# Dynamical heterogeneity in aging colloidal glasses of Laponite

**ABSTRACT** Glasses behave as solids due to their long relaxation time; however the

origin of this slow response remains a puzzle. Growing dynamic length scales

due to cooperative motion of particles are believed to be central to the

understanding of both the slow dynamics and the emergence of rigidity. Here, we

provide experimental evidence of a growing dynamical heterogeneity length scale

that increases with increasing waiting time in an aging colloidal glass of

Laponite. The signature of heterogeneity in the dynamics follows from dynamic

light scattering measurements in which we study both the rotational and

translational diffusion of the disk-shaped particles of Laponite in suspension.

These measurements are accompanied by simultaneous microrheology and

macroscopic rheology experiments. We find that rotational diffusion of

particles slows down at a faster rate than their translational motion. Such

decoupling of translational and orientational degrees of freedom finds its

origin in the dynamic heterogeneity since rotation and translation probe

different length scales in the sample. The macroscopic rheology experiments

show that the low frequency shear viscosity increases at a much faster rate

than both rotational and translational diffusive relaxation times.

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**ABSTRACT:**The stiction properties of a star polyisoprene (PIP) melt (having 22 arms and an arm molecular weight of around 5000, M(w) ≈ 110 000) confined between mica surfaces were investigated using the surface forces apparatus. Stop-start experiments were carried out and the stiction spike was measured as a function of surface stopping (aging) time t and applied pressure P; the time constants of the phase transitions in the stiction dynamics (freezing on stopping and melting on starting) were obtained from the force relaxation behaviors. The results were compared with those of a confined linear-PIP melt (M(w) ≈ 48 000) and other confined fluid systems; the effect of star architecture on the phase transitions in confinement during aging is discussed. Estimation of the molecular size gives that the confined star-PIP films consist of three molecular layers; a non-adsorbed layer sandwiched between two layers adsorbed on opposed mica surfaces. There are (at least) four time constants in the freezing transition of the confined star-PIP melt; fast (τ(1)) and slow (τ(2)) time constants for lateral force relaxation on stopping, critical aging time for freezing (τ(f)), and the logarithmic increase of the spike height against t. The three time constants on stopping, τ(1), τ(2), and τ(f), increase with the increase of P (decrease of the thickness D). As regards the melting transition on starting, spike force decay was fitted by a single exponential function and one time constant was obtained, which is insensitive to P (D). Comparison of the time constants between freezing and melting, and also with the results of linear-PIP reveals that the stiction dynamics of the star-PIP system involves the relaxation and rearrangement of segmental-level and whole molecular motions. Lateral force relaxation on stopping is governed by the individual and cooperative rearrangements of local PIP segments and chain ends of the star, which do not directly lead to the freezing of the system. Instead, geometrical rearrangements of the soft star-PIP spheres into dense packing between surfaces (analogous to the concept of a colloidal glass transition) are the major mechanism of the freezing transition (stiction) after aging. Interdigitation of PIP segments∕chain ends between neighboring star molecules also contributes to the spike growth along with aging, and the melting transition on starting.The Journal of Chemical Physics 11/2012; 137(19):194702. · 3.12 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Polyamorphism and dynamical heterogeneities in network-forming liquids (SiO2, GeO2, Al2O3) at 3200 K and in a wide pressure range are investigated by molecular dynamics simulation. Results show that their structure comprises three structural phases: TO4-, TO5-, and TO6-phases (T = Si, Ge, or Al). The size of structural phase regions significantly depends on compression. Besides, the mobility of atoms in different structural phases is different. For SiO2 and GeO2 systems, the TO5-phase forms mobile regions. For Al2O3 system, AlO6-phase forms mobile regions. The coexistence of TOx-phases (x = 4, 5, 6) in the network-forming liquids is origin of the spatially dynamical heterogeneity.Applied Physics Letters 05/2013; 102(19). · 3.52 Impact Factor

Page 1

Dynamical heterogeneity in aging colloidal glasses of Laponite

S. Jabbari-Farouji,abR. Zargar,bG. H. Wegdamband Daniel Bonnbc

Received 23rd January 2012, Accepted 9th March 2012

DOI: 10.1039/c2sm25171j

Glasses behave as solids due to their long relaxation time; however the origin of this slow response

remains a puzzle. Growing dynamic length scales due to cooperative motion of particles are believed to

be central to the understanding of both the slow dynamics and the emergence of rigidity. Here, we

provide experimental evidence of a growing dynamical heterogeneity length scale that increases with

increasing waiting time in an aging colloidal glass of Laponite. The signature of heterogeneity in the

dynamics follows from dynamic light scattering measurements in which we study both the rotational

and translational diffusion of the disk-shaped particles of Laponite in suspension. These measurements

are accompanied by simultaneous microrheology and macroscopic rheology experiments. We find that

rotational diffusion of particles slows down at a faster rate than their translational motion. Such

decoupling of translational and orientational degrees of freedom finds its origin in the dynamic

heterogeneity since rotation and translation probe different length scales in the sample. The

macroscopic rheology experiments show that the low frequency shear viscosity increases at a much

faster rate than both rotational and translational diffusive relaxation times.

I.Introduction

Glasses are non-equilibrium solids with no long-range structural

correlations. Due to their long relaxation time, glasses behave as

solids on experimental time scales. This slow response is often

attributed to cooperative motion of particles in which a subre-

gion of the liquid relaxes to a new local configuration.1,2

However, the size of cooperatively rearranging regions has never

been observed to exceed a few particle diameters, and the

observation of long-range correlations that are signatures of an

elastic solid has remained elusive. The universal dynamical

feature in a variety of such systems, including molecular,

colloidal, metallic glasses is a dramatic slowing down of the

diffusion of particles concomitant with a huge increase of

viscosity upon approaching the glass-transition.3,4The slowing

down of dynamics expresses itself, among other things, in the

non-exponential relaxation of translational and rotational

degrees of freedom and non-decaying plateaus of the associated

correlation functions in the glassy phase.

Duetotheintrinsicnon-equilibriumnatureofallglassysystems,

theyexhibitaging.Agingmeansthatthephysicalpropertiesofthe

system evolve with waiting time i.e., the time elapsed since the

quenchintotheglassyphase.Theagingisoftenattributed tolocal

rearrangements,i.e.,locallyanenergybarriercanbeovercomeand

the system falls into a new configuration in phase space. Cooper-

ativemotionrequiresspatiallyandortemporarilycorrelatedlocal

rearrangements, therefore suggests the existence of heterogenous

dynamics for the particle motion in both space and time.

Indeed, various theoretical and experimental studies have

provided evidence for the existence of dynamical heterogeneity in

both molecular and colloidal glass formers5,6,8despite the

differences between the two systems. The building blocks of

molecular glass formers are in the range 0.1–1 nm and the glass

transition is these systems is generally driven by a temperature or

pressure quench. Colloidal glasses, on the other hand, consist of

large particles with sizes in the range 10–1000 nm that undergo

Brownian motion in a suspending liquid. The key control

parameter in colloidal systems is the volume fraction or density.

These systems have the advantage that they can be viewed

directly using microscopy, so that we can have a microscopic

picture of how particles rearrange at the glass transition.7In

computer simulations8,9and confocal microscopy experiments

on colloidal glasses7,10–14the heterogenous regions were visual-

ized directly. One way of quantifying the spatio-temporal

correlations is to determine the four-point dynamic correlation

functions that measure the correlations of motion in space–

time.15Such four-point correlation functions have been used to

quantify the dynamical heterogeneity and its associated length-

scale in several glass formers;14,16however this correlation func-

tion is not always easily accessible in measurements of the glass-

forming systems, especially for molecular glass-formers and

nanometer sized colloidal particles. Therefore, other probes that

reveal the nature of heterogenous dynamics are helpful.

aLPTMS, CNRS and Universit? e Paris-Sud, UMR8626, Bat. 100, 91405

Orsay, France

bVan der Waals-Zeeman Institute, Institute of Physics (IoP) of the Faculty

of Science (FNWI) University of Amsterdam, 1098 XH Amsterdam, the

Netherlands

cLaboratoire de Physique Statistique de l’ENS, 24 rue Lhomond, 75231

Paris Cedex 05, France

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One important consequence of dynamical heterogeneity is the

break-down of mean-field dynamical relations such as the well-

known Stokes–Einstein relation.17–21If spatially correlated

mobile regions exist, it seems plausible that there should be

a difference between the globally measured viscosity that results

from the cooperative motion and the local diffusion dynamics of

the particles that probe the local ‘viscosity’. Therefore, a scale-

dependent diffusion coefficient can be considered as a signature

of dynamical heterogeneity and provides us with some indication

of the dynamic correlation length scale.

The translational diffusion coefficient and rotational relaxa-

tion time of particles suspended in a (low-viscosity) liquid follow

the Stokes–Einstein and Debye–Stokes–Einstein equations,

respectively, that relate the translational diffusion coefficient and

rotational relaxation time to the temperature and the viscosity as

follows:22,23

Dtr¼ akBT

hR

Stokes ? Einstein relation (1)

sr¼ a0hR3

kBT

Debye ? Stokes ? Einstein relation (2)

Here R is the size of the largest dimension of the particles and

a and a0are prefactors that depend on the shape of particles and

are known in principle. These relationships predict that trans-

lational and orientational relaxation times should be directly

proportional to the macroscopically measured viscosity.

Here, by studying the aging dynamics of relaxation time and

viscosity of a colloidal glass of Laponite,24–26we investigate the

validity of the relationships in eqn (1). Laponite is a synthetic

clay that is widely used as a rheology-modifier in industrial

materials such as paints, household cleaners and personal care

and house construction products. Furthermore, it is considered

as a model system for charged colloidal disks and its rich

complex phase behavior has attracted a lot of attention in recent

years and has been studied extensively in from the fundamental

point of view as discussed in a recent review paper.27Here, our

aim is to find a link between slowing down of translational and

rotational motion and increase of viscosity and investigate the

possible development of dynamical heterogeneity in Laponite

colloidal glass. The insights obtained from this study are possibly

useful for other soft glassy fluids such as red blood cells or

anisotropic colloids.28

At sufficiently high concentrations, Laponite particles sus-

pended in water, form a glassy phase that ages.24–26During aging

the viscosity and the relaxation times associated with trans-

lational and rotational relaxation increase.29–32The anisotropic

shape allows us to study both the translational and rotational

diffusion of the particles.31,33Although the aging of the trans-

lational and rotational diffusion has been already investigated in

a previous work,31their connection to the development of

viscosity in relation to Stokes–Einstein and Debye–Stokes–Ein-

stein relations is missing. The current study is complemented

with rheology and microrheology experiments. Thus we have

examined the problem from the more fundamental perspective of

dynamical heterogeneity. We observe how the aging from

a liquid-like to a disordered solid-like state influences the

orientational and translational degrees of freedom, and how

these are related to the increase of the global viscosity. Further,

by comparing the translational diffusion of Laponite particles

with that of a probe much bigger than their size, we can inves-

tigate how dynamical heterogeneity develops in the system as

a function of waiting time.

II.Materials and methods

The Laponite grade that we used for our experiments is Laponite

XLG that consists of platelets of an average diameter of 25 nm

and 1.2 nm thickness with an estimated polydispersity index of

about 30%.34Laponite can absorb water, increasing its weight by

up to 20%. Therefore, we first dried it in an oven at 100?C for

one week and subsequently stored it in a desiccator. Laponite

dispersions are prepared in ultra pure Millipore water and are

stirred vigorously by a magnetic stirrer 1.5 h to make sure that

the Laponite particles are carefully dispersed. The dispersions are

filtered using Millipore Millex AA 0.8 mm filter units to obtain

a reproducible initial state. This instant defines the zero of

waiting time, tw¼ 0.24

Our dynamic light scattering setup (ALV) is based on a He–Ne

laser (l ¼ 632.8 nm, 35 mW) with avalanche photodiodes as

detectors. An ALV-60X0 correlator directly computes the

intensitycorrelation functions

gðq;tÞ ¼

?Iðq;tÞIðq;0Þ?

?Q

?Iðq;0Þ?2

2

, at

a scattering wave vector number q ¼4pn

the scattering angle and n is the refractive index of scattering

sample. For Laponite suspensions used in this study with

a concentration around 3 wt%, the refractive index is 1.33535and

the measurements were always performed at angle Q ¼ p/2,

leading to a scattering wave number q ¼ 1.87 ? 107m?1.

The total electric field scattered by the colloidal particles with

an axially symmetric optical anisotropy has a vertically polarized

component EVVwith an amplitude proportional to the average

polarizability, g¼ (gk+ 2gt)/3 and a horizontal depolarized

component EVHproportional to the intrinsic particle anisotropy

b ¼ gk? gt, the difference between the polarizabilities parallel

and perpendicular to the optical axis.33Depolarized dynamic

light scattering (DDLS) measures the correlation functions of the

scattered light intensity whose polarization (horizontal) is

perpendicular to the polarization of incident light (vertical), i.e.,

the VH mode, as opposed to the VV mode for which the polar-

ization of scattered and incident light are both vertical.

The viscoelastic moduli during the aging process were also

measured using a conventional Anton Paar Physica MCR300

rheometer with a Couette geometry. To avoid perturbing the

sample during the aging process we performed the oscillatory

shear measurements with a small strain amplitude of 0.01 at

a fixed frequency 0.05 Hz. In orderto preventevaporation during

the long time measurements, we installed a vapor trap. The

experimental setup for performing microrheology consists of

optical tweezers formed by two polarized laser beams. Details of

this experimental setup can be found in ref. 36.

l

sin

?

, in which Q is

III. Results and discussion

We measured the VV and VH intensity correlation functions at

a fixed scattering angle Q ¼ 90?regularly during the aging of the

glassy suspension. The VV intensity correlation functions mainly

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reflect the dynamics of the translational degree of freedom as the

contribution of the rotational motion to the VV correlations

is rather small as has been demonstrated before.31The VH

correlation functions, on the other hand are determined by both

translational and rotational degrees of freedom. Indeed, for

dilute enough suspensions, the VH correlations can be written as

a product of VV correlation function and a purely orientational

correlation.31,33

Studying the aging dynamics of VV correlations for an

extensive range of samples, it has been well established that one

can observe two regimes of aging.24,25,37In the first regime of

aging, the translational correlation functions decay to zero

within the experimental accessible time scales and the average

relaxation time grows exponentially with waiting time. In the

second regime of aging, the ensemble-averaged correlations

do not decay to zero and a plateau in intermediate scattering

function appears (see Fig. 1(a)). The waiting time for which the

correlation functions (i) no longer decay to zero within the

experimental time-scale and (ii) the time-averaged correlation

functions are not equal to their ensemble-averaged values defines

the ergodicity-breaking time teb. Studying the aging dynamics of

VV and VH correlation functions simultaneously, we find out

that the VH correlations become non-ergodic at nearly the same

time as the VV correlations. Therefore, we take the same tebvalue

for both VV and VH correlations. To be able to quantify the

relaxation times for rotational and translational diffusion, here

we only focus on aging dynamics in the first, ergodic, regime of

aging.

In Fig. 1, we show the fVVand fVHintermediate scattering

functionsextractedfrom intensity

measured at different stages of aging. We already know that for

the VV intermediate scattering functions fVVa two-step relaxa-

tion can be observed;29we observe a similar relaxation for the

VH correlations as well.31In the ergodic regime, we can quantify

the relaxation times by fitting the VV intermediate scattering

functions with the sum of an exponential and a stretched

exponential.29

correlation functions

f(q,t) ¼ Aexp(?t/s1) + (1 ? A) exp(?(t/s2)b).(3)

where A determines the relative contribution of the exponentially

decaying fast relaxation time s1and b is known as the stretching

exponent characterizing the broadness of slow relaxation modes

that contribute to the correlation function. The smaller b is,

the broader the distribution of the slow relaxation times. s2

gives us an idea of the mean relaxation time that is determined as

sm¼ s2/bG(1/b), where G is the gamma function.

The VV mode data reflects the aging ofthe translational degree

of freedom, and the relaxation times from the fit are therefore

a direct measure of the relaxation times of translational diffu-

sion. Fitting the VV correlations with eqn (3), we find that A is

constant with waiting time with a value 0.215 ? 0.02. The same

holdsfor the fast relaxation timewhose inverse givesus the short-

time diffusion coefficient Ds, i.e. s1¼ 1/(Dsq2). As was shown in

previous work13,31s2is growing exponentially with waiting time

concomitant with the decrease of the stretching exponent

b whose value decreases linearly from 0.606 ? 0.005 at twz 0 to

0.358 ? 0.005 at twz teb. In Fig. 3(a), we have presented the

evolution of both fast (s1) and slow (s2) relaxation times; the fast

relaxation time is usually interpreted as a ‘cage rattling’ motion,

whereas the longer relaxation time corresponds to a cage

reorganization.

As mentioned before, both translational and rotational

degrees of freedom contribute to the VH correlations. In order to

gain a more direct insight into the rotational dynamics, we

extract the orientational correlation functions defined as the

ratio for ¼ fVH/fVV as shown in Fig. 2. These orientational

correlations can also be fitted with the sum of a single and

a stretched exponential according to eqn (3). The corresponding

fast s1 f 1/Dsand slow s2 rotational relaxation times can

be extracted as depicted in Fig. 3(b). We find that for the

orientational correlations, A decreases with waiting time from

A ¼ 0.12 ? 0.03 at early stages of aging to A ¼ 0.012 ? 0.01

at tw z teb. The stretching exponent b also decreases from

b ¼ 0.30 ? 0.01 at early stages of aging to b ¼ 0.15 ? 0.02 upon

approaching the ergodicity-breaking time, pointing to a broader

distribution of rotational slow relaxation times as compared to

the translational ones.

We also measured the macroscopic complex shear modulus

with the rheometer; the complex viscosity magnitude can be

obtained as jh?j ¼

are then in a position to discuss the translational and rotational

relaxation times and compare them to the measured macroscopic

viscosity.

As can be observed from Fig. 3 there are important differences

between the aging rates of the different measured quantities, i.e.,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

G

02þ G002

p

=u. With these measurements, we

Fig. 1

scattering functions (symbols) and their corresponding fits (solid lines)

for a glass (Laponite 3 wt%, pure water). The waiting times are shown

in the legends in terms of the ergodicity-breaking time teb¼ 450 min.

The last three VV data-sets are in the non-ergodic state obtained by

ensemble-averaged measurements.

Evolution of polarized (VV) and depolarized (VH) intermediate

Fig. 2

different waiting times in a glass (Laponite 3 wt%, pure water). The lines

show the fits with the sum of a single and a stretched exponential

according to eqn (3).

The orientational correlation functions defined as fVH/fVVat

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translational and rotational diffusion and viscosity. The first

important observation is that the slow relaxation time of the

orientational degree of freedom grows faster than the corre-

sponding one for the translational degree; this is the first indi-

cation of the decoupling of the translational degree of freedom

from the rotational degree during the aging of Laponite

suspensions. The second important observation comes from the

comparison of the light scattering results with the low-frequency

complex viscosity modulus as a function of waiting time, as also

shown in Fig. 3.

In Fig. 3(a (b)) the viscosity is scaled in such a way that if

Stokes–Einstein(–Debye) relation were valid, one would obtain

the translational (rotational) relaxation time from the corre-

sponding viscosity according to the Stokes–Einstein(–Debye)

relation. From this figure, we interestingly find that the growth of

both translational and rotational relaxations occurs at a different

rate than that of viscosity. In other words, not only translational

and rotational degrees are decoupled but also the structural

relaxation time of the system characterized by viscosity is

decoupled from these degrees of freedom. Looking at Fig. 3, we

can recognize three regimes of aging for the evolution of

viscosity. The first regime of aging (tw < 0.4teb) where the

viscosity grows at a similar rate as that of the slow translational

relaxation time and the Stokes–Einstein relation is obeyed but

the slow rotational relaxation time grows at a much faster rate,

although short-time rotational diffusion shows a good agreement

with the viscosity. Thesecond regime (0.4teb< tw< 0.75teb) where

the viscosity growth goes beyond the slow translational relaxa-

tion time and the slow rotational relaxation time still remains

much higher than the other two relaxations. In the third regime

(tw> 0.75teb), upon approaching the ergodicity-breaking time,

the structural relaxation time becomes comparable to the slow

rotational relaxation time, however the slow translational

relaxation time still remains considerably lower, signalling an

enhanced translational diffusion compared to rotational diffu-

sion and structural relaxation.

The observed behavior in Fig. 3 represents the central result of

our work and can be understood in light of spatially heteroge-

nous dynamics, i.e., formation of mobile regions in which the

translational (rotational) dynamics of particles are correlated.5

The observation of a stretched exponential for both translational

and orientational correlations points to a broad distribution of

relaxation times supporting the existence of underlying spatially

heterogenous domains. The decrease of the stretching exponents

with waiting time31shows that the width of distribution of

relaxation times also increases with waiting time. Particularly,

the different behavior of translational and the rotational corre-

lations is very interesting and shows that the rotational degree of

freedom becomes glassy at a faster rate. This points to the fact

that translationally correlated mobile regions do not necessarily

correspond to the domains where the rotational motion of

particles are correlated. Indeed, recent measurements of

a monolayer of ellipsoidal particles have revealed that trans-

lational and rotational cooperative motions are anti-correlated

in space and translationally mobile regions correspond to

pseudo-nematic domains where the orientational degree of

freedom is frozen.10

The much slower growth of the translational relaxation time

compared to rotational and structural ones points to an

enhanced translational diffusion that becomes more pronounced

with increasing waiting time. However, one should note that the

average translational diffusion is not simply the inverse of the

average relaxation time because of a broad distribution of

relaxation times. The average translational diffusion that

involves the passage of particles through different domains and is

determined by hs?1i rather than 1/hsi.38

A more quantitative picture for the translational diffusion can

be obtained by extracting the frequency-dependent diffusion

from dynamic light scattering measurements. The frequency-

dependent diffusion can be obtained from the Laplace transform

of intermediate scattering function according to the following

relation:39

ðN

Fitting f(q,t) with eqn (3), we can obtain its Laplace transform

analytically using Mathematica and therefore extract D(q,u).

We can now compare this frequency-dependent diffusion

coefficient with that of a probe particle whose size is much larger

than the Laponite particles. Such large particles probe the global

viscosity, and it has been shown previously that for a probe size

much larger than the Laponite size, the diffusion coefficient of

the probe obeys the Stokes–Einstein relation:42

frequency-dependent complex viscosity that is equal to the bulk

viscosity.32Utilizing the microrheology technique, we obtained

Sðq;uÞ ¼

0

fðq;tÞexpðiutÞ ¼

1

?iu þ Dðq;uÞq2

(4)

it gives

Fig. 3

and the scaled viscous modulus at f ¼ 0.05 Hz versus scaled waiting time

measured for a Laponite suspension of C ¼ 3 wt% (glass). The viscosity is

scaled in such a way that if Stokes–Einstein relation Dtr¼kBT

Stokes–Einstein–Debye relation Dr¼3kBT

obtain the translational (rotational) relaxation time from the corre-

sponding viscosity. The dashed lines shown are just for a guide to the

eyes.

The fast and slow translational and orientational relaxation times

12hRand

32hR3were valid, one would

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the displacement fluctuations of a micron-sized probe particle

trapped by optical tweezers and directly computed the power

spectrum of displacements, DðuÞ ¼1

In order to compare the diffusion coefficients from micro-

rheology and DLS experiments, we have scaled their values with

the factors 6pRprobeand 12RLaponite, respectively to remove the

trivial dependence of diffusion coefficient on particle size and

shape. In Fig. 4, we have depicted the frequency-dependent

diffusion of Laponite particles and the probe particle at two

different waiting times. At early stages of aging the two diffusion

coefficients agree at high frequencies, while the diffusion of

Laponite particles is significantly lower at low frequencies.

Interestingly, this is in line with our earlier observation of the

agreement of the short-time rotational diffusion with the

viscosity at early stages of aging. For small tw, the viscosity is

frequency-independent;32

therefore

at high frequencies has the same value as that measured at

f ¼ 0.05 Hz. This suggests that for high frequencies both Stokes–

Einstein and Stokes–Einstein–Debye relations hold at the early

stages of aging. As the sample ages, this difference becomes

stronger and the two diffusion coefficients no longer coincide at

any frequency. This demonstrates that the diffusion of the larger

particle, i.e. the viscosity, evolves at a faster rate than the

Laponite particles themselves, in line with the results presented

in Fig. 3.

To demonstrate the scale-dependence of diffusion with waiting

time, in Fig. 5, we have plotted the ratio of the scaled diffusion

coefficient of the Laponite particles to the scaled diffusion

coefficient of the probe particle as a function of waiting at a low

u ¼ 21 rad s?1and a high frequency u ¼ 6 ? 106rad s?1. As can

be seem from this figure this ratio is close to 1 at high frequencies

while it grows with waiting time at low frequencies. The

comparison between the two methods clearly shows the devel-

opment of a scale-dependent diffusion coefficient, which in turn

demonstrates the development of a spatially heterogenous

dynamics that becomes more pronounced as the waiting (aging)

time increases. The microrheology measurements, performed

with probe particles of diameters of 0.5 and 1 mm, showed that

beyond 0.5 mm the translational diffusion of probe particles no

longer shows a size-dependence and the Stokes–Einstein relation

is obeyed: the local viscosity measured by micron-sized particles

agrees with that macroscopic viscosity.32This means that at this

2u2??xðuÞ??2.

themeasuredviscosity

length scale the probe is large and slow enough to average out the

influence of dynamical heterogeneity and consequently the

Stokes–Einstein relation is valid.32,42Our findings also agree with

other experiments that investigated the diffusion of tracer

particles of different sizes as a function of waiting time.40,41

Measuring the diffusion of 50, 100 and 200 nm size tracer

particles in aging Laponite suspensions, Strachan et al. found

that their scaled diffusion coefficients are identical at the begin-

ning of aging, however as time proceeds, the correlation func-

tions of the larger tracer particles evolve at a faster rate.40Also in

line with our findings, Petit et al.41investigated the dependence of

translational diffusion on probe size at late stages of aging and

report that the diffusion coefficient becomes independent of

probe size for length scales larger than the interparticle distance.

The developments of scale-dependent diffusion in the later

stages of aging shows that the glass formation is accompanied by

growing dynamic correlation length scales during the aging

similar to the trend found in hard sphere glass suspensions upon

increasing the volume fraction.16The length scale beyond which

there is no scale-dependent diffusion can provide us with an

estimate of the size of the spatially correlated mobile regions in

the sample at late stages of aging. Our results set an upper bound

of 500 nm for the dynamic correlation length. On the other hand,

results of ref. 40 and 41 give us a rough estimate for the size of

dynamical correlation length 100 # xd< 200 nm which is between

3 to 7 particle diameters. Of course, a more systematic study of

probe size-dependent diffusion is required to determine the exact

waiting time-dependence of such a dynamical correlation length.

The growth of the average rotational relaxation time

concomitant with the broadening of the width distribution of

relaxation times suggests that there should be a distinct dynamic

correlation length associated with the characteristic slow time for

the orientational degree of freedom, that also grows with

increasing waiting time. Interestingly, our data show that the

orientational relaxation time at late stages of aging agrees with

the macroscopic viscosity. How the orientational relaxation

depends on the size of probe particle and how the translationally

Fig. 4

particlesat ascatteringvector corresponding toqR ¼0.3 comparedtothe

diffusionof a probe particle of diameter 1.16 mm at two different stages of

aging in a colloidal glass of Laponite 3.2 wt%. The waiting times are

shown in the legend. Note the diffusion of probe and Laponite particles

are scaled with their radii to be comparable.

The frequency-dependent diffusion coefficient of Laponite

Fig. 5

Laponite particles to the diffusion of a probe particle of diameter 1.16 mm

at two different frequencies (shown in the legend) as a function of waiting

time during the aging in a colloidal glass of Laponite 3.2 wt%. Note the

diffusion of probe and Laponite particles are scaled with their radii to be

comparable.

The ratio of the frequency-dependent diffusion coefficient of

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and rotationally mobile domains develop with waiting time, is an

issue that merits further investigation. However, our data does

support the existence of dynamically heterogenous domains for

both degrees of freedom that are not strongly correlated.

At this point it is instructive to compare our results of aniso-

tropiccolloidalglasswith

formers5,43,44where both translational and rotational relaxations

grow concomitant with the increase of viscosity upon super-

cooling. Studies on supercooled liquids have found out that the

rotational correlation time follows the temperature dependence

of the Debye–Stokes–Einstein relation to within a factor of 2 or 3

while the viscosity changes by 12 orders of magnitude43,44upon

cooling. To the contrary, the translational diffusion does not

follow the temperature-dependence expected from Stokes–Ein-

stein relation5,43,44upon decreasing the temperature. Similarly to

what happens for our colloidal glass, in these systems (i) a faster

translational diffusion compared to rotation and (ii) a depen-

dence of the diffusion coefficient on probe-size have been

observed.5Therefore, our observations are qualitatively in

agreement with findings on molecular glass-formers, although as

we study here the waiting-time dependence rather than temper-

ature dependence, some features are particular to our aging

colloidal glass.

To summarize, the observed decoupling between rotational

and translational relaxation times and viscous relaxation, can be

understood in terms of dynamical heterogeneity. Particularly,

our data suggests that the development of heterogenous domains

for translational and orientational degrees of freedom are

distinct in accordance with the recent findings on a glass of

ellipsoidal colloidal suspensions.10The observed behavior of

rotational diffusion at early stages of aging and its strong

decoupling from translational and structural relaxation is

different from the reported measurements in the literature and

requires further experimental and theoretical investigations.

findingsonmolecularglass

Acknowledgements

We would like to thank Giulio Biroli and Gilles Tarjus for

stimulating and fruitful discussions. This research has been

supported by the Foundation for Fundamental Research on

Matter (FOM), which is financially supported by the Nether-

lands Organization for Scientific Research (NWO). S. J-F. was

further supported by the foundation of ‘‘Triangle de la

Physique’’. LPS de l’ENS is UMR8550 of the CNRS, associated

with the universities Paris 6 and 7.

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