Bryant, G

Martinez, V

Van Megen, W

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PLEASE DO NOT REMOVE THIS PAGE
 Martinez, V, Bryant, G and Van Megen, W 2009, 'Power law dependence of ageing in a
colloidal hard sphere glass', in John Cashion and Trevor Finlayson (ed.) Proceedings of the
33rd Annual Condensed Matter And Materials Meeting, Wagga Wagga, Australia, 4-6 Feb
2009, pp. 1-4.
https://researchbank.rmit.edu.au/view/rmit:11518
Accepted Manuscript
2009
 N/A 
Power Law dependence of Ageing in a Colloidal Hard Sphere Glass 
 
V.A Martineza, G. Bryanta and W. van Megena 
 
a Applied Physics, Royal Melbourne Institute of Technology, Melbourne, VIC 3000, Australia. 
 
Measurements of the Intermediate Scattering Function are performed for a 
colloidal hard sphere glass as functions of scattering vector and waiting time, 
tw. We identify a fast process which is independent of tw, and a slow process 
which shows significant ageing behaviour. We quantify this behaviour via the 
stretching index, defined at the crossover between fast and slow process. This 
analysis shows that the approach of the sample to an ideal “aged” glass, the 
state in which particles perform only limited motions about fixed average 
positions, is algebraic in nature. 
 
 
1. Introduction 
At high volume fractions, colloidal suspensions can exist in out-of equilibrium states 
known as colloidal glasses, which exhibit slow dynamics and ageing effects 1. Recent 
investigations of the ageing process have concentrated on colloidal gels and glasses (e.g. 
laponite) as functions of the waiting time, tw, the time elapsed since the quench. The 
intermediate scattering functions (ISF) of these soft solids are generally found to decay in two 
stages as a function of delay time τ: an initial decay onto a plateau, followed by a final, age 
dependent decay. In some cases the plateau height, or Debye-Waller factor, is observed to be 
non-stationary 2. 
In order to shed more light onto the mechanisms responsible of the ageing process, we 
perform experiments on a simple model system system: a colloidal hard sphere glass. In this 
paper we present the results of an extensive series of measurements on the ageing behaviour 
of hard sphere glasses as functions of scattering vector and waiting time, and provide some 
explanations for the observed behaviour. Some of this data has been presented elsewhere 3. 
 
2. Sample preparation 
The particles used in the experiments reported here consist of cores of 
methymethacrylate (MMA) and trifluoroethyl acrylate (TFEA). Particles were coated by an 
approximately 10nm thick layer of poly(12-hydroxystearic acid). The preparation and 
characterisation of these particles have been extensively described in previous work 4, 5. Their 
average hydrodynamic radius (R=200nm) and polydispersity (6%), were determined by DLS 
on very dilute samples. The particles are suspended in a solvent of cis-decalin providing good 
optical matching and thus removing multiple scattering effects over the range of wave 
vectors, 1.5<qR<4.15, studied here. These suspensions represent a good approximation to a 
perfect hard-sphere system. Accordingly, after mapping the suspension’s equilibrium phase 
behaviour onto that known for the perfect hard-sphere system, the volume fractions of the 
freezing and melting points can be identified as φf=0.493 and φm=0.545. In addition these 
suspensions exhibit a glass transition at φg~0.57 6, 7. 
 
The sample studied here has an effective volume fraction of 0.58 and thus is in the 
glassy region. Colloidal glasses are non-ergodic (ie time and ensemble averages are not the 
same), and specialized experimental methods are needed to measure dynamics in these 
systems. To make statistically valid measurements, nonergodic averaging techniques must be 
employed 1, 8 and separate procedures must be adopted to capture the fast and slow processes. 
We measure the fast process, or initial decay of the ISF, via the well-established procedure 
developed by Pusey & van Megen 8. The slow process, or long-time decay of the ISF, is 
measured by the Echo DLS method recently developed by Pham et al. 9. The results of the 
two methods are combined to yield dynamical measurements over several decades in time. 
Details of the methods are described elsewhere 3. 
 
3. Results 
Figure 1 shows the ISF as a function of delay time at φ=0.58 for several waiting times, 
measured at 4 different scattering vectors. In all cases an inflection point separates a fast 
decay, with no apparent dependence on tw, and a slow, non-stationary, process that stretches 
out as the waiting time increases. The fast process is associated with local motions of the 
particles around average fixed positions and the slow process with structural rearrangements 
of the particles. Ageing is manifest by the non stationary behaviour observed at long delay 
times, which is caused by irreversible structural changes. 
 
1.0
0.9
0.8
0.7
0.6
0.5
f(q
,τ)
 2.01
 2.93
 3.57 (Peak)
 4.15tw = 10h
 0h40
 4h
 10h
 24h
 60h
 108h
 f(∞)
qR=2.01
f(q
,∞
) S(q)
 f(q,∞)
 S(q)
 0h40
 4h
 10h
 24h
 60h
 108h
 f(∞)
qR=2.93
1.0
0.9
0.8
0.7
0.6
0.5
f(q
, τ)
420-2
log(τ / τB)
 0h40
 4h
 10h
 24h
 60h
 108h qR=3.57
420-2
log(τ / τB)
 0h40
 4h
 10h
 24h
 60h
 108h
 f(∞)
qR=4.15
 
FIG. 1: ISF as function of delay time 
€ 
τ τB  for the waiting times indicated. Measurements are made at several 
scattering vectors, q, around the structure factor peak (qR=3.57). Times are expressed in terms of the Brownian 
time 
€ 
τB = R2 6Do = 0.0207s ,  where R~200nm is the particle radius and Do the free diffusion coefficient. 
The upper limit of the delay time window is 
€ 
τmax τB ~ 4 ×105  and corresponds to 
€ 
τmax ~ 8200s . The inset 
shows the ISF for an age of 10h at several values of qR. Some of the data has been presented elsewhere [3]. 
 
 
Note that the ISF’s do not decay to zero in our time window, which spans 
approximately nine decades. This contrasts with the results of more complex soft systems, 
which show a full, or almost full, decay of the measured intensity correlation functions in the 
experimental time window. Despite the limited decay of the ISF, the ageing behaviour can be 
quantified in a number of ways. 
As seen from figure 1, the tw -dependency of the ISF increases as delay time is increased 
– it is most exposed at τ=τmax but almost imperceptible at the crossover between the fast and 
slow process (ie the inflection point). One way of  quantifying the ageing is by plotting the 
actual value of the ISF at the upper time limit of our experimental time window as a function 
of waiting time 3, but this is somewhat arbitrary as it depends on the time chosen. An 
alternative method, with no ambiguity, is to calculate the “width” function 
€ 
w q,τ( ) = −q2 ln f q,τ( ) , defined by analogy with the mean square displacement for single 
particle statistics. If the relaxation responsible of the decay of the ISF were diffusive, w(q,τ) 
would grow linearly with the delay time τ. Figure 2 shows w(q,τ) as function of delay time for 
several the same values of qR and waiting times as figure 2. It is clear that w(q,τ) is linear 
(diffusive) at short times, but non-diffusive, or non-Fickian, processes emerge at intermediate 
times. 
-5
-4
-3
-2
-1
lo
g[
w 
(q
,t)
/R
2 ]
-1.8
-1.6
-1.4
-1.2
-1.0
lo
g[
n(
q,
τ )]
4.03.0
qR=2.01
-1.8
-1.6
-1.4
-1.2
-1.0
lo
g[
n(
q,
τ)]
4.03.0
qR=2.93
-5
-4
-3
-2
lo
g[
w 
(q
,t)
/R
2 ]
420-2
log(τ / τB)
-1.5
-1.4
-1.3
-1.2-1.1
-1.0
lo
g[
n(
q,
τ)]
4.03.0
qR=3.57
420-2
log(τ / τB)
-1.6
-1.4
-1.2
-1.0
lo
g[
n(
q,
τ )]
4.03.0
qR=4.15
 
 
FIG. 2: The width function [Eq. ()] as a function of delay time for several values of qR at the waiting times 
indicated. Insets: n(q,t) [Eq. ()] in the Echo DLS range, where the inflection is observed. Lines are polynomial fit 
in the minima region to better identify the minima. Some of the data has been presented elsewhere [3]. 
 
The crossover between the fast and slow processes (the inflection point) corresponds to 
the maximum deviation from diffusive behaviour - ie this point most clearly exposes the non-
Fickian processes. We identify this crossover by calculating the logarithmic derivative n(q,t): 
€ 
n q,τ( ) =
d log w q,τ( )[ ]
d logτ  
which is shown in insets of figure 2. It is obvious that n(q,t) has a minimum, 
€ 
ν q( ) =min n q,τ( )[ ], that deepens with age, and the delay time of this minimum is designated 
τm. This corresponds with the crossover between the fast and slow processes, and the 
stretching index 
€ 
ν q( ) as a measure of the coupling between them. The stretching index, as a 
measure of the deviation from Fickian behaviour, could also be called the Fickian index. For 
an infinitely dilute suspension, where all particles follows random processes, 
€ 
ν q( ) =1 and τm 
is undefined (no inflection point). For the perfect glass, where all particles are “caged”, 
€ 
ν q( ) = 0  and τm is infinite. We represent, in figure 3, 
€ 
ν q( ) as function of waiting time for 
several qR values. 
10-2
2
3
4
5
6
7
8
910
-1
St
re
tc
hin
g 
ind
ex
 ν(
q)
10-1 100 101 102waiting time tW (h)
5
4
3
2
1
0
S(
q)
54321
qR
0.30
0.20
0.10
0.00
tx
 2.01
 2.93
 3.57 (peak)
 4.15
 
FIG. 3: Stretching index as functions of waiting time, for the indicated values of qR. Lines through the data are 
power law fits. The inset shows that tx scales with the structure factor S(q). Some of the data has been presented 
elsewhere [3]. 
 
It is clear that the decay of 
€ 
ν q( ) follows a power law, for all qR, of the form: 
€ 
ν q,tw( ) =
tw
tx
# 
$ 
% 
& 
' 
( 
β
 
where the amplitude tx scales approximately with S(q), as shown in inset of figure 3, and the 
exponent b does not show any systematic q dependency. 
 
To conclude, we identify a fast process, with no ageing behaviour, and a slow process with 
strong ageing behaviour. We quantify the ageing behaviour at the crossover between the fast 
and slow processes, which corresponds to the maximum deviation from Fickian behaviour. 
The non-fickian aspects of the density fluctuations, quantified through the stretching index 
€ 
ν q( ), is found to follow a power law decay with tw. In other words, following the quench, the 
colloidal glass approaches its final state algebraically. This final state, the ideal “aged” glass, 
defined by 
€ 
f q,t→∞( ) = f q,τm( )  and 
€ 
ν q( ) = 0 , the state in which particles perform only local 
motions about fixed average positions, with no structural changes. 
 
 
Acknowledgments 
 
 
 
 
References 
 [1] W van Megen, T C Mortensen, S R  Williams, et al, Phys. Rev. E 58, 6073 (1998). 
[2] V A Martinez, G Bryant and W van Megen, Phys. Rev. Lett. 101, 135702 (2008). 
 
 
 
 
 
 
 
 
 
 
 
 
1 W. van Megen, T. C. Mortensen, S. R. Williams, et al., Physical Review E 58, 6073 
(1998). 
2 S. Jabbari-Farouji, G. H. Wegdam, and D. Bonn, Physical Review Letters 99 (2007). 
3 V. A. Martinez, G. Bryant, and W. van Megen, Physical Review Letters 101 (2008). 
4 S. M. Underwood and W. vanMegen, Colloid And Polymer Science 274, 1072 (1996). 
5 G. Bryant, T. Mortensen, S. Henderson, et al., Journal Of Colloid And Interface 
Science 216, 401 (1999). 
6 W. Vanmegen and S. M. Underwood, Physical Review Letters 70, 2766 (1993). 
7 P. N. Pusey and W. Vanmegen, Physical Review Letters 59, 2083 (1987). 
8 P. N. Pusey and W. Vanmegen, Physica A 157, 705 (1989). 
9 K. N. Pham, S. U. Egelhaaf, A. Moussaid, et al., Review Of Scientific Instruments 75, 
2419 (2004). 
 
 


Power law dependence of ageing in a colloidal hard sphere glass

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