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Brownian Dynamics Simulations of Aging Colloidal Gels

Rodolphe J.M. d’Arjuzon∗

Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, U.K.

William Frith and John R. MelroseColworth House, Unilever Research, Bedford, U.K.

(Dated: October 25, 2018)

Colloidal gel aging is investigated using very long duration brownian dynamics simulations. TheAsakura Oosawa description of the depletion interaction is used to model a simple colloid polymermixture. Several regimes are identified during gel formation. The Intermediate scattering functiondisplays a double decay characteristic of systems where some kinetic processes are frozen. Theβ relaxation at short times is explained in terms of the Krall-Weitz model for the decorelationdue to the elastic modes present. The α relaxation at long times is well described by a stretchedexponential, showing a wide spectrum of relaxation times for which the q dependence is τα = q−2.2,lower than for diffusion. For the shortest waiting times, a combination of two stretched exponentialsis used, suggesting a bimodal distribution. The extracted relaxation times vary with waiting time asτα = τ 0.66

w , slower than the simple aging case. The real space displacements are found to be stronglynon-Gaussian, correlated in space and time. We were unable to find clear evidence that the gel agingwas driven by internal stresses. Rather, we hypothesise that in this case of weakly interacting gels,the aging behaviour arises due to the thermal diffusion of strands, constrained by the percolatingnetwork which ruptures discontinuously. Although the mechanisms differ, the similarity of some ofthe results with the aging of glasses is striking.

PACS numbers: 82.70.Dd,61.43.Hv,61.20.Lc

I. INTRODUCTION

Like other non-equilibrium phenomena, colloidal gela-tion has attracted considerable interest in the last twentyyears. Colloids are found in many forms and are highlyrelevant for industry. Muds, paints, cosmetics, and manyfood products are examples where understanding of col-loid science is essential in the design of product proper-ties. Studies can be conducted over a wide range of den-sities, and the interactions can be finely tuned in termsof range and depth of potential by altering the particlecoating, solvent conditions, or solutes present.

Importantly, under certain conditions it is possible tomake colloid particles form a macroscopic gel. Althoughthere is no universal definition, a gel is usually under-stood to be a percolating viscoelastic network that hasno long range order (although it may have some local or-der), but a characteristic length scale is present; gels areusually fractals at low volume fractions but this is nota necessity. Common examples of colloid gels includeyogurt, toothpaste, and clays.

The system under study here is a colloid polymer mix-ture. Such systems have been known to have interest-ing equilibrium and non-equilibrium properties for manyyears [1]; they are also a stepping stone to understandingthe more complicated mixtures that make up many ordi-nary products: shampoo and paint are examples whichinvolve complex polymer colloid mixtures. Such systems

∗rjmd2@phy.cam.ac.uk

can display depletion flocculation, and the resulting gelsare often used as one of the simplest models of weaklyattractive system due to their relative simplicity arisingfrom the absence of charge effects.The properties of non-equilibrium systems as they age

are not only relevant for applications, but are also a cru-cial step to understanding the very nature of gels. Indeedif slow aging were not a feature of the gels, phase sepa-rations would only result in equilibrium phases.The dramatic slow down of the dynamics in gels has

prompted investigations into their possible link withglasses. While gels are usually found at lower volumefractions and when strong interparticle interactions arepresent, colloid glasses are usually found at high volumefractions, above φ= 0.58, or for deep quenches. Theyshare many properties such as being out of equilibrium,non-ergodic, the absence of long range order and slowdynamics. Since work on the glasses and their aging isextremely topical in soft condensed matter, this work willlook for similarities between the aging of colloid gels andglasses.There is abundant literature on gels and on colloid

polymer systems in particular. Asakura and Oosawawere the first to describe the attraction between colloidparticles when non-adsorbing polymer is present in solu-tion [2]. The depletion mechanism which is responsiblefor this can be understood in different ways. The essen-tial point is the presence of a region around the colloidsfrom where the polymer coil centre of mass is excluded.When two colloid particles come together the regions ofexcluded volume overlap resulting in an increase of thefree volume available to the polymer, causing an increaseof their entropy and an overall decrease of the free en-

2

ergy of the system. The problem can also be analysed interms of the net osmotic pressure on a pair of colloids asthey come together.

Theoretical work on the system was done by Vrij [3],who independently rediscovered the Asakura and Oosawamodel, and demonstrated the presence of spinodal insta-bility in the system. Gast et al. treated the polymer asa perturbation to the hard sphere results thus producingthe first phase diagrams [4, 5]. Lekkerkerker et al. [6]used a free volume approach to treat this problem whileDjikstra et al. formally integrated out polymer degreesof freedom [7]. It has been shown that for size ratiosbelow k = 0.154 there can be no three body interactionsand that the two body Asakura and Oosawa (AO) poten-tial can therefore be an adequate description. A reviewof the details of the different approaches to the colloidpolymer system can be found in [7]. It is important tonote that the depletion potential give well controlled sys-tems where the interparticle interaction potential wellis usually in the range of 1 − 10kT , very different fromemultions, for example, where interaction energy can beof order 100kT .

Felicity et al. [8] and Lodge et al. [9] modelled col-loid gels interacting via a Leonard-Jones potential andstudied their viscoelastic properties [9, 10]; Bijsterboshet al. investigated fractal colloidal gels for various in-teractions using brownian dynamics simulation methods[11]. Dickenson et al. [12], and Bos et al. [13] haveworked on stress relaxations after shear stresses were ap-plied. Soga et al. have worked on percolation [14] andphase transitions [15] in the colloid polymer system. Theyielding behaviour of 2D colloid gels was investigated byWest et al. [16].

Experimentally, most notable was the work of Grantand Russel [17], Poon et al. [1], Lekkerkerker et al. [18],and Verdin and Dhont[19]. An important result fromall these works was that at moderate volume fractions(0.1 < φ < 0.4) gelation can be expected to occur forhigh polymer concentrations (deep quenches). Also, forranges of interaction of less than a third of the particlediameter no liquid phase formed; the liquid gas binodalbecoming metastable with respect to fluid solid phaseseparation. In light scattering, the gelation is markedby the ‘freezing’ of the low q peaks demonstrating thatthe dynamics become arrested as the sol gel transition isreached.

The following experiments are explained in more detailsince they contain the most interesting data for compar-ison with our simulations. Cipelletti et al. [20] have pro-duced experimental results on the aging of polystyrenecolloid gels at low volume fractions (10−4 < φ < 10−3).The gels where formed by aggregation due to the vander Walls forces, and were found to be fractal. The ag-ing behaviour was sampled for waiting time going fromhours to ten days, the relaxations where stretched withthe exponent greater than one. The unusual q depen-dence of the relaxation times, τf ∝ q−1, was explained interms of a dipole elastic field caused by the high internal

stresses of the gels. The waiting time dependence of therelaxation times was nearly linear, τf ∝ τ0.9w , followinga period of exponential growth (this exponential regimewas also observed in an aging clay suspension [21]).

The same group, in a series of rheology and dynamiclight scattering experiments, also found the same be-haviour in a system composed of multilamellar vesicles,where, once again, it was thought to be the strong inter-nal repulsive stresses that dominate the response [22].

Experiments on the dynamics of colloidal gels havebeen performed by Krall and Weitz [23] for the shorttime (or β, see below) relations. They also presented amodel for the elastic modes present in a fractal gel, fromwhich the mean squared displacements of strands con-strained by the network can be calculated. Working ongel at densies conparable with this work, Romer et al.

[24], showed that the predictions of this model held evenfor non-fractal structures.

Both short and long time scale dynamics where ob-served by Solomon et al. [25] in their study of adhe-sive colloids at low volume fractions and for interparti-cle interactions strengths close to that used in this work( 10kT ). The Krall-Weitz model was used at short times,which a simple exponential was observed at long times.

A commonly calculated quantity is the intermediatescattering function (ISD), a time dependent correlationfunction. When looking at glasses and gels the self partof the ISD will display a double decay. At short times afast decay often referred to as the β relaxation, and atlong times a second decay called the α relaxation. Struc-tural arrest, brought about by increased density for hardspheres or by quenching for a Leonard-Jones system, willlead to a wide separation of these time-scales and the ap-pearance of a plateau at intermediate times see reference[26] and references therein. Since the scattering functionis wavevector dependent, spatial information can also beobtained.

The α and β relaxations are often explained in termsof the cage effect. For a system of purely repulsive parti-cles, a particle is confined by its neighbours which forma cage around it; the particle is free to diffuse within itscage, thus setting the β time scale. The time requiredfor the particle to diffuse out of the cage is the α timescale. As a system becomes denser, cage rearrangementsbecome increasingly difficult, causing the α relaxationsto occur on longer time scales. For systems of attrac-tive particles, one can talk about an open cage, wherebya particle would be confined to its neighbours becauseof the attractive forces; two relaxation time scales alsoresult from this. On this basis it has been argued thatthe gel transition should be described in terms of ModeCoupling theory (MCT) as an ergodicity breaking tran-sition [27, 28, 29, 30]. MCT has considerably helpedto understand supercooled liquids [26], and high volumefraction colloid glasses [31], and its predictions are nowbeing tested for gels as well [25]. The kinetic arrest ofgels has also been reported by Segre et al. , who com-pared it to the features of the hard sphere colloid glass

3

transition [32].Ideal MCT assumes closeness to equilibrium and is

therefore only useful for the fluid-gel transition. Non-equilibrium MCT has also been developed by Bouchaudet al. [33] which predicts aging behaviour. Some of thesepredictions have been qualitatively tested on a Laponiteglass [21]. However, earlier theoritical models have inter-esting aging properties. Cugliandolo and Kurchan havesolved the spin glass model for the out of equilibrum dy-namics [34]. Crucially, these equations are identical tothose produced by some MCT models.Another approach to explain the behaviour of glasses

has been the study of the properties of the PotentialEnergy Surface (PES) as proposed by Goldstein [35].One model which uses this concept is the successful trapmodel of Bouchaud [36] which has been widely appliedfor spin glasses and of which many variants exist, thiswas one of first models to make useful predictions aboutaging. Another formalism is now introduced to facilitatediscussion. The Inherent Structures (IS) of a system aredefined as the local minima of the PES; all points in con-figuration space which end up in the same IS following asteepest decent path define the basin of that IS [37]. Sci-otino and Tartaglia distinguish between four dynamicalregimes: a high temperature region where the system isfluid; an intermediate region where the system is alwaysclose to the ridges separating the different basins, thisbeing the regime in which MCT is successful; the thirdregime is found at low T where the system populates ISwith lower and lower energy. When the time of escapefrom a basin becomes the longer than the experimentaltime scale the system no longer equilibriates, which is thefourth regime, normally called a glass [38]. In this for-malism the two step relaxation of the correlation functionis seen as decoupling of the fast (intra-well) vibrationalexcitations and the slower rearrangements correspondingto the system sampling different basins which for low Tmay require activated processes.It is useful to set up a few hypothesises about the possi-

ble causes of the aging in gels. Firstly, very slow crystalli-sation about some locally highly ordered regions is a pos-sibility although this is not normally seen in experiments.Secondly, the aging may be stress driven as argued to bein some experimental systems [20, 22]. Thirdly, thermalfluctuations and random breaking of bonds to allow therelaxation of the network as suggested by Solomon et al.

[25]. Compactification of strands up to a point wherethey might dislocate has also been suggested as a possi-bility to explain the collapse of transient gels [39].

II. METHODS

A. The model

The Brownian Dynamics model is based on a Langevintype equation of motion for each colloid particle witha fluctuating random force accounting for the thermal

collisions of the solvent molecules with the particle. Thepolymer coils are not simulated, their presence only beingfelt through the attractive potential between the colloids.

MX = FC + FB + FD = 0 (1)

Where M and X are the inertia and position matricesrespectively; FC are the conservative forces, FB are therandom brownian force, and FD are the dissipative dragforce. This last term encompasses all the hydrodynamicinteractions, which couples the forces on the particlesthrough the solvent. It is these forces which oppose therelative motion of the particles, and are most importantin flow. They will affect non-equilibrium phenomena atrest, but in common with many others we neglect themany body interactions in this study. In this model, thisterm is reduced to Stoke’s drag for a spherical particle(α), applied individually: this is the free draining ap-proximation. So the drag force on each particle satisfies:

fiD = −αvi = −6πησvi (2)

where η is the viscosity of the liquid, σ is the particlediameter and vi is is the velocity of particle i. The iner-tia terms are neglected so this becomes a force balanceequation, which is solved for the displacements of theparticles. The conservative forces are the sum of a steepcore potential, and the depletion force obtained via thefollowing two body potential energy term.

U =1

2

∑

ij

(

UHC(rij) + UAO(rij))

, (3)

where the first term, UHC , is the power law repulsivehard core,

UHC

kT=(rijσ

)−n

. (4)

n = 36 is used to avoid anomalies that may result fromhaving a softer potential [40] while retaining computertime efficiency. The two-body depletion potential usedwas:

UAO

kT= QH(L, rij)

(

L2 rijσ

−1

3

(rijσ

)3

−A

)

, (5)

Where rij is the separation between spheres i and j. Ifk is the size (diameter) ratio between a polymer and aparticle at the polymer volume fraction φp, then

L = 1 + k

A =2L3

3

Q =3φp2k3

H(

L,rijσ

)

=

{

1,rijσ < L

0,rijσ > L.

(6)

4

Strictly speaking, the parameter φp = 16πk

2σ2nR, wherenR is the polymer number density in a reservoir of purepolymer which would match the chemical potential of thepolymer solution with colloid particles at volume fractionφ [6]. It useful that the potential used here is of finiterange, as this avoids truncation errors and allows us todefine nearest neighbours and interacting particles pre-cisely.

The Brownian forces represent the action of the liquidmolecules averaged out over one time step. They averageout to zero, and are uncorrelated in time.

< FB > = 0 (7)

< FB(t)FB(t′) > = 2παkBTδ(t− t′) (8)

The fluctuation-dissipation theorem relates the magni-tudes of FB and α by enforcing that every squared sep-arable degree of freedom satisfies the equipartition theo-rem.

In order to simplify the calculation reduced units areused in the simulation, kBT becoming the energy unit.The polymer volume fraction, φp, is the only control pa-rameter for the depth of potential, and the ratio of thecolloid to polymer size k controls the range of the poten-tial. This was fixed at k = 0.1 for all simulations shownhere.

All distances are measured in units of the particle di-ameter σ.

The natural time unit to use is the Brownian relaxationtime. It is defined as the time for a particle to diffuse itsown length scale by Brownian motion.

τR =ησ3

kBT(9)

For a micron size particle in water, τR is of order 0.2seconds. Note however the strong dependence on parti-cle size. The typical length of runs showed here was of5000 τR and the longest run was twice that. This is agreat increase compared to previous studies which weretypically run for about 100 τR.

The system being simulated is a three dimensional cu-bic box with 4000 particles in Periodic Boundary Condi-tions.

B. Quantities of interest

As mentioned earlier the self part of the intermediatescattering function (also called incoherent intermediatescattering function) gives information about the relax-ation of density fluctuations in the sample. It lets usassess whether particles are displaced on a length scaleset by 2π

qor whether kinetic processes of that wavelength

are ’frozen out’, it is a commonly calculated quantity as

it can be measured in light scattering experiments [20].It is calculated from simulation data by

Fs(q, t, tw) =1

N

∑

i

eiq·(ri(tw)−ri(tw+t)), (10)

where tw is the waiting time for that sample. The wavevector q is discretised to correctly take into account theperiodic boundary conditions. Therefore, for each axiswe have:

q =2πn

L, n = 0, 1, 2, ...

with L the linear dimension of the box. To improve thestatistics the results are averaged over the x, y and z axes.Since it depends both on t and tw, FS is a so-called two

time quantity, which are most useful in determining andcharacterising any aging behaviour if it is present.

In order to calculate particle correlations in real spacethe radial distribution function g(r) is calculated, as wellas the self part of the Van Hove correlation functionwhich samples particle displacements in time.

g(r) =1

< n >

N∑

i6=j

δ(|ri − rj | − r)

where < n > is the average number density,

Gs(r, t) =1

N

N∑

i

< δ(|ri(t)− ri(0)| − r) >

where < ... > is an average over time origins. The quan-tity

S(r, t) = 4πr2Gs(r, t)

gives the probability of finding a particle at a distance rafter a time t, and Gs(r, 0) = g(r).

In the Gaussian approximation,

Gs(r, t) =1

(4πDt)3

2

exp

(

−r2

4Dt

)

which is a solution of the diffusion equation.

Our results are divided into two parts; the first partwill be concerned with ‘macroscopic’ quantities, i.e. cor-relation functions and quantities that have been aver-aged over several systems; the second part will look moreclosely at the ‘microscopics’ of individual systems.

5

0 1000 2000 3000 4000 5000Time / τR

−8

−7

−6

−5

−4

2U/N

Um

in /

kT

1 10 100 1000 10000Time / τR

0

1

10

[2U

/NU

min +

7.3

] / k

T

FIG. 1: Potential Energy of gel, from formation to aging. Thefit for the inset is a power law.

III. RESULTS

A. Macroscopics

As in previous simulations [14, 15] the samples wereequilibrated at high temperature (φ= 0) and instanta-neously quenched to an appropriate value of φp. Work in[15] determined that for φp≥ 0.3 crystals were no longerobtained. For these relatively deep quenches, amorphous,non-equilibrium structures were obtained; these networkscome under the general name of gels or transient gelswhen seen experimentally before they collapse gravita-tionally. They are qualitatively similar to created ex-perimentally gels and LJ gels from simulations. All gelsconsidered here have φ = 0.30 and φp = 0.40, giving apotential minimum of Umin = −8kT . Properties are av-eraged over five samples unless otherwise stated.

In figure 1 the potential energy of the system, U , isscaled by U → 2U

NUmingiving the approximate number

of neighbours assuming inter-particle distances are nearthe potential minimum. The gel particles have approxi-mately seven neighbours. It is immediately obvious thatwe are dealing with a non-equilibrium system since, af-ter the initial very fast drop in energy after quenching,a long regime of slow decay is entered; no equilibriumvalue was ever reached in the simulations we ran. Suchsmall decays can be fitted with a number of functions,however for comparison with Barrat and Kob [41], it wasfitted with a power law U = a · t−b, as shown in the insetof figure 1, yielding b = 0.3.

It is revealing to look at the total number of bonds inthe system as gelation occurs. When plotted on log-linearscale (figure 2) three different growth regimes are evi-dent, the first up to 10τR , another from 10τR to around100τR , and the last one from there on. Note that sincethe ordinate range is small, a log-log plot shows practi-cally the same result, hence one cannot easily distinguishbetween a low power law growth and logarithmic growth.

1 10 100 1000 10000Time / τR

5

6

7

8

Nbo

nds /

2N

part

.

~ 1.5

~ 0.66

~ 0.21

FIG. 2: Total number of bonds in the gel. The three straightlines and their gradient are superposed to highlight the threedifferent logarithmic regimes.

0 0.5 1 1.5 2time / τR

0

0.2

0.4

0.6

0.8

1

Fs(

q,t)

0 0.5 1 1.5

time / τR

0

0.002

0.004

0.006

<∆r

(t)2 >

/d2

FIG. 3: Beta relaxation at short times for qd =23.0, 32.9, 46.0, with stretched exponential fits given in text(white line). Inset: Mean Squared Displacement (black cir-cles) for the same time scale with fit to equation 11 (whiteline).

The fits are of the form N = c.log(t)+d where the valuesof c are shown on the plot. As explained in the discus-sion we are here mostly concerned with behaviour overthe last of these regimes.

The ISF was found to display the expected two steprelaxation at short (β relaxation) and long (α relaxation)times. They are treated separately here since the aver-aging cannot be done in the same way. For the waitingtime independent β relaxation an average over time ori-gins can be used, whereas for the α relaxation the non-ergodic nature of the system means only an ensembleaverage can be performed. Figure 3 shows this initial de-cay for increasing values of q. As the gel forms, the meansquared displacements of the particles is Fickean, but asaggregation proceeds the behaviour becomes subdiffusive(< ∆r(τ)2 > ∝ te, e < 1). Finally, for measurements

6

0 20 40 60qd

0

0.2

0.4

0.6

0.8

1

f q(t=

1.5τ

R)

FIG. 4: Non-ergodicity parameter as a function of q, withexponential fit.

in the region of interest the mean squared displacementtakes on the form of figure 3 (inset) where a plateau isreached at displacements that are small compared to theparticle size. This is interpreted in terms of the Krall-Weitz model for the initial decay of the correlation func-tion [23]. This model considers the relaxation due tothe elastic modes present in the gel, with the particlesassumed to be confined to the gel strands. Their con-clusion was that the mean squared displacement shouldobey:

< ∆r(τ)2 >= δ2[1− e(− τ

τβ)p

] (11)

The particle displacements are constrained to an am-plitude δ; the stretched exponential form comes aboutfrom the sum of the relaxations on all length scales.Consistently with the model we are using, the β decay

was fitted with the following law which is closely relatedto a stretched exponential,

Fs(q, t) = exp(q2 < ∆r2(t) > /6) (12)

for all q vectors, with < ∆r2(t) > from equation 11.The results are shown in figure 3 for three different valuesof q. The value of the exponent was determined fromthe fits to data in figure 3 with equations 11,12 to bep = 0.4 ± 0.1. Similarly, δ = 0.075 ± 0.004d and τβ =0.26± 0.04τR.δ2 being the maximum mean squared displacement

(excluding any α processes), therefore sets the height of

the plateau in Fs via Fs(q, t) =1N

∑

i e−q2(∆r2i )/6 = f s

q ,where f s

q is the non-ergodicity parameter for the incoher-ent scattering. Figure 4 shows the non-ergodicity param-eter decays exponentially for increasing q; δ2 can then be

extracted since fq = e−(qδ)2/6 in the gaussian approxi-mation. The previously extracted value of δ was used forthe fit. However there is a small ambiguity about whereexactly fq should be measured, since α processes are also

1 10 100 1000 10000time / τR

0.5

0.6

0.7

0.8

0.9

1

Fs(

q,t)

0 1 2 3 4 5

(t/tr)0.62

−0.6

−0.4

−0.2

0.0

ln(Fs)

FIG. 5: q vector dependence of the intermediate scatteringfunction for long times and fits to equation 12. From top curveqd=2.30, 3.28, 4.27, 5.26, 6.25. The waiting time is 1000 τR.Inset collapse of the same data when scaled as described intext and plotted against stretched time.

1 10 100 1000 10000Time / τR

0

0.2

0.4

0.6

0.8

1

FS(q

,t,t W

)

FIG. 6: Intermediate scattering function at fixed qd = 6.25,and increasing waiting times, from left tw = 20, 100, 500,1500, 2250 τR.

present there is no true plateau; in this case the valuesof fq were measured after 1.5τR.

We now turn to the long time dynamics; the second de-cay (alpha relaxation) is the part of the dynamics whichis expected to show signs of aging. Because the system isnon-ergodic, an ensemble average cannot be replaced bya time average. Unfortunately, very long runs are neededin order to get to the time scales that are relevant for thisstudy, and therefore only a limited number of runs canbe performed due to computer time limitations. The av-erages were therefore performed over a set of five runs.Information about the dynamics on different wavelengthscan be extracted by looking at data from a fixed waitingtime and sampling at different q. For a given q the wait-ing times were then varied to observe the aging in thedynamics.

7

Figure 5 shows the alpha relaxations for a single wait-ing time and the q vectors qd = 2.3, 3.28, 4.27, 5.26, and6.25. The relaxations were fitted to a stretched exponen-tial law:

Fs(q, t) = A exp(−((t− tw)/τα)µ) (13)

where A where was a constant chosen to be the plateauheight and µ was kept constant for different q. The datacollapses onto a master curve when plotted as a functionof stretched time (t− tw)

µ once normalised by A and τq(inset of figure 5). The relaxation times τalpha(q) whichare extracted from the fits for different q where found tofollow a power law relationship

τα ∼ q−ν

with ν = ν(tw). The best fit values extracted wereslightly age dependent as summarised in Table I.

tw/τR µ ν Double Exp. µ′

1 µ′

2 ν′

2

100 - - Y 0.7 0.7 2.2

200 - - Y 0.85 0.7 2.1

500 0.45 3.0 Y/N 0.9 0.7 2.2

750 0.50 2.7 N - - -

1000 0.62 2.5 N - - -

1250 0.70 2.3 N - - -

1500 0.70 2.3 N - - -

2000 0.72 2.2 N - - -

TABLE I: Summary of the exponents as a function of age ofthe gel

For waiting times tw ≤ 1000 τR, as well as fast de-creasing values of µ, it was noticed that the quality ofthe fits decreased becoming impossible for tw < 500 τR.The α relaxation at these early times displayed a rela-tively fast decay followed by a slower one (see curves fortw = 20, 100 τR in figure 6. These relaxations were fittedwith a double stretched exponential:

Fs(q, t) =fq

(1 +A1)

{

exp

(

−

(

t− twτ ′α1

)µ′

1

)

+A1 exp

(

−

(

t− twτ ′α2

)µ′

2

)}

(14)

The justification for this law is given in the discussionsection. Once again the relaxation times contain wave-length dependent information, τ ′α2 ∼ qν

′

2 ; from the fitsν′2 ≈ 2.2 for all appropriate waiting times as summarisedin table I. τ ′α2 was found to depend only very weakly onq. The exponent µ′

2 = 0.7 was held constant and fq wasused to give the correct plateau height.

1e−04 1e−02 1e+00 1e+02Time / τR

0.4

0.5

0.6

0.7

0.8

0.9

1

FS(q

,t,t W

)

FIG. 7: Intermediate scattering curves made to coincide atFs = 0.71.

The significance of these exponents is discussed in thenext section.Figure 6 shows the aging behaviour of the gels. The

data was all collected at a single q value, qd = 6.25. Thesmallest waiting time considered here is 20 τR. As thewaiting times were increased from 20 τR up to 2250 τRa lengthening of the β plateau was observed, with therelaxation of the system becoming slower with increasingwaiting time. This is classic aging behaviour for non-equilibrium system and glasses and has been seen forsupercooled LJ systems by Barat and Kob [41].The collapse of the α decay is often observed in exper-

iments on glasses where one often finds that each set ofdata can be scaled using a single time scale which encom-passes the whole parameter dependence [42]. Hence thefollowing scaling:

Fs(q, t, tw) ∼ F (t/τγ)

with τγ = τγ(tw). Using an arbitrary value of Fs =0.71 to define a relaxation time for each curve τγ =t(Fs = 0.71), the results were rescaled and reploted onfigure 7 using waiting times ≥ 100 τR. The lower limitwas chosen to be at the start of the last regime seen infigure 2. As can be seen the data in the range consideredappears to collapse well. This relatively good collapse issurprising in that the exponent of the stretched exponen-tial µ varies with waiting time.Figure 8 shows the dependence of τγ on tw. The inter-

pretation of the results in figure 8 is slightly ambiguous.It is possible to interpret this data in two ways. Firstly interms of a single power law regime τγ ∼ tξw with ξ = 1.26.Secondly in terms of two power law regimes, the first withξ = 1.33 and the second with ξ = 0.62, the separationbetween the regimes being around 1000 τR. All of thesepossible values for ξ are not far from 1, and similar toaforementioned results. It was also checked that the samebehaviour resulted if values of τα were extracted directly

8

100 1000 10000tW / τR

10

100

1000

10000

tγ / τR

gradient = 1.0

gradient = 1.33

grad. = 0.62

FIG. 8: Relaxation times extracted from the rescaling of fig-ure 6.

0 1000 2000 3000t / τR

0

0.05

0.1

0.15

ψx

0 4 8 12

Coordination

0

0.1

0.2

0.3

Pro

b. d

ensi

ty

FIG. 9: Crystalline Order Parameter. Inset: distribution ofparticle coordinations in gel; solid line: tw = 3000 τR; dot-dashed line: tw = 100 τR

from the fitted stretched exponentials. The issue of oneor two regimes will be discussed further on.Another scaling was proposed by Mussel and Rieger

for the LJ supercooled fluid results [43] and was at-

tempted for comparison. Fs(q, t, tw) ∼ F{ln[(t +tw)/µ]/ln(tw/µ)}, where µ is a fit parameter and playsthe role of an affective microscopic length scale. Thiswas proposed by Fisher and Huse in the context of spinglasses [44]. This scaling was also applied to our data,however it did not produce a successful collapse.

B. Microscopics

Over the period of time considered, it is seen that al-though the average numbers of bonds in the system in-creases, there is little increased crystalinity as shown bythe local order parameter, denoted ψx (main part of fig-ure 9). The local crystaline order parameter is defined

0 0.2 0.4 0.6 0.8 1 1.2r / d

0

1

2

3

4

Ss(

r,t)

0 0.1 0.2 0.3r / d

0

5

10

15

Ss(

r,t)

FIG. 10: Self part of the Van Hove correlation function. Maingraph: long time interval, tw = 3000 τR. Inset: short timeinterval, tw = 10 τR. The peak area is fitted with a gaussian,shown as a dot-dashed line.

0 2 4 6 8r / d

0.0

0.2

0.4

0.6

0.8

1.0

Cv(

r)

FIG. 11: Displacement correlation function, with an expo-nential fit.

as the fraction of of particles which are 12 coordinatedor neighbouring a 12 coordinated site. The inset of fig-ure 9 shows the probability density distribution of thecoordination of the particles in the system. The initialcrystalinity, which is present very soon after the quench,is just the result of the high initial colloid volume frac-tion together with the strong local order, which is itselfdue to the lack of angular rigidity of the bonds. There istherefore no ambiguity about a gel with a non-zero crys-talline local order parameter. At later times an increaseof the average position of the peak is seen but no sig-nificant increase in crystalinity. This already rules outone of the possible hypothesises for the aging of the gel,namely slow growth of crystals around the crystallitespresent from formation.In order to obtain a better understanding of the re-

laxation mechanisms at work the particle dynamics aresampled in real space through the Van Hove correlationfunction. S(r,t) gives the probability of finding a particle

9

at time t + δt at a distance r from its position at timet. Due to the non-ergodicity it is not possible to timeaverage this correlation function either. However the ac-curacy of this result was checked by averaging over a timeinterval that was short compared by the typical α relax-ation time scale. Figure 10 show a pair of typical results;they were calculated for tw = 3000 τR and δt = 50 τRand 3000 τR. For these parameters we expect to be inthe β plateau for the former time interval and in the αrelaxation regime for the latter. Indeed, the peaks of thedistributions are around 0.05d and 0.16d which are con-sistent with earlier deductions about those regimes. Alsoshown on figure 10 are the Gaussian fits for the peaksand the regions around them. The difference in the be-haviour of the dynamics in those two regimes can herebe seen more intuitively. For times up to and includ-ing the β plateau, the gel particles’ displacements arenearly Gaussian, consistent with a diffusion-like mecha-nism. The main part of the plot shows the displacementsduring the α relaxation regime, the distribution is verymuch broader with the longer displacements not fittingwithin a Gaussian. The displacements found after thepeak follow a power law S(r, t) ∼ r−f , with f = 2.8±0.4.The strongly non-Gaussian displacements rule out a

diffusive mechanism, and hint at cooperative motion.This can be seen visually when colour coding particledisplacements on a 3D representation of the gel (notshown here). Clusters of particles are observed to bedisplaced by similar amounts, in fact this was also truewhen colour coding displacements which were less thenthe peak value. In order to show the presence of collec-tive displacements clearly, a velocity correlation functionwas calculated:

Cv(r) =1

NW

N∑

ij

< δ(r− (ri − rj))v(ri) · v(rj) >, (15)

Where W =∑N

i < v(ri) ·v(ri) > is used to normaliseCv. This is shown in figure 11 for tw = 4000τR and ve-locities calculated over an interval of ∆t = 50 τR. Thedata are fitted with a an exponential allowing a char-acteristic correlation length rv = 1.9d to be extracted.An interesting point was that the results for the velocitycorrelation function, for all times beyond the β relax-ation, were practically unchanged, thereby showing thatwhatever the relaxation mechanism, the same degree ofcooperativity is present.Now it is known that particle displacements are spa-

tialy correlated, their temporal distribution can be stud-ied. For this we calculated the intermediate scatteringfunction for a single gel, with no averaging procedure. Infigure 12 the β relaxation and the plateau are not ap-parent due to the linear time scale which is useful forcomparison with other microscopic results. It is appar-ent that the relaxation processes at work are not evenlydistributed in time. A succession of plateaus and sharpdrops are observed. Although the displacements are co-

0 2000 4000 6000 8000 10000 12000t / τR

0.55

0.65

0.75

0.85

0.95

Fs(

q,t) 0 1000 2000

t / τR

0.96

0.98

Fs(

q,t)

FIG. 12: Close up the intermediate scattering function; qd =6.25, tw = 500 τR. Inset: same for a 500 particles system.

0 2000 4000 6000 8000 10000 12000time / τR

7.73

7.83

7.93

2Nbo

nds /

Npa

rtic

les

Total number of bonds in gel10 pts running average

FIG. 13: Close up of the scaled total number of bonds in thesystem and a 10 point running average.

operative that this function is averaged over all particlesin the system, therefore a sharp drop in the averagedfunction is a sign of major cooperative rearrangement.For a smaller sized system where a greater fraction ofthe particles is included in any given strand the plateausare closer to the horizontal and the fast decorrelations aresharper (see inset of figure 12 for a 500 particle system).

Following on from the localisation in time and in spaceof cooperative motion, we found similar non-uniform be-haviour for the total number of bonds in the system, aclose up of which is shown on figure 13 (scaled to showthe average number of neighbours). A running averagehas been used to better show the underlying trends ofthe data. Here again fast increases are interspaced withplateaus and short time decreases. Qualitatively we cansee that both the size and gradient of the rapid increasestend to decrease as time goes on. This was also true ofFs and the same behaviour is apparent in closeups of theenergy (not shown here). It is consistent with a slowingdown of relaxation processes as the gel ages.

10

Considering that the particles are strongly localised,it was surprising to see substantial decreases in the to-tal number of bonds in the system. The lifetime of abond based on a simple diffusion over a barrier argumentis τbond ≈ τR e−U/kBT ≈ 3100 τR for a pair of parti-cles interacting with this potential. When taking intoaccount the average number of neighbours within inter-action range, one would not expect to find many brokenbonds in any given time interval. Due to the finite rangeof the potential, it is possible to define bonds exactly asbeing within the interaction distance; therefore, we useda procedure that kept track of all particles that havehad broken or created bonds within a given time inter-val. Remarkably, this shows that a high fraction of theparticles break and create bonds during the evolution ofthe system. Nine percent of the particles were found tohave broken bonds (slightly higher for the created bonds)when comparing configuration files 500 τR apart, this ismuch higher than what would be expected from figure13. This number is probably an underestimate since itdoes not include bonds broken and recreated between thesame pair of particles over that time interval. Over anysizable time interval there is always a surplus of createdbonds. In order to illustrate the effect of the dynam-ics we show on figure 14 the proportion of particles thatchange their near neighbour environments compared to areference configuration chosen every 1000 τR. There arethree points of importance. There is a short time intervalwhere about 5− 7% of the particles change environmentwithin a couple of τR. This regime is independent ofthe reference configuration and corresponds to the β de-cay. Its independence suggests that there is a continuousprocess of breaking and creating bonds at short times.This will be further discussed in the next section. Fig-ure 14 also has a second regime which is waiting timedependent and mostly follows the trends of the agingbehaviour, the rate at which new particles change envi-ronment slowing down with increasing waiting time. Thethird point to notice is the absolute values. For the topcurve the changes in the gel involve 27% of the particles,a huge number considering the relatively small distancesmoved by the particles.

The presence of particle displacements that are greaterthan the contribution from the elastic modes, of corre-lated dynamics and a high fraction of particles changingenvironments, meant we expected to find some correla-tions in the position of those particles which have bro-ken bonds or changed environment. However breakingbonds in a concentrated system is always going to lead tosome correlation between the particles; to see this imag-ine pulling away one of the particles of a tetrahedron, thethree particles left behind have all had a broken bondbut are highly correlated in space. A visual inspection ofthe gel with highlighted particles where bonds had beenbroken confirmed the suspected correlation; we also cal-culated a pair correlation function gb(r) by selecting theparticles that had broken bonds in a given time interval(figure 15), and comparing it to the normal pair correla-

0 200 400 600 800 1000t − tw / τR

0

0.1

0.2

0.3

0.4

0.5

Fra

ctio

n of

par

ticle

s w

ith c

hang

ed e

nviro

men

ts

tw = 1000 τR

tw = 2000 τR

tw = 3000 τR

tw = 4000 τR

FIG. 14: Fraction of particles which change their nearestneighbour environment for increasing waiting times.

0.5 1 1.5 2 2.5r / d

0

5

10

g(r)

particles with broken bondswhole gel

FIG. 15: Radial Distribution function done with all the allthe particles in the gel and by selecting particles with brokenbonds.

tion g(r). The correlation shows up in gb(r) as a higherfirst neighbour peak and a large peak beyond it which isfound at the limit of the interaction potential, which isno surprise since it is by definition the distance at whichone considers a bond to be broken. In terms of the previ-ously given argument about pulling apart a tetrahedron,the higher first peak is due to the strong correlation ofthe particles left behind, while the second peak is dueto the particles that are just beyond the interaction dis-tance, i.e. the ones that have just broken off. This secondpeak at the limit of the potential is probably related tothe initialy rapid increase seen in figure 14, since it mostlikely shows that a lot of particles are hovering near theedge of the potential well, accounting for much of thebond breaking as they pop in and out when the strandis deformed by thermal motion.

Exactly the same observations were made about theparticles that had created bonds in any reasonable timeinterval.

11

0 2000 4000 6000 8000 10000 12000time / τR

0

2

(σxx

+ σ

yy +

σ zz )

/ 3

FIG. 16: Averaged normal stress tensor components (negativeof pressure); raw data in black, 20 point running average inwhite.

Since it was the case in other studies [20], one pos-sible driving force for the aging of the gel could be theinternal stresses. It is known that gels formed in PeriodicBoundary Condition have non-zero normal stresses [45],and experimentaly a gel which has been detached fromthe walls of its container shrinks macroscopicly [20]; fig-ure 16 shows the average of the normal stresses, raw datain black and a running average in white. The raw datashow the high frequency noise due to the brownian fluctu-ations, but the presence of underlying low frequency fluc-tuations is revealed by the running average. The generaltrend is that of a decrease, and there are large fluctua-tions in the average, although not as large as any of thefluctuations in the raw data. A spatial stress correlationfunction, Cσ(R) =< σ(r)σ(r + R) > was calculated butwas found to show only very low spatial correlations.The distribution of bond lengths in the gel was found

to be quasi-equilibrium with only a small discrepancy atdistances beyond the potential minima due to the normalstresses present in the system, as can be seen on figure17.

IV. DISCUSSION AND CONCLUSIONS

This study of the dynamics of gels formed from col-loids interacting via a depletion potential has showed thatcomplex behaviour arises during the aging.It was found that the total potential energy of the sys-

tem could be described in terms of a power law with anexponent b = 0.3 which is an intermediate value to whatwas obtained by Barrat and Kob for a Leonard Jonesglass (b=0.14) [41] and by Parisi [46] (b=0.7) for a MCgenerated glass. However, the total number of bonds inthe system was found to grow as three separate logarith-mic regimes. Some of this three regime structure was alsopresent in the energy plot demonstrating the difficultyin describing accurately these very slow growths/decays.

0.9 0.95 1 1.05 1.1r / d

0

0.5

1

P(r

ij) /

arb.

uni

ts

P(rij) = exp(−Uij/kT)P(rij) from simulation

FIG. 17: Distribution of bond lengths in gel (circles). Blackline is P = exp(−Uij/kT ). Recall that range of the potentialis 1.1d.

The impossibility in calculating these quantities in ex-periments make verification difficult if not impossible.However, whether interpreting the slow growth of thenumber of bonds as a power or logarithmic law, the pres-ence of three distinct regimes is of much interest anddemonstrates the complex behaviour associated with theformation and consolidation of the gel. We know fromprevious work that the time to percolate is less than 1 τR[14]. Following this one would expect some fast consoli-dation of the network during which thin strands coarsen.Since diffusion processes occur, by definition, on the timescale of 1 τR, one would expect complex and fast rear-ranging at relatively short times and short length scalesas was seen by Haw et al. for 2D colloid simulationswhere compactification was obvious at short times [47].However, given that we know that the end product ofthis is not at thermodynamic equilibrium, these fast re-arrangement processes must be inactive at intermediateto long times (otherwise the equilibrium phases wouldresult). Although all the regimes seen on figure 1 occurafter percolation, it is beyond the scope of this paper toexplain all three, our attention is here focused on the lastone for several reasons: it is expected that the behaviourmaybe less complex in this region; these time scales aremore accessible to experiments thereby providing inter-esting comparisons on the simulation results and insightsthat may be tested on real systems. In order to study theaging and not the formation/consolidation behaviour thework showed here focuses on the behaviour in the regimefrom 100 τR after quenching onwards.

As an aside it is worth mentioning that the noise inthe energy of the gel goes as FT (U)

2≈ 1/f1.4, this value

being closer to the 1/f flicker noise than the more usual1/f2 white noise. This has been linked in the literature toSelf Organised Criticality and to cooperative behaviourin complex systems. However this needs to be studiedfurther before conclusions can be drawn from it.

The self part of the ISF calculated for this sys-

12

tem displayed the classic two step relaxation. Both ofthese relaxations were found to be well approximated bystretched exponentials with a plateau between them.

The age invariant short time dynamics, correspondingto the beta relaxation was interpreted in terms of theKrall Weitz model for the decorrelation due to the elasticmodes present in a gel. Although they first used it fordescribing low volume fraction fractal gels, the physicalreality of the model is still present even though the gelis not a fractal object. This model has been shown towork well in describing gels at higher φ by Romer et al.[24] with reduced values of δ2 and τc reflecting on thecompactness of the gels.

The mean square displacements and the beta relax-ation were fitted using equations 11 and 12; the valueof the exponent was determined to be p = 0.4 ± 0.1which is distinctly smaller than Krall and Weitz (p = 0.7)for their low φ polystyrene gel [23], Romer et al. alsofound p = 0.7 for a high φ polystyrene gel [24]. How-ever Solomon and Varadan found p = 0.5 for a low φthermoreversable gel of silica particles [25]. In the KrallWeitz model p is related to the length scale dependentspring constant for fractal clusters via κ(s) ∼ s−β and

p = ββ+1 . From computer simulation β ∼ 3.1 for DLCA

[48]; from Krall and Weitz, and Romer et al. p = 0.7,β = 2.3; whereas from Solomon and Varadan p = 0.5,therefore β = 1.0 for thermoreversable gels which theauthors believed had some angular rigidity on the locallevel due to surface roughness and a different aggregationmechanism leading to a locally denser gel. Furthermore,previous studies by one of the authors on 2D gel net-works where trimers provided some angular rigidity [16],showed that for such systems β ∼ 1.1 − 1.3. All of theprevious suggests, as proposed by Solomon and Varadan,the the elasticity exponent β is strongly dependent on theangular rigidity of the local network. Here p = 0.4 there-fore β = 0.66 suggesting a structure that only slowlybecomes more compliant with increasing length scale.There are several factors influencing this in these sim-ulations. Firstly there is no imposed angular rigidity forthe colloid interaction (which is different from experi-mental particles which may be rough or deform underthe strength of the interactions). Secondly the colloidvolume fraction used here is φ=0.3, much higher thanmost experiments; and thirdly, the interaction range isvery small (0.1d), this has been shown by Heyes and co-workers [45] to affect the structures obtained when form-ing the gels. Short interaction distances resulted in thecreation of gels where the local structure was very com-pact, for example it can be seen as a split second peakin g(r), whereas when longer ranged potentials were used‘fluffier’ structures resulted. The former is the case inthis study, and it is obvious that a thicker strand of col-loids has a higher angular rigidity. Furthermore, thisthird point is strengthened by the high φ, and probablynegates the first point since the lack of angular rigidityof a single bond becomes irrelevant if there are no singlyconnected particles in the strand. This explains the high

angular rigidity of this structure and hence the low val-ues of p and β. An interesting point that arises out ofthis discussion is that for a given maximum depth of po-tential, the rigidity of the gel may be influenced stronglyby the range of the potential. This may explain why forthe higher volume fraction gels of Romer et al. p = 0.7,the same as for the Krall and Weitz’s low volume frac-tion gels, although what the range of the interactionspresent in the aforementioned experimental systems wasnot clear.

This physical picture is confirmed by direct visualisa-tion of the gel which shows the clusters of particles mov-ing relative to each other on time scales of order 1 τR.Furthermore this is consistent with the high number ofparticles which change environment during such time in-tervals. This is a direct consequence of clusters in relativemotion within a relatively stiff structure, thereby causingmany bonds to break and reform.

The length scale dependent plateau height as charac-terised by f s

q was found to have a very large q-width.This result is very similar to Puertas et al. [30], whosimulated the approach to gels in concentrated fluids nearthe glass transition. They obtained a similar broad de-pendence showing again the strong confinement of theparticles due to the short range potentials. This extendstheir results to a lower φ than they have used. A furtherdistinction is that this result is at a point on the phasediagram deep in the gel state whereas they were in thefluid leading up to the gel state.

The α relaxation for waiting times ≥ 500 τR wasfound to be well described by a stretched exponentialFs(q, t, tw) = A exp(−((t−tw)/τα)

µ) where the exponentwas always µ < 1.0. This type of law is often observedand results from the presence of relaxation modes with awide distribution of time scales. This can be written asa Laplace transform over the distribution of relaxationmodes [42]: F (t) =

∫∞

0ρ(γ)e−γtdγ. The exponent of the

stretched exponential is strongly dependent on the shapeof the distribution ρ(γ), wider distributions leading tolower exponents. In this study, the variation of the ex-ponent of the stretched exponential, µ = 0.45 → 0.72as tw = 500 → 2250τR clearly indicates a reduction ofthe width of the modes distribution. Furthermore theobservation that the variation in µ is fast for small wait-ing times and levels to µ ∼ 0.7 for longer waiting timesand also the observation of the change in behaviour fortw < 500 where the α relaxation appeared to have a ‘fast’decay followed by a ‘slow’ one, making it impossible to fita stretched exponential, all point to the possibility of ρ(γ)being a bimodal distribution where the short time modesare widely separated from the long time modes. Hencethe success of using a function containing two stretchedexponentials to fit the decay of the alpha relaxation upto waiting times of tw = 100 τR. In this scheme each ofthe exponentials describes one part of the distribution.This behaviour was reproduced by numerical integrationof the Laplace transform for distributions of modes madeof two widely separated Gaussians. All of the correlation

13

function data can then be explained. µ and µ′2 describe

the distribution of the ‘slow’ modes, µ = µ′2 = 0.7 ± 0.2

for all waiting times larger than the time for the slow-est of the ‘fast’ modes to relax. µ′

1 which describes thedistribution of ‘fast’ modes increases towards 1.0 as thedistribution of the modes that have not yet relaxed nar-rows. For intermediate to long waiting times, a singlestretched exponential is recovered.

The q dependence of the characteristic relaxation timesτα, τ

′α1, τ

′α2, extracted from the fits also contain informa-

tion about the processes at work. ν at long waiting times,and ν′2 = 2.2± 0.1; for the latter exponent this was truefor all the waiting times back to tw = 100 τR. This isvery close to a diffusive law, τ ∼ q−2, yet with a con-sistent deviation. ν′1 was found to vary only very weaklywith q highlighting the fact that different mechanisms areresponsible for the ’fast’ and ’slow’ relaxations. It alsoshows that these ’fast’ modes are not related to the KrallWeitz elastic modes which showed up in experiments asτ ∼ q−2.

Comparing these relaxation processes with the muchmore studied glasses, the α relaxation behaviour is morecomplex in this case as opposed to homogeneous super-cooled liquids [41]. It may not be surprising that thereshould be some processes that would be specific to gels,considering their spatial inhomogeneity.

This framework also explains why in experiments aconstant value of the stretched exponential exponent isusually found, since the ‘fast’ modes observed here wouldbe inaccessible.

When considering the ISF in non-equilibrium systems,aging can be observed to occur as a lengthening of theplateau at intermediate times. The response becomesstrongly dependent on waiting time after the quench.This was seen clearly in our data. By making the curvescoincide at one point (or equivalently using the extractedparameters from the fits), a collapse within a fairly shorttime window was observed. It is obvious that since theexponents of the stretched exponentials are slightly wait-ing time dependent, the collapse cannot be perfect overa long time scale, however it is an important feature dis-played by many gels and glasses [30]. The dependenceof τγ on the the waiting time is explained from the pre-vious deductions about our system, taking into accountthe fact that τγ is nearly equivalent to τα for a singlestretched exponential describing the whole decay. Thepresence of slow and fast modes, and therefore a changeof behaviour once the short modes are no longer present,was observed again in what appeared to be two differentaging regimes in figure 8. It is in fact a confirmation ofthat change in behaviour due to the distribution of modesgoing from bimodal to single peaked. Therefore the cor-rect value of ξ to compare to experiments is ξ = 0.66.Previously τ ∼ tξw with ξ = 0.88 [41] and ξ ∼ 1.1 [44]have been found for glasses, close to ξ = 1 which wouldbe the simple aging case.

The experimental studies of Cipelletti et al. have re-ported ν = 1, i.e. τα ∼ q−1, meaning the length scale dis-

placed scaling linearly with time. This was done for bothattractive and repulsive systems, where it was thought inboth cases that the relaxation of internal stresses dom-inate the dynamics. The theoretical argument from thethe same studies also explained their value of the expo-nent µ = 1.5. The aging behaviour found was τα ∼ tξwwith ξ = 0.9 after a period of exponential growth for thegels [20] and ξ = 0.78 for the repulsive system [22]. Thevery different values of µ and ν obtained here point tomajor difference the mechanisms of the aging even if thescaling of the relaxation times may look similar.

The Van Hove correlation function gives us a more in-tuitive idea of the particle displacements in real space.For large observation times and long waiting times, as isthe case in figure 10, the system is in a regime where thedisplacements are greater than the β plateau regime. Forcomparison with a system with diffusion-like displace-ments a Gaussian was fitted to the peak. The very largetail present shows just how non-Gaussian the dynamicsare within the α relaxation regime. The displacementsaround the peak and up to 0.25d appear to be Gaussiandistributed, leaving the aforementioned large tail. Not alldisplacements within this tail are due to the same phys-ical processes. We can distinguish between two types ofbehaviour. Firstly single particle movements: particlesthat break away from the network, becoming part of thegas phase before rejoining the network somewhere elseor moving along the gel by surface diffusion. These par-ticles make up all of the displacements greater than 1d,but the numbers concerned are small (due to the highcoordination of the particles in the gel), i.e. they couldnot account on their own for the α relaxation. Secondly,there are particles which have moved distances of around0.25d < r < 1d, but which never leave the network, andfor which the displacements are correlated in time andspace. We also note that the part of the displacementsfound after the peak follows a power law with an expo-nent f = 2.8 ± 0.4. This is, within errors, what wasdeduced by Cipelletti et al. and predicted by Bouchaudand Pitard [49]. However, as we have just seen, theirsystem is quite different being strongly attractive, andother parameters such as the wavevector dependence aredifferent from this work. Further work is necessary beforeconclusions can be drawn from this.

The microscopic data presented also indicates that theaging does not occur as continuous process but ratheras a succession of ’events’ which occur on a time scalewhich increases with increasing waiting time, as was ob-served on a close up of the number of bonds per particle.Indeed it seemed that for some short time intervals thenumber of bonds in the system decreased. These eventsinvolve a relatively large number of particles moving co-operatively since they can even be seen on a correlationfunction averaged over 4000 particles, and dominate thegel dynamics for time scales greater than the duration ofthe beta plateau.

While the ideas of cooperative dynamics in non-equilibrium systems is not new, string-like motion of

14

chains of particles moving cooperatively having been ob-served in supercooled liquids [50], the behaviour seen hereis more reminiscent of the work of Sanyal and Sood [51]who studied a dense binary colloid mixture in which steplike displacements were observed. This was accompaniedby some hop back motion as well. It is likely that a sitehopping scenario is responsible for this, whereas back andforth displacements were not observed in the gel case, al-though such motion may be present at shorter times.

Unsurprisingly, these correlated dynamics were linkedto a correlation in the localisation of the particles thathave broken or created bonds for a given time interval.The relatively high number of bonds broken (and cre-ated) in the gel when comparing configurations separatedby a fixed time interval is an indication of the ease withwhich thermal fluctuation of the strands disrupt the net-work, causing a high fraction of particles to change envi-ronment. This may be a result of the spatially anisotropicnature of the gel, whereas large thermal flutuation aremore unlikely to occur in an isotropic glassy system.

Although the low frequency trend in the stress datafrom figure 16 is the result of the microscopic aging‘events’ that occur in the gel it is now thought by theauthors to be unlikely to be the driving force. Whileit was a surprise to find that the stresses were not spa-tially correlated, this may point to the fact that forcesthat can be sustained by the inter-colloidal bonds are notvery high in comparison to the stresses occasioned in thestrands by the thermal fluctuations, even though theysuffice to confine the individual colloid particles to thegel. Under the influence of thermal fluctuations strandscan break bonds relatively easily; this is the only expla-nation for the the large number of bonds broken in thesystem. A further clue which supports this argument isthe relative size of the high frequency stress fluctuationscompared to both the average value of the stress, and thesize of the low frequency fluctuations. In both cases thestresses caused by the thermal fluctuations are of muchhigher magnitude. This supports the idea that the inter-nal stresses cannot be responsible for the aging, and thatthe low frequency fluctuations and the general decreasingtrend of the average stress are consequences of the aging,not driving factors.

It is not yet clear why the relaxation times increasewith waiting time, however the mechanisms invoked hereinvolve breaking the network under the thermal fluctua-tions. Presumably, the network will break preferentiallyat the point at which it is weakest. These weak points be-tween the strands that make up the network would tendto disappear in time (they will keep breaking until theydo). As the network become devoid of such weak points,we can expect the dynamics to slow down, thereby caus-ing the aging behaviour. Further work is underway toclarify this issue.

Comparison of this work with that of Barrat and Kob,reveals great similarities in the overall properties of therelaxation dynamics in these gels and glasses, the sameaging was seen in the correlation functions, as well as

superposition of the α relaxation regime. However it isclear from this work that the microscopic mechanisms bywhich the aging occurs in the gels are different to theones observed in homogeneous glasses, the dynamics inthe latter involving cooperative motion of strings of par-ticles [50]. Similarly, comparison with Puertas et al. [30],shows the same overall behaviour even though their sim-ulations focused on the fluid states leading up to the geltransition. The relaxation mechanisms (although theydid not report any details) were very probably similar towhat was observed in [50].

The aging behaviour predictions given by non-equilibrium MCT are power law regimes either side of theβ plateau, a waiting dependent α relaxation time and vi-olation of the fluctuation dissipation theorem. Althoughthe predictions are not specifically tested here, our data,at least qualitatively is not in disagreement with them.We found a stretched exponential fit to be better than apower law for the α relaxation but within a certain regionnear the plateau, a power law is certainly possible.

The Potential Energy Surface approach lends itself wellto the interpretation of our data, and comparison withaging glasses. In this picture, the gel would be in a regimein configuration space where the system samples basinsof IS with ever decreasing energy, but with the barriersbetween Inherent Structures becoming greater than thethermal energy in the system. The slow hopping overbarriers manifests itself in the gel as network rearrange-ments, and hence the behaviour of quantities such as thetotal number of bonds in the system.

Based on the work of Kob et al. [52], who have showedthat an effective time dependent temperature can be de-fined for an aging system, Sciortino and Tartaglia [38]have proposed an out of equilibrium thermodynamicsframework based on the IS description of the PES. Inthis picture, the free energy is separated into a basin termand a configuration term reflecting the separation of timescales in supercooled fluids. In the case of a temperaturejump the system equilibrates to the bath temperaturein the same basin rapidly. The exploration of configura-tion space takes place on a much longer time scale, hencethe aging observed as the system lowers its configura-tional energy. This description is fitting for our systemwhere figure 17 shows that the system is in quasi thermalequilibrium while of course not being in thermodynamicequilibrium. For the correlation functions calculated fora Leonard Jones Binary Mixture, Sciortino and Tartagliaobserved stretched exponentials for the equilibrium fluid,but logarithmic decays for the out of equilibrium LJ fluid.Note that this is in fact an MCT prediction for quenchesnear an A3 point. Once again superposition of the α re-laxation held and relaxation time scales increasing withtw were found.

The different regimes seen the total number of bonds inthe gel (figure 1), can then be interpreted in terms of thePES, much in the way of [38]and [52]. After the instan-taneous quench the system is still fluid as aggregationbegins (0 to 10 τR), this is followed by ridge dominated

15

dynamics (10 to 100 τR) and finally the aging behaviourof the last regime dominated by the slow sampling of ISof lower and lower energy.The similarities of this work on supercooled fluids with

ours is striking and the only difficult point concerns theshape of the α relaxations. Whereas studies of the PESof liquids found logarithmic decays, we had two regimesof stretched exponentials. It is possible that is it morestrongly influenced by the microscopic mechanisms thanwhat was previously thought. Those mechanisms couldwell be different due to the localisation of the parti-cles within stands rather than being distributed nearlyisotropicaly. The thermal fluctuations in strands beingenough to break the network thereby causing the agingeven though thermal fluctuations of a single particle arevery rarely enough for them to break away from a strand.This highlights a problem concerning general theories

of the glass transition, if the microscopic details thatbring about the final relaxation effect the macroscopicobservations, as was seen when comparing our resultswith the stress driven systems of Cipelletti et al. [20]and Solomon et al. [25] and the aforementioned resultsfor supercooled fluids, then a theory like MCT whichworks with averaged quantities is unlikely to distinguishbetween them. On the other hand, there seems to be agreater chance to incorporate different microscopic relax-ation theories when using the PES approach in order toestimate the time to explore different basins, and under-stand its relation with the aging time.

V. CONCLUSIONS

To finally conclude this study, we can say that simu-lations now appear to be a reasonable way to study theshort term aging of soft materials. It has allowed us tostudy the intermediate scattering function of an aginggel in order to compare it to experimental studies whileat the same time to understand the microscopic mecha-nisms responsible for the aging. The conclusion is thatfor this weakly attractive, gel-forming, system the ag-ing is caused by discontinuous network rearrangementsdue to the thermal fluctuations. It was also possible toinvestigate and further strengthen the relationship be-tween gels and glasses. A possible extension of this studywould concern the possibility of reproducing experimen-tal results for a more stress driven system to check the re-lationship between the α relaxation and the microscopicmechanisms.

VI. ACKNOWLEDGEMENTS

RA acknowledges the support of EPSRC and Colworthlabs (Unilever Research) for funding; he would also likethank H. Lekkerkerker, W. Poon, A. Puertas, M. Fuchs,L. Starrs, L. Cipelletti, and members of the Polymersand Colloids group for many useful discussions.

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