Dynamical susceptibility of glass formers: Contrasting the predictions of theoretical scenarios
Cristina Toninelli,
1
Matthieu Wyart,
2
Ludovic Berthier,
3
Giulio Biroli,
4
and Jean-Philippe Bouchaud
2,5
1
ENS 24 rue Lhomond, 75231 Paris Cedex 05, France
2
Service de Physique de l’État Condensé Orme des Merisiers—CEA Saclay, 91191 Gif sur Yvette Cedex, France.
3
Laboratoire des Verres UMR 5587, Université Montpellier II and CNRS, 34095 Montpellier, France
4
Service de Physique Théorique Orme des Merisiers—CEA Saclay, 91191 Gif sur Yvette Cedex, France
5
Science & Finance, Capital Fund Management 6-8 Bd Haussmann, 75009 Paris, France
Received 15 December 2004; published 14 April 2005
We compute analytically and numerically the four-point correlation function that characterizes nontrivial
cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-
coupling theory MCT, collectively rearranging regions CRR’s, diffusing defects, and kinetically constrained
models KCM’s. Some features of the four-point susceptibility
4
t are expected to be universal: at short
times we expect a power-law increase in time as t
4
due to ballistic motion t
2
if the dynamics is Brownian
followed by an elastic regime most relevant deep in the glass phase characterized by a t or
t growth,
depending on whether phonons are propagative or diffusive. We find in both the and early regime that
4
t
, where is directly related to the mechanism responsible for relaxation. This regime ends when a
maximum of
4
is reached at a time t = t
*
of the order of the relaxation time of the system. This maximum is
followed by a fast decay to zero at large times. The height of the maximum also follows a power law
4
t
*
t
*
. The value of the exponents and allows one to distinguish between different mechanisms. For
example, freely diffusing defects in d =3 lead to =2 and = 1, whereas the CRR scenario rather predicts
either = 1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic
behavior of
4
t
*
. MCT leads to = b and =1/ , where b and are the standard MCT exponents. We
compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system.
Within the limited time scales accessible to numerical simulations, we find that the exponent is rather small,
1, with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple
diffusive defect models, KCM’s with noncooperative defects, and CRR’s. Experimental and numerical deter-
mination of
4
t for longer time scales and lower temperatures would yield highly valuable information on the
glass formation mechanism.
DOI: 10.1103/PhysRevE.71.041505 PACS numbers: 64.70.Pf
I. INTRODUCTION
The idea that the sharp slowing down of supercooled liq-
uids is related to the growth of a cooperative length scale
dates back at least to Adam and Gibbs 1. But it is only a
few years back that this idea has started being substantiated
by convincing experiments 2–6, numerical simulations
7–14, and simple microscopic models 15–25,27. One of
the basic problems has been to find an observable that allows
one to define and measure objectively such a cooperative
length scale. An interesting quantity, proposed a few years
ago in the context of mean-field p-spin glasses 28see 29
for an important early insight and measured in simulations,
is a four-point density correlator, defined as
G
4
r
, t = 0,00, tr
,0r
, t
- 0,00, tr
,0r
, t , 1
where r
, t represents the density fluctuations at position r
and time t. In practice one has to introduce an overlap func-
tion w 28 to avoid a singularity due to the evaluation of the
density at the same point or consider slightly different corre-
lation functions 30. This quantity measures the correlation
in space of local-time correlation functions. Intuitively, if at
point 0 an event has occurred that leads to a decorrelation of
the local density over the time scale t, G
4
r
, t measures the
probability that a similar event has occurred a distance r
away within the same time interval t see, e.g., 31. There-
fore G
4
r
, t is a candidate to measure the heterogeneity and
cooperativity of the dynamics. The best theoretical justifica-
tion for studying this quantity is to realize that the order
parameter for the glass transition is already a two-body
object—namely, the density-density correlation function
Ct = 0,00, t—which decays to zero in the liquid
phase and to a constant value in the frozen phase. The four-
point correlation G
4
r
, t therefore plays the same role as the
standard two-point correlation function for a one-body order
parameter in usual phase transitions. Correspondingly, the
associated susceptibility
4
t is defined as the volume inte-
gral of G
4
r
, t and is equal to the variance of the correlation
function 28,32,33. The susceptibility
4
t has been com-
puted numerically for different model glass formers and in-
deed exhibits a maximum for t = t
*
, the relaxation time
of the system 11–14. The peak value
4
t
*
is seen to in-
crease as the temperature decreases, indicating that the range
of G
4
r
, t
*
increases as the system becomes more sluggish.
The dynamical correlation length
4
t
*
extracted from
G
4
r
, t
*
in molecular dynamics simulations grows and be-
comes of the order of roughly 10 interparticle distances when
the time scale is of the order of 10
5
microscopic time scales
0
with
0
0.1 ps for an atomic liquid. In experiments close
PHYSICAL REVIEW E 71, 041505 2005
1539-3755/2005/714/04150520/$23.00 ©2005 The American Physical Society 041505-1