Universal non-diffusive slow dynamics in aging soft matter
Luca Cipelletti,
a
Laurence Ramos,
a
S. Manley,
b
E. Pitard,
c
D. A. Weitz,
b
Eugene E. Pashkovski
d
and Marie Johansson
d
a
GDPC cc 26, Universite
´
Montpellier II, Place E. Bataillon, 34095, Montpellier Cedex 5,
France. E-mail: lucacip@gdpc.univ-montp2.fr
b
DEAS, Harvard University, Cambridge, MA, USA
c
LPM, Universite
´
Montpellier II, Montpellier, France
d
Colgate-Palmolive Co., 909 River Rd, P.O. Box 1343, Piscataway, NJ 08855-1343,
USA
Received 9th May 2002, Accepted 7th June 2002
First published as an Advance Article on the web 25th September 2002
We use conventional and multispeckle dynamic light scattering to investigate the dynamics
of a wide variety of jammed soft materials, including colloidal gels, concentrated
emulsions, and concentrated surfactant phases. For all systems, the dynamic structure
factor f (q,t) exhibits a two-step decay. The initial decay is due to the thermally activated
diffusive motion of the scatterers, as indicated by the q
2
dependence of the characteristic
relaxation time, where q is the scattering vector. However, due to the constrained motion of
the scatterers in jammed systems, the dynamics are arrested and the initial decay terminates
in a plateau. Surprisingly, we find that a final, ultraslow decay leads to the complete
relaxation of f (q,t), indicative of rearrangements on length scales as large as several microns
or tens of microns. Remarkably, for all systems the same very peculiar form is found for the
final relaxation of the dynamic structure factor: f (q,t) exp[(t/t
s
)
p
], with p 1.5 and
t
s
q
1
, thus suggesting the generality of this behavior. Additionally, for all samples the
final relaxation slows down with age, although the aging behavior is found to be sample
dependent. We propose that the unusual ultraslow dynamics are due to the relaxation of
internal stresses, built into the sample at the jamming transition, and present simple scaling
arguments that support this hypothesis.
I Introduction
Disordered, solid-like materials are ubiquitous in soft condensed matter. They range from foams to
polymer or particle gels, concentrated emulsions or colloidal suspensions. These materials, whose
applications are countless, are typically obtained by a fluid-to-solid transition, which often
quenches the system in a far-from-equilibrium configuration. Recent work has focused on the
shared features of the solid–fluid transition in disordered materials, leading to the introduction of
the concept of jamming. Liu and Nagel
1
have proposed a 3-dimensional jamming transition phase
diagram, which unifies a wide range of fluid–solid transitions; Trappe et al.
2
have shown that a
jamming phase diagram can indeed be established for a large variety of attractive colloidal systems.
Other recent investigations have also pointed out the similarities between jammed systems such as
gels and glasses.
3,4
In jammed systems, the mobility of the constituents is extremely reduced, due to
DOI: 10.1039/b204495a Faraday Discuss., 2003, 123, 237–251 237
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their crowding, and/or to the presence of strong attractive or repulsive interactions. Conversely, a
jammed system may be fluidized by applying an external stress s that exceeds a critical yield stress.
Despite the advances in the rationalization of the jamming transition, the behavior of soft
materials in the jammed phase itself is still poorly understood. One of the most striking features of
jammed, out-of-equilibrium colloidal systems, as well of other glassy materials,
5,6
is their aging
behavior: sample properties as measured by correlation or response functions change continuously
with time, as the sample evolves very slowly towards equilibrium. In particular, these systems
exhibit ultraslow relaxations, whose characteristic time increases with age. Dynamic light scattering
(DLS) provides a unique tool to probe the slow dynamics and the aging without perturbing the
system; moreover, the introduction of new techniques, such as the multispeckle method,
7,8
over-
comes the difficulties traditionally posed by ultraslow, non-stationary dynamics and non-ergodi-
city.
9
In this paper, we present light scattering measurements of the ultraslow dynamics and the aging
of several soft matter jammed systems. Remarkably, we find that for all systems the dynamic
structure factor exhibit the same very unusual behavior: at long times an ultraslow, ‘‘ compressed-
exponential’’ relaxation, whose characteristic time scales as the inverse scattering vector, leads to
the complete loss of correlation of the scattered light. This behavior is in sharp contrast with the
diffusive or sub-diffusive, slower-than-exponential relaxation typically observed when approaching
the jammed phase from the fluid side. We propose a simple model to explain these uncommon
dynamics, based on the relaxation of internal stresses, which are built in the sample at the jamming
transition. The observation of the very same dynamics in systems ranging from tenuous colloidal
fractal gels to concentrated emulsions, and from lamellar gels to micellar polycrystals suggests the
generality of this behavior in disordered, jammed, soft materials, underlying the central role of
stress relaxation on the system evolution.
The paper is organized as follows: in Section II we present the four experimental systems
investigated and we recall the main features of the multispeckle light scattering technique, while in
Section III we show the experimental results and briefly discuss the fast dynamics. The ultraslow
dynamics, which represents the main focus of the paper, is discussed in Section IV.
II Materials and methods
The colloidal gels are obtained by salt-induced aggregation of polystyrene spheres of radius
a ¼ 10.5 nm, suspended in a buoyancy-matching mixture of H
2
O and D
2
O. The amount of salt is
chosen in such a way that the aggregation follows the diffusion-limited cluster aggregation (DLCA)
regime, resulting in the formation of a percolated network of closely-packed fractal clusters, whose
size is roughly monodisperse and of the order of several tens of microns.
10
Typical particle volume
fractions range from 10
4
to 10
3
; more details on the sample can be found in ref. 11.
The concentrated emulsions are formed by water droplets in oil (cyclomethicone) and were
prepared using a standard laboratory homogenizer. In order to match the refractive index of the oil
phase, the water phase contains 56.25% and 1.25% w/w propylene glycol and NaCl, respectively.
The droplets are stabilized by a surfactant (Copolyol, Dow Corning 5225C, at 0.25% w/w); their
average size and standard deviation are 1 mm and 0.4 mm, respectively, as determined by optical
microscopy. Prior to light scattering measurements, the samples are centrifuged for 15 min at a rate
between 1500 and 1900 rpm, in order to eliminate the air bubbles that are inevitably trapped when
loading the scattering cell. No significant changes in the droplet size distribution are found after
centrifugation, as checked by optical microscopy. Oscillatory rheology measurements exhibit the
typical behavior for concentrated emulsions:
12
for frequencies 0.1–100 rad s
1
, the elastic modulus
G
0
is essentially frequency-independent and scales as f f
c
, where f is the volume fraction of the
dispersed phase and f
c
0.6 is the critical volume fraction for the fluid–solid or jamming transi-
tion. All experiments reported here were done at f ¼ 0.777.
The micellar polycrystal is formed by a mixture of water and a commercially available triblock-
copolymer (Synperonic F108, (ethylene oxide)
127
–(propylene oxide)
48
–(ethylene oxide)
127
,by
Serva, used without any further purification), to which a small amount of oil is added (p-xylene,
from Aldrich). At low temperature (T ¼ 4
C), both the poly(ethylene oxide) (PEO) ends and the
central poly(propylene oxide) (PPO) section of the block-copolymer are soluble in water. As the
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temperature is raised, PPO becomes increasingly hydrophobic and the polymers start forming
micelles, whose PPO core is swollen by p-xylene
13
(the purpose of the oil is to increase the refractive
index contrast between the micelles and water, in order to obtain a high enough light scattering
signal). Depending on the final temperature and the sample composition, different liquid crystalline
phases can be obtained. Typically, samples are prepared by thoroughly mixing all components at
T ¼ 4
C (the system is fluid at this temperature); the temperature is then increased and the sample
solidifies. For all experiments reported here, the sample composition is 75.8/22.3/1.9 water/F108/
p-xylene (w/w %), and T is increased from 4 to 30
C at a rate of about 1
Cs
1
. Under these
conditions, the micelles arrange themselves on a face centered cubic (fcc) lattice, whose cell size is
51 nm, as revealed by small-angle X-ray scattering (SAXS) experiments.
14
The lamellar gel is formed by closely-packed polydisperse multilamellar vesicles (MLV, char-
acteristic size about 5 mm), whose bilayers are composed of a mixture of cetylpyridinium chloride
(CpCl) and octanol (Oct) (CpCl/Oct ¼ 0.95 w/w) diluted in brine ([NaCl] ¼ 0.2 M) at a weight
fraction of 16%.
15
The smectic periodicity of the lamellae is 13 nm, as measured by small-angle
neutron and X-ray scattering.
16,17
An amphiphilic block-copolymer (Synperonic F68, (ethylene
oxide)
76
–(propylene oxide)
29
–(ethylene oxide)
76
, by Serva) is added to the system at a copolymer-
to-bilayer weight ratio of 0.8. The central hydrophobic section of the copolymer adsorbs to the
bilayers, while the hydrophilic ends are swollen in water and decorate the membrane. Upon
copolymer addition, a marked and continuous hardening of the system is observed, resulting in a
so-called lamellar gel.
16–18
Similarly to the micellar polycrystals, lamellar gels exhibit a transition
from a fluid state to solid-like behavior when increasing T from about 4 to 20
C.
For all systems, we define the sample age t
w
as the time elapsed since the sample was first
quenched in the jammed phase. For the colloidal gels, the jamming is due to the growth of the
effective volume fraction as the fractal cluster size increases, eventually leading to gelation; we thus
identify t
w
¼ 0 with gelation, as detected by the arrest of the shift towards smaller scattering
vectors of the peak in the scattered intensity vs. q.
10,11
We recall that the concentrated emulsions are
centrifuged prior to scattering measurements. Since the stress induced by centrifugation provides a
means to erase any memory of the sample’s previous history,
19–22
for these samples we take t
w
as
the time elapsed since the end of centrifugation. Finally, for both the micellar polycrystals and the
lamellar gels, the jamming transition is controlled by varying T and thus t
w
is defined as the time
elapsed after the temperature (inverse) quench. For all systems, the uncertainty in the determi-
nation of the sample age is of the order of a few tens of seconds, small compared to the typical
values of t
w
that can be as long as several tens of days.
We use dynamic light scattering as a non-invasive technique to probe the sample dynamics: by
measuring the time autocorrelation function of the fluctuations of the scattered intensity, the
dynamic structure factor f (q,t) can be obtained via the Siegert relation.
23
Here, q ¼ 4pnl
1
sin(y/2)
is the scattering vector, with n the refractive index of the medium, l the in vacuo laser wave length,
and y the scattering angle. The dynamics of the fractal gels and that of the concentrated emulsions
were studied for a wide range of scattering vectors q (1.5 10
2
mm
1
< q < 1 mm
1
) by using
ultralow angle multispeckle DLS. The set-up, which is described in detail in ref. 8, is based on a
charge coupled device (CCD) detector; it allows us to measure relaxation processes as long as a few
tens of hours on length scales as large as several tens of microns. For the micellar polycrystals and
the lamellar gels, we used a combination of conventional and multispeckle wide-angle DLS in order
to access the sample dynamics on time scales ranging from a fraction of microsecond to a few tens
of hours, and at scattering vectors between 3.0 10
2
and 0.33 mm
1
(corresponding to
10
< y < 150
).
A more complete discussion of the multispeckle technique can be found in ref. 7 and 8; here we
simply recall that intensity autocorrelation functions are measured in parallel for several thousands
of speckles,
23
or coherence areas, by using a CCD camera and custom designed software. The
correlation functions thus obtained are averaged over sets of speckles with the same temporal
statistics. For isotropic systems and in the ultralow angle apparatus (colloidal gels), these sets of
speckles are associated with rings of pixels centered about the incident beam direction, corre-
sponding to scattering vectors q whose magnitude spans a small interval (q
0
< q < q
0
+ Dq, with
Dq/q
0
0.01), and whose azimuthal angle varies from 0 to 2p. For the concentrated emulsions
studied by ultralow angle light scattering, the averaging is done over small solid angles DO centered
about scattering directions corresponding to q either parallel or perpendicular to the direction of
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the acceleration imposed on the sample during centrifugation. Finally, for wide angle measure-
ments (micellar polycrystals and lamellar gels), the averaging is done over a small solid angle DO
centered about a given scattering direction lying in the scattering plane. Thanks to the speckle
averaging procedure, the measurement time can be as short as the longest relaxation time of the
system, several thousand times less than in traditional DLS. It is therefore possible to measure very
slow dynamics and to fully characterize their time evolution. Moreover, multispeckle DLS directly
yields ensemble-averaged correlation functions, so that no special precautions are needed to
investigate non-ergodic samples such as the jammed systems presented here. On the contrary,
special care was taken for the traditional DLS measurements, since the samples studies in this
paper are non-ergodic on the time scale of a conventional light scattering experiment. Therefore,
the measured time-averaged intensity autocorrelation function is not simply related to the desired
ensemble-averaged dynamic structure factor f (q,t) and special techniques are required to collect
and interpret the data.
24
We used the so-called ‘‘ brute force method’’, where raw correlation
function are averaged over several tens of runs (typically 100) prior to normalization, the sample
being rotated or translated between every run to probe different scattering volumes.
III Results
In this section we present the multispeckle and conventional DLS measurements on the four
systems we investigated: colloidal fractal gels, concentrated emulsions, micellar polycrystals, and
lamellar gels. For each system, we first describe and briefly discuss the fast dynamics, then we
report the experimental results for the slow dynamics and its evolution with sample age.
Colloidal fractal gels
A detailed study of the fast and slow dynamics of DLCA colloidal fractal gels, as well as of their
evolution with age, has been published in ref. 11 and 25. In view of the striking similarities with the
other systems described here, we recall briefly the main findings of these works. The dynamic
structure factor f (q,t) exhibits a two-step decay. The faster relaxation has been studied by con-
ventional DLS:
25
at short times, it can be very well described by a stretched exponential,
f (q,t) ¼ exp(D
b
q
2
t
b
), with a stretching exponent b ¼ 0.7 and a q
2
dependence that is the signature
of the diffusive nature of the fast dynamics. This initial decay is followed by a plateau, whose height
decreases with increasing q. The physical origin of this dynamics is the thermally excited internal
elastic modes of the gel, which span a wide range of length scales, due to the fractal morphology of
the system: Krall and Weitz developed a model which allows the elastic properties, namely the
elastic modulus, to be obtained from the DLS data. Note that no significant age dependence is
found for the short time dynamics. At time scales of several thousands of seconds, much longer
than those accessible to traditional DLS, multispeckle experiments
11
have revealed, totally unex-
pectedly, the existence of a second, ultraslow relaxation, which leads to the full decay of the
correlation function. The complete loss of correlation of the scattered light indicates that rear-
rangements occur on length scales of the order of 2p/q; these rearrangements thus involve dis-
placements over distances as large as tens of microns, comparable to the cluster size. The slow
relaxation has the very peculiar shape f (q,t) / exp[(t/t
s
)
p
], with p 1.5 and a very unusual q
dependence of the characteristic decay time: t
s
q
1
. Because the stretching exponent is larger than
one, the final relaxation is faster than exponential; in the following we shall refer to it as a
‘‘compressed’’ exponential and indicate by p the ‘‘compressing’’ exponent. Moreover, we note that
the inverse-q scaling of t
s
is in sharp contrast with the q
2
scaling typical of a diffusive process, thus
ruling out diffusion as a possible mechanism for the slow dynamics. Contrary to the fast dynamics,
the slower decay of the dynamic structure factor is found to be strongly dependent on sample age
t
w
, increasing by almost three orders of magnitude over 10 days. The initial growth of t
s
is
approximately exponential, while at large t
w
the aging is almost linear (t
s
t
0:90:1
w
).
Concentrated emulsions
The short time dynamics of concentrated monodisperse emulsions has been studied by both dif-
fusing wave spectroscopy (DWS)
26
and traditional DLS.
27
At early times ( t < 1 ms), the dynamics
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is dominated by shape fluctuations of the droplets, as revealed by DWS. At longer times, the
dynamics crosses over to the subdiffusive (because of hydrodynamic interactions) local motion of
the droplets within the cage formed by their neighbors. This motion results in the initial decay of
the dynamic structure factor measured by conventional DLS. For concentrated emulsions, the
initial relaxation is followed by a well developed plateau that extends over almost all the accessible
time delays, until some decay is observed for the longest time delays (t > 200 s), in good qualitative
and quantitative agreement with the predictions of the extended mode coupling theory.
Interestingly, the incipient decay observed at the longest time delay has been ascribed to the
deformability of the liquid droplets and to the relaxation of internal stresses built up as the sample
is loaded in the cell; moreover, a slow increase of the final relaxation time with sample age has been
observed.
To better investigate the slower relaxation of f (q,t) in concentrated emulsions, we performed
ultra-low angle multispeckle DLS measurements on a similar (but polydisperse) system. The
dynamic structure factors measured simultaneously for several q parallel to the direction of the
acceleration imposed by the centrifugation and for a sample age of 300 s are shown in Fig. 1(a).
Surprisingly, we find the same behavior as for the colloidal gels: the dynamic structure factor
decays with a compressed exponential shape, f(q,t) / exp[(t/t
s
)
p
], with an average exponent
p 1.50 0.08 (the lines are compressed exponential fits to the data; for f (q,t) < 0.25, the
experimental data are affected by stray light contributions and were not included in the fit). The q
dependence of the slow relaxation time is found to be very close to that of the colloidal gels:
t
s
q
0.900.3
(see inset), once again ruling out diffusive motion as the physical mechanism
responsible for the decay of f (q,t). Interestingly, at small sample ages we find a similar form for the
correlation functions measured for q perpendicular to the centrifugation acceleration, with a
relaxation time larger than that measured for q parallel. This anisotropy in the dynamics is con-
sistent with the hypothesis that the slow relaxation is related to the internal stress built in as the cell
is loaded and centrifuged, similarly to what was noted in ref. 27. Fig. 1(b) shows the age depen-
dence of the characteristics relaxation time t
s
measured at q ¼ 0.22 mm
1
for samples that were
centrifuged for the same time (15 min), but at different rates. As can be seen, t
s
increase drama-
tically with t
w
; similarly to the initial regime of the colloidal gels, the growth is faster than linear,
although more data will be needed to better characterize the aging. Interestingly, as a general
trend the relaxation time decreases when increasing the centrifugation rate, further suggesting
that the slow dynamics is due to the relaxation of built-in stress, which is likely to be higher
Fig. 1 (a) Dynamic structure factor measured by ultralow angle multispeckle DLS for a concentrated
emulsion of age t
w
¼ 300 s. Lines are compressed exponential fits to the data (see text), with average com-
pressing exponent p ¼ 1.50 0.08. Inset: q dependence of the characteristic relaxation time of the final decay of
f (q,t). The line is a power law fit to the data, yielding an exponent 0.90 0.03, indicative of ‘‘ballistic’’ motion
on long time scales. (b) Age dependence of the characteristic time of the final decay of the dynamic structure
factor at q ¼ 0.22 mm
1
for a compressed emulsion. Curves are labeled by the rate at which the samples were
centrifuged prior to measurements.
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