Dept. BL0111780
Research Paper
| Materials Chemistry Research
Dept. BL0111780 Research Paper | Frank H. Stillinger |
|
Srikanth Sastry1, P. G. Debenedetti2, F. H. Stillinger 3, Thomas B. Schrfder4;5, Jeppe C. Dyre5, and Sharon C. Glotzer4;6
1
Jawaharlal Nehru Centre for Advanced Scientific Research Jakkur Campus, Bangalore
560064, INDIA
2
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544
3
Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, and
Princeton Materials Institute, Princeton University, Princeton, NJ 08544
4
Center for Theoretical and Computational Materials Science, National Institute of
Standards and Technology, Gaithersburg, Maryland, USA 20899
5
Department of Mathematics and Physics (IMFUFA), Roskilde University, PO Box
260, DK-4000 Roskilde, Denmark.
6
Polymers Division, National Institute of Standards and Technology, Gaithersburg,
Maryland, USA 20899
January 1999
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Most liquids, when cooled to low temperatures under equilibrium conditions,
freeze at a well de ned temperature to form a crystalline solid. It is, however, rou-
tinely possible to cool a liquid below its freezing temperature without an alteration
of state, and thereby study properties of the liquid under metastable, supercooled,
condidtions. Though supercooled liquids eventually attain the lower free energy
solid state through the processes of homogeneous or heterogeneous nucleation, it is
possible to maintain the supercooled liquid state over wide ranges of temperatures
for long intervals of time. Such liquids, upon progressive lowering of the tempera-
ture below the feezing temperature, display strong temperature dependence of their
relaxation properties, such that they no longer equilibrate below a nite tempera-
ture, which depends on the rate at which the liquid is cooled and the experimental
probes employed to determine equilibration. This `falling out of equilibrium' at a
nite temperature constitutes the laboratory glass transition [1]. A detailed under-
standing of the strong, apparantly divergent, temperature dependence of relaxation
properties and their possible origin in a thermodynamic, ideal, glass transition is
at present incomplete, and forms the motivation of the work presented here.
The approach we consider is the characterization of the con guration space ex-
plored by the system in terms of the potential energy minima to which instanta-
1
neous con gurations of the liquid map under a local minimization of the potential
energy. The properties of such local minima, termed inherent structures [2], have
been studied by many workers as a useful approach to understanding glassy behav-
ior [3{15]. Particular issued addressed are the relationship between the distribution
of inherent structures as a function of their energy and the nature of the thermody-
namic transition that may underlie glassy behavior [4,7,9], the role of transitions be-
tween inherent structure basins and dynamics [5,6,10,12{15], and structural change
upon lowering temperature [8]. Similar studies have also been conducted to anal-
yse local `free energy' minima for the hard sphere system [16]. Here, we discuss
results concerning the onset of `slow dynamics' (the onset of stretched exponential
relaxation and deviation from the Arrhenius temperature dependence of relaxation
times) and the transition to a temperature regime wherein relaxation has been
argued to be dictated by activated transitions between potential energy minima
across signi cant energy barriers. The results are obtained from computer simula-
tions of two model liquids. Each is a binary mixture of atomic species, interacting
via the Lennard-Jones potential. The rst model liquid (referred to henceforth as
the `80:20 binary mixture') consists of 80% of particles of type A and 20% of par-
ticles of type B, with Lennard-Jones parameters AA = 1:0, AB = 1:5, BB = 0:5,
AA = 1:0, AB = 0:8, and BB = 0:88. The second model liquid (referred to as
the `50:50 binary mixture') consists of 50% each of particles of type A and B, with
parameters BB=AA = 5=6, AB = (AA + BB)=2, and AA = AB = BB, and
a ratio of masses mB=mA = 1=2. In each case, molecular dynamics simulations
are performed at a series of temperatures. Inherent structures are generated by
local minimization of the potential energy of selected instantaneous con gurations.
Further details have been presented elsewhere [12,14,15].
0.4 0.9 1.4 1.9
Temperature
-6.975
-6.925
-6.875
Inherent Structure Energy
N = 256
N = 1372
N = 10976
N = 19652
0.5 1 1.5 2 T 2.4
2.6
2.8
3
k B T ln( t/t 0 )
FIG. 1. Shown are average inherent structure energies per particle for a range of tem-
peratures for the 80:20 binary mixture, for system sizes N = 256; 1372; 10976; 19652. Inset
shows transformed values of relaxation times plotted against temperature.
Figure 1. shows the potential energies of inherent structures as a function of
temperature for the 80:20 binary mixture, averaged over 100 con gurations for
2
sizes N = 256; 1372, over 20 con gurations for N = 10976 and 10 con gurations
for N = 19652. Even though the data shown display dependence on system size,
the qualitative behavior is the same: for temperatures above T = 1, the energies are
seen to approach a constant value, while at lower temperatures they display a clear
temperature dependence. The saturation at higher temperatures is con rmed by
calculations over a wider range of temperature values (data not shown). It is seen
that the biggest change with system size occurs between N = 256 and N = 1372,
while the energies are not signi cantly altered for larger system sizes. A systematic
study of the size dependence, however, remains to be performed. The comparison
of the temperature dependence of inherent structure energies with the relaxation
behavior of the system is performed by rst calculating relaxation times from the
self intermediate scattering function Fs(q; t) (Fourier transform of the van Hove self
correlation function Gs(r; t) which describes the probability of nding a particle a
distance r away at time t from its position at time t = 0); the wavevector magnitude
q is chosen to be close to the rst peak of the structure factor). These relaxation
times (T ) are plotted, in the inset of Figure 1, in such a manner than if they obey
an Arrhenius dependence = oexp(E=kBT ), the plotted numbers are temperature
independent. It is apparent from the inset of Figure 1 that (T ) depart from
Arrhenius behavior below T = 1, in correspondence with the temperature behavior
of the inherent structure energies. Figure 2 shows the average inherent structure
energies for the 50:50 binary mixture, which displays behavior similar to the 80:20
binary mixture. Shown in the inset of Figure 2 is the temperature dependence
of the stretching exponent obtained from stretched exponential ts of Fs(q; t),
which also shows temperature dependence corresponding to that of the inherent
structure energies, deviating from a high temperature value of 1:0 to lower values
at low temperatures. It must be noted that the temperature dependence of the
inherent structure energies and correspondingly those of (T ), (T ) are gradual
despite clearly identi able qualitative changes from high to low temperatures.
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Temperature
-6.95
-6.9
-6.85
-6.8
Inherent Structure Energy
0.4 0.6 0.8 1 1.2 T
0.5
0.6
0.7
0.8
0.9
1
b
FIG. 2. Shown are average inherent structure energies per particle for a range of tem-
peratures for the 50:50 binary mixture, for system size N = 500. Inset shows the stretching
exponent .
3
Based on power law ts to the ts of relaxation times and di usion coecients,
the mode coupling critical temperature Tc 0:45 for the 80:20 binary mixture and
Tc 0:592 for the 50:50 binary mixture. The power law singularity in relaxation
times as predicted by ideal mode coupling theory [17] is not in fact observed;
it has been argued that this singularity is instead `rounded out' due to single
particle `hopping' processes, unaccounted for in ideal mode coupling theory. It has
been argued [6,18] that mode coupling Tc coincides with a crossover temperature
discussed earlier by Goldstein [5] below which activated energy barrier crossing
become important for the relaxation of the system. A similar picture has also been
discussed in the context of mean eld theories of certain spin glass models [19{21].
Indeed, a qualitative change in the dynamics is observed near the estimated Tc
value for both models considered here. Figure 3 shows the van Hove self correlation
function Gs(r; t) for the 50:50 binary mixture, where for each temperature, the time
t is chosen such that the mean squared displacement < r
2
(t) >= 1. While at high
temperatures a Gaussian form is a good approximation of Gs(r; t), a progressive
deviation from the Gaussian shape is seen going to low temperatures. At the
two lowest temperatures, one has clearly identi able secondary peaks, interpreted
usually as indicating single particle hopping [22,23].
0 1 2 3 4
r
0
0.5
1
1.5
4 p r 2 G sA (r, t)
T = 1.06
T = 0.88
T = 0.73
T = 0.69
0 1 2 3 4
r
0
0.5
1
1.5
4 p r 2 G sA (r, t)
T = 0.66
T = 0.63
T = 0.61
T = 0.59
FIG. 3. Distribution of particle displacements, for the A particles in the 50:50 binary
mixture, 4r
2
GsA(r; t 1 ), where t 1 is de ned by hr
2
(t 1 )iA = 1. At high T the Gaussian
approximation (thick curve) is reasonable, whereas at the lowest T a second peak is
present, indicating single particle hopping.
In seeking evidence of the changes in the sampled potential energy landscape,
we rst note that the average inherent structures show no clear signature of change
in the relevant temperature range. There are, however, signatures to be found in
the topography of potential energy basins, and in transitions between basins, as
we now discuss.
4
0 0.1 0.2 0.3 0.4
Root Mean Squared Distance
0
0.5
1
1.5
Path Length
0 0.5 1
Temperature
0
0.5
1
1.5
Path Length
FIG. 4. The length of the path taken by the system during potential energy minimiza-
tion, from the instantaneous con guration to the inherent structure, showing marked
deviation from low temperature behavior above T = 0:5. The inset shows the dependence
of the path length on the root mean squared distance between instantaneous and inherent
structures, which con rm that the excess increase in the path length at T > 0:5 is not
simply due to an increased in the Euclidean distance between instantaneous and inherent
structure con gurations. The straight line in the inset is drawn to guide the eye.
It has been shown in Ref. [12] that the mean squared distance between instan-
taneous con gurations and the corresponding inherent structures displays a devia-
tion from linearity (the expected behavior for a harmonic system) for temperatures
above the estimated Tc for the 80:20 binary mixture. The mean squared distance
is the Euclidean distance in con guration space between the location of an instan-
taneous con guration R (t) and the corresponding inherent structure con guration
R I
(t). Alternately, one can consider the `path length' travelled by the system in
the course of nding the local energy minimum starting from an instantaneous
con guration, which provides information about the `ruggedness' of the potential
energy basins. Figure 4 shows the temperature dependence the path length for the
80:20 binary mixture for a cooling rate (see Ref. [12] for details) of 8:33 10
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