Materials Chemistry Research Dept. Bl011178 Technical Memorandum

Frank H. Stillinger

Identification of Potential Energy Landscape Signatures of Distinct Dynamical Regimes in a Glass Forming Liquid

Srikanth Sastry (1) , Pablo G. Debenedetti (1) ,
and Frank H. Stillinger (2;3) 

(1) Department of Chemical Engineering 
Princeton University, Princeton, NJ 08544 
(2) Bell Laboratories, Lucent Technologies 
Murray Hill, NJ 07974 
(3) Princeton Materials Institute 
Princeton University, Princeton, NJ 08544
BL0111780-971223-16-TM

Abstract The vast majority of natural and man­made materials attain a glassy state at low temperatures under suitable methods of preparation, exhibiting mechanical properties of a solid and microscopic structural disorder [1,2]. A comprehensive understanding of the glassy state is lacking at present, and forms a major challenge in the physical sciences [3]. A powerful idea, explored in diverse areas such as spin glasses, protein dynamics, protein folding, and population dynamics in addition to glasses is that non­exponential relaxation and other manifestations of complex dynamics in these systems are strongly determined by features of the underlying energy landscape. Concrete evidence for this picture from studies of glass formation, however, has been scarce. We present such evidence for a model glass former, obtained from computer simulations. We demonstrate that the onset of non­exponential relaxation corresponds to a well defined temperature below which the depth of the potential energy minima explored by the liquid increases with decreasing temperature, and above which it does not. A sharp transition is also observed at lower temperature when the liquid gets trapped in the deepest accessible energy basin. This transition temperature depends on the cooling rate, analogous to the experimental glass transition. We also present evidence suggesting that an abrupt increase occurs in the barrier heights separating potential energy minima at a temperature above the glass transition temperature and well below the temperature of onset of non­exponential relaxation.

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Identification of Potential Energy Landscape Signatures of 
Distinct Dynamical Regimes in a Glass Forming Liquid
 
Srikanth Sastry (1) , Pablo G. Debenedetti (1) , and Frank H. Stillinger (2;3)
 
(1) Department of Chemical Engineering 
Princeton University, Princeton, NJ 08544 
(2) Bell Laboratories, Lucent Technologies 
Murray Hill, NJ 07974 
(3) Princeton Materials Institute 
Princeton University, Princeton, NJ 08544 

The vast majority of natural and man­made materials attain a glassy state at 
low temperatures under suitable methods of preparation, exhibiting mechanical 
properties of a solid and microscopic structural disorder [1,2]. A comprehensive 
understanding of the glassy state is lacking at present, and forms a major chal­ 
lenge in the physical sciences [3]. A powerful idea, explored in diverse areas such 
as spin glasses, protein dynamics, protein folding, and population dynamics in 
addition to glasses is that non­exponential relaxation and other manifestations 
of complex dynamics in these systems are strongly determined by features of the 
underlying energy landscape. Concrete evidence for this picture from studies 
of glass formation, however, has been scarce. We present such evidence for a 
model glass former, obtained from computer simulations. We demonstrate that 
the onset of non­exponential relaxation corresponds to a well defined tempera­ 
ture below which the depth of the potential energy minima explored by the liquid 
increases with decreasing temperature, and above which it does not. A sharp 
transition is also observed at lower temperature when the liquid gets trapped 
in the deepest accessible energy basin. This transition temperature depends on 
the cooling rate, analogous to the experimental glass transition. We also present 
1 

evidence suggesting that an abrupt increase occurs in the barrier heights sep­ 
arating potential energy minima at a temperature above the glass transition 
temperature and well below the temperature of onset of non­exponential relax­ 
ation. 
Glasses can be formed in numerous ways [4]. The term `glass transition', however, is 
commonly used to designate the phenomenon in which materials maintained in the liquid 
state, upon cooling to sufficiently low temperatures, exhibit the disappearance of struc­ 
tural relaxation processes characteristic of liquids, thereby becoming rigid, while retaining 
microscopic structural disorder. Though it is conventional to speak of a glass transition 
temperature T g where this change of behavior occurs, it is widely recognized that the ex­ 
perimentally determined T g of a material depends on the experimental protocol, e.g., how 
fast the liquid is cooled [1]. Thus, unlike phase transitions such as melting or vaporization, 
the glass transition is kinetically controlled, and the possible existence of a thermodynamic 
transition underlying the experimental kinetic transition remains a subject of controversy 
[5--7]. Part of the reason for the essential role played by kinetics in the glass transition phe­ 
nomenology is the fact that relaxation rates in the liquid become progressively sluggish as T g 
is approached, changing by over a dozen orders of magnitude between temperatures near T g 
and a temperature twice as high [1]. Apart from the rapid change in the time scales involved, 
a glass­forming liquid at low temperatures exhibits `complex dynamics' such as nonexponen­ 
tial relaxation [1], breakdown of the Stokes­Einstein relation [8,9] and translation­rotation 
decoupling [10] . The features of microscopic disorder, complex dynamics and loss of relax­ 
ation at finite temperature observed in glass­forming liquids find analogies in various other 
`complex systems' most notably spin glasses [11] and protein dynamics [12] and protein 
folding [13]. 
An appealing paradigm that has found fruitful application in qualitative understanding of 
complex dynamics is that of the influence of a system's `Energy Landscape' on the relaxation 
processes it displays [14,15]. In the context of glass forming liquids, the overall picture is as 
follows: the dynamics of the system is viewed as the motion of the `state point' (described 
2 

by the coordinates of all particles) in the 3N­dimensional configuration space, where N 
is the number of particles assumed here to possess no internal degrees of freedom. The 
potential energy of the system defined as a function of particle coordinates then defines 
a 3N­dimensional surface or `landscape' with respect to this configuration space, and is 
in general a very complicated function. We may partition the configuration space into 
basins for each of the many potential energy minima present in this space, such that a 
local minimization of the potential energy maps any point in a basin to the corresponding 
minimum. The minima then serve to define a coarse­grained description of the configuration 
space. The properties of the system at a given temperature is dictated by the potential 
energy basins typically sampled by the system and their mutual accessibility (as given by 
the energy barriers separating them). At high temperatures, the available kinetic energy 
permits access to most potential energy basins, and the typical basins sampled are simply 
determined by the relative number of basins at different energies. At lower temperatures, 
the sampling of basins shifts to lower energy values and the mutual access of typical basins 
becomes subject to considerable `activation', the necessary energy fluctuations becoming 
rarer as the temperature is lowered. Qualitatively, such a picture can lead to a stronger 
dependence on temperature of the system's dynamics than the expectation based on the 
dependence seen at high temperatures. 
A purely kinetic description of glassy dynamics and the glass transition which has recently 
generated much interest, the mode coupling theory (MCT) [16], makes no formal contact 
with the potential energy landscape view described above (though such a connection is 
implicit in some studies in the mode coupling regime, e. g. [17]). In its idealized version, 
MCT predicts a critical temperature T c where various dynamical quantities diverge. This T c 
has subsequently been shown to lie above the experimental glass transition temperature, and 
at present is understood to be a consequence of the approximations of the idealized theory. 
It is currently believed that T c represents the temperature at which the kinetic factors 
accounted for in MCT cease to dominate the system's dynamics. It has been suggested that 
T c corresponds to the temperature where activated dynamics begins to be the dominant 
3 

relaxation mechanism [18]. Further, it has been proposed that the MCT T c marks the 
separation between temperatures where the system explores deeper regions of the potential 
energy surface and those at which the system has access to all regions of the potential energy 
surface. Figuratively, above T c the system reaches the 'top of the landscape', while below T c 
it explores deeper regions of the potential energy surface and hence the system's dynamics 
is `landscape dominated' [19]. 
Despite the importance of the landscape paradigm, direct quantitative measures of the 
manner in which a glass­forming liquid samples the potential energy landscape are scarce. 
In this letter, we report on an extensive set of computer simulations of a model glass­forming 
liquid, which precisely quantify the features of the potential energy landscape sampled upon 
supercooling, and correlate these features with observed dynamical behavior. Our principal 
result is to identify a direct correspondence between the temperature dependence of the 
depth of the potential energy minima explored and the existence of three distinct dynamical 
regimes in the liquid: a high temperature regime; a `landscape­influenced' regime where 
non­exponential relaxation sets in; and a `landscape­dominated' regime where activated 
relaxation occurs 1 . Furthermore, we show that the temperature range where the system 
begins to explore progressively deeper regions of the potential energy landscape, i. e. the 
`landscape­influenced' regime corresponds to the onset of slow dynamics as described by 
MCT, rather than to activated dynamics, as considered previously [19]. Different dynamical 
regimes studied recently with the application of replica theory to glasses [20] appear to be 
similar to the ones we identify in this work; however, further study is required to elucidate 
this comparison. 
The system we study by computer simulations is a binary mixture of particles interacting 
1 We use the term `activated relaxation' here as commonly used in the supercooled liquids litera­ 
ture, e. g. [18], to denote relaxation dominated by episodic particle motions that require overcoming 
appreciable energy or activation barriers. 
4 

via a Lennard­Jones potential, which was originally proposed as a model for simulating 
Ni 80 P 20 [21] and has recently been widely studied as a convenient model glass former for 
computer simulations [22,23]. We simulate this system at a wide range of temperatures, 
ranging from very high to deeply supercooled temperatures. Each set of simulations is 
carried out at constant volume, and at a given cooling rate. After obtaining configurations 
from molecular dynamics simulations, we perform a local minimization of the potential 
energy of selected configurations. These energy minimum configurations (called inherent 
structures [24]) serve as markers of the configuration space explored by the system at any 
given temperature. 
In Figure 1(a) we show the temperature dependent energies of these energy minimum 
configurations, obtained by averaging over roughly a hundred configurations at each tem­ 
perature. Data are shown for four different cooling rates, spanning over two orders of 
magnitude. These curves show three distinct regimes: At high temperature (T ? 1:0), the 
energies do not change with temperature, which we confirm further by simulations done at 
temperatures T = 3:0; 4:0; 5:0 (data not shown). Between T = 1:0 and a lower value in the 
range of T = 0:3 to 0:4, the energies are found to decrease progressively. Below this range 
of temperatures, the energies are constant once again. The manner in which these average 
inherent structure potential energy values are attained also proves insightful. We show in 
Figure 1(b) the set of energies obtained at selected temperatures. We see that at high and 
intermediate temperatures, these minimum potential energies cover a broad range, and only 
show a gradual trend towards lower values as the temperature is lowered. This observation 
confirms the picture that at high temperatures the system has access to the entire range 
of energy minimum basins, and that intermediate temperatures are characterized by a pro­ 
gressive biasing of the manner in which the landscape is sampled towards lower energies. 
We notice at low temperatures, however, a qualitative difference, with the sampled energies 
becoming quite narrowly distributed around the average values. 
We focus first on the crossover seen around T = 1:0. Based on a qualitative picture of the 
influence of the energy landscape on the system's behavior, we expect that below T ¸ 1:0, a 
5 

qualitative change should occur in the dynamics. To evaluate this expectation, we consider 
the self intermediate scattering function F s (k; t) [25], which is the Fourier transform at given 
wavevector k of the van Hove self correlation function defined as 
G s (r; t) = 1 
N 
N 
X 
i=1 
hffi(jr i (t) 
reaches the 
value 1=e. The temperature dependence of the relaxation time ø (T ) at high temperatures 
displays a simple Arrhenius form, ø = ø o exp(E=kB T ). Data displayed in Figure 2(a) shows 
that this simple behavior breaks down around T = 1:0, below which the measured ø are 
progressively higher than the expectation based on high temperature behavior. We note that 
this behavior corresponds to that of `fragile liquids' in the strong­fragile classification [4]. At 
high temperatures, the time dependence of F s (k; t) is expected to follow a simple exponential 
function, while at lower temperatures, numerous studies show that the Kohlrausch­Williams­ 
Watts stretched exponential form, 
F s (k; t) = Aexp( 
time behavior, we consider the functions Fs (k;t 
(2). We adopt this procedure as a convenient way of eliminating from consideration the 
short time decay of F s (k; t), which has a Gaussian dependence on time t [25]. The resulting 
fits as well as the values for fi(T ) are shown in Figure 2(b), which show that fi(T ) drops 
to values considerably below 1 as the temperature decreases below T = 1:0. The change 
from values close to 1 to lower values, however, occurs over a broader temperature range 
compared to the corresponding deviation from high temperature behavior of ø (T ). These 
results demonstrate unequivocally the connection between the onset of `slow dynamics' and 
a change in the manner of exploration of the potential energy landscape. Such a connection 
6 

has been discussed in the literature in a qualitative fashion for many years [1,4,15]. Our 
results constitute the first clear demonstration of this connection for a model glass forming 
liquid. 
We consider next the crossover in behavior at low temperatures, where the inherent 
structure energies become constant. It is clear from Figure 1(a) that the zero temperature 
values of the energies depend on the cooling rate. Examination of the internal energy 
and pressure of the liquid from which the inherent structures are obtained reveals that 
the low temperature break in the inherent structure energy curve roughly corresponds to 
the temperature at which the liquid's properties attain the constant low temperature slope, 
which one identifies as the glass transition temperature T g [26]. Thus, calculation of inherent 
structure energies offers an alternate way for determining the glass transition temperature 
in computer simulations. The distinct advantage of our procedure is that the signature 
of the transition is very sharp. Our procedure, however, also requires considerably more 
computational effort. 
The temperature range slightly above T g where the inherent structure energies show a 
gradual change with temperature includes the temperature where mode coupling theory 
predicts a divergence in relaxation times, as determined by previous simulations on a model 
system closely related to the one we study [22]. As mentioned earlier, the extrapolated 
divergence of relaxation times at this temperature (T c ¸ 0:435 in this system) is not in fact 
present. Instead, one observes deviations from MCT predictions as T c is approached. It 
has been suggested that the reason for the observed deviation is the fact that relaxation 
mechanisms not included in the MCT description become dominant below T c . In particular, 
it has been argued that `activated processes' becomes relevant near T c [18,14]. Evidence 
for such activated processes has previously been sought [22,27] by studying the van Hove 
self correlation function. The presence of activated processes leads to secondary peaks in 
the van Hove self­correlation function, resulting from rare jumps of particles over distances 
roughly equal to inter­particle separations. Such secondary peaks have previously not been 
observed for T ? 0:466 in the model we study [22], but have been observed in a related model 
7 

[27] at low temperatures. Figure 3 shows G s (r; t) for a range of values of t for T = 0:425, 
which clearly shows the presence of secondary peaks at positions r ¸ 1:0; 1:44; 2:0; 3:0, while 
no such distinct peaks are found at higher temperatures. Thus, at T = 0:425, the system 
exhibits activated dynamics. Secondary peaks are also observed for T = 0:375. The location 
of secondary peaks roughly at integral multiples of the average particle separation suggests 
their possible origin in the cooperative motion of recently demonstrated string­like clusters 
[28]. 
Seeking indications in the potential energy topography of such a change in dynamical 
behavior, we calculate the mean squared distance \Delta 2 
R in configuration space between a 
typical liquid configuration and the corresponding energy minimum, which we define as 
\Delta 2 
R = 1 
N 
P 
i (r i 
t cooling rates studied. Two remarkable features 
that are apparent are (a) the temperature dependence of the mean squared distance \Delta 2 
R is 
independent of the cooling rate employed, and (b) a crossover from roughly linear behavior at 
low temperatures to a more rapid increase occurs at T ¸ 0:45. In addition, the dependence 
on \Delta 2 
R of the path length traversed during energy minimization also displays a break at 
T ¸ 0:45 (data not shown), indicating an increased roughness of the potential energy surface 
at higher temperatures. These observations indicate a sharp change in the local topography 
of the potential energy surface at T ¸ 0:45. 
Further insight into the changes in the local potential energy topography is obtained by 
exciting individual inherent structures obtained from a given starting temperature to various 
degrees (with kinetic energies corresponding to very low excitation temperatures, T f ) and 
monitoring the distribution of distances from the starting inherent structure configuration. 
Such distributions are shown for two starting temperatures in Figure 4(b). At the lower 
temperature of T = 0:4, the location of the peaks of the distribution are roughly propor­ 
tional to T f . At the higher temperature T = 0:5, on the other hand, we find that even at 
very low excitation, the system finds other nearby basins, as evidenced by multiple peaks 
8 

and substantially higher sampled distances Such a difference indicates that while at high 
temperature the system explores a part of the landscape with low barriers between energy 
minima, at lower temperatures the system explores minima with substantially higher energy 
barriers. While a definitive interpretation of this crossover must await a more detailed study 
over a wider range of cooling rates and longer simulations, it merits attention to the extent 
that (a) for the cooling rates we have studied, the behavior is cooling rate independent, and 
(b) this crossover occurs at a temperature very close to previous estimates of T c . 
In summary, results presented here identify for the first time the temperature below which 
a glass forming liquid begins to sample deeper regions of the potential energy landscape, and 
above which the sampling no longer depends on temperature. This constitutes a significant 
advance since it eliminates a central ambiguity that has prevailed regarding the role of 
the energy landscape in determining various stages of slow dynamics. We show a clear 
topographic signature of the glass transition, namely the temperature below which the depth 
of the basins sampled doesn't change. Finally, we present evidence suggesting that as a liquid 
is cooled through the MCT T c , an abrupt increase occurs in the height of barriers between 
energy minima. 
9 

REFERENCES 
[1] P. G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton, 1996). 
[2] M. D. Ediger, C. A. Angell and S. R. Nagel, J. Phys. Chem. 100, 13200 (1996). 
[3] P. W. Anderson, Science 267, 1615 (1995). 
[4] C. A. Angell, Science 267, 192 (1995). 
[5] W. Kauzmann, Chem. Rev. 43, 219 (1948). 
[6] J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28, 373 (1958). 
[7] F. H. Stillinger, J. Chem. Phys. 88, 7818 (1988). 
[8] G. Heuberger and H. Silescu, J. Phys. Chem. 100, 15255 (1996). 
[9] I. Chang, F. Fujara, B. Geil, G. Heuberger, and H. Silescu, J. Non­Cryst. Solids 172­ 
174, 248 (1994). 
[10] M. T. Cicerone, F. R. Blackburn, and M. D. Ediger, J. Chem. Phys. 102, 471 (1995). 
[11] R. V. Chamberlin, G. Mazurkevich, and R. Orbach, Phys. Rev. Lett. 52, 867 (1984). 
[12] A. Ansari, J. Berendzen, D. Draunstein, B. Cowen, H. Freuenfelder, M. K. Hong, I. E. 
T. Iben, J. B. Johnson, P. Ormos, T. B. Sauke, R. Scholl, A. Schulte, P. J. Steinbach, 
J. Vittitow, and R. D. Young, Biophys. Chem. 26, 337 (1987); A. Ansari, J. Berendzen, 
S. F. Bowen, H. Freuenfelder, I. E. T. Iben, T. B. Sauke, E. Shyamsunder, and R. D. 
Young, Proc. Natl. Acad. Sci USA 82, 5000 (1985). 
[13] J. D. Bryngelson, J. N. Onuchic, N. D. Socci and P. G. Wolynes, PROTEINS: Structure, 
Function and Genetics 21, 167 (1995). 
[14] M. Goldstein, J. Chem. Phys. 51, 3728 (1969); J. Chem. Phys. 67, 2246 (1977); 
[15] F. H. Stillinger, Science 267, 1935 (1995). 
10 

[16] W. G¨otze and L. Sj¨ogren, Rep. Prog. Phys. 55, 241 (1992); U. Bengtzelius, W. G¨otze 
and A. Sj¨olander, J. Phys. C 17, 5915 (1984); E. Leutheusser, Phys. Rev. A 29, 2765 
(1984). 
[17] F. Sciortino and P. Tartaglia, Phys. Rev. Lett. 78, 2385 (1997). 
[18] C. A. Angell, J. Phys. Chem. Sol. 49, 863 (1988). 
[19] C. A. Angell, in Complex Behavior in Glassy Systems M. Rubi and C. Perez­Vicente 
(Eds.) (Springer­Verlag, Berlin Heidelberg, 1997). 
[20] G. Parisi, in Proceedings of International workshop on The Morphology and Kinetics of 
Phase Separating Complex Fluids (to be published). 
[21] T. A. Weber and F. H. Stillinger, Phys. Rev. B 49, 1954 (1985). 
[22] W. Kob and H. C. Andersen, Phys. Rev. E 51, 4626 (1995). 
[23] W. Kob, C. Donati, S. J. Plimpton, P. H. Poole and S. C. Glotzer, Phys. Rev. Lett. 79, 
2827 (1997). 
[24] F. H. Stillinger and T. A. Weber, Science 225, 983 (1984). 
[25] J.­P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, London, 1986). 
[26] J. R. Fox and H. C. Andersen, J. Phys. Chem. 88, 4019 (1984). 
[27] G. Wahnstr¨om, Phys. Rev. A 44, 3752 (1991). 
[28] C. Donati, J. F. Douglas, W. Kob, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Phys. 
Rev. Lett. (submitted). 
[29] S. D. Stoddard and J. Ford, Phys. Rev. A 8, 1504 (1973). 
11 

Acknowledgements: PGD acknowledges the support of the United States Department of 
Energy and the Petroleum Research Fund. 
Correspondence should be addressed to Pablo G. Debenedetti. 
12 

FIGURES 
FIG. 1. The results shown are obtained from molecular dynamics simulations of a binary 
Lennard­Jones mixture [22], with 80% of the particles are of type A, 20% of the particles of 
type B, with Lennard­Jones parameters, ffl AA = 1:0, ffl AB = 1:5, ffl BB = 0:5, oe AA = 1:0, oe AB = 0:8, 
and oe BB = 0:88. All quantities are reported in reduced units, length in units of oe AA , temperature 
in units of ffl AA =kB , and time in units of (oe 2 
AA m=ffl AA ) 1=2 , where m is the mass of the particles. The 
density ae in all cases is 1:2. The Lennard­Jones potential, with a quadratic cutoff and shifting of 
the potential at r fffi 
c = 2:5oe fffi [29], ff; fi 2 A; B is used. (a) Local energy minimization is performed 
for a subset of configurations generated in the simulations at each temperatures. The average of 
the minimum energies per particle obtained are shown. (b) Individual minimum energies are shown 
here to demonstrate the range of energies sampled for the configurations obtained at cooling rate 
= 8:33 \Theta 10 
part of the van 
Hove correlation function (self intermediate scattering function) are shown. The infinite temper­ 
ature relaxation time ø 0 is obtained by fitting ø(T ) values for T = 1:5 
vector is 
k = 7:21oe 
The van Hove self­correlation function for T = 0:425 shown for different values of t. 
Inset shows a magnification of the secondary peaks corresponding to activated events. 
13 

FIG. 4. (a) The average mean squared distance per particle \Delta 2 
R between a typical liquid con­ 
figuration and the corresponding inherent structure. The straight line is drawn as a guide to the 
eye. (b) Distribution of squared distances sampled when an inherent structure is excited in en­ 
ergies which are here labeled by the temperature value to which the kinetic energy of excitation 
corresponds. Note that all such temperature values T f are substantially lower than the glass tran­ 
sition temperature T g . For inherent structures obtained from the higher temperature T = 0:5 the 
system finds basins of nearby energy minima for all but the lowest level of excitation. For the 
lower temperature T = 0:4, this is not the case, and the system remains confined to the basin of 
the initial inherent structure. 
14