Chemistry Reference
In-Depth Information
where
j
y
ð
p
x
Þ
is the momentum flux,
q
v
x
=
q
y
is the velocity gradient or shear
rate, and
is the shear viscosity. The most widely used NEMD approach
for viscosity calculations is the so-called SLLOD algorithm
113
in which a shear
rate is imposed on the system and the resulting stress is computed. The shear
viscosity is found at a given shear rate from the following expression:
Z
P
xy
g
Z
¼
½
24
where
P
xy
is the
xy
component of the stress tensor and
is the shear rate.
There are several major advantages of using NEMD. Unlike equilibrium
MD methods that rely on small natural fluctuations in quantities such as the
stress, the ''signal'' in a NEMD simulation is often strong due to the external
perturbation: the stronger the perturbation, the stronger the signal. In addi-
tion, the computed quantity is determined by averaging the relevant response
variable for as long as required. This means that simulations much longer than
any correlation time can (in principle) be carried out. Finally, equilibrium MD
only gives a linear response value. For example, only the Newtonian (zero
shear rate) viscosity is obtained from equilibrium MD. In contrast, NEMD
enables one to compute the shear-dependent viscosity so that shear thinning/
non-Newtonian behavior can be studied. There are downsides to the NEMD
methods, however. Application of the SLLOD method for viscosity requires
special ''sliding brick'' boundary conditions and a modification to the Ewald
sum (if it is used to compute long-range Coulombic interactions), for example.
Aside from these minor technical difficulties, a bigger problem with the
SLLOD method is that it requires the calculation of the stress, a quantity
that is difficult to converge in a simulation. Finally, because applied shear rates
in simulations are typically much larger than experimental shear rates (due to
the need to obtain an adequate signal), some method of extrapolating results
to the zero shear rate limit is required if the linear response transport coeffi-
cients are to be calculated.
An alternative NEMD method has been developed that is much simpler
to implement than is the SLLOD method, particularly for charged systems
such as ionic liquids. The method is called reverse nonequilibrium molecular
dynamics (RNEMD) and was first developed as a means for computing ther-
mal conductivity
114
but has also been applied to viscosity.
115
It differs from
conventional equilibrium and nonequilibrium methods where the ''cause'' is
an imposed shear rate and the measured ''effect'' is a momentum flux/stress.
RNEMD does the opposite; it imposes the difficult to compute quantity
(the momentum flux or stress) and measures the easy to compute property
(the shear rate or velocity profile). The method is very simple to implement
because it only requires periodic swapping of momenta between atoms at dif-
ferent positions in the box. These swaps set up a velocity profile in the system
(i.e., a shear rate). By tracking the frequency and amount of momentum
g
