Interdisciplinary Applied Mathematics

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11.3 Dynamic Behavior


Understanding the dynamic behavior of water is critical to many biological and engineering applications. For example, the study of the diffusion of water molecules through nanochannels can help explain the operating mechanisms of the water channels, which are responsible for many important biological processes in the cell (Sui et al., 2001; Hummer et al., 2001; Beckstein and Sansom, 2003). In this section, we first review the research on the basic concepts of dynamic behavior of water and then summarize some of the simulation results on the diffusion transport of water through nanochannels. Finally, the filling and emptying kinetics of water in nanopores is discussed.

11.3.1 Basic Concepts


In this section, we focus our attention on the properties of the motion of a single water molecule in confined states, such as the reorientation dynamics, residence time, dipole correlation, and the velocity distribution. Understanding these properties can provide insight into the dynamic properties of water in more complicated scenarios such as the diffusion transport.


H


Reorientation Dynamics


Molecular reorientational motions in liquids are usually analyzed through the time correlation functions


Cl,a(t) = (Pl(ua(t) • Иa(0))} ,


where Pl refers to /th Legendre polynomial and ua is a unit vector along a given direction (a = p, HH, OH, T). As shown in Figure 11.19, four different unit vectors are considered: a unit vector uM(t) = j(t)/p(t) along the molecular dipole moment direction, a unit vector uhh(t) = rHH(t)/rHH(t) along the H-H direction, a unit vector uoH(t) = rOH(t)/rOH(t) along the O-H direction, and a unit vector u±(t) = uM(t) x uHH(t) perpendicular to the molecular plane. The correlation functions associated with the Legendre polynomials are

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