Crossover between short- and long-time behavior of stress fluctuations and viscoelasticity of liquids
Siegfried Hess,
1
Martin Kro
¨
ger,
1,2
and Denis Evans
3
1
Institut fu ¨r Theoretische Physik, Technische Universita ¨t Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany
2
Polymer Physics, Materials Science, ETH Zu ¨rich, CH-8092 Zu ¨rich, Switzerland
3
Research School of Chemistry, The Australian National University, Canberra ACT 0200, Australia
Received 15 November 2002; published 28 April 2003
An effective viscosity coefficient is introduced based on definite time averages of equilibrium stress fluc-
tuations rather than stress correlations. Analysis of this quantity via molecular dynamics of a simple model
liquid reveals a crossover between the expected short-time elastic and the long-time viscous behavior with
increasing averaging time. The procedure allows us to extract the zero-rate shear viscosity when the averaging
time becomes one order of magnitude larger than the relevant relaxation time. A relationship between this
effective viscosity and the dynamic viscosities is established.
DOI: 10.1103/PhysRevE.67.042201 PACS numbers: 66.20.+d, 05.20.Jj, 62.20.Dc
I. INTRODUCTION
It has been demonstrated that the shear viscosity of a fluid
can be inferred from ensemble averages of the mean square
of time averages of the stress fluctuations 1,2. In particular,
the long-time behavior of the stress fluctuations was ana-
lyzed, where the time averages are taken over times large
compared with a stress Maxwell relaxation time. Then a
simple relation exists with the Green and Kubo 3 expres-
sion for linear transport coefficients. In this note we study the
crossover between the behavior of stress fluctuations aver-
aged over short times to that where the averaging times are
long compared with relevant relaxation time. We present re-
sults from molecular dynamics MD computer simulations
and compare them with analytical considerations for a model
liquid where both limiting cases, viz., short and long averag-
ing times, are accessible in the numerical calculations. An
effective viscosity coefficient is introduced which depends
on the averaging time. Analysis of this quantity reveals the
transition from the short-time elastic to the long-time viscous
behavior. The relation of the effective viscosity to the fre-
quency dependent viscosity coefficient is discussed. Under-
standing the crossover behavior of the stress fluctuations is
also important for the calculation of the viscous behavior of
fluids with long relaxation times, where it may be difficult to
reach the long-time limit. Expressions used here are similar
to those applied in an analysis of the statistical dependence
of data extracted from a dynamical interpretation of Monte
Carlo simulations 4. The present approach is also some-
what akin to the calculation of elastic constants from strain
fluctuations 5, where their dependence on a spatial length
rather than on the duration of a time–interval plays a crucial
role. In Refs. 6–8 a ‘‘hard’’ interaction 1/r
n
—rather than
the short range attractive ‘‘SHRAT’’ potential to be consid-
ered in the present work—with a large n had been used. The
crossover from hard to soft spheres had been discussed, e.g.,
in Refs. 6,9, where it is argued that there are two relevant
time scales: the Enskog mean free time, and the mean tra-
versal time for a particle to cross the steep part of the poten-
tial.
II. STRESS FLUCTUATION FORMULA
Consider a system composed of N spherical particles with
mass m and position vectors r
i
, i =1, . . . , N in a volume V.
The number density is n =N / V . In the MD simulations, pe-
riodic boundary conditions and the ‘‘minimum image con-
vention’’ are used in order to avoid boundary layer effects
10,11. In a streaming fluid, the stationary rheological prop-
erties such as the non-Newtonian viscosity and the normal
pressure differences are obtained from long-time averages of
the Cartesian components of the stress tensor
, which is
the negative of the pressure tensor p
, which in turn is the
sum of kinetic and potential contributions: p
= p
kin
+ p
pot
, p
kin
=V
-1
i
mc
i
c
i
, p
pot
=V
-1 1
2
ij
r
ij
F
ij
. Here c
i
is the peculiar velocity of particle i, i.e., its velocity relative
to the flow velocity v( r
i
), r
ij
=r
i
-r
j
is the relative position
vector of particles i , j , and F
ij
is the force acting between
them. The Greek subscripts , , which assume the values
1,2,3, stand for Cartesian components associated with the
x , y , z directions.
In an equilibrium situation where one has v=0, the shear
stress, i.e., the off-diagonal components of the stress tensor,
e.g., =- p
12
and the normal stress differences, e.g., p
22
- p
11
fluctuate about zero and their long-time averages van-
ish. The mean square average of these fluctuating quantities
depends on the averaging time t
av
12. More specifically, the
definition of a time-interval average
¯
( t
av
) =t
av
-1
0
t
av
( t ) dt
is introduced. The time dependence of ( t ) =- p
12
( t ) stems
from the time dependence of the positions and momenta of
the particles. It is understood that the integration limits 0 and
t
av
can be replaced by t
0
and t
0
+t
av
provided that these
times are also within the time span for which the phase space
trajectory is available. The mean square average is given by
¯
t
av
2
=t
av
-2
0
t
av
dt
1
0
t
av
dt
2
t
1
t
2
. 1
The angular brackets ••• indicate an ensemble average,
¯
( t
av
) =0 has been assumed. In a stationary situation
( t
1
) ( t
2
) depends on the time difference t =t
1
-t
2
only.
Upon the assumption that the equilibrium fluctuations cannot
distinguish between ‘‘past’’ and ‘‘future,’’ i.e.,
( t ) (0)
=
( -t ) (0) , the integral over t, ranging from -t
av
to t
av
is replaced by two times the integral from 0 to t
av
. Further-
more, one has to take into account that the time variable t
m
=( t
1
+t
2
)/2, for fixed t 0, lies between t /2 and t
av
-t /2.
PHYSICAL REVIEW E 67, 042201 2003
1063-651X/2003/674/0422014/$20.00 ©2003 The American Physical Society 67 042201-1