Crossover from Fragile to Strong Glassy Behaviour in Kinetically Constrained Systems

Abstract

We show the existence of fragile-to-strong transitions in kinetically constrained systems by studying the equilibrium and out-of-equilibrium dynamics of a generic constrained Ising spin chain which interpolates between the symmetric and fully asymmetric cases. We find that for large but finite asymmetry the model displays a crossover from fragile to strong glassy behaviour at finite temperature, which is controlled by the asymmetry parameter. The relaxation in the fragile region presents stretched exponential behaviour, with a temperature dependent stretching exponent which is predicted. Our results are confirmed by numerical simulations. Comment: 4 pages, 3 figures

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arXiv:cond-mat/0104340v1 [cond-mat.stat-mech] 18 Apr 2001
Crossover from Fragile to Strong Glassy Behaviour in Kinetically Constrained Systems
Arnaud Buhot∗and Juan P. Garrahan†
Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP, U.K.
(April 18, 2001)
We show the existence of fragile-to-strong transitions in kinetically constrained systems by study-
ing the equilibrium and out-of-equilibrium dynamics of a generic constrained Ising spin chain which
interpolates between the symmetric and fully asymmetric cases. We find that for large but finite
asymmetry the model displays a crossover from fragile to strong glassy behaviour at finite tem-
perature, which is controlled by the asymmetry parameter. The relaxation in the fragile region
presents stretched exponential behaviour, with a temperature dependent stretching exponent which
is predicted. Our results are confirmed by numerical simulations.
PACS numbers: 64.70.Pf, 75.10.Hk, 05.70.Ln
Glasses are everywhere in nature. It is difficult to find a
liquid which when supercooled does not form the amor-
phous, microscopically disordered solid we call a glass.
The salient feature of supercooled liquids is their dramat-
ical slowing down with decreasing temperature, signaled
by the increase of their relaxation times and viscosities
by several orders of magnitudes in a temperature range
of a few decades. For reviews see [1–3].
Given the generic nature of the glassy state, universal
principles for the classification of glass-forming materi-
als are of paramount importance. Central to this is the
concept of fragility [4,1], which measures the speed with
which viscosity and relaxation times grow as a system ap-
proaches the glass transition temperature. Liquids which
display Arrhenius behaviour, that is, the logarithm of
their viscosity or relaxation time grows linearly with in-
verse temperature, are classified as ‘strong’ [4,1], as for
example network liquids like SiO2and GeO2, and define
one of the extremes of the classification by fragility. Most
liquids, however, behave in a non-Arrhenius manner, and
the larger the departure from Arrhenius behaviour, the
more ‘fragile’ [4,1], the most fragile liquids, which define
the second extreme in fragility, being polymeric in nature.
An exception to this classification is supercooled water
[5]: at temperatures close to its melting point it behaves
as extremely fragile, while near the glass transition it is
very strong. Supercooled water has a fragile-to-strong
transition.
Some of the simplest systems which display the slow co-
operative relaxation characteristic of glasses are the facil-
itated kinetic Ising models, first introduced by Fredrick-
son and Andersen [6], in which glassiness is not a con-
sequence of either disorder or frustration in the interac-
tions, but of the presence of kinetic constraints in the
dynamics of the system. Depending on whether the con-
straints are isotropic [6], or fully directed [7], these mod-
els may behave as strong or fragile glasses, and the low
temperature dynamics can be understood in terms of ac-
tivation over energy barriers [8–11]. They are particu-
larly useful in the study of activated processes, which
become highly relevant for supercooled liquids near the
glass transition, but are not taken into account by ap-
proximations like mode-coupling theory [12] or mean-
field models [13,14].
The purpose of this Letter is to show that fragile-to-
strong transitions also occur in kinetically constrained
systems. We prove this for the simplest case of the one-
spin facilitated Ising chain. We study a generalization of
the kinetically constrained Ising chain which interpolates
between the cases of symmetric [6] and fully asymmetric
constraints [7]. We show that for large but finite asym-
metry the model displays a fragile-to-strong crossover at
finite temperature. The crossover temperature is con-
trolled by the asymmetry parameter, which also deter-
mines the largest timescale and energy barrier of the
problem. The relaxation in the fragile region presents
stretched exponential behaviour, with a temperature de-
pendent stretching exponent which is obtained analyti-
cally. We performed extensive numerical simulations to
confirm our results.
We consider a chain of Ising spins σi
(i = 1,...,N), with periodic boundary conditions, and
Hamiltonian H =?
single flips of spins which have at least one nearest neigh-
bour in the up state. The rates for the possible transi-
tions are in general different depending on whether the
up neighbour is on the right or left, and are given by:
∈{0,1}
iσi. The dynamics is restricted to
11
b
−→ 01, 01
bǫ
−→ 11, 11
1−b
−→ 10, 10
(1−b)ǫ
−→ 11, (1)
where b ∈ [0,1] and ǫ ≡ exp(−1/T). Detailed balance
is obeyed, and the stationary distribution is the Boltz-
mann distribution at temperature T for the Hamiltonian
H. The parameter b sets the degree of asymmetry of
the kinetic constraints. The limiting values b = 1/2 and
b = 0 (or 1) correspond to the Fredrickson–Andersen
(FA) model [6] and the asymmetrically constrained Ising
chain (ACIC) [7], respectively.
Due to the non-interacting nature of the Hamiltonian
the thermodynamical properties of the model are triv-
ial, and are the same for any b. The energy density is
given by the concentration c of up spins (or ‘defects’),
which in equilibrium becomes ceq = ǫ/(1 + ǫ). At low
temperatures c is very small, and since defects facilitate
the dynamics, the system slows down: isolated up spins
1

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are locally stable and the system has to overcome energy
barriers to evolve.
In contrast with the statics, the low temperature dy-
namics depends strongly on the value of b (except at T
strictly zero [11]). In the symmetric limit b = 1/2, which
corresponds to the FA model, isolated defects can dif-
fuse to the left (resp. right) by means of processes (ii)
and (iii) [resp. (iv) and (i)] of Eq. (1). Each move re-
quires the temporary creation of one defect, so there is
a single activation barrier to diffusion ∆E = 1. This
constancy of the energy barriers implies that relaxation
times follow the Arrhenius law τFA∼ exp(∆E/T), char-
acteristic of strong glass behaviour [8,11]. The decay of
the concentration of up spins, in the out of equilibrium
regime, is well approximated by the coagulation process
A + A → A, which means that the typical lengthscale
grows as l ∼ (t/τ)1/2[8]. The situation in the asym-
metric limit b = 0 or 1, which corresponds to the ACIC
model [7], is very different. Here the activated diffusion
mechanism of the symmetric case is absent: a defect at
a distance 2n−1< d ≤ 2nfrom the nearest defect (in the
direction of the constraint) has to cross a barrier ∆E = n
to move [10], i.e., barriers grow with the logarithm of the
size of relaxing regions. This means that typical length-
scales grow as l ∼ tT ln2, which leads to a relaxation time
at low temperatures of the form τACIC∼ exp(1/T2ln2)
[10]. This is the B¨ assler law [15,2] used as an alternative
to the Vogel-Fulcher equation [1,2] to represent fragile
behaviour. The absence of a finite temperature singu-
larity is consistent with the trivial statics of the ACIC
model.
We now consider the behaviour at intermediate val-
ues of the asymmetry b. We first focus on the relaxation
towards equilibrium after a quench from infinite tempera-
ture. When b is not far from the symmetric limit b = 1/2,
the rates of Eq. (1) for reactions to the left and right are
comparable and the symmetric diffusive mechanism is
still effective: the behaviour is essentially that of the FA
model. The region of large but finite asymmetry, that is,
when b (or 1−b) is small, is more interesting. In this case
rates (i) and (ii) of Eq. (1) are very much suppressed re-
spect to rates (iii) and (iv) (or vice-versa), the system can
only make use of the asymmetric mechanism to relax, and
behaves like the ACIC. The timescales associated with
the asymmetric process, however, grow as the relaxing
regions become larger: the timescale for relaxation of a
region of length 2n−1< d ≤ 2nis τ(n)
dynamics takes place in stages labeled by n [10]. The
timescale for the symmetric process, τS∼ [b(1 − b)ǫ]−1,
is large, but not infinite, except in the ACIC limit, and
in contrast to the asymmetric one does not grow with in-
creasing length. This means that eventually τSbecomes
comparable to τ(n)
A
for some value of n. This defines the
last stage of asymmetric relaxation n∗before the system
switches to symmetric behaviour:
A
∼ ǫ−n, and the
n∗∼ 1 − T ln[b(1 − b)].(2)
10
−2
10
2
10
6
10
10
time
0.0
0.1
0.2
0.3
0.4
0.5
defect concentration
FA
10
10
10
10
ACIC
−2 = b
−3
−4
−5
−101234
T ln t
10
0
10
1
10
2
l
FIG. 1.
after a quench from infinite temperature to T = 1/6, for val-
ues of the asymmetry b = 1/2 (FA model), 10−2, 10−3, 10−4,
10−5, and 0 (ACIC model). Inset: typical lengthscale l ≡ 1/c
against scaled time variable T lnt. The upper dashed line cor-
responds to t1/2and the lower one to tT ln 2. Simulations were
performed using a continuous time Monte Carlo/Metropolis
algorithm [16,17] for a system of 105spins.
Defect concentration c as a function of time t
In Fig. 1 we show the decay of defect concentration c
(the energy density of the system) as a function of time
t after a quench from an initial state at T0= ∞ to a low
temperature T = 1/6, for several values of the asymme-
try b. For the b = 0 ACIC limit, at least four plateaus
in the concentration are visible, which correspond to the
different stages in the dynamics. The number of plateaus
decreases with increasing b in accordance with Eq. (2):
there are three plateaus for b = 10−5(n∗= 2.9), two for
b = 10−4and 10−3(n∗= 2.5 and 2.2), and only one for
b = 10−2and 1/2 (n∗= 1.7 and 1.1), the latter being
the FA limit. The Inset shows how the typical length-
scale l ≡ 1/c grows with time. While in the ACIC case
it follows tT ln2(lower dashed line), for all the nonzero
values of b the system eventually switches to the diffusive
t1/2behaviour of the FA case (upper dashed line).
Let us turn to the implications that a finite asym-
metry has on the relaxation time of the system.
arbitrary b, the symmetric and asymmetric processes
compete.The relaxation timescale is given by τ ∼
?τ−1
add up. The factor b(1 − b) in the symmetric relaxation
timescale τS can be interpreted as an entropic barrier:
τS∼ exp{[∆E − T lnb(1 − b)]/T}, where ∆E = 1. No-
tice that this ‘free energy’ barrier is precisely n∗of Eq.
(2). Thus, the suppression of the symmetric mechanism
due to b decreases with decreasing temperature. The con-
sequence of this on the relaxation time for a fixed value
of b is the following. At higher temperatures the asym-
metric process dominates, τ ∼ τACIC, and the system
For
S
+ τ−1
ACIC
?−1, assuming that the corresponding rates
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displays fragile relaxation. At lower temperatures, the
symmetric process becomes dominant, τ ∼ τS, and the
behaviour is strong. The crossover temperature for this
fragile-to-strong transition is determined by b,
Tc∼1 −?1 − 4ln[b(1 − b)]/ln2
2 ln[b(1 − b)]
,(3)
corresponding to τS∼ τACIC.
In Fig. 2 we show the equilibrium relaxation time τ
as a function of inverse temperature 1/T. Given that
the static properties of the system are known exactly
it is simple to construct low temperature equilibrium
configurations.This allows to study the equilibrium
dynamics down to really low temperatures in contrast
with most glassy systems where only out of equilibrium
quantities are accessible. We obtain the relaxation time
through the connected equilibrium autocorrelation func-
tion, C(t) ≡ N−1?
fined by C(τ) = e−1C(0). Alternative definitions of the
equilibrium relaxation timescale give similar results. The
figure shows the following features: (i) in the b = 0
ACIC limit, the relaxation time has the fragile behaviour
τACICfor all temperatures, as expected; (ii) for small b, τ
crosses over from fragile τACICat higher temperatures to
strong τS behaviour at lower ones, the crossover taking
place around Tc given by Eq. (3) (e.g., 1/Tc = 3.2 for
b = 10−5and 1/Tc= 2.9 for b = 10−4); (iii) for larger b
the fragile region shrinks, and disappears completely at
the FA limit b = 1/2. The displacement of the curves in
the strong regime is due to the entropic barrier. Rescal-
ing the relaxation time by τ → b(1 − b)τ makes them
collapse, as shown in the top-right panel of Fig. 2. The
bottom-right panel gives the effective activation barrier
∆E ≡ dlnτ/d(1/T). The crossover here appears as a
change from the linear growth in the ACIC to the con-
stant barrier of the FA model. The crossover is sharper
the smaller b. The high temperature behaviour of τ is
exponential in 1/T, which gives the offset in the straight
line for b = 0, and becomes irrelevant at low tempera-
tures. The slope of ∆E is about 1.7 rather than 2/ln2.
This difference is due to the non-exponential nature of
the autocorrelation, which does not affect the 1/T2be-
haviour of the log of τ.
i?σi(t)σi(0)? − c2
eq, where τ is de-
The fragile-to-strong crossover can also be observed
in the behaviour of equilibrium dynamical quantities.
The simplest one to study is the equilibrium persistence,
P(t) ≡ N−1c−1?
fraction of defects of the initial configuration which have
never flipped between times 0 and t. It is closely related
to the equilibrium autocorrelation C(t), but is free from
the problem of the recurrence of defects which makes the
analysis of the latter more tricky, particularly in one di-
mension.
i??t
t′=0σi(t′)?, which measures the
0246810
inverse temperature
10
0
10
2
10
4
10
6
10
8
10
10
10
12
relaxation time
ACIC
10
10
10
10
10
FA
−5 = b
−4
−3
−2
−1
0246810
10
−5
10
−1
10
3
scaled τ
024
0
2
4
6
8
∆Ε
FIG. 2.
verse temperature 1/T, for values of the asymmetry b = 1/2
(FA model), 10−1, 10−2, 10−3, 10−4, 10−5, and 0 (ACIC
model). Top-right: rescaled time b(1− b)τ against 1/T. Bot-
tom-right: effective activation barrier ∆E as a function of
1/T. Simulations were performed for system sizes which var-
ied from 105for high temperatures to 5 × 106for low ones,
and data points were averaged over twenty runs.
Equilibrium relaxation time τ as a function of in-
As mentioned before, in the asymmetric limit, the
probability to flip a defect depends on the distance to the
nearest defect in the direction of the constraint. In equi-
librium, the probability distribution of these distances
is independent of time. The persistence may be then
approximated by the sum of the independent exponen-
tial relaxation of defects at different distances from their
neighbours, P(t) ∼ ceqe−t/τ0+?∞
τ0∼ 1 is the timescale associated with the initial T inde-
pendent transient, and pn= (1−ceq)2n−1−(1−ceq)2nis
the probability for a defect to have a chain of spins 0 of
length 2n−1≤ d < 2nnext to it, which takes into account
the fact that the relaxation time for the corresponding
distances is τ(n)
A. The equilibrium condition is crucial to
assume an independent relaxation of the different length-
scales. If the initial configuration is an out of equilib-
rium one, the probability distribution of distances evolves
in time and the approximation considered is no longer
valid. For short times the persistence is dominated by the
fastest exponential decay, leading to −lnP ∼ t/˜ τ for low
temperatures, where ˜ τ ≡ τ0/ceqdefines the timescale for
short times. The long time behaviour may be estimated
replacing the sum by an integral which is evaluated in
the saddle point approximation, in a manner similar to
that of Ref. [18]. As a result we obtain a stretched ex-
?
exponent
n=1pne−t/τ(n)
A , where
ponential P(t) ∼ exp−(t/τACIC)β?
, with a stretching
β = (1 + 1/T ln2)−1.(4)
Notice that this is an alternative method to obtain τACIC
to the one of Ref. [10]. In the symmetric limit the persis-
tence is simply the sum of the relaxation of defects with
and without an up neighbour. At low temperatures it
3

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reads P(t) ∼ 2ceqe−t/τ0+ (1 − 2ceq) e−t/τS. The small
time behaviour is similar to the asymmetric case, while
the long time one is given by P(t) ∼ exp(−t/τS).
The behaviour of the persistence for b = 10−5and
b = 10−3at different temperatures is shown in Fig. 3.
We present the data in a double log scale for P and a log
scale for t to display the different stretching exponents.
As in the case of the out of equilibrium relaxation, the
decay is first dominated by the asymmetric process cor-
responding to the fragile regime. For short times it is ex-
ponential with a characteristic timescale ˜ τ ∼ ǫ−1, as de-
scribed above. At longer times we see a change of slope in
the plot, which corresponds to stretched exponential be-
haviour, with an exponent given by Eq. (4). The stretch-
ing region is larger for smaller b, as expected. The fragile-
to-strong crossover then takes place, and the persistence
becomes exponential again, now with a timescale τS. In
the Insets we rescale time by a factor of ǫ to superimpose
the curves in the exponential regimes of short and long
times. The two limiting lines correspond to exp(−t/˜ τ)
and exp(−t/τS).
We conclude with a comment on the explicit spatial
asymmetry in the definition of the model studied here.
It seems that to obtain other than strong behaviour in
systems with kinetic constraints it is necessary to con-
sider cases in which spatial isotropy is explicitly broken,
like in the ACIC [7] or its generalizations, which is a
rather unphysical feature. This is also the case of sys-
tems with interactions, but which display a dynamical
behaviour similar to the spin facilitated models [19]. The
system considered in this work, however, can be defined
in an alternative but explicitly spatially symmetric for-
mulation, which also clarifies the relation between the
asymmetry b and the timescale and lengthscale at which
the fragile-to-strong crossover takes place [20]. Consider
b as a collective field b(t) which takes values 0 and 1
with a characteristic timescale for flipping τb. The sim-
plest physical choice for this dynamics would be a Poisson
process. b(t) may also have a spatial dependence, but on
lengthscales larger than the typical ones for the spin sys-
tem. Since ?b(t)? = 1/2 spatial symmetry is unbroken.
The instantaneous value of b(t) is either 0 or 1, so for
times smaller than τbthe behaviour is that of the asym-
metric case. For times much larger than τb the system
effectively sees the average of b(t) and the behaviour is
the symmetric one. τbsets the timescale for the fragile-
to-strong crossover. This argument is easily generalized
to higher dimensions by considering a collective vector
field instead.
The authors would like to thank David Chandler, An-
drea Crisanti, F´ elix Ritort, Andrea Rocco, and Peter Sol-
lich for useful discussions. The work of AB is supported
by EU Grant No. HPMF-CT-1999-00328and that of JPG
by a Violette and Samuel Glasstone Research Fellowship.
10
−4
10
1
10
6
time
10
−5
10
−3
10
−1
10
1
− ln ?persistence
2=T
3
4
5
−1
10
−4
10
1
10
6
10
−6
10
−4
10
−2
10
0
2=T
4
6
8
−1
FIG. 3.
(left panel) and b = 10−3(right panel). Notice that we plot
−lnP(t) in a log-log scale. The dotted lines are the expected
stretching exponents β. Insets: same as main panels with
time rescaled as t → tǫ. Details of simulations are the same
as in Fig. 2.
Persistence P as a function of time t, for b = 10−5
∗
Email: buhot@thphys.ox.ac.uk
Email: j.garrahan1@physics.ox.ac.uk
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