Direct observation of stretched-exponential relaxation in Lennard-Jones glasses using the cage
correlation function
Eran Rabani, J. Daniel Gezelter, and B.J. Berne
Department of Chemistry
Columbia University
3000 Broadway, New York, New York 10027
(December 14, 1998)
We report on the direct observation of stretched exponential relaxation in Lennard-Jones glasses via the decay
of the cage correlation function, which measures changes in atomic surroundings. We discuss this behavior in
terms of the roughness of the potential energy surface, and obtain a distribution of hopping rates assuming that
the origin of the Kohlrausch-Williams-Watts law is from static disorder in the distribution of barrier heights.
raphy of the multidimensional potential energy surface of a
liquid with the different relaxation times in the glass.
I. INTRODUCTION
The nature of the glass transition is one of the long-standing
problems in solid state theory. Many relaxation phenomena
in fragile glass-forming materials that are just above their
glass transition temperatures ( ) differ from those same relaxations in the liquid phase. For example, relaxation times
in fragile glass-formers cross over from Arrhenius to superexponential behavior at a critical temperature which is just
above . Also well-known are the breakdown of the StokesEinstein relation between self-diffusion and viscosity and the
emergence of a two step relaxation of correlation functions
characterized by and relaxation processes.
In recent years, much of the attention has been focused
on understanding the primary cause of the various dynamical
anomalies, particularly the change in relaxation times when
the liquid is rapidly cooled below the melting temperature.
Models which have contributed to our understanding of the
dynamics of fragile glasses include the cooperative rearranging regions of Adam and Gibbs, and Goldstein’s activated
low temperature “jump” model.
We have recently developed a method to obtain estimates of
the hopping rate between basins for Zwanzig’s model of selfdiffusion, in atomic and molecular liquids. We approached
the problem by introducing the cage correlation function
which measures the rate of change of atomic surroundings in a
computer simulation. The hopping rate was associated with
the long-time exponential decay of this function. The cage
correlation concept has proven itself to be a useful tool in understanding the mechanisms of self-diffusion, and as such was
used to test the limits of the Zwanzig model, which is built
on the foundation of the activated jump and the rearranging
regions models.
What we report in this note is the use of the cage correlation function to study the hopping rate in Lennard-Jones systems that are deep within the glassy regime, and the kinetics
of the hopping process at temperatures where the roughness
of the potential energy surface is no longer a small perturbation. Unlike the decay observed in liquids, the cage correlation function shows a long time stretched exponential decay near , which is consistent with the predictions made by
mode-coupling theory for arbitrary correlation functions.
We note that our approach is closely related to the approach
taken by Stillinger and collaborators,
who link the topog-
II. CAGE CORRELATIONS
An atom’s immediate surroundings are best described by
the list of other atoms in the liquid that make up the first solvation shell. When a diffusive barrier crossing involving the
atom has occurred, the atom has left its immediate surroundings. Following the barrier crossing, it will have a slightly different group of atoms surrounding it. If one were able to paint
identifying numbers on each of the atoms in a simulation, and
kept track of the list of numbers that each atom could see at
any time, then the barrier crossing event would be evident as
a substantial change in this list of neighbors.
FIG. 1. A sketch of the idea behind the cage correlation function.
The black atom’s cage radius is denoted by the dotted line. The grey
atom was inside the black atom’s cage at time (left side), but has
exited the cage at time (right side). The value of the cage correlation function is therefore in the right figure even though four of the
original five atoms stayed within the cage radius.
The cage correlation function uses a generalized neighbor
list to keep track of each atom’s neighbors. If the list of an
atom’s neighbors at time is identical to the list of neighbors
at time , the cage correlation function has a value of for that
atom. If any of the original neighbors are missing at time , it
is assumed that the atom participated in a hopping event, and
the cage correlation function is . The atom’s surroundings
can also change due to vibrational motion, but at longer times,
the cage will reconstitute itself to include the original members. Only those events which result in irreversible changes
to the surroundings will cause the cage to decorrelate at long
1
times. The mathematical formulation of the cage correlation
function is given in our previous work, and is not repeated
in this note. A graphical representation of the cage correlation
idea is shown in Fig. 1.
Averaging over all atoms in the simulation, and studying
the decay of the cage correlation function gives us a way to
measure the hopping rates directly from relatively short simulations. We have used the cage correlation function to predict
the hopping rates in liquids, and below we discuss the results for this correlation function when the liquid is cooled
below the melting temperature.
widths in the perfect crystal are narrower in comparison with
these configurations.
There are some distinctions that can be made within each
broad family. The “liquid-like” configurations (the solid and
dotted lines in Fig. 2) can be further split based on the details of the bifurcation of the second solvation peak, while the
two groups of “defective crystals” (the dashed and dot-dashed
lines in Fig. 2) show some differences in the third solvation
peaks. Since the “liquid-like” configurations are not stable,
and since we wanted to collect data on trajectories that were
stable during the entire 7.2 ns data collection runs, we studied
only those trajectories that belong to the more stable defective crystal phase, i.e. those trajectories that belong to
described by the dot-dashed line in the figure.
III. COMPUTATIONAL DETAILS
<U>
We performed molecular dynamics simulations on systems
of particles interacting via the Lennard-Jones potential
(1)
with parameters chosen to approximate the interactions between argon atoms (
kcal/mol
Å).
is the standard Lennard-Jones potential evaluated at the
cutoff radius (
) outside of which the potential energy is set to . The simulations were carried out with 256
atoms at a reduced density of
.
The simulations were started with the atoms in the facecentered cubic (fcc) lattice, and with velocities sampled from
a Maxwell-Boltzmann distribution with a temperature that
was four times the melting temperature. Following a 100 ps
period of equilibration at this temperature, we quenched the
liquid system to a temperature just above the melting point
(
) with velocity rescaling performed every 100 fs.
The trajectory was allowed to stabilize at the new temperature
for 100 ps, and following this stabilization we quenched the
system in steps of approximately
until the system reached
. For each temperature we allowed an
equilibration of 100 ps before we performed the next velocity quench. At the reduced temperature
we started
collecting data for
ns runs, and at the end of each run we
reduced the temperature by
, stabilizing the system for 100 ps at the new temperature. This procedure was
done separately for each member of an ensemble of trajectories which had different initial configurations.
We calculated the pair distribution function,
, for each
trajectory (and at each temperature).
was computed during the 100 ps equilibration period, and we classified the configurations according to the shape of this function. We find
that there are two broadly defined types of
: one set of
configurations are “liquid-like” and have first and second solvation peaks which are characteristic of liquid simulations.
These configurations are very unstable (i.e. they often crystallize if allowed to evolve for about 200 ps). The other family
of configurations are “defective crystals” which have an additional peak between the first and second solvation peaks. This
peak is the signature of the fcc structure, although the peak
0.00
0.25
*
0.50 T
0.75
1.00
1.25
g(R)
6.0
4.0
2.0
0.0
0.5
1.5
R / Re
2.5
3.5
FIG. 2. Plots of the pair distribution function (at
) of
the four groups of glasses (lower panel) and of the averaged potential
energy classified according to these groups (upper panel). The solid
and dotted lines correspond to “liquid-like” configurations (note the
bifurcation of the second solvation peak), while the dashed and dotted-dashed lines correspond to “defective crystals” (which have an
additional peak between the first and second solvation shells). The
full circle in the upper panel is the potential energy of the perfect fcc
crystal structure. There is a small discontinuity in the average potential of the structures represented by the solid line. This is due to the
change in the cooling rate that is required to observe these unstable
structures.
In the upper panel of Fig. 2 we show the potential energy at
each temperature, averaged over all configurations that can be
grouped in the same
families. At a reduced temperature
of about
(which is much higher than
for Argon)
the averaged potentials for these structures deviates from the
averaged liquid potential energy (as extrapolated from the liquid regime) for all four groups. This indicates that there is
a structural change that happens to the supercooled liquid at
this temperature. We note that this is the same temperature
at which the supercooled liquid begins to show the anoma2
lous slow-down in the dynamics, and is also the temperature
at which some of the trajectories crystallize.
Below this temperature it is hard to obtain a “liquid-like”
configuration described by the solid line in Fig. 2, and we
were only able to generate these configurations using an increased cooling rate. The configurations corresponding to the
dotted line have an average potential energy which is closer
to the defective crystals than to that of the supercooled liquid.
However, when extrapolated to
, the average potential
energy of the this group is still substantially greater than the
potential energy of the perfect fcc crystal.
We also note that the defective crystal configurations studied here are far from having a perfect fcc structure. Their average potential energy when extrapolated to
is higher
than the fcc potential energy, and the peaks in
are much
broader than those of a perfect fcc structure at the same temperature. By examining several configurations we have observed atomic level defects as well as large-scale fault planes
and twists between planes in the local packing arrangement.
The clearest evidence that these are not perfect fcc crystals is
that we observe a self-diffusion process on the time scale of
the simulation, which is absent for the perfect fcc structure.
(2)
with
for all three temperatures shown. Since the cage
correlation function is averaged over all atoms in the simulation, it cannot distinguish non-exponential decay that is due
to a static distribution of local environments in the glass from
non-exponential decay that is due to dynamically fluctuating
local environments. What the cage correlation function does
provide is a direct observation of the dynamics of the hopping between basins. In the glassy regime, these hops are not
characterized by uniform jumps that give rise to a simple exponential relaxation behavior. We note that the analysis of the
van Hove correlation function provides a very similar picture
for binary mixtures of Lennard-Jones glasses,
but it does
not directly measure the hopping between basins.
2.0
(k)
1.5
0
10
1.0
Ccage(t)
0.5
0.0
0.0
10
10
1.0
2.0
3.0
4.0
5.0
k (ns )
10
0
10
10
FIG. 4. The distribution of rates for three different temperatures
assuming a Kohlrausch-Williams-Watts law for the decay of the cage
correlation function, and a static mechanism for the non-exponential
decay. The solid line is for a temperature of
, the dotted
line is for
, and the dashed line is for
.
1
10
time (ns)
If the correlation time of the fluctuating barriers is on the
same order as the decay of the cage correlation function, then
dynamic disorder could conceivably be the cause of the nonexponential decay. Barriers that fluctuate much more rapidly
than the hopping time will result in exponential decay at long
times,
but we currently have no estimate of the correlation times for the fluctuating barriers. We thus make a simplifying assumption that the stretched behavior has only static
causes. Once we make this assumption, the cage correlation
functions that were fit to the Kohlrausch-Williams-Watts law
(Eq. (2)) can be used to obtain the distribution of rates by inverting the integrated expression for the correlation function,
FIG. 3. A log-log plot of the decay of the cage correlation function for crystalline Lennard-Jones system at three different temperatures. The open circles are for
, the open triangles are for
, and the open squares are for
. The dashed
curve is an exponential decay with a rate that equals the stretched
rate at
.
IV. RESULTS
The primary result of this paper is the observation that the
cage correlation functions in the glass cannot be fit with a
simple exponential function at long times, as was done for
the liquids. The cage correlation functions shown in Fig. 3
were fit with a stretched exponential function (KohlrauschWilliams-Watts law),
(3)
where
is the distribution of rates. For
distribution of rates is known analytically,
3
, this
V. CONCLUSIONS
(4)
In this note we have shown that the cage correlation function can be used to study the complex dynamics of glassy materials. The cage correlation function measures the rate of
change in atomic surroundings, and is a direct probe of the
hopping event between basins on the multidimensional potential energy surface. We have found that when the liquid is
cooled far below the melting temperature, the cage correlation function decays with a Kohlrausch-Williams-Watts law,
with a stretched factor
for all three temperatures
studied. This is the first direct observation of such complex
dynamics for the rearranging regions in a molecular dynamics
simulation. We also obtained the distribution of hopping rates
assuming that the origin of the stretched decay is static, and
used this distribution to predict the self-diffusion constant by
extending Zwanzig’s model to include a distribution of hopping rates.
(We note that the distribution of rates can also be obtained numerically when
.)
The distribution of rates,
,
is shown in Fig. 4 for three different temperatures. In table I,
we show the best-fitting values for assuming the Kohlrausch
law with
. We also show values for and from a
non-linear fit to
that allowed both parameters to vary.
The fit values of are almost exactly
. The origin of this
value for is still unexplained.
Reduced
Temperature
0.5
0.41
0.33
ns
(fit with
1.87
0.83
0.43
)
full non-linear fit
ns
0.49
1.89
0.50
0.83
0.49
0.43
TABLE I. Best fits of the parameters in the Kohlrausch law
(Eq. (2)) to the decay of the cage correlation function for three different temperatures of the Lennard-Jones glass.
VI. ACKNOWLEDGMENTS
The distribution of rates provides a statistical view of the
hopping dynamics in glassy materials. It is also a useful
tool to obtain averaged properties such as the self-diffusion
constant. Again, if we assume that the non-exponential decay has static origins, we can obtain an expression for the
self-diffusion constant by integrating Zwanzig’s self-diffusion
model over the distribution of hopping rates
ER thanks David Reichman for discussions, and the Rothschild and Fulbright foundations for financial support. This
work was supported by a grant to BJB from the National Science Foundation.
(5)
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where
is the distribution of vibrational frequencies in
the basins. In the above equation we have introduced a maximum value for the hopping rate,
, to properly limit the
rates to those which are associated with diffusive hopping.
Rates higher than this threshold are an artifact of assuming
static barrier heights, and beyond a maximum value are unphysical. The agreement between the self-diffusion constant
calculated directly from the Einstein relation and from Eq. (5)
is very good using a single cutoff value of the rate at all temperatures, and the results are given in Table II.
Reduced
Temperature
0.5
0.41
0.33
Diffusion Constant
Einstein Relation
Cage Correlation
0.657
0.53
0.515
0.61
0.352
0.35
TABLE II. Comparison of diffusion constants measured via
molecular dynamics with those calculated using Eq. (5).
has
been set to
ps for all of these calculations, and the diffusion
constants reported are in units of Å ns .
4