High-bandwidth viscoelastic properties of aging colloidal glasses and gels.

Abstract

We report measurements of the frequency-dependent shear moduli of aging colloidal systems that evolve from a purely low-viscosity liquid to a predominantly elastic glass or gel. Using microrheology, we measure the local complex shear modulus G;{*}(omega) over a very wide range of frequencies (from 1Hzto100kHz ). The combined use of one- and two-particle microrheology allows us to differentiate between colloidal glasses and gels-the glass is homogenous, whereas the colloidal gel shows a considerable degree of heterogeneity on length scales larger than 0.5microm . Despite this characteristic difference, both systems exhibit similar rheological behaviors which evolve in time with aging, showing a crossover from a single-power-law frequency dependence of the viscoelastic modulus to a sum of two power laws. The crossover occurs at a time t_{0} , which defines a mechanical transition point. We found that the data acquired during the aging of different samples can be collapsed onto a single master curve by scaling the aging time with t_{0} . This raises questions about the prior interpretation of two power laws in terms of a superposition of an elastic network embedded in a viscoelastic background.

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arXiv:0710.5459v1 [cond-mat.soft] 29 Oct 2007
High-bandwidth viscoelastic properties of aging colloidal glasses and gels
S. Jabbari-Farouji1,2, M. Atakhorram3,D. Mizuno3,4, E. Eiser5,6,
G.H. Wegdam1, F.C. MacKintosh3, Daniel Bonn1,7, C.F. Schmidt4,8
1van der Waals-Zeeman Institut, Universiteit van Amsterdam, 1018XE Amsterdam, The Netherlands
2Group Polymer Physics, Department of Applied Physics,
Technische Universiteit Eindhoven 5600MB Eindhoven, The Netherlands
3Divisie Natuur- en Sterrenkunde, Vrije Universiteit Amsterdam, 1081HV Amsterdam, The Netherlands
4Organization for the Promotion of Advanced Research,
Kyushu University, Higashi-ku, Hakozaki 6-10-1, 812-0054 Fukuoka, Japan
5van ’t Hoff Institute for Molecular Sciences, Universiteit van Amsterdam, 1018WV Amsterdam, The Netherlands
6University of Cambridge, Department of Physics,
Cavendish Laboratory,J J Thomson Avenue, Cambridge CB3 0HE, UK
7Laboratoire de Physique Statistique de l’ENS, 75231 Paris Cedex 05, France
8Physikalisches Institut, Georg-August-Universitt, 37077 G¨ ottingen, Germany
(Dated: February 2, 2008)
We report measurements of the frequency-dependent shear moduli of aging colloidal systems that
evolve from a purely low-viscosity liquid to a predominantly elastic glass or gel. Using microrheology,
we measure the local complex shear modulus G∗(ω) over a very wide range of frequencies (1 Hz- 100
kHz). The combined use of one- and two-particle microrheology allows us to differentiate between
colloidal glasses and gels - the glass is homogenous, whereas the colloidal gel shows a considerable
degree of heterogeneity on length scales larger than 0.5 micrometer. Despite this characteristic
difference, both systems exhibit similar rheological behavior which evolve in time with aging, showing
a crossover from a single power-law frequency dependence of the viscoelastic modulus to a sum of
two power laws. The crossover occurs at a time t0, which defines a mechanical transition point. We
found that the data acquired during the aging of different samples can be collapsed onto a single
master curve by scaling the aging time with t0. This raises questions about the prior interpretation
of two power laws in terms of a superposition of an elastic network embedded in a viscoelastic
background. keywords: Aging, colloidal glass, passive microrheology
PACS numbers:
I.INTRODUCTION
Soft glassy materials are ubiquitous in everyday life.
A common feature of all such materials is their relatively
large response to small forces (hence soft) and their dis-
ordered (glassy) nature. Pertinent examples of such sys-
tems are foams, gels, slurries, concentrated polymer so-
lutions and colloidal suspensions. These systems show
interesting viscoelastic properties; depending on the fre-
quency with which they are perturbed, they can behave
either liquid- or solid-like. In spite of their importance for
numerous applications, the mechanical behavior of such
soft glassy materials is still incompletely understood [1].
In recent decades, colloidal suspensions have been used
extensively as model systems for the glass transition in
simple liquids [2, 3, 4] and gel formation [5]; since the
diffusion of the particles can easily be measured using,
e.g., light scattering or confocal microscopy [6]. The vis-
coelasticity of such systems, especially its development
during the aging of glassy systems or the formation of a
gel, has, however, received relatively little attention.
Another issue that deserves attention is differentiating
between colloidal gels and glasses in terms of their rhe-
ological properties [7]. The main difference between col-
loidal gels and glasses stems from their structure. While
the glass has a homogenous liquid-like structure with no
long-range order, a gel can have a heterogeneous struc-
ture whose characteristic length is set by the mesh size
of the gel network (see Fig. 1) [12].
As an example of a soft glassy system, we here fo-
cus on the viscoelasticity of Laponite suspensions for
which a very rich phase diagram has been reported [8, 9].
When dissolved in water, Laponite suspensions evolve
from a liquid-like state to a non-ergodic solid-like state
[3, 4, 10, 11]. During this process the mobility of the
particles slows down and viscoelasticity develops. This
system is an interesting one to study since both colloidal
gels and glasses can be obtained depending on Laponite
concentration and added salt content [12]. Therefore, it
provides us with the possibility to investigate the sim-
ilarities and differences in the viscoelastic properties of
the two types of non-equilibrium states.
To study the mechanical properties of gels and glasses,
we used microrheology (MR), which allows us to measure
frequency-dependent shear moduli over a wide range of
frequencies. This technique is based on the detection of
small displacements of probe particles embedded in the
soft glassy material, from which we obtain the mechanical
properties of surrounding matrix [21]. Considering the
fragility of soft materials, this technique is ideally suited
for our studies, since it is less invasive than conventional
rheometry.
Here we have used a combination of one and two-
particle MR Measurements [15, 24] to probe the mechan-

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FIG. 1: Light scattering intensity relative reduced with re-
spect to scattered intensity from toluene for two samples
a)Laponite 3.2 wt% and b) Laponite 0.8 wt%, 6 mM NaCl.
The data points are taken at late stages of aging when the
scattering intensity has stabilized.
ical properties and possible inhomogeneities of colloidal
gels and glasses on length scales of the order of the parti-
cle size 1µm and separation distances of the order of 5-20
µm.
II. EXPERIMENTAL
A. Materials
We have studied charged colloidal disks of Laponite
XLG, with an averageradius of 15 nm and a thickness of 1
nm. Laponite can absorb water, increasing its weight up
to 20%. Therefore, we first dried it in an oven at 100oC
for one week and subsequently stored it in a desiccator.
We prepared a number of Laponite samples with differ-
ent concentrations and salt contents. Laponite solutions
without added salt were prepared in ultra pure Millipore
water (18.2 MΩ cm−1) and were stirred vigorously with a
magnet for 1.5 hours to make sure that the Laponite par-
ticles were fully dispersed. The dispersions were filtered
using Millipore Millex AA 0.8 µm filter units to obtain
a reproducible initial state [4]. This instant defined the
zero of waiting time, tw= 0.
The Laponite solutions with pH=10 were obtained
by mixing the Laponite with a 10−4mole/l solution of
NaOH in Millipore water. The samples with non-zero salt
content were prepared by diluting the Laponite suspen-
sions in pure water with a more concentrated salt solution
[13]. For instance, a sample of 0.8 wt %, 6mM NaCl was
prepared by mixing equal volumes of 1.6 wt% Laponite
solution in pure water with a 12mM salt solution.
For the microrheologymeasurements, we added a small
fraction, below 10−4vol %, of silica beads with a diame-
ter of 1.16 µm ±5% [14] immediately after the prepara-
tion of the sample. Subsequently we infused the solution
into a sample chamber of about 50 µl volume, consisting
of a coverslip and microscope slide separated by spacers
of double-sided tape with a thickness of 70 µm, sealed
with vacuum grease at the ends to avoid evaporation
of the sample. All the experiments were performed at
room temperature (21 ± 1◦C). After placing the sample
chamber into the microscope, we trapped two beads and
moved them to about 20 µm above the bottom glass sur-
face.
B. Microrheology
The experimental setup for performing one- and two-
particle MR consists of two optical tweezers formed by
two independent, polarized laser beams λ1 = 1064 nm
(Nd:YVO4, CW) and λ2 = 830 nm (diode laser, CW)
which can trap two particles at a variable separation r.
Details of the experimental setup can be found in [15, 16].
Stable trapping is achieved using a high numerical aper-
ture objective lens which is part of a custom-built in-
verted microscope. Two lenses in a telescope configura-
tion allow us to control of the position of the beam foci
in the plane perpendicular to the beam directions. The
two beams are focused into the sample chamber through
a high numerical objective of the microscope (100×, NA
1.3).
Back-focal-plane interferometry is used to measure the
position fluctuations of the probe bead away from the
trap center [17].The signals emerging from each of
the traps are separately projected onto two independent
quadrant photodiodes, yielding a spatial resolution for
the particle position that is better than 1 nm.
During our measurements, the power of each laser was
typically less than 10 mW. Labview software was used
to acquire time series data of particle positions from the
QPDs for a minimum time of 45s. The data were digi-
tized with an A/D converter at 195 kHz sampling rate.
C.Macrorheology
The viscoelastic moduli during the aging process were
also measured using a conventional Anton Paar Phys-
ica MCR300 rheometer in Couette geometry. To avoid
perturbing the sample during the aging process we per-
formed the oscillatory shear measurements with a strain
amplitude of 0.01 in the frequency range of 0.1-10 Hz. In
order to prevent evaporation during the long time mea-
surements, we installed a vapor trap.
D.Light scattering
Our light scattering setup (ALV) is based on a He-Ne
laser (λ = 632.8 nm , 35 mW) and avalanche photo-
diodes as detectors. Static light scattering experiments
were performed (scattering angle range 20-150oon non-
ergodic samples at late stages of aging when the scattered
intensity had stabilized. Samples were rotated to average
over different positions in the sample.

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III.THEORY AND DATA ANALYSIS
There are two classes of microrheology (MR) tech-
niques: active (AMR) and passive (PMR) [18]. In the
first of these, the response of a probe particle to a
calibrated force is measured. In the second approach,
only passive, thermal fluctuations are monitored, from
which one can infer the response function and the rhe-
ological properties of the surrounding medium using
the fluctuation-dissipation theorem (FDT). Applying the
FDT assumes thermal equilibrium.
be potentially problematic to apply PMR to such non-
equilibrium systems as aging glasses. Nevertheless, in a
prior study [19], we not only directly confirmed the va-
lidity of the FDT, but also found excellent agreement
between active and passive methods in the slowly aging
Laponite glass. Therefore, in what follows we will use
only the passive method, which has the significant advan-
tage that, with a single measurement of the fluctuation
power spectrum, one can determine the complex shear
modulus simultaneously over a wide range of frequencies
[20, 21, 22, 23].
It may therefore
A. One-particle Microrheology
In one-particle MR, we first extract the complex com-
pliance from the position fluctuations of one particle.
The time-series data of the bead displacement measured
by the quadrant photodiode is Fourier transformed to
calculate the power spectral density of displacement fluc-
tuations:
?|x(ω)|2? =
?∞
−∞
?x(t)x(0)?eiωtdt(1)
This is done for x and y directions in the plane normal to
the laser beam. The power spectral density of the ther-
mal fluctuations of the probe is related to the imaginary
part of the complex compliance α(ω) = α′(ω) + iα′′(ω)
via the FDT:
α′′(ω) =ω?|x(ω)|2?
2kBT
. (2)
Provided that α′′(ω) is known over a large enough range
of frequencies, one can recover the real part of the re-
sponse function from a Kramers-Kronig (principal value)
integral:
α′(ω) =2
πP
?∞
0
ω′α′′(ω′)
ω′2− ω2dω′. (3)
Before calculating the shear modulus from the response
function, we calibrate the setup and correct for the trap
stiffness that shows up at low frequencies as explained in
detail in [17, 24].
The complex shear modulus G∗(ω) = G′(ω) + iG′′(ω)
can be obtained from the corrected complex compliance
through the generalized Stokes relation, valid for incom-
pressible and homogenous viscoelastic materials [20, 21]
G∗(ω) =
1
6πRα(ω), (4)
where R is the radius of the probe bead.
B. Two-particle microrheology
In two-particle MR, we calculate the correlated fluc-
tuations of two probe beads inside the material. Such
measurements probe the viscoelastic properties of the
medium on length scales comparable to the interparticle
separation. In general, with more than one probe parti-
cle, the displacement of particle m in direction i is related
to the force applied to particle n in direction j via the
complex response tensor u(m)
i
the case of two particles, the response tensors α(1,1)
α(2,2)
ij
describe how each of the particles 1 and 2 respond
to the forces applied to the particle itself, while α(1,2)
scribes how particle 1 responds to the forces on particle
2.
In thermal equilibrium and in the absence of external
forces, the FDT again relates the imaginary part of the
response tensor to the spectrum of displacement fluctu-
ations of the particles.
(ω) = α(m,n)
ij
(ω)F(n)
j
(ω). In
ij
and
ij
de-
α(m,n)
ij
(ω) =
ω
2kBTS(m,n)
ij
(ω), (5)
where the spectra of thermal fluctuations S(m,n)
fined as
ij
are de-
S(m,n)
ij
(ω) =
?∞
−∞
?u(m)
i
(t)u(n)
j(0)?eiωtdt.(6)
The problem of two hydrodynamically correlated parti-
cles in a viscoelastic medium and the relation between
the response tensor and the rheological properties of the
medium has been worked out in [25]. The self-parts of
response tensor α(1,2)
ii
are the same as the ones obtained
from one-particle microrheology.
The cross component part of the response tensor α(1,2)
can be decomposed into two parts α?parallel to the vec-
tor r separating the two beads and α⊥perpendicular to
r: α(1,2)
ij
= α?ˆ riˆ rj+ α⊥(δij− ˆ riˆ rj). For incompressible
fluids each of the components are related to the complex
shear modulus via a generalization of the Oseen tensor:
ij
α?(ω) = 2α⊥(ω) =
1
4πrG∗(ω).(7)
Similarly to the one-particle method, the measured re-
sponse function must be corrected for the trap stiffness.
The trap correction for two-particle microrheology has
been explained in detail in reference [24].

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IV. RESULTS
We carried out the measurements on a variety of
Laponite concentrations and salt contents (2.8, 3.2 wt%,
in pure water, 3 wt% in pH=10, 1.5 wt %, 5mM NaCl,
0.8 wt %, 6mM NaCl, 0.8 wt %, 3mM NaCl ). We have
chosen these samples to ensure that their rate of aging
is slow enough to guarantee that no significant aging oc-
curs during each measurement. On the other hand they
evolve fast enough to allow us to follow the whole evo-
lution within a few hours. The samples 2.8, 3.2 wt%, in
pure water, 3 wt% in pH=10 and 1.5 wt %, 5mM NaCl
showed the properties of a glassy sample according to our
light scattering data and samples 0.8 wt %, 6mM NaCl,
0.8 wt %, 3mM NaCl behaved like a colloidal gel [26, 27].
In addition, we find that a pH of 10 did not affect the
aging dynamics qualitatively. It merely acted as an elec-
trolyte that slightly accelerated the aging. The same held
for Laponite 1.5 wt%, 5mM NaCl. In this case salt just
accelerated the aging, but it did not change the underly-
ing dynamics of the aging process. As we demonstrated
elsewhere [26] both samples belong to the glass region of
the phase diagram.
For the detailed discussion of our results, we focus on
two samples that are representative of the others: one
sample that behaves as a glass (C= 3.2 wt%), and one
sample (C=0.8 wt%, 6mM NaCl) that behaves like a gel.
In the latter case, the structure factor shows a strong
q-dependence, as illustrated in Fig. 1. This suggests a
more heterogenous, gel-like structure, in contrast to the
more homogenous samples that we identify as a glass. In
the following, we shall characterize our samples as gels
or glasses in this way.
To follow the aging of the systems we trapped a bead in
a single laser trap and measured the displacement power
spectral densities (PSD) as a function of waiting time.
Since the system evolves towards a non-ergodic state,
the time average may not necessarily be equal to the
ensemble average for the measured PSDs. However, in
our range of frequencies (1 − 105Hz) we confirmed that
our results did not depend on the time interval used to
compute the time average. Thus, we can use the time-
averaged PSD without averaging over several beads in
our study.
Fig. 2 shows the measured displacement PSDs as a
function of frequency during the aging of the glass and
the gel respectively. We normalized the PSDs with
the diffusion coefficient (D0= kT/(6πηwaterRbead) of a
same-size bead measured in water, so that the normal-
ized PSDs will be independent of bead size. It is evi-
dent that in both systems the particle motion progres-
sively slows down with increasing aging time tw, reflect-
ing the increase of viscosity in the system. The PSDs in
both samples start from a state close to water, for which
?|x(ω)|2?/2D0 = 1/ω2. Gradually their amplitudes as
well as the absolute values of their power-law slopes de-
crease with time. There is a crossover time t0such that
for tw< t0, the PSDs can be described by a single power
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FIG. 2: Normalized displacement power spectral densities
?|x(ω)|2?/2D0 of silica probe particles in a glassy sample (3.2
wt%, bead diameter 1.16 µm ) and a gel-like sample (0.8 wt%,
6mM NaCl, bead diameter 0.5 µ) in the x direction) with in-
creasing age after preparing the sample. Waiting times are
given in the legend. The filled squares show the PSD of a
bead in pure water for comparison.
done at 21oC.
All experiments were
law. At longer aging times tw> t0, two distinct slopes
appear in the log-log plots (Fig. 2).
The evolution of the local shear moduli G∗obtained
from PSDs are shown in Fig. 3 (glass) and 4 (gel). The
shear moduli are derived from single particle MR accord-
ing to Eq. (4). It is evident that the systems evolve from
an initially completely viscous to a strongly viscoelas-
tic fluid. At the early stages of aging, the loss modulus
is still much larger than the storage modulus (G′′≫ G′)
representing a more liquid-like state. With time the sam-
ples become more solid-like: the elastic modulus becomes
larger than the loss modulus (G′′≪ G′). We observe
also that the changes in G′are more dramatic than the
changes in G′′. While G′′almost saturated after 170 min
for the glass and 100 min for the gel, G′continues to
grow with time.
Visual inspection showed that the gel was “softer” than
the glass. When we mechanically shook similar tubes
containing gel or glass, the gel liquefied at a clearly
smaller stress: it appears that gels had a lower yield
stress compared to glasses. Therefore it is reasonable
to expect that gels also have a lower viscoelastic mod-
ulus than glasses, as comparison of Fig. 3 and Fig. 4
indeed confirms. Furthermore, the ratio G′/G′′at low
frequencies is higher in the gel (G′/G′′= 30 for Laponite

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FIG. 3: Glass data: The symbols show the shear moduli G′(ω)
and G′′(ω) (absolute magnitude) as a function of frequency
measured using 1.16 µm silica probe particles in a 3.2 wt%
Laponite solution in pure water with increasing aging time
after preparing the sample. Aging times are given in the leg-
end. The lines show the fits of G′(ω) and G′′(ω) according
to C1(−iω)a+ C2(−iω)bin which C2 = 0 for aging times
tw < 120 min.
0.8 wt%, 6mM) compared to the glass (G′/G′′= 12 for
Laponite 0.8 wt%), at late stages of aging when G′′has
almost saturated and G′evolves very slowly.
From microrheology we conclude that the aging be-
haviors of gels and glasses are qualitatively similar. We
know, however, from light scattering measurements that
the underlying structures of gels and glasses are very dif-
ferent [12]. Since spatial heterogeneity is the defining
feature of gels, we thus set out to investigate if local
measurements of microrheology across the samples can
detect the difference between gels and glasses.
A.Heterogeneity
Heterogeneities within a sample can be explored by
measuring the PSDs of multiple beads at different po-
sitions in the sample. A discrepancy between the shear
moduli obtained from one- and two-particle MR can also
be used as an indicator of a heterogeneous structure. A
further test of heterogeneity in a material is provided
by comparison of MR with bulk rheology, as will be dis-
cussed below. To investigate the homogeneity of colloidal
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FIG. 4: Gel data: The symbols show the shear moduli G′(ω)
and G′′(ω) (absolute magnitude) as a function of frequency
measured using a 0.5 µm silica probe particle in a 0.8 wt%
Laponite solution in 6 mM NaCl water with increasing aging
time after preparing the sample. Aging times are given in the
legend. The lines show the fits of G′(ω) and G′′(ω) according
to C1(−iω)a+ C2(−iω)bin which C2 = 0 for aging times
tw < 100 min.
gels and glasses of Laponite, we performed two types
of measurement. First, we made simultaneous measure-
ments of PSDs of two independent beads in two inde-
pendent traps at different stages of aging. In another set
of experiments, we measured PSDs of multiple beads in
aged gels and glasses. The results of our experiments for
both gels and glasses will be discussed below.
1.Glass
For the glassy samples the displacement PSDs turned
out to be independent of the bead position, as was con-
cluded from a comparison of simultaneous measurements
of PSDs of two independent beads in two independent
traps during aging.Furthermore, the comparison be-
tween one- and two-particle MR reveals that within the
experimental error, the complex shear moduli are iden-
tical to within the experimental error between the two
methods for all stages of aging as shown in Fig. 5.
This was further verified by measuring the PSDs of
several beads at different positions of an aged sample. As
can be seen in Fig. 7, the measured shear moduli were
independent of the position of the bead in the sample,
verifying the homogeneity of the glassy sample, as shown

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FIG. 5:
(magnitude) at two different stages of aging in a 3.2 wt%
Laponite solution derived from one- (lines) and two-particle
(symbols) MR using 1.16 µm silica probe particles. The dis-
tance between the two particles was 6 µm. The aging times
are shown in the figure. Note that in the late stages of ag-
ing the material becomes too stiff to obtain a good cross-
correlation signal between the two beads over the background
noise.
Glass data: The shear moduli G′(ω) and G′′(ω)
also in Fig. 6a.
These results suggest that the Laponite glass has a ho-
mogenous viscoelasticity, at least on length scales larger
than half a micrometer, which is the length scale one-
particle MR intrinsically averages over. An additional
check on this can be obtained from a comparison between
microrheology and macrorheology which should yield the
same results if the sample is homogeneous.
Figure 8 shows the shear moduli extracted from MR
and macrorheology experiments at a fixed frequency of
(f = 0.7 Hz) during the course of aging.
all agreement between macrorheology and MR is good.
For the early stages of aging, the G′′measured by the
macrorheometer appears slightly higher, but this can be
attributed to the large moment of inertia of the rheome-
ter bob; macrorheology does not provide accurate mea-
surements of the shear moduli when G∗< 1 Pa. MR,
on the other hand, has other sources of errors at low
frequencies, especially for the late stages of aging, when
the material becomes very rigid. In this case, the signal
detected by the photodiode becomes small compared to
the noise level; 1/f laser pointing noise dominates at low
frequencies. This is the most plausible explanation for
the slight discrepancy between the results from the two
methods at long aging times.
The over-
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FIG. 6: The displacement PSDs of 0.5 µm silica beads mea-
sured at different positions within an aged glass (Laponite 3.2
wt % in pure water, tw ≈ 5 h ) and within an aged colloidal
gel (Laponite 0.8 wt% in 6mM NaCl solution, tw ≈ 10 h ).
2.Gel
Measuring the displacement PSDs of several beads at
different positions of a gel at the late stages of aging
revealed a considerable degree of inhomogeneity (Fig. 6).
Not only were the PSDs position dependent, but at some
positions in the sample the measured PSDs were also
anisotropic, i.e. fluctuations in x and in y direction gave
different results.
This result is consistent with the static light scatter-
ing measurements for this sample shown in Fig. 1 that
suggest inhomogeneities of the gel on a length scale com-
parable to the inverse scattering vector, i.e. micrometers.
Therefore, exploring such a gel using microrheology with
a probe on the order of the mesh size of the network, one
can detect these characteristic inhomogeneities. In (Fig.
9) we have plotted the shear moduli seen by the beads
at different positions. It can be seen that there was an
order of magnitude difference between the smallest and
largest elastic moduli measured in the same sample and
at the same time.
It is intriguing to ask when the heterogeneity starts
to develop in the aging samples. It is likely to appears
as a network-like structure is building up in the gel. To
answer this question, we measured the PSDs of two beads
at different positions of a gel as a function of aging time.
We performed two sets of experiments: in the first one the

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FIG. 7:
measured at different positions in an aged glassy sample of
Laponite 3.2 wt % in pure water (tw ≈ 5h)
Local elastic modulus G′and loss modulus G′′
two beads were positioned at a relatively close distance
r =4.66 µm (Fig. 10) and in the other one at a large
distance of r = 19 µm.
In both experiments, the responses, i.e. the PSDs at
different positions were equal in the early stages of ag-
ing. However, as time progressed, the PSDs measured
at different positions began to differ.
later stages of aging, the displacement PSDs measured
for some of the beads became anisotropic, meaning that
the PSDs in the x and y directions were not equal any-
more. In some measurements the anisotropy survived
the latest measurement. For some other measurements,
the anisotropy disappeared after some time (look at Fig.
10 for example). This suggests that the building up of
structure in the gel is a dynamic process; at some points
and times more particles join to the network and at some
other points and times some particles disintegrate from
the network.
Furthermore, our experiments showed that immedi-
ately after preparation, shear moduli obtained from two-
particle MR and one-particle MR were equal. But al-
ready at relatively early stages of aging, the two-particle
MR results differed from 1PMR results as demonstrated
in Fig. 10. This deviation appeared long before the local
shear moduli of the two beads in one-particle MR started
to differ. For more details, see also [27].
Our measurements on several bead pairs at varying dis-
tances suggest that these inhomogeneities extend over a
range of at least 100 micrometers. Therefore the macro-
scopic bulk shear modulus is not necessarily expected to
In addition, at
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FIG. 8: Elastic and loss modulus as a function of aging time
for a sample of Laponite 3.2 wt % in pure water obtained from
macrorheology and one-particle MR at f = 0.7 Hz. The strain
amplitude in the macrorheology measurements was 0.01.
be equal to that measured by single particle MR. In Fig.
11, we compare the shear moduli obtained from one- and
two-particle MR with the results of macrorheology at late
stages of aging (tw≈ 8.5 h) when the changes in the loss
and elastic moduli are slow. It is evident that the local
shear modulus measured at one of the positions in the
sample was equal to the bulk value, while the others re-
ported a considerably lower shear modulus. Notably, the
shear modulus obtained from the cross correlation of two-
particles is lower than both bulk and local shear moduli.
This suggests that two-particle MR can be used to detect
inhomogeneities as long as they occur on length scales
below the distance between the particles, but the results
may still not reflect bulk properties if heterogeneities ex-
tend beyond the scale of the inter-particle distance.
B.Model for the viscoelastic behavior
It has been noted in the context of weakly attractive
colloids [28] and biopolymer networks [29] that the addi-
tion of two power law contributions describes the shear
modulus very well. This result appears to reflect the ex-
istence of two distinct contributions to the viscoelasticity
of the system and can be interpreted as a superposition
of a more elastically rigid network (weakly frequency-
dependent) and viscoelastic background (with a strong

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FIG. 9: Gel data: local elastic modulus G′and loss modulus
G′′measured at different positions in an aged gel sample of
Laponite 0.8 wt % in 6mM NaCl solution (tw ≈ 10h)
frequency dependence).
Our data ( Fig. 3 and Fig. 4) can be interpreted in
a similar manner: In addition to a strongly frequency-
dependent viscoelastic response at high frequencies, a
more elastic (weakly frequency-dependent) response ap-
pears after some aging time t0 and slowly increases in
amplitude during the aging process. To be more precise,
we see that the complex shear modulus of both gels and
glasses crosses over from a single power law to a super-
position of two power laws around a certain waiting time
t0 which depends on the sample (t0 ≈ 155min for the
glass sample of Laponite 3.2 wt % and t0 ≈ 95min for
the gel sample of Laponite 0.8 wt %,6mM NaCl ) [19].
The local shear moduli of both samples turn out to be
well-described by the following expression:
G(ω) = G′(ω) + iG′′(ω)
?C1(−iω)a
C1(−iω)a+ C2(−iω)b
≡
:
:
tw< t0
tw> t0
(8)
Physically, this model implies that to two distinct stresses
arise under a common imposed strain — in the way
forces add for springs in parallel, as opposed to displace-
ments/compliances that would add for springs in series.
This response would be expected, for instance, for two in-
terpenetrating structures/systems that displace together
under strain, at least on the scale of our probe particles,
which are large compared to the individual Laponite par-
ticles. This can, in principle, be the case whether or not
the material appears to be homogeneous on this scale.
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FIG. 10: Gel data: The shear moduli G′(ω) and G′′(ω) at
different stages of aging derived from one- and two-particle
MR of 0.5 µm silica probe particles in a gel of Laponite 0.8
wt % in 6mM NaCl solution. The distance between the two
beads was 4.66 µm. For ta = 129 min the shear moduli at
two different positions are equal. At ta = 205 min the shear
moduli at the two positions are not equal. Furthermore shear
modulus at position 2 shows anisotropy. At a later time ta =
510 min the shear moduli at the two positions are not equal
but the anisotropy observed earlier at position of bead 2 has
been disappeared.
A tenuous elastic network structure immersed in a more
fluid-like background, such as we might expect for a gel,
would behave in this way, provided that the network and
the background medium are strongly coupled hydrody-
namically, and that the network spans length scales corre-
sponding to the imposed strain. Such a gel could appear
to be either homogenous or heterogeneous, depending on
the length scale probed.
We find that the exponent of the single power law de-
creased from 1 to a value of about 0.7 before the second
component becomes visible. The exponent and ampli-
tude of the first component C1(−iω)ado not change fur-
ther with aging time for tw> t0while the amplitude of
the other one C1grows appreciably over the same times.
In Fig. 12(a) and (c), we have plotted the evolution
of the fitting parameters as a function of aging time for
different samples. As can be seen in the figure, the de-
velopment of the two viscoelastic components for differ-
ent samples is qualitatively similar, although the rate of
change depends on sample concentration and salt con-
tent.

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FIG. 11: Complex shear modulus at a late stage of aging
tw ≈ 8.5h obtained from single-particle MR at two different
positions of the sample, two-particle MR and bulk rheology in
a sample of Laponite 0.8 wt%, 6mM NaCl. The circles show
G′and triangles show G′′values.
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FIG. 12: The complex shear moduli of Laponite suspensions
can be described as the sum of two power laws C1(−iω)a+
C2(−iω)bin which C2 = 0 for waiting times tw < t0. The
crossover times are t0 = 155,95,120,105,95 min for Laponite
concentrations 2.8 wt%, 3 wt%,pH=10, 3.2 wt% , 1.5 wt%,
5mM NaCl and two different positions of 0.8 wt%, 6mM NaCl,
respectively.(a) The evolution of power-law exponents a
(filled symbols) and b (open symbols) as a function of aging
time for different concentrations of Laponite. (b) The expo-
nents a (filled symbols) and b (empty symbols) as a function of
scaled aging time (c) The amplitude of viscoelastic contribu-
tions C1 (filled symbols) and C2 (open symbols) as a function
of aging for different samples. (d) The same as panel (c) but
plotted versus scaled aging time. The sample concentrations
are shown in the legend.
Interestingly, the evolution curves of the exponents a
and b for the different samples superimpose if we scale
the aging time as t′
a= (tw− t0)/t0. The crossover times
are t0= 155,95,120,105,95 min for Laponite concentra-
tions 2.8 wt%, 3 wt%, pH10, 3.2 wt%, 1.5 wt%, 5mM and
0.8 wt%, 6mM NaCl, respectively. For the amplitudes on
the other hand, the data do not collapse. Especially the
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FIG. 13: Bold symbols: The elastic modulus obtained from
macrorheology at late stages of aging at the aging time that
there is a crossover from a fast regime of aging to a slower
regime as a function of concentration measured for different
glassy samples and a gel sample of 0.8 wt%, with 6mM salt
at f = 0.05 Hz . Empty symbols: An estimate of the plateau
value G′
P∝ kT/D3is shown for comparison.
amplitudes of the second (viscoelastic) component sys-
tematically decreases as the Laponite content is reduced.
Furthermore, for the gel the amplitude depends on the
position and we can see some fluctuations in the ampli-
tude of the second component C2, at later stages of evo-
lution. This can be understood in terms of the dynamic
process of gel formation in which Laponite particles still
can join or detach from the network.
V.DISCUSSION AND CONCLUSION
We have studied the evolution of the viscoelastic prop-
erties of a variety of Laponite suspensions including both
gel-like and glassy states over a wide range of frequencies
using macro- and micro-rheology techniques. Our mea-
surements reveal the differences between the mechanical
properties of gels and glasses.
The glassy samples are homogenous on all length scales
probed in our experiments (l > 0.5µm). This is further
confirmed by comparing microrheology and conventional
macrorheology results. We find that measurements at
different scales all give the same results.
is no evidence for spatial inhomogeneity as expected for
glassy systems in general.
In the gels, however, along with the evolution from a
liquid-like state to a viscoelastic state, inhomogeneities
develop in time. These inhomogeneities are detected by
measuring the local shear moduli at different positions
within the samples at nearly equal waiting times.
Another difference between gels and glasses that we ob-
served is that the change in the ratio G′/G′′is much more
rapid for a gel than for a glass as the material evolves
from a purely viscous liquid to solid-like system.
When we track the values of the elastic modulus at
a fixed frequency (here 0.05 Hz) at late stages of aging,
when there is a crossover from a fast aging rate to a
slower aging rate, we find that the elastic modulus scales
Thus, there

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linearly with concentration. One can roughly estimate
the plateau value for glasses as G′
is the characteristic structural length of the system. We
have taken D as half of the interparticle distance. This
estimate predicts the order of magnitude fairly well. As
shown in Fig. 13, the extrapolation of our data suggests
that no glassy samples exist at concentrations lower than
1.6 wt%.
Therefore the elasticity for samples with lower concen-
trations should stem from a different mechanism. Indeed,
in such samples the aging proceeds through gel forma-
tion. For comparison, we have shown the elastic modulus
of a gel sample of 0.8 wt%, 6mM in Fig. 13.
Despite the differences between gels and glasses , we
find a similar frequency dependence of the visco-elastic
moduli for gels and glasses. The local viscoelastic moduli
for both gels and glasses cross over from a single power
law to the sum of two power laws around a certain time
t0. These results demonstrate the existence of two dis-
tinct contributions in the viscoelasticity of the system
in the later stages of aging. In addition to a strongly
frequency-dependent viscoelastic shear modulus at high
frequencies∼= ω0.7, we also observe the slow develop-
ment of a more elastic (only weakly frequency-dependent)
shear modulus during the aging. The exponents of the
P∝ kT/D3, where D
power laws follow exactly the same time course of evo-
lution for different concentrations if we scale the aging
time as t′
a= (tw− t0)/t0. This result is independent of
the sample being a gel or a glass.
The crossover from a single frequency-dependent com-
ponent to a superposition of a strongly frequency-
dependent viscoelastic component plus a weakly fre-
quency dependent (elastic) component was previously in-
terpreted in the context of polymer networks as being due
to large inhomogeneities [29, 30]. Here the sum of two
power-laws describes both gel (heterogenous) and glass
(homogenous) local shear moduli, suggesting that locally
the underlying physical process responsible for the evolu-
tion of gels and glasses is similar. This poses the rather
puzzling question what the physical origin of the two
power-laws in the viscoelasticity is.
Acknowledgments This research has been supported
by the Foundation for Fundamental Research on Mat-
ter (FOM), which is financially supported by Nether-
lands Organization for Scientific Research (NWO). LPS
de l’ENS is UMR8550 of the CNRS, associated with the
universities Paris 6 and 7. C.F.S was further supported
by the DFG Center for the Molecular Physiology of the
Brain (CMPB).
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