Interdisciplinary Applied Mathematics

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1


■ H г n!



d



n-1



ri



dr



+



(16.22)


Thus, from the knowledge of the positions of the atoms, velocities of the atoms, and forces acting on the atoms obtained from a typical “MD run,” the stress (or pressure) tensor in the fluid medium can be computed.


Shear Viscosity


Each molecular-dynamics method for calculating the shear viscosity of a fluid falls into one of two main categories: equilibrium molecular dynamics (EMD) or nonequilibriummolecular dynamics (NEMD) techniques. The EMD techniques involve either the calculation of time correlation functions by measuring the decay of near-equilibrium fluctuations in properties of the fluid (Green-Kubo methods) or by accumulating displacements in properties over time (Einstein methods). For example, the Green-Kubo relation for shear viscosity, n, is given by (Arya et al., 2000)


VOT fx


V = ~квТ J0


where Pxz    is    the xz    component    of the    pressure    tensor    P    given    by    (Arya


et al., 2000)








where p* is    the    momentum    vector    for    atom    i, and    rij    = r — rj    is the


vector joining the centers of molecules i and j. A weakness of these EMD methods is that the shear viscosity suffers from substantial nonmonotonic system size dependence (Hess, 2002).


The NEMD techniques usually involve measuring the macroscopic steady-state response of the system to a perturbing field and relating the linear


response to a transport coefficient. One of the earliest NEMD techniques, which maintains conventional periodic boundary conditions, involves imposing a spatially periodic external force on the molecules to generate an oscillatory velocity profile (Arya et al., 2000). The amplitude of this velocity profile at steady state is inversely related to the shear viscosity, and hence the viscosity can be calculated. The shear viscosity is wavelength dependent, however, and the Newtonian shear viscosity is obtained only in the long wavelength limit, i.e., in the limit к ^ 0, where к is the wave vector of the oscillatory perturbation. This means that a very large simulation box is required to get reasonably accurate values of shear viscosity, which limits the usefulness of this technique.

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