Interdisciplinary Applied Mathematics

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Such interaction with the wall material can be studied with molecular dynamics (MD) simulations; see Section 16.1. In a typical molecular dynamics simulation, a set of molecules is introduced initially with a random velocity for each molecule corresponding to a Boltzmann distribution at the temperature of interest. The interaction of the molecules is prescribed in the    form    of    a    potential    energy,    and    the    time    evolution    of    the    molec


ular positions is obtained by integrating Newton’s equations of motion. Realistic intermolecular potentials are constructed by modeling the atom-atom interaction potential using relatively simple equations, such as the Lennard-Jones potential


V (r) = 4e


written for a pair of two atoms separated by distance r. The Lennard-Jones potential incorporates the shape effects by an anisotropic repulsive core and anisotropic dispersion interactions. For an appropriate choice of these parameters a reasonable description of real liquids is possible. For example, using e/kB ~ 120 K, where kB is Boltzmann’s constant and a « 0.34 nm, a reasonable description of liquid argon can be obtained; see Section 16.1 for more details.


In Figure 1.7 we plot results from a molecular dynamics simulation of (Koplik et al., 1989) that shows a large density fluctuation very close to the wall. Specifically, the fluid is governed by a Lennard-Jones potential, and there is no net flow. The total number of atoms in this simulation is 27,000. The geometry is a three-dimensional periodic channel made of two atomic walls with 2,592 atoms each of FCC lattice type. The size of the channel    is 51.30    x    29.7    (in    the    plane)    x 25.65 (out    of plane),    with    all


dimensions in molecular units; i.e., the atom diameter is 1.0. The density profile is obtained by binning the atomic positions in 170 slabs parallel to the    walls,    and    the    overall    density is    0.8    units,    while    the temperature

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