Echo Dynamic Light Scattering experiments were performed on a colloidal hard sphere glass to measure the intensity autocorrelation function in the range 1 to 104 s. The first derivative of the intensity autocorrelation function is used to determine the inflection point, and isolate the two processes characterizing the decay of the intensity autocorrelation function - the &quot;fast&quot; process (often designated β decay) and the &quot;slow&quot; process (α decay). The behaviour of both processes are studied as a function of the age of the colloidal glass, defined as the time since the quench. The end of the fast process follows a stretched exponential law at early ageing times, and becomes more and more stretched, tending eventually towards a power law. The slow process follows a power law for all sample ages

Bryant, G

Martinez, V

Van Megen, W

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Martinez, V, Bryant, G and Van Megen, W 2006, 'Slow dynamics and ageing of
colloidal hard sphere glasses', in R. Sang and J. Dobson (ed.) Proceedings of the
17th National Congress of the Australian Institute of Physics, Brisbane, Australia,
3-8 December 2006.
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2006 Australian Institute of Physics
https://trove.nla.gov.au/version/244570581
Please	cite	as:	
	
Martinez, V, Bryant, G and Van Megen, W 2006 
'Slow dynamics and ageing of colloidal hard sphere glasses' 
in R. Sang and J. Dobson (ed.) Proceedings of the 17th National Congress of the 
Australian Institute of Physics, Brisbane, Australia, 3-8 December 2006. 
	
Australian Institute of Physics 17th National Congress 2006 –  Brisbane, 3-8 December 2006  
RiverPhys 
Paper No. XXX      1 
SLOW DYNAMICS AND AGEING OF COLLOIDAL HARD SPHERE GLASSES 
V. Martinez
A
, G. Bryant
A
 and W. van Megen
A
 
A
  Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne, Australia. 
 
Abstract 
Echo Dynamic Light Scattering experiments were performed on a colloidal hard sphere glass to measure the 
intensity autocorrelation function in the range 1 to 10
4 
s. The first derivative of the intensity autocorrelation 
function is used to determine the inflection point, and isolate the two processes characterizing the decay of 
the intensity autocorrelation function – the “fast” process (often designated β decay) and the “slow” process 
(α decay). The behaviour of both processes are studied as a function of the age of the colloidal glass, defined 
as the time since the quench. The end of the fast process follows a stretched exponential law at early ageing 
times, and becomes more and more stretched, tending eventually towards a power law. The slow process 
follows a power law for all sample ages. 
 
Introduction 
Colloidal hard sphere suspensions are a widely used model for the fundamental study of crystallization and the 
glass transition. Such suspensions exhibit crystallization for volume fractions exceeding the freezing volume 
fraction φf=0.494, in agreement with molecular dynamics simulations of ideal hard spheres. Once the volume 
fraction exceeds ~0.565, however, crystallization is suppressed, yielding long lived metastable colloidal glasses. 
These systems exhibit slow dynamics and ageing effects, which are of great fundamental interest, as well as 
being relevant to a range of problems in applied materials science. Colloidal hard sphere have been extensively 
used for the last two decades as the simplest model for investigating the traditional glass transition in 
condensed state matter. There are also many soft materials, such as gels, pastes, blood and foams that exhibit 
similar out-of-equilibrium behaviour. However, this behaviour is still poorly understood due to a range of 
experimental obstacles. First, colloidal glasses exhibit slow dynamics and ageing behaviour as function of the 
elapsed time since the quench. Second, colloidal glasses are non-ergodic, which means that statistically 
meaningful results can only be obtained by making a large number of equivalent measurements on different 
regions of the sample. The recently developed echo dynamic light scattering technique makes it possible to 
study the slow dynamics and ageing of colloidal glasses and achieve a better understanding of the glass 
transition and colloidal glasses. Here, we present the initial results of such experiments. 
 
Materials and Methods 
Particles 
The preparation and characterization of suspensions used here have been described in previous work 
(Underwood and van Megen, 1996; Bryant et al., 1999). Here we mention briefly the most important points. 
The particles consist of a core of methylmethacrylate (MMA) and trifluoroethyl acrylate (TFEA). Particles 
were coated by an approximately 10nm thick layer of poly(12-hydroxystearic acid) to prevent any 
coagulation. The hydrodynamic radius Rh=200 nm and polydispersity 6% of the particles were determined by 
Dynamic Light Scattering (DLS) on very dilute samples. The particles are suspended in a solvent of cis- 
decalin, which has a refractive index very close to that of the particles. Due to this index matching there is no 
significant multiple scattering at wave vectors in the range 1.5<qR<4 (e.g. Williams and van Megen, 2001). 
The effective hard-sphere volume fraction was determined by scaling the observed freezing volume fraction to 
the known freezing volume fraction for hard-sphere particles φf=0.494 (Hoover and Ree, 1969). This gives a 
glass transition volume fraction φg=0.565±0.005. The sample studied here has an effective volume fraction of 
0.58 and thus is in the glassy region. 
 
DLS 
Dynamic Light Scattering (DLS) has been extensively used for the last two decades to measure particle 
dynamics, and details can be found elsewhere (Berne and Pecora, 1990). Here we describe the main points. A 
laser with a wavelength λ=632.8 nm is focussed onto a small region within the particle suspension. The 
incident light is scattered by the particles and creates a time varying speckle (diffraction) pattern. A single 
photon detector measures the intensity scattered by the suspension in the far field at the scattering vector q: 
 
! 
q =
4"n
#
sin
$
2
% 
& 
' 
( 
) 
*      (1) 
 
Where n and θ are the refractive index and scattering angle respectively. As the particle positions evolve due 
to Brownian motion, the speckle pattern fluctuates. The intensity fluctuations provide information about the 
particle dynamics. The quantity of interest is the ensemble average autocorrelation function (ACF) of the 
Australian Institute of Physics 17th National Congress 2006 –  Brisbane, 3-8 December 2006  
RiverPhys 
Paper No. XXX      2 
scattered intensity fluctuations 
! 
gE
2( ) q,"( ) . Normal DLS measures the time average intensity ACF 
! 
gT
2( ) q,"( )  of 
the scattered intensity fluctuations, which is related to the electric field ACF 
! 
f q,"( ), also known as the 
intermediate scattering function (ISF), by the Siegert relation: 
 
! 
gT ,E
2( ) q,"( ) =
I q,t( ).I q,t + "( )
T ,E
I q,t( )
T ,E
2
 
! 
g
2( ) q,"( ) =1+ c f q,"( )
2
   (2) 
 
Where 
! 
T
 and 
! 
E
 denote time and ensemble averages respectively. For a colloidal fluid at low volume 
fractions, φ<φf, fluctuations in the scattered intensity decay in less than about 1s. Thus, for a measurement 
lasting several hundred seconds, the time and ensemble average of the intensity ACF are equivalent, i.e. 
 
! 
gE
2( ) q,"( ) = gT
2( ) q,"( )      (3) 
 
In others words, the particles are able to evolve through a representative fraction of the spatial configurations 
on the experimental time scale. Such a system is called ergodic. 
 
Increasing the volume fraction above the glass transition, φ>φg, the particle motions are restricted and only 
small scale motions are permitted. Only a sub-ensemble, rather than a full ensemble, of spatial configurations 
is accessed on the experimental time scale. Thus the fluctuation time of the scattered intensity is of the same 
order as the measurement time once the glass transition is reached. In other words, the time and ensemble 
averages of the intensity ACF are no longer equivalent. 
 
! 
gE
2( ) q,"( ) # gT
2( ) q,"( )       (4) 
 
Such a system is called non-ergodic, and ensemble averaging needs to be performed to measure the particle 
dynamics. To achieve this we have implemented a newly developed technique, called echo DLS (Pham et al., 
2004). This technique allows the measurement of the ensemble average intensity ACF for non-ergodic 
systems. 
 
Echo DLS 
The echo DLS method is an efficient technique for the measurement of the ensemble average intensity ACF. 
Details of the technique can be found elsewhere (Pham et al., 2004). The method involves a continuous 
rotation of the sample while measuring the scattered intensity. At a certain time t, a speckle i illuminates the 
detector. At every revolution of the sample nT, where n and T are respectively the number and period of the 
revolution, the same speckle illuminates the detector. The time average intensity ACF of the speckle i, 
! 
gi,T
2( ) q,"( ) , is calculated at every delay time τ=nT and results in peaks, or echoes. The maximum height of the 
peak corresponds to the speckle i. Either side of the peaks corresponds to multiple speckles. During one 
rotation, thousands of speckles illuminates the detector and thus thousands of time average intensity ACFs are 
calculated over a measurement time scale corresponding to the last delay time of interest, typically 10
4 
s for 
our colloidal glasses. Pham et al. found that the shape of the peak is independent of the samples and depends 
only on how many speckles are measured, and the rotation quality. The area under the peak is independent on 
the quality of the rotation, and is used to correct for imperfect rotation. For colloidal glasses, every speckle 
corresponds to a sub ensemble i of the full spatial configurations. Averaging the independent intensity ACF of 
the sub ensembles gives the intensity ACF of the full spatial configuration. 
 
! 
gE
2( ) q,"( ) = gi,T
2( ) q,"( )
i
      (5) 
 
The bracket 
! 
i
 denotes an average over thousands of independent speckles providing the required ensemble 
average. Due to the limitations of the accuracy of the rotation period, and to minimize the risk of disturbing 
the particle dynamics, the period of the rotation is limited to T=1 s. This places a lower limit on the measured 
delay time. The maximum delay time is fixed by the measurement time, typically 10
4 
s for our samples. The 
Echo DLS method must be combined with conventional DLS to measure the particle dynamics at shorter 
delay times (van Megen et al., 1998). 
 
Australian Institute of Physics 17th National Congress 2006 –  Brisbane, 3-8 December 2006  
RiverPhys 
Paper No. XXX      3 
Analysis 
The aim of this study is to try to characterise the ageing of a colloidal hard sphere glass. The dynamics of 
colloidal glasses are characterized by an initial decay at short delay times, typically less than about 1s, called β 
decay or the “fast” process, following by a plateau and another decay at longer delay time, typically more 
than about 5000s, called α decay or the “slow” process. The objective here is to separate and study both 
processes as functions of the age of the sample, which is quantified by the waiting time tw, also named ageing 
time, from the end of the quench to the start of the measurement. The sample is tumbled gently to destroy 
any crystals inside the sample in order to erase the memory of particles in term of particle motions and 
particle positions. The quench is achieved by stopping the tumbling. The plateau or inflection point between 
the fast and slow processes was observed to occur at delay times between 50s and 1000s. Thus, in this study we 
restrict our measurement to Echo DLS experiments and delay times in the range 1-10
4
s. The intensity ACF 
was normalized to the point τ=1s. Analyses of the autocorrelation function were made in several steps. 
(i) First, we determine the position of the inflection point, and the corresponding delay time τm, by calculating 
the minimum of the first derivative of the intensity ACF (van Megen et al., 2005): 
 
! 
d
d log"
log #log gE
2( ) q,"( )( )( )[ ]       (6) 
 
(ii) Second, we calculate the quantity P(q,τ), which translates the ACF so that 
! 
gE
2( ) q,"m( ) = 0  at the inflection 
point. 
 
! 
P q,"( ) =
gE
2( ) q,"( ) # gE
2( ) q,"m( )
1# gE
2( ) q,"m( )
     (7) 
 
We refer to 
! 
P q," < "m( )  as the “fast” component of the fluctuations and 
! 
P q," > "m( )  as the “slow” 
component. 
(iii) Third, we apply appropriate fits to the “fast” and “slow” processes as a function of the sample age. 
 
The purpose of this paper is to determine what information the quantities 
! 
P q," < "m( )  or 
! 
P q," > "m( )  reveal 
about the ageing process. 
 
Results and Discussion 
The experiments were all performed at a scattering angle of 60°, near the main peak of the structure factor, and at a 
temperature of 23°C. The ACFs are shown in fig. 1a. 
1.00
0.95
0.90
0.85
0.80
g
E
(2
) (
q
,!
)
10
0
10
1
10
2
10
3
10
4
! (s)
 0h40
 4h
 10h
 24h
 48h
 84h
(a)
10
-2
10
-1
10
0
P
(q
,!
)
10
0
10
1
10
2
10
3
10
4
! (s)
 0h40
 4h
 10h
 24h
 48h
 84h
 
(b)
10
1
10
2
10
3
! m
10
-1
10
0
10
1
10
2
tw (h)
(c)
 
Figure 1: (a) Ensemble average of the intensity ACF and its first derivative as functions of the delay time for 
φ=0.58 at different ageing times. The inset shows the derivative (eq. 6) from which the inflection point is 
determined. (b) 
  
! 
P q,"( ) as function of the delay time for φ=0.58 at different ageing time. The lines are a 
stretched exponential a fit 
! 
P " < "
m( ) = exp # tw / t0( )
$[ ]  and a power law fit 
! 
P " > "
m( ) = tw / ts( )
#
 for the 
“fast” and “slow” process respectively at an ageing time of 40 min. (c) Delay time of the inflection point, as a 
function of the ageing time tw. The straight line is power law fit. 
 
Figure 1b clearly shows that the quantity 
  
! 
P q,"( ) (fig 1b) highlights the behaviour around the inflection point, and 
allows a quantitative comparison of the fast and slow processes as functions of the ageing time. The inflection 
point is shown to follow a power law as a function of the ageing time, as shown in figure 1c.  
 
Australian Institute of Physics 17th National Congress 2006 –  Brisbane, 3-8 December 2006  
RiverPhys 
Paper No. XXX      4 
We turn now to the parameters extracted from the fits to the fast and slow processes. The decay time to (fig 2a) for 
the stretched exponential fit shows no significant change as a function of ageing time. The stretching parameter β 
(fig 2b) on the other hand shows a small but steady reduction as ageing time increases – in other words, the 
stretching is becoming more pronounced as a function of ageing. However, it is in the power law fits to the slow 
process that the ageing process is highlighted. The characteristic time ts (fig 2c) behaves as a power law with 
ageing time, while the exponent λ (fig 2d) is linear with ageing time. By extrapolating to zero, we can estimate the 
delay time which would be needed for the plateau in fig 1a to become completely flat – ie for the sample to become 
a perfect glass. For the current sample this occurs at a delay time of  ~300 hours.  
 
20
15
10
5
0
t 0
0.1 1 10 100
tw (h)
1.0
0.8
0.6
0.4
0.2
0.0
!
0.1 1 10 100
tw (h)
10
2
10
3
10
4
10
5
t s
0.1 1 10 100
tw (h)
1.0
0.8
0.6
0.4
0.2
0.0
"
100806040200
tw (h)
(a) (b) (c) (d)
 
Figure 2: Fit parameters for the “fast” and “slow” processes as functions of the ageing time tw: (a) to and (b) 
β for stretched exponential fits to the fast process; (c) ts and (d) λ for the power law fits to the slow process. 
 
Although there are no quantitative experimental studies of ageing of colloidal hard spheres in the glassy state, 
several ageing studies has been performed on other systems such as gels formed by the aggregation of colloids 
(Cipelletti et al., 2000) and glassy laponite samples (Abou et al., 2001; Bellour et al., 2003). The three authors use 
different models to characterize the decay, so comparison is difficult. However, both Cipelletti et al. and Bellour et 
al. find that the characteristic decay time decays at long times as a power law with an exponent of ~1. In the case of 
the hard spheres studied here, this exponent is 0.8±0.05. Clearly the hard sphere glass is dynamically slower than 
these systems. 
 
Conclusions 
These results demonstrate that this technique and analysis provides a viable method for the analysis of ageing in 
colloidal glasses. Further experiments are currently underway to study ageing as functions of both volume fraction 
and scattering vector. This work will shed new light on our understanding of ageing processes in colloidal glasses, 
and will have wider application to the study of ageing in a range of glasses and gels. 
 
Acknowledgements 
The authors would like to thank Khoa Pham, Ben Kent and Hans Joachim Schöpe for invaluable advice and 
discussions, and Phil Francis for technical assistance. 
 
References 
Abou, B., Bonn, D. and Meunier, J. (2001). Aging dynamics in a colloidal glass. Phys. Rev. E 64, 021510. 
Bellour, M., Knaebel, A., Harden, J.L., Lequeux, F. and Munch, J.-P. (2003). Aging processes and scale dependence in soft glassy colloidal 
suspensions. Phys. Rev. E 67, 031405. 
Berne, B.J. and Pecora, R. (1990). Dynamic Light Scattering. Robert E. Krieger Publishing Company, New York. 
Bryant, G., Mortensen, T., Henderson, S., Williams, S. and van Megen, W. (1999). Optical contrast variation study of nonaqueous 
suspensions of polymer particles. Journal of Colloid and Interface Science 216, 401-408. 
Cipelletti, L., Manley, S., Ball, R.C. and Weitz, D.A. (2000). Universal aging features in the restructuring of fractal colloidal gels. Phys. Rev. 
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Hoover, W.G. and Ree, F.H. (1969). Melting transition and communal entropy for hard spheres. J. Chem. Phys. 49, 3609-3617. 
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Rev. of Sci. Instrum. 75 (7), 2419-2431. 
Underwood, S.M. and van Megen, W. (1996). Refractive index variation in nonaqueous sterically stabilized copolymer particles. Colloid 
Polym. Sci. 274, 1072-1080. 
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Slow dynamics and ageing of colloidal hard sphere glasses

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Slow dynamics and ageing of colloidal hard sphere glasses

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