Frustration-limited
clusters
in liquids
Steven A. Kivelson
Department of Physics, Universiq of California, Los Angeles, California 90024
Xiaolin Zhao, Daniel Kivelson, Thomas M. Fischer, and Charles M. Knobler
Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024
(Received 10 January 1994; accepted 13 April 1994)
We present a continuum theory of frustration-limited clusters in one-component glass-forming
liquids that accounts, in part, for the recently reported [Fischer et al., J. Non-Cryst. Solids, 131133, 134 (1991)], and quite unexpected, presence in simple glass-forming liquids of stable clusters
at low temperatures (T) and the even less expected persistence for very long times of these clusters
at higher Ts. The model is based on the idea that there is a local structure that is energetically
preferred over simple crystalline packing, which is strained (frustrated) over large distances;
although in a curved space the preferred packing could lead to “ideal” crystallization at
temperatures that are usually above the actual freezing temperature, in “flat” space this transition is
narrowly avoided. We are led to a new ansatz for the T dependence of the viscosity, which permits
us to collapse data for many liquids onto a universal curve.
I. INTRODUCTION
There is still no definitive understanding of glass formation. However, there is a body of thought that suggests that it
is a consequence of geometric frustration.‘-5 In this view, a
glass-forming substance has a preferred local structure into
which it could crystallize in an appropriately curved space,
but in three-dimensional space the strain associated with this
“ideal structure” would diverge with the size of the system.
This notion of geometric frustration has been useful for understanding the appearance of large-scale structures in a
number of materials, such as Frank-Kasper phases in
transition-metal alloys and the blue phases in liquid crystal~.~
However, no direct evidence for the role of geometric frustration has been reported for glass-forming materials. In this
communication we show that this model can lead to
“frustration-limited cluster” formation in the liquid: In equilibrium the concentration of clusters is low at high temperatures, T, and high at low T, but metastable high concentrations can be found even at high T’s. We further show that the
model provides a simple and natural explanation for at least
some of the observed6*7 light scattering anomalies in glassforming liquids that have been attributed by Fischer et aL6 to
the presence of clusters; in particular, it provides a rationalization for the existence of finite clusters in equilibrium, and
for the very long persistence of metastable cluster concentrations at temperatures well above the freezing point. Moreover, in turn, the success of this explanation provides support
for the idea that frustration, and perhaps clusters, play major
roles in glass formation.
In their light scattering studies of the glass-feting
liquid orthoterphenyl (OTP), Fischer et ~1.~ observed that the
turbidity and Landau-Placzek ratio were greatly enhanced in
the deeply supercooled liquid. The growth in the scattering is
not observed immediately, but develops over hours. These
observations suggest the formation of large “clusters,” a
supposition that is further supported by experiment8 that
show that the integrated intensity and the Landau-Placzek
ratio dependupon scatteringangle,i.e., upon wave number
q. Such behavior could be explained by a number of altema-
tive models. However, Fischer et al. also observed that the
enhanced scattering that develops at low temperatures persists for days after the system is heated well above the melting point. This latter observation was unexpected and remained unexplained. In our view, this metastability reflects
the fact that even above the physical freezing point, the system can be well below the “ideal freezing temperature”, Tif,
at which it would freeze into the preferred local structure in
the appropriately curved space.8
Motivated by the results of Fischer ef ~1.~ on OTP, and
initially somewhat sceptical of their findings (particularly the
long persistence of clusters above the melting point), we
have carried out detailed and comprehensive studies on a
second good glass-forming neat fragile liquid, triphenyl
phosphite (TPP);7 our results are in most ways similar to
those of Fischer et al., but because of serious hysteresis effects and slow equilibration, we do not yet consider our work
definitive. Nevertheless, we include here a description of
some of our new experimental findings that seem particularly
pertinent to the understanding of molecular glass-forming
liquids. [A fragile liquid’ is defined to be one whose viscosity has an activation energy, E7( T), that increases greatly as
the temperature is lowered from the melting point to the
glass transition temperature.]
II. THEORY
We postulate that in a good molecular glass-forming liquid there exists a locally preferred structure, such as a closepacking configuration, that cannot be extended throughout
space. This concept has been investigated by a series of
workers, among them Bernal,” Boerdijk,” Frank and
Kasper,12 Hoare,13 and Stillinger,5 who have recognized that
the local-preferred structure of spherical particles is icosahedral, but that the extension of this structure is thwarted by
frustration. Said differently, the inability to extend the locally
preferred configuration throughout space is a kind of frustration. The formation of locally preferred clusters is analogous
J. Chem. Phys. 101 (3), 1 August 1994
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Kivelson et a/.: Frustration-limited
2392
clusters in liquids
to the formation of nuclei in the crystallization process, but
whereas in crystallization the nuclei can grow and ultimately
extend over the entire system, in our model the growth is
limited by the frustration.
To incorporate this physics we write the free energy F of
a single cluster of characteristic size L as
F=aL’--
+L3+.s(L)L3.
0)
The first two terms are those commonly encountered in classical theories of nucleation,14 the first representing the surface free energy required to construct a cluster. The second
term represents the bulk free energy difference between the
liquid and an “ideal crystal” which can be formed in a reference system in which the locally preferred structure can be
extended globally; it has been shown that for spheres the
reference system can be taken as a properly curved space,
and that frustration arises from the mapping of the crystal in
curved space onto flat space.lT4 The third term represents the
effect of the frustration; it can be thought of as the strain
energy associated with forcing this structure into “flat”
three-dimensional space; this term is absent if the solid phase
is an ordinary crystal. Equation (1) is a continuum representation of a cluster free energy in which the physics that takes
place on a molecular length scale is neglected and which,
therefore, is not likely to be applicable to small clusters; the
model is also restricted to dilute solutions of clusters where
cluster-cluster interactions can be ignored.
The coefficient 4, is a measure of the free energy density
gained by ordering in the locally preferred structure; it is
positive at low temperatures and vanishes as the temperature
is raised towards a temperature Z”, the “ideal freezing” temperature in the absence of frustration. Typically we expect Tif
to be greater than the observed freezing temperature because
the ideal crystal has, by assumption, lower free energy than
the ordinary crystal. The surface-energy coefficient (T is positive and should, we believe, be less temperature dependent
than 4. The strain coefficient s(L)=s,L2
for L small compared to the radius of curvature of the ideal space because at
short distances any space is locally flat; this leads to a superextensive dependence of the strain energy on L. At sufficiently large L the strain coefficient s(L) must be relieved by
defects, which will result in a positive saturation value s(m);
this may result in some complex structure, perhaps a crystal
of defects.4 Because the preferred local and the ideal crystal
structures are, by assumption, quite similar, the freezing transition in the absence of frustration (e.g., in the curved space)
is likely to be either weakly first order or continuous.
It is convenient to rewrite Eq. (1) in terms of dimensionless variables:
f=E2-d3+P,
(2)
where f = (F/F,,)
is the reduced free energy, 1= L/L,,,
Lo= ( C7/sop3, Pa= mL& and r= ( &a)Lo . A related expression for the free energy of frustrated domains has been
discussed by Stillinger.5
At I= 0, the reduced free energy f(O) = 0, which is a
local minimum at all values of 3-,i.e., at all temperatures. In
our model, l=O represents the liquid. Below a temperature
T, , which, in turn, lies below Tif, a second minimum in f( I)
I
FIG. 1. f vs I at different T’S[See Eq. (Z)].
appears at 1>O, and, of course, there is a maximum between
the two minima. We take the value of 1=I, at the second
minimum to be the reduced size of the cluster, and f(l,) is
the reduced local free energy of formation of the cluster. As
T is lowered, f( 1,) decreases, and for temperatures less than
T2 one finds f(1,) <f(O) (see Fig. 1). As T is lowered below
T, , the reduced cluster size 1, increases. Although the free
energy of the cluster relative to that of the liquid has been
calculated only along a specific growth/decomposition coordinate (1), the model suggests that the free energy must indeed have a local minimum at I= 1,. The model thus implies
the existence of finite-sized equilibrated clusters, in contrast
to kinetically limited clusters.
The model indicates that just below T, , equilibrated liquids should contain a small fraction of (presumably small)
clusters. As T is lowered towards T, , the size and number of
clusters should increase. However, if the activation energy
for cluster formation is large at temperatures above T,, the
equilibrium concentration of clusters may be difficult to
achieve. Below T, the equilibrium concentrations of the now
larger clusters should be high, and since the activation free
energy for cluster formation is then relatively small, these
high concentrations should be readily achieved. In other
words, surprisingly, equilibrium cluster concentrations are
more rapidly obtained at low temperature than at high temperature. Moreover, if now the low-temperature system containing clusters is heated to temperatures above T,, but below T1, one should be able to achieve large metastable
cluster concentrations which, because of the high cluster-toliquid barrier, decay only slowly. All this is, in fact,
observedFT7
The very fact that a molecular fluid is a good glassformer suggests that our assumption that Tif lies above the
freezing point is reasonable; in our model the liquid super-
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Kivelson et al.: Frustration-limited clusters in liquids
I
1
0.14
I
I
I
I
;I
t
I
- --
transition from metaslable io stable
transition from stable to metastable
:
I
:
0.10
T
WV 0.08
I
:
0
.’
%
iii.
0.06 -
High T,.
:
:
:
:
Low -I
,,”
2393
dynamics of the cluster-to-liquid process could be described
as a first order decomposition, one would expect exponential
decay governed by a decay rate with Arrhenius dependence
on T. Since the rate is very slow, one would expect a large
activation free energy; in fact, however, it appear8 that
above the melting point the cluster concentration decays with
time approximately as (tell) 1’3 over 2.5 orders of magnitude,
with a rather mild decrease of to with increasing T. Later we
speculate on a mechanism for this decay, keeping in mind
that this particular power-law decay may be specific to the
TPP system we have studied.
IV. SUMMARY OF CLUSTER STUDY
0.04 -
0.02 -
0.00
1.6
1.7
1.0
1.9
2.0
2.1
2.2
2.3
2
FIG. 2. Reducedactivation energy.
cools because the local-preferred structure is locally much
more stable than the crystal structure: this suggests that the
activation energy for transformation from finite clusters to
the infinite crystal is very high. One might expect that the
better the glass forming potential of the material, the greater
the difference between Ti, and the physical freezing point.
The same reasoning suggests that for a very good glass
former, T, , and even T, , may lie above the freezing point. In
many cases the locally preferred structure should be closepacked, in which case cluster formation should be accompanied by a density increase; this is the case, for example, for
the frustrated icosahedral close-packing of spheres, and is
what we observe experimentally for TPP.7
III. DYNAMICS
The model represented by Eqs. (1) and (2) provides a
reasonable picture for the thermodynamic properties of clusters in a one-component liquid, but it cannot yield complete
information concerning the dynamics. It does yield free energies of activation for the liquid-to-cluster and cluster-toliquid processes (see Fig. 2), but these free energies are relevant only to a particular reaction coordinate, one in which
the size (L) of dilute clusters changes according to the rules
of preferred packing. The fact that this activation free energy
for the liquid-to-cluster process can be large at Ts above T2
can account for the fact that clusters do not readily form at
high temperatures. The fact that the activation free energy for
the cluster-to-liquid process can also be quite high can account for the fact that clusters, once formed at low temperatures, can persist in high metastable concentrations at temperatures above the melting point. Attractive as this picture
may be, one can only say that these high free energy barriers
block easy passage ‘along one likely reaction coordinate,
thereby encouraging the cluster-liquid process to proceed
along alternate, presumably less direct paths. Actually, if the
The concept of clusters of one kind or another in onecomponent liquids, particularly in supercooled liquids, is not
new.5,9-13**5However, the experiments of Fischer et al. on
OTP, and the further experiments on TPP in our laboratory,
represent the most convincing and tangible evidence to date
for their existence. A number of theories have postulated the
existence of clusters, and the consequent inhomogeneity of
the liquid, in order to explain the observed stretched exponential relaxation behavior and consequent inhomogeniety of
the fluid, but our theory, we believe, is the first to provide a
theoretical description for the observed clusters in a onecomponent liquid. The model is based on the idea that there
is a local structure that is energetically preferred over simple
crystalline packing but which is strained (frustrated) over
large distances. The model is consonant with pictures of the
glass built upon the concept of just such locally preferred but
macroscopically strained structures.1-5
Our theory has the central feature that the finite size of
the clusters is an equilibrium, not a kinetic property. The
principal achievement of the theory, besides its simplicity
and physical motivation, lies in the fact that it not only accounts for the formation of clusters at low temperatures and
their existence at high T (apparently even at equilibrium7),
but it also rationalizes the astonishingly long-lived (days)
metastable cluster concentrations observed at temperatures
well above the freezing temperature. The stability of clusters,
even above the freezing temperature, can be understood by
noting that it is the ideal freezing temperature, Tif, or the
cluster-tbreshhold temperature, T, , and not the actual freezing temperature, that is relevant to cluster formation, and
clusters can exist provide T is below T, .
The theory also explains why an equilibrium distribution
of clusters cannot be readily prepared in a liquid that has not
been previously supercooled, and why such a distribution
can apparently be prepared by allowing the clusters to form
slowly at low T. If the liquid is cooled to temperatures below
T2 and then reheated before large clusters have time to form,
no high-T clusters are observed.7 Or if the liquid is deeply
supercooled, allowed to crystallize, and then heated, no
high-T metastable clusters are observed.7 The model explains
these observations by specifying that the activation barrier
for cluster formation is high at temperatures above T, and
low below T,.
As the clusters grow in a low-T, high-viscosity system,
the observed density increases7 which suggests that indeed
the stable clusters represent a locally close-packed configu-
J. Chem. Phys., Vol. 101, No. 3, 1 August 1994
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Kivelson et a/.: Frustration-limited
2394
ration. The rate at which these clusters grow is observed to
slow down as T is lowered. This can be understood by noting
that according to the model the activation free energy for
cluster growth should be small and the dynamics of cluster
growth should then be controlled largely by the preexponential factor, which should be inversely proportional to the viscosity. The experimental data seem to confirm this.7
The simple theory presented here is based on a continuum model (not likely to be applicable to small clusters
where molecular effects dominate); furthermore, it applies to
dilute clusters in liquid, not to liquids with high concentrations of clusters such as one finds at low T and high 7,~
V. SPECULATION ON SUPERCLUSTERS
(Phenomena
above T2)
Having described our model and the limits of its applicability, we now extend it to account for a number of observations not properly incorporated by the model alone. We
speculate that the large clusters (several hundred nanometers) detected by light scattering may actually be agglomerates (superclusters) of smaller primary clusters described by
the theory: we adduce this by noting that, as reported by
Fischer et al.,6 the scattering correlation length of the clusters detected at high T (i.e., above T2 but below T,) decays
at about the same rate as does the metastable concentration
of clusters measured by the light scattering intensity. This
can be understood if equilibration among clusters is rapid
compared to the decay of the primary clusters.
We propose a mechanism, based on the concept of rapidly equilibrating superclusters, in order to rationalize the
slow and nearly temperature-independent nonexponential decay of the observed cluster concentrations at temperatures
above T,. We assume that interactions between primary
clusters (i.e., the presence of superclusters) reduce the surface energy of the primary clusters [i.e., reduce CTin Eq. (l)],
thereby reducing the activation energy for cluster decomposition into liquid. This then suggests alternate paths to decomposition into liquid via bicluster, tricluster, tetracluster
mechanisms, paths with successively lower activation energies. If one assumes rapid equilibration among dilute superclusters, these mechanisms then give rise, respectively, to
1.-1 ,t - 1127 t -1’3 asymptotic decays for the primary cluster
concentration (and for the cluster-supercluster mean size)
with successively reduced temperature dependences. And,
indeed, our experiments on TPP seem to indicate power-law
(approximately t - 1’3) decay over 2.5 orders of magnitude
with only slight temperature dependence. In the proposed
mechanism the slowness of the dynamics is associated with
the low primary cluster concentrations.
It is difficult to establish the uniqueness, or even the
applicability of any reaction mechanism. That difficulty is
operative here. But the proposed mechanism rationalizes the
power-law decay in time of cluster concentrations, the weak
temperature dependence along with the slowness of the rate,
and the fact that the observed mean size of clusters decays at
about the same rate as the number of clusters. To elevate this
picture from rationalization to theory, it would be necessary
to understand the dominance of a tetracluster mechanism.
clusters in liquids
VI. SPECULATION ON VISCOSITY
We speculate that there exists a connection between the
physics of geometric frustration that leads to cluster formation at high temperature and the dramatic low-temperature
increase in viscosity that is characteristic of the “glass transition.” This speculation leads us to an intriguing fitting
function for viscosity.
That simple cluster formation at low temperature is not
the direct cause of the high viscosity is suggested by the
observations of Fischer et aL6 that the viscosity (measured
by rotational relaxation) does not change markedly as the
enhanced scattering develops slowly with time at low T, and
that at high T the viscosity appears to be similar in liquids
both with and without clusters.6 Although this high viscosity
could be the consequence of a rapidly created high concentration of small primary clusters (too small to be observed),
which then agglomerate slowly into observed superclusters
that have little effect upon viscosity, this model would not
explain why clusters are found at high temperature (above
T,) only if the liquid is first supercooled and kept at low T
until low-temperature clustering is observed. It is not yet
clear whether the large, detected superclusters can be obtained in the absence of impurities; Fischer and
co-workers6(b) suggested that supercooling and cluster formation were promoted by doping of the liquid, and we have
found evidence that impurities may be needed to inhibit
crystallization.7
Despite the fact that we have not established any direct
correlation between clusters and viscosity, it is possible,
nonetheless, that the scale length associated with frustration
(e.g.,’that implicit in the projection of the ideal crystal into
flat space) could govern the viscosity, though not necessarily
through the presence of clusters.‘6 Thus we consider the possibility that the physics of supercooled liquids can be understood in terms of a narrowly avoided high temperature transition at Tif, rather than the approach to a low temperature
critical point To. In this context, it is, therefore, not unreasonable to assume that the activation free energy,
EJ T) = R T ln[ v( 7’)l v(m)] , for shear motion might also be
a function of T,- T or T, - T. In fact, for the eleven simple
liquids (both fragile and strong) and three polymeric liquids
for which we have collected data from the literature, excellent fits to the viscosity can be obtained over the entire experimentally accessible temperature range (down to the glass
temperature Tg) with the simple formula
E,(T)-E,@J)=BT*~~
=o
(820)
(6<0),
(3)
where E,&m), B, and T* depend upon the material, and
S= (T* - T)IT* (see Figs. 3 and 4 and Table I). T* might be
taken as T, or Ti, or some related temperature; it need not
necessarily be larger than the actual melting temperature, but
except for cresol and salol, it is. (Note that the fractional
change in T is usually small in the relevant temperature
range.)
The fits to Eq. (3) require adjustment of the temperutureindependent parameters rl(~), E,(m),
T*, and B for each
liquid. If one argues that the first two are obtained “indepen-
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Kivelson et a/.: Frustration-limited
2395
pirically; 3 was taken as the integer exponent that gave the
best results, although for some substances best fits yield exponents that are about 10% smaller. To elevate this fit from
numerology to theory, one would need to understand the origin of the exponent; indeed, we have recently made progress
on a scaling theory of the glass transition which, among
other things, produces an exponent of about 813.
Equation (3) suggests that it is neither the empirical Tg
nor the Vogel-Fulcher critical temperature, To, that is fundamental to the description of supercooled liquids, but the
high-temperature T*. The experimental realization of Eq. (3)
presented in Fig. 4 suggests that viscosity data from all supercooled liquids, fragile and strong, can be scaled and superimposed on a single universal curve. Furthermore, in accord with the conclusions of Nagel and co-workers,20’2’Fig.
100
Fitted to E(m) + ET* ( 1 -T/T*)
..-,..-,Vogel-Fulcher TD/(T - TO)
clusters in liquids
3
6
4
4 suggests that there is no turnover to Arrhenius-like
behavior at temperatures just above Tg. However, to check this
250
300
350
400
Temperature
450
picture one would like to see some associated avoided critical behavior around T* in thermodynamic quantities.
500
[K]
VII. DISCUSSION OF CLUSTER FORMATION
FIG. 3. E(T) vs T for orthoterphenyl(Refs. 17 and 18) with T,,,=328 K,
r,=243 K. and 6=(T*-T)IT*.
E,(m)=3150 K and 7+)=1.O3X1O-3
cl? (a) (0 0 0) experimental E,,(T)=T ln[ all];
(b) (---) VogelFulcher .E,(T)=DT,T/(T-T,),
with D=6.8, To=206 K, and TC,==308
K: (Ref. 19) (c) (-) cluster model fitted to Eq. (3) with B=488 and
T*=361 K.
dently” from the high-temperature asymptotic behavior of
the viscosity, then the low-temperature behavior fitted by Eq.
(3) requires only two adjustable parameters.” It is true, however, that the “universal” exponent 3 has also been set em-
0.12
0 crasol
X dibutyl phthalate
0 glycerol
I isopropyl benzene
H n-butyl benzene
X npropanol
0 naphthyl benzene
W o-terphenyl
A propylene carbonate
i
salols
0.08’
2%
1
w
m
-
0.10,
*&
Throughout this article we have attributed enhanced
angle-dependent light scattering to the presence of clusters.
Such scattering could arise from the presence of microcrystals, but for a number of reasons we do not believe this to be
the case: (1) At temperatures well below melting, crystallized
and allegedly heavily clustered systems look very different,
(2) the q dependence of the scattering is not characteristic of
particles with sharp boundaries, and (3) most convincingly,
the large enhanced scattering at temperatures above the melting point is achieved only from samples in which clusters
0.06
I
b
w
0.04,
0.02
-0.8
:
0.00
FIG. 4. [E,(T) - E,$m)]lBT*
vs s’, with &O,
where, presumably,
-0.4
-0.2
0.0
0.2
0.4
1 -T/T*
0.02
0.04
0.06
(1 -T/T*)3
vs 6, and extended to 6~0,
-0.6
0.10
0.12
“‘*
for a number of glass-forming liquids (SeeTable I). In the inset the [E,(T) - EJm)]IBT*
En(T) =E,(m). Note that TB falls at different values of 6 for different liquids.
data are plotted
J. Chem. Phys., Vol. 101, No. 3, 1 August 1994
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Kivelson et al.: Frustration-limited
2396
clusters in liquids
TABLE I. Material parameters.
B
n-butyl benzene’ab)
isopropyl benzene’ab)
propylene carbonatelc)
salol’d)
dibutyl phthalate’a.b)
o-terphenyl’.’
s-trinaphthyl benzene(‘)
n-propanol’b’
n-phenyl-cresol”’
glycerol(s)
boron oxide’h’
poly(p-chlorostyrene)
yco:;’
poly (propylene-gl
(i)
poly(viny1 acetate)“)
331
210
338
478
169
488
334
61.4
543
98.9
68.1
566
741
342
T* W
E,,(=J) (IO3 K)
204
216
241
317
298
361
531
200
301
335
1020
559
271
467
1.43
1.72
2.12
3.22
3.31
3.15
5.07
2.22
6.09
5.18
10.5
6.08
4.52
2.76
“Reference22.
bReference23.
‘Reference24.
dSeeRef. 20 (dielectric and heat capacity data).
‘Reference 17.
have previously been allowed to grow at low temperature
and not from samples that have previously been merely crystalized at low temperature.7 The enhanced scattering could
also arise from large critical-like fluctuations that are not
characterized by an appreciable activation barrier, but we do
not believe this to be the case because one would then expect, contrary to observation, that this enhanced scattering
would be observed above the melting point whether or not
the system had been previously taken to low temperature and
given time to form clusters.
Fragile liquids are often associated with small values of
the Vogel-Fulcher parameter, D, which is obtained from
low-T fits of the viscosity.” Curiously, D correlates quite
well with our high-T asymptotic viscosity activation energy
E,(a). In any case, our model suggests that the degree of
clustering should not be associated with D but with the ratio
BT*lE,(~)
(see Table I). To date, direct experimental observation of clustering has been reported for only two liquids, OTP and TPP;6,7 OTP has the largest BT*lE,(~)
of all
the liquids in Table I, and the clustering parameter of TPP
has yet to be determined.
BT*/(E,P))
T,,, (K)
47.1
26.5
38.5
47.2
5.2
56.0
35.9
5.53
26.9
6.40
6.63
52.0
44.4
57.8
185
174
218
318
238
331
472
147
327
293
723
D
2.6
4.5
5.3
5.4
6.4
6.8
8.7
15.7
15.7
17.5
6.3
4.7
6.9
To WI
112
112
136
189
127
206
278
48.3
160
130
35
338
176
260
‘Reference18.
gReference25.
hReference26.
‘Reference27.
Noting that clusters do actually seem to form, we have
also postulated that the physics of cluster formation is related
to the great increase in viscosity with decreasing temperature
that is observed in supercooled liquids. Here we do not pursue this correlation in depth, but because the high viscosity is
observed even before superclusters are formed, and also in
liquids where superclusters have not been detected at all, we
postulate the existence of smaller, primary clusters which do
directly affect the viscosity. The large superclusters may or
may not have a large effect on the viscosity (the final answer
is not yet in), but in any case, according to the model, they
are not central to the phenomenon of high viscosity in supercooled liquids. This approach is supported by the fact that it
leads us to Eq. (3) which gives excellent fits for the viscosity
over the entire accessible temperature range for both fragile
and strong simple liquids and for polymeric liquids, as well.
The fact that the viscosity (and so presumably the primary
clusters) do not exhibit hysteresis, whereas the superclusters
do, has been addressed, but it has not yet been satisfactorily
resolved.
ACKNOWLEDGMENTS
VIII. COMMENTS
We have, by means of Eq. (l), introduced a model of
frustration-limited cluster formation in order to explain the
existence and behavior of the clusters inferred from light
scattering. Not only does the model give a rationale for the
growth of clusters at low temperatures, but it explains why
such clusters can persist for long periods when heated to
temperatures above the melting point. Although previous
cluster models5”0”3 have incorporated the concept of frustration by studying the stability of specific arrangements of
spherical particles, we believe that our continuum or collective approach is the first to generalize these ideas in such a
way so as to explain all the phenomena reported by Fischer
et al. Our treatment of frustration has been based closely on
that of Sethna.3*4
This work was supported in part by the National Science
Foundation under Grants Nos. CHE 91-17192, DMR 9312606, and CHE 89-02354 at UCLA T.M. Fischer would like
to thank the Humboldt Foundation for providing a fellowship. We would also like to thank Dr. Anders Karlhede and
Professors Sidney Nagel and James Sethna for their valuable
comments.
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J. Chem. Phys., Vol. 101, No. 3, 1 August 1994
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the spacein which the locally preferred structure is unfrustratedis finite
(as when it has negativecurvature). However, in this case we can take T,
to be a crossover temperaturebetween liquidlike and ideal-solidlike behavior.
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“When at low T the concentrationof clusters gets large, the problem may
not be one of stable clusters in a liquid, but rather of growing clusters
interfering with eachother. Our model doesnot incorporatethesephenomena.
clusters in liquids
2397
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cut-off, Tcr] (seeFig. 3). If the temperaturerangeof the fitted data is very
restricted,the results are not sensitiveto the choice of T,, . However,if, in
addition, one wishes to fit the data at temperaturesabove Tcr, then one
needs,at least, a fifth parameter,E?(m), where E,(T) = E(m) for T> Tcr.
This analysis requiresone parametermore than does Eq. (3) and it yields
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fitted to the high-temperaturedata with a low-T cut off; these fits work
poorly at low temperatures.
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J. Chem. Phys., Vol. 101, No. 3, 1 August 1994
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