusions. Mao and Sinnott further showed that the diffusion of ethane and ethylene in a (9,0) carbon nanotube (diameter: 0.72 nm) follows the single-file mode, while the diffusion of methane is still in the normal mode. Such differences are caused by the fact that the size of ethane and ethylene are larger compared to methane; therefore, the methane molecules can pass one another in a


0.72-nm diameter, while ethane and ethylene molecules cannot pass each other in such small-diameter tubes.

10.4 Validity of the Navier-Stokes Equations


In the continuum fluid transport theory governed by the Navier-Stokes equations, it is assumed that the state variables (e.g., density and tempera-


1.30    1.80    2.30    2.80    3.30    3.80    4.30

























In (time)


3 30








2.10 1.50


30    1.80    2.30    2.80    3.30    3.80    4.30    4.80


In (time)


FIGURE 10.12. Log-log plot of the mean-square displacement for methane (a), ethane (b), and ethylene (c) in a (10, 0) carbon nanotube (diameter: 0.8 nm). (Courtesy of S. Sinnott.)

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