ms2: A Molecular Simulation Tool for Thermodynamic Properties

3.3.6. Transport properties
In ms2, transport properties are calculated using equilibrium MD. Here, the fluctuations of a system around
its equilibrium state are evaluated as a function of time. The Green-Kubo formalism relates these microscopic
fluctuations to the respective transport properties.
Diffusion coefficients. The self-diffusion coefficient Di is related to the mass flux of single molecules within a
fluid. Therefore, the relevant Green-Kubo expression is based on the individual molecule velocity autocorrelation
function [44]. Since all molecules contribute to the self-diffusion coefficient, the autocorrelation function is
averaged over all Ni molecules of component i in the ensemble to achieve better statistics. In binary mixtures,
the Maxwell-Stefan diffusion coefficient − Dij is defined by [45]
−Dij = xj
Λii +
xi
xi
Λjj −Λij −Λji , (8)
xj
wherexi = Ni/N andΛij can be written in terms of the center of mass velocity
Λij = 1
3N
∞
0
Ni
Nj
dt 〈 vi,k(0)· vj,l(t)〉 . (9)
k=1
From the expressions above, the collective character of the Maxwell-Stefan diffusion coefficient is evident. This
leads to significantly less data for a given system size and time step and therefore to larger statistical uncertainties
than in case of the self-diffusion coefficient. Note that equations for the Maxwell-Stefan diffusion coefficient are
implemented in ms2 both for binary and ternary mixtures [45].
Shear viscosity. The shear viscosityη, as defined by Newton’s ”law” of viscosity, is a measure of the resistance
of a fluid to a shearing force [46]. It is associated with the momentum transport under the influence of velocity
gradients. Hence, the shear viscosity can be related to the time autocorrelation function of the off-diagonal
elements of the stress tensor Jp [44]
η = 1
VkBT
∞
Averaging over all three independent elements of the stress tensor, i.e. J xy
p ,Jxz
0
l=1
dt J xy
p (t)·J xy
p (0) . (10)
p andJyz p , improves the statistics.
The componentJ xy
p of the microscopic stress tensor Jp is given in terms of the molecular positions and velocities
by [46]
J xy
N−1
p =
i=1
mv x i v y
i −
N−1
N
n
n
i=1 j=i+1 k=1 l=1
r x ij
∂uij
∂r y . (11)
kl
Here, the lower indicesl andk indicate theninteraction sites of a molecule and the upper indicesxandy denote
the spatial vector components, e.g. for velocityvx i or site-site distancerx ij . The first term of Eq. (11) is the kinetic
energy contribution and the second term is the potential energy contribution to the shear viscosity. Consequently,
the Green-Kubo integral (10) can be decomposed into three parts, i.e. the solely kinetic energy contribution, the
solely potential energy contribution and the mixed kinetic-potential energy contribution [46].
11