Aging in Dense Colloids as Diffusion in the Logarithm of Time

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by  S. Boettcher

Aging in Dense Colloids as Diffusion in the Logarithm of Time
Stefan Boettcher1,∗ and Paolo Sibani2,†
1Department of Physics, Emory University, Atlanta, GA, USA
2Institut for Fysik og Kemi, SDU, DK5230 Odense M, Denmark
The far-from-equilibrium dynamics of glassy systems share important phenomenological traits.
A transition is generally observed from a time-homogeneous dynamical regime to an aging regime
where physical changes occur intermittently and, on average, at a decreasing rate. It has been
suggested that a global change of the independent time variable to its logarithm may render the
agingdynamicshomogeneous: forcolloids,thisentailsdiffusionbutonalogarithmictimescale. Our
1 novelanalysisofexperimentalcolloiddataconfirmsthatthemeansquaredisplacementgrowslinearly
1 in time at low densities and shows that it grows linearly in the logarithm of time at high densities.
0 Correspondingly,pairsofparticlesinitiallyinclosecontactsurviveaspairswithaprobabilitywhich
2 decaysexponentially in either time or its logarithm. The form of the Probability Density Function
ofthedisplacementsshowsthatlong-rangedspatialcorrelationsareverylong-livedindensecolloids.
n
A phenomenological stochastic model is then introduced which relies on the growth and collapse
a
of strongly correlated clusters (“dynamic heterogeneity”), and which reproduces the full spectrum
J
of observed colloidal behaviors depending on the form assumed for the probability that a cluster
8
collapses during a Monte Carlo update. In the limit where large clusters dominate, the collapse
2
rate is ∝ 1/t, implying a homogeneous, log-Poissonian process that qualitatively reproduces the

experimental results for dense colloids. Finally an analytical toy-model is discussed to elucidate
]
t the strong dependence of the simulation results on the integrability (or lack thereof) of the cluster
f
o collapse probability function.
s
. PACSnumbers:
t
a
m
I. INTRODUCTION not applicable to colloids, since the dichotomy between
-
d equilibrium-like fluctuations and off-equilibrium dynam-
n ics is absent. Reconciling differences and similarities
Aging in amorphous materials has attracted
o between aging in colloids and other glassy systems thus
widespread experimental, simulational and theoreti-
c requires a novel and broader approach.
[ cal interest for more than thirty years [1–6]. As a
In this work, the colloidal particle tracking experi-
spontaneous off-equilibrium relaxation process, aging
2 ments of Courtland et al. [13] are re-analyzed, lead-
entails a decrease of the free energy and, correspond-
v ing to the conclusion that particle motion is diffusive
ingly, a slow change of thermodynamic averages. E. g.,
0
in either time or logarithmic time, at low or high den-
7 in numerical studies of models for disordered mag-
sities, respectively. Dynamics properties, such as the
4 nets, the thermal energy decreases intermittently and,
Mean Squared Displacement (MSD) and the persistence
5 on average, at a decelerating rate during the aging
. process [6–9]. Like these low-temperature materials, of tracer particles, appear homogeneous when viewed as
0
a function of an appropriately re-scaled time variable.
1 colloidal suspensions at high density (i. e. high volume
As the MSD, for instance, is a sum of independent dis-
9 fraction) exhibit intermittent dynamics and a gradual
0 slowing down, here, in the rate at which particles move placements, the correct choice of time variable, hence,
v: during light scattering [10, 11] and particle tracking trivializes colloidal dynamics[30]. To explain why that
is, a novel stochastic model is introduced with a sin-
i experiments [12–15]. While the phenomenology of
X gle free parameter controlling the transition from diffu-
colloidal aging is broadly similar to that of thermally
r activated aging, spatially averaged quantities as energy siveto log-diffusivebehavior. Inthe log-diffusiveregime,
a thedynamicsisdrivenbyincreasinglyrareevents,called
and particle density hardly change in colloids. Further-
“quakes”,whosetemporaldistributionishomogeneousin
more, no external field is required to elicit a measurable
logarithmic time. This links the dynamical behavior of
response. Finally, the motion of colloidal particles is
dense colloidsto the notionof“recorddynamics”[18,19],
time-homogeneousfor sufficiently low density,but for no
a relaxation paradigm already applied to a number of
accessibletimescalesandfornovalueofthedensitydoes
complex systems[3, 9, 20, 21].
it look stationary or equilibrium-like. Generalizations of
the Fluctuation Dissipation Theorem and the concept OurexperimentaldataanalysisconsiderstheMSDand
of effective temperature[16, 17], developed to describe the persistence, i. e. the probability that a pair of parti-
thermally controlled aging in disordered magnets, are cles initially close to each other remain close for at least
timet,underanumberofdifferentconditions. TheProb-
ability Density Function (PDF) of the particle displace-
mentislikewisemeasuredtoillustrateintermittencyand
∗www.physics.emory.edu/faculty/boettcher/ the long range of the spatial correlations. The model
†[email protected] encodes the spatial heterogeneity in particle mobility,
2
0.5 100
1.6
(MSD/5)
0.4 1.4 Thr.=0.8
1.2 ability 100 Thr.=0.6
00..23 2µMSD ()00..681 Survival prob Thr.=0.4
air
0.4 P 10−1
0.1 5 10 15 20 25 30 35 40
t (s)
0.2
0 0.5 1 1.5 2 2.5 30 1000 101 102 103 104 10−1100 101 102
t (hrs) t/tw t/tw
2
Figure 1: (Color online) Left: Mean Square Displacement (in µ ) for colloids at low density, volume fractions ρ ≈
0.46,0.52,0.53,0.56, top to bottom, all plotted versus time, in seconds. The top-most data set has, for graphical reasons,
been scaled down by a factor of five. Middle: same quantity for dense colloids, where for all data ρ ≈ 0.62. Waiting times,
in seconds: tw ≈ 1,10,20, and 150 for pluses, squares, circles, and stars, respectively. Right: At high density, (ρ ≈ 0.62) the
persistencedecaysasapowerlawdependentonaseparation-thresholdθ. Theinsetshowstheexponentialdecayatlowdensity,
ρ≈0.46, for thesame thresholds.
which is ubiquitous in glassy systems [12, 15, 22, 23], ing variable t/t , where t ,t ≤ t is the age at which
w w w
in terms of the formation and evolution of clusters of data collection commences. Although a more precise
strongly correlated spatially contiguous particles. These knowledge of t would facilitate a better data collapse,
w
clusterssignalemerging“dynamicheterogeneities” (DH), it is apparentthat the particles ina dense colloidclearly
i. e. a broad distribution of time-scales for the sur- diffuse in logarithmic time.
vival and break-up of correlation patterns in different The rightmost panel of Fig. 1 displays data for the
regions [12]. The stochastic model abstracts the under- persistence defined above. To ensure statistical indepen-
lying dynamics solely into a cluster survival probability: dence, we partition the system in 20×20×5 identical
complexbehaviorreplacessimplediffusivebehavioronce sub-volumes. At time t , a particle centrally located in
w
clustersattainnon-zeroprobabilitytosurviveindefinitely each sub-volume and its closest neighbor are tagged as
under the update rule. When the probability per up- a pair. If at time t > t the distance between the ele-
w
date to break up falls off exponentially with cluster size, ments of a pair increases beyond a given threshold value
events in the model become well-separatedin time, with θ and subsequently remains above θ, the pair is deemed
a log-Poissonian statistic. This reproduces the qualita- to have broken up at t. In the plot, three θ-values are
tive features of the experimental data: the dynamics is considered. Measured in units of particle radius, these
intermittent and homogeneous in logarithmic time. θ are large compared to the typical range of the relative
motion. The persistence decaysas a powerof t for dense
colloids,i.e.exponentiallyinlog(t), while atlowdensity
II. EXPERIMENTAL DATA ANALYSIS it relaxes exponentially, as shown in the insert.
Finally, Fig. 2 shows the empirical PDF (also known
Obtained using particle tracking techniques, the col- as the self-part of the Van Hove distribution function)
loidal data of Courtland et al. [13] comprise the 3d- of the absolute value of the particle displacement (in
trajectories of a set of tagged colloidal particles. The any direction) in the liquid (left panel) and glassy (right
particle radius is 1.18 microns, and the position coordi- panel) regimes of the colloids. The vertical scale is log-
nates are given in microns, a length unit nearly equal to arithmic. The three data sets displayed correspond to
the particle radius. Colloids were initially centrifuged to different values of the time interval ∆t over which the
obtain the desired density, and then briefly stirred. The displacement is observed. The inserts include more val-
end of the stirring phase is considered as the origin of ues of ∆t, and merely confirm the diffusive, respectively
the time axis, i.e., t = 0. Without time-translational log-diffusive, character of the dynamics previously dis-
invariance, this choice impacts the description of dense cussed. Note that the MSD does not cross the origin.
colloids. The particles already have acquired some dispersion af-
The MSD, relative to any overall drift in the system, ter the first sweep of the confocal microscope, probably
is calculated from the trajectories as a function of time. due to a residual stirring motion.
Fig.1,leftmostpanel,showsthewell-knowndiffusivebe- In similar analyses performed on a number of differ-
havior of low density colloids. The diffusion coefficient ent systems [24, 25], a central Gaussian part is gener-
decreasesmonotonicallywithincreasingvolumefraction. ally flanked by an exponential ‘non-Fickian’ tail. The
MSD pertaining to high-density colloids are plotted in left-most part of the PDF should, by analogy, have a
themiddlepanelonaloghorizontalscaleversusthescal- parabolic shape, and some curvature is indeed visible in
3
102 102
∆ t = 360 ∆ t = 100
∆ t = 180 0.1 ∆ t = 60 0.2
100 ∆ t = 18 2µD ()0.05 100 ∆ t = 20 2µSD ()0.15
S M
F M F
D D
P 00 500 1000 P 0.1101 102 103
10−2 ∆ t (s) 10−2 ∆ t (s)
10−4 10−4
0 0.5 1 1.5 0 0.5 1 1.5 2 2.5
|∆ r| (µ) |∆ r| (µ)
Figure 2: (Color online) Theleft panelpertains totheliquid,and theright oneto theglassy regime of acolloid. Both panels
display on a log-scale thePDF for theabsolute value |∆r| of theparticle displacement (in any direction). Three data sets are
shown, corresponding to different durations (in seconds) of the time interval ∆t over which the displacement is measured. In
theinserts,theMeanSquareDisplacement(MSD)overalargersetoftimeintervalsisplottedversustheintervallength,using
a linear (left panel) and logarithmic (right panel) abscissa.
the left panel at small values of the abscissa. However, eragenumberofparticlesboundinclustersPLi=1hi/L=
the exponential ‘tail’ extends right up to |∆r| = 0 in h¯ remains conserved. Note that these clusters do not
the glassyregime. Thus,long-rangedspatialcorrelations per-se represent individual colloidal particles and their
survive over long times in dense colloids. This very fea- motion, since particles typically occupy space uniformly,
ture is incorporatedas a centralhypothesis in the model with density fluctuations difficult to discern. Instead,
introduced next. ranges in cluster sizes represent DH in particle mobil-
ities, which can shift dramatically even though actual
particles hardly move. Here, heterogeneities arise dy-
III. CLUSTER MODEL namically as largerclusters,once formed, aremorelong-
lived than smaller ones, and the particles which belong
Theclustermodeldiscussedbelowassumesthatstrong to them correspondingly are less mobile.
kinematic constraints bind the particles together inside We update t → t+1 in parallel at all even and odd
clusters that form a spatially heterogeneous patchwork sites in alternating order. Each site i with h > 0 is
i
of particle properties. The growth and destruction of updated by drawing a univariate random number r . If
i
these clusters are the only mechanisms allowed for the r < Pα(h ), the cluster on that site breaks up into two
i i
netdisplacementsofparticles. Theupdatedynamicsisa randomly split fractions that are respectively added to
Markovchainwherearandomlyselectedclusterof(inte- the neighboring sites i±1, leaving i empty. To simulate
ger)sizeheithersurvivesintactorisdestroyedwithprob- the actual (activated) motion of particles, we add (mu-
ability P(h). By assumption, 0 ≤ P(h) ≤ 1 decreases tually non-interacting) tracer particles which reside on
with h, i. e. larger clusters are more stable than smaller latticesites. Thosewalkrandomlyi→i±1iff thereisa
ones. The particles released by a collapsed cluster join cluster on their site i that happens to break up. In par-
neighboring clusters, and are activated to move in real ticular,atracerparticlecanbestrandedonacluster-free
space by a unit step in a random direction. Model vari- site for a long time until a nearbycluster breaks up such
antsarecharacterizedbydifferentformsofP(h). Merely that its scattered “debris” can reactivate that particle.
the integrability of P(h) in h discriminates between the
Fig.3showstheMSDaveragedoveralltracerparticles
behaviors similar to those of low and high density col-
forα=1andα=∞. Theformerbehavesdiffusively(up
loids,andwithoutlossofgeneralitywecanlimitourselves to a cutoff scale t ∝ L2). In the latter, no stationary
st
to
state is approachedand the behavior is diffusive in loga-
1 rithmic time,asinthecorrespondingexperimentalresult
Pα(h)= , α=1,2..., (1)
α hk shown in Fig. 1. Fig. 3 also depicts persistence curves,
P
k=0 k! i. e. probabilities that pairs of tracer particles initially
an α-family of models with Pα=∞(h)=exp(−h). located at the same grid point remain within a separa-
In our Monte Carlo simulations the clusters are ar- tion of θ sites after t sweeps. As in the experiments, the
rangedinaperiodiclatticeoflengthL. Initially,eachlat- (asymptotic) decay goes from being exponential in time
tice site holdsuniformlya clusterofsizeh =h¯. The av- for α = 1 (not shown [26]) to being exponential in log-
i
4
104 LLL===521152268 P1(h)~1/h 3 LLL===521152268 Poo(h)~e-h 1100--01 Poo(h)~e-h
103 L= 64 L= 64 nce10-2 L=512, θ=2
MSD111000021 LL~t== 3126 MSD21 LL== 3126 Int. Persiste11110000----6453 LLLLLLLL========55111 1122233322888222,,,,,,,, θθθθθθθθ========44824882
10-1 101 102 103 104 105 106 0 101 102 103 104 105 106 101 102 103 104 105 106 107 108
Sweep time t Sweep time t Sweep time t
Figure 3: (Color online) Simulation results for different system sizes L for P1(h) ∼ 1/h (left) and P∞(h) ∼ exp(−h) (center
and right), to be compared with Fig. 1. For α = 1 (left) the MSD behaves diffusively, represented by the dotted line of unit
slope, while for α = ∞ the MSD grows linearly in logarithmic time, with no sign of saturation. The persistence for α = ∞
(right) exhibits a power-law, independentof the threshold θ.
-1 -2
10 10
P1(h)~1/h L=512 Poo(h)~e-h
-3
10
L=256
L L
/ L=128 /
s s -4
e L= 64 e10 L=512
k k
a L= 32 a L=256
u u
Q L= 16 Q10-5 L=128

of of L= 64
e e 10-6 L= 32
t t
a a
R R L= 16
-7 ~1/t
10
-2 -8
10 10
1 2 3 4 5 6 1 2 3 4 5 6
10 10 10 10 10 10 10 10 10 10 10 10
Sweep time t Sweep time t
Figure 4: (Color online) Plot of the rate of cluster collapse vs. system age t for P1(h) (left) and P∞(h) (right).
arithmic time when the cluster collapse probability goes tions. For P1(h) the rate approaches a constant on a
fromhyperbolictoexponentialatα=∞,independentof time scale independent of system size. For P∞(h), the
θ. Unliketheexperiments,changingθ doesnoteffectthe stationaryregimeisinfactinvisibleevenforthesmallest
power-law exponent but modifies the time for its onset. systems, andthe dynamics is thus purely non-stationary
with a decelerating quake rate decaying as 1/t. Assum-
ing that successive quakes are statistically independent
IV. RECORD DYNAMICS events (demonstrated elsewhere [26]), they provide the
trueclockofthe dynamics,andthe particleMSDsimply
becomes proportional to the time integral of the quake
As the collapse of large clusters controls all aspects
rate: linearintimeinthestationary(liquid)regime,and
of the model in the aging regime, the statistics of such
∼ log(t) in the aging regime. Clearly then, the MSD in
quake-events is key to the analysis, a situation identi-
theagingregimebetweentimest andtscalesast/t ,a
w w
cal to aging in quenched glasses [7]. In turn, the fast
so-calledfull aging behavior which is consistentwith the
collapse of small clusters provides a background of mo-
experimental findings.
bile particles diffusing within regions (“cages”) bounded
by large clusters. The divide between small and large Themodeldynamicsfortheexponentialcollapseprob-
clusters was empirically taken to be max(h¯,σ(t)), where abilitycanbelinkedtorecorddynamics [9,18–21,27,28],
σ(t) is the (growing) standard deviation of the cluster using the wide separation of time scales associated with
size distribution. The quake rate versus time, scaled by the collapse of clusters of different sizes. Let us first
the system size L, is shown in Fig. 4 for our simula- note that the empirically estimated boundary between
5
∞
small(i.e. irrelevant)andlarge(i.e. important)clusters, equallysharedbetweenallclusters,Nt =Pk′=0nk′,t. Its
h(t) ≈ lnt can in this case also be obtained by equat- form ensures the conservation of P∞k′=02k′nk′,t(= h¯L),
ing to the time t the number of queries that a cluster of as used in our simulations above.
height h on average survives. The pace of the dynamics Defining a =2kp n , we rewrite Eq. (2) as
k,t k k,t
isalwayscontrolledbythesmallestamongthelargeclus-
ters,henceforththeSLcluster. Thelatteristhefirstone 1
(a −a ) = a −a +a K T . (3)
to collapses with overwhelmingly high probability. Let p k,t+1 k,t k+1,t k,t k−1,t k−1 t
k
thecorrespondingquakehappenonatimescalet(0). The
newSLclustercaneitherbepre-existingorcanbeformed Amazingly, our analysis finds that one of the factors,
through the accretion process of the debris produced by Kk = 1/(cid:0)2kpk(cid:1) or Tt = n0,t/Nt, is always sufficiently
the last quake, thus recapitulating all previous history, smallasymptoticallyastoignorethelasttermofEq.(3).
an accretion process which is now instantaneous on the The remaining linear equation has stationary solutions
timescalet(0). Sincetheclustersizedistributionisessen- (for t ∼ t+1 → ∞) entailing ak+1 ∼ ak ∼ const, i. e.
tiallycontinuous,thenewSLclusterwillonlybeslightly nk ∼1/(cid:0)2kpk(cid:1). Sincethestationaryclustersizedistribu-
larger than its predecessor, a hallmark of “marginal sta- tion must have a finite integral (after all, Nt ≤L<∞),
bility increase” [8]. Hence, a record-sized random num- nk must decrease sufficiently, demanding pk ≫ 2−k or
ber (corresponding to a record sized re-arrangement in α<1. In this regime, Tt →T∞ ≤1 but ak−1Kk−1 ≪ak
the colloid) will suffice to destroy the new SL cluster. justifiesdismissalofthelastterminEq.(3). Conversely,
At time-step t the probability is 1/t that such event will for 2kpk ≪ 1 (α > 1), the aging regime, numerics sug-
happen. Statistical independence is assured by the wide gests that nk,t is highly localized at some increasing
separation of break-up events. Thus, inter-event times k = k(t): clusters grow and attain a typical size h(t).
τ(n) =lnt(n)−lnt(n−1)=log(cid:2)t(n)/t(n−1)(cid:3)becomePoisso- Unit-size clusters hardly ever occur, n0,t = 0 almost al-
nian distributed [27, 29]. To ascertain the log-Poisson ways, and although Kk grows exponentially, Tt ≈ 0 an-
nature of the quake time distribution, we collect the nihilates the last term. Rescaling τ = pkt then turns
quake times t(n) and show (elsewhere [26]) that the se- Eq. (3) into a “wave” equation, [∂τ −∂k]a(k,τ) ≈ 0,
ries (τ(1),τ(2),...,τ(n),...) of logarithmic waiting times with characteristic k ∼ τ = pkt. Using pk = pαk and
τ(n) is iid with the same exponential distribution. In h ∼ 2k, we extract the dominant growth-law for clus-
contrast, when the cluster collapse probability becomes ter sizes, h(t) ∼ t1/α, or for the event-rates (quakes) of
long-tailed, α < ∞, break-up events overlap and long- cluster-size increases:
lived clusters are deemphasized.
∂h ∼tα1−1, (α>1). (4)
∂t
V. MEAN-FIELD DESCRIPTION
The scaling reproduces for α → 1+ and for α → ∞ the
simulation results in Fig. 4 (and, correspondingly, those
Finally, we provide a mean-field model of the cluster- for any α in-between [26]). While the equations do not
ing dynamics that elucidates the preceding results. The describe the displacement of tracer particles explicitly,
fundamental quantity considered is the average number the discussionabove implies that the quake rate sets the
of clusters nk,t of binary size h=2k (k =0,1,2,...) ex- scale for spread in time for any observable, be it h(t) or
tant at time t. For k > 0, those k-clusters are immobile MSD: logarithmic for α→∞ and linear (with a cut-off)
and break up with probability pk ∼ P(2k), but merely for α≤1.
into two k−1-clusters. Eq. (1) reduces to pα ∼(cid:0)2k(cid:1)−α.
k
Integrability of p is conferred via P(h)dh ∼ p d(2k) ∝
k k
2kpkdk, i. e. Pk2kpαk < ∞ for α > 1. To write down a VI. CONCLUSIONS
self-consistentequationforn ,westipulate thatduring
k,t
eachsweep thenumberofk-clustersnk,t forallk >0de- Experimentaldataareanalyzedshowingthatcolloidal
creases through break-ups by a factor 1−pk, yet, it also motionisdiffusiveineithertime orlogarithmictime,de-
increases throughthe break-upof k+1-clustersinto two pending on colloidal density. A model explaining these
k-clusters (with weight2pk+1). Unit (k =0)-clustersare findingsispresented,basedonthesurvivalprobabilityof
indivisiblebutmobile. Largerk-clustersfurthergrowvia clusters, i. e. highly correlated groups of particles, rep-
accretion of those small mobile clusters, but (somewhat resenting dynamically heterogeneous regions of different
artificially)onlyingroupsof2k withprobability∝1/2k. mobility: a net displacement of a particle in the model
During a sweep, all unit clusters attach somewhere, i. e. requires the collapse of the cluster to which the particle
n0,t+1 is entirely unrelated to n0,t. Hence, we obtain belongs, i. e. a re-arrangementof (possibly large and in-
creasing) correlation patterns. The survival probability
n n
k−1,t 0,t
nk,t+1 = nk,t(1−pk)+2pk+1nk+1,t+ 2kN (2) ofaclusterincreaseswithitssize. Oncelargeclustersob-
t tain the capacity to survive indefinitely, P(h)≪1/h for
for k >0, and n =2p n . The last term describes h→∞,non-stationary,agingbehaviorarises,character-
0,t+1 1 1,t
growth through the accretion of all unit clusters n , ized by intermittency and memory effects. Remarkably,
0,t
6
the t/t scaling of the experimental data (full aging) is
w
only achievedin the modelinthe limitofanexponential
P(h), corresponding to the extreme value α = ∞ of its Acknowledgments: WethankEricWeeksforhelpful
parameter. In that sense, tuning a parameter such as discussionandaccesstohisdata. SBissupportedbythe
α, say, by changing the volume fraction ρ in a colloid, U. S. National Science Foundation through grant DMR-
doesnotdescribeasharpphasetransition. But,presum- 0812204. PS thanks the Physics department of Emory
ably,the transitionbetweenboth extremesoccursovera University for its hospitality and its Emerson Center for
nearly unobservable interval of ρ. financial support.
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