Interdisciplinary Applied Mathematics

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As mentioned above, simple fluids can be described using a Lennard-Jones (LJ) potential (LJ and other potentials are discussed in detail in Section 16.1; here we quickly recap the salient features of the LJ potential and introduce the reduced units that are used in this chapter) of the form


Vlj = 4e



cri2


r )




where e, a are the Lennard-Jones parameters that depend on the atoms involved in the interaction. Note that:


1.    e is related to the interaction strength, and a higher e corresponds to a higher interaction energy between the atoms.


2. a corresponds to the distance at which the potential between the two atoms goes to zero, which can be approximately taken as the diameter of a fluid atom.


Since the Lennard-Jones potential describes the interactions between non polar molecules quite well (Talanquer, 1997) and the force corresponding to the Lennard-Jones potential can be evaluated efficiently numerically, it is the most popular interaction potential used in MD simulations. In the MD simulation of Lennard-Jones fluids, the physical quantities are typically computed using reduced units. Table 10.1 summarizes the units for various quantities, e. g., length, temperature, and density. In the table, e and are as defined in equation (10.1), kB is the Boltzmann constant, and is the mass of a Lennard-Jones atom. Unless otherwise mentioned, all the quantities are measured in reduced units in the next two sections.


The studies on Lennard-Jones fluids have indicated that depending on the critical length scale of the channel (typically the channel width/height or the diameter), the fluidic transport behavior (e.g., convection and diffusion phenomena) can either deviate significantly from the classical continuum theory prediction or be very similar to the transport of a bulk fluid described by the classical theory. These observations follow from the fact that when the fluid atoms are confined to molecular channels, the fluid can no longer be taken to be homogeneous, and strong oscillations in fluid density occur near the solid-fluid interface. Therefore, the dynamic behavior of the fluid becomes significantly different from that of the bulk. Some

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