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Dynamics of a colloidal glass during stress-mediated structural

arrest

A. Negi1 and C. Osuji1

1 Department of Chemical Engineering, Yale University, New Haven CT 06511

PACS 82.70.Dd – ColloidsPACS 81.05.Kf – Glasses (including metallic glasses)PACS 83.85.Cg – Rheological measurements - rheometry

Abstract. - We employ parallel superposition rheology to study the dynamics of an aging colloidalglass in the presence of a mean field stress σm. Over a range of intermediate stresses, the lossmodulus exceeds the storage modulus at short times but develops a maximum concomitant witha crossover between the two as the system ages. This is attended by a narrowing of the loss peakon increasing stress. We show that this feature is characteristic of the structural arrest in thesematerials, which is made observable on reasonable timescales by the activating influence of thestress. The arrest time displays an exponential dependence on inverse stress. These results provideexperimental validation of the role of stress as an effective temperature in soft glassy systems ashas been advanced in recent theoretical frameworks.

The thermalizing effects of shear have been well rec-ognized in colloidal systems [1–4] where an applied shearstress lowers energetic barriers to particle motion, or animposed shear rate provides higher particle mobility thanexpected based on the system (bath) temperature. In suchscenarios, accelerated dynamics are found in equilibriumsystems under shear such as non-Brownian particulate sus-pensions [5] and polymers in non-linear creep flow wheretime-stress superposition is observed [6]. However the ef-fects of imposed stress on the dynamics of out of equilib-rium glassy materials, that is, in systems which displayaging, has not been explored experimentally in great de-tail. In the presence of an applied stress, such systems, forexample gels, pastes, emulsions and slurries, are known todisplay a viscosity bifurcation [7,8]. Application of a sub-critical stresses σ < σc results in an eventual immobilestate as the shear rate developed in response to the stressdecreases due to an ever increasing viscosity as the systemages or structures over time. By contrast, application ofa larger than critical stress produces a quasi-steady stateresponse with a well defined shear rate and finite viscos-ity. These respective displays are due to the dominance ofeither aging or rejuvenation in the presence of the appliedstresses, where rejuvenation in this sense should be under-stood simply as a decrease of the characteristic relaxationtime of the system, rather than a strict path reversal of

aging.

Here, we report on the evolution of stress activated dy-namics in an aging repulsive colloidal glass. Considerableattention has been paid to the questions of aging and re-juvenation in these and analogous systems, as often stud-ied by creep [9] or dynamic scattering methods [10] wheretime-elapsed time rescaling is obtained. By contrast, ex-plicit observations of the loss and storage modes of thesystem and their evolution during structural arrest afterrejuvenation have not been made. Specifically, we focuson the regime of stresses below the viscosity bifurcationwhere physical aging of the system brings it to a dynamicalarrest on timescales that are comparable to the observa-tion time. We find that the time evolution of the complexmodulus is strongly dependent on the magnitude of theapplied shear stress. At small stresses, the storage mod-ulus increases and the loss modulus decreases, both withweak power laws, with G′ > G′′. The system shows an en-tirely viscous response in the presence of large stresses. Atintermediate stresses, G′′ > G′ at short times but devel-ops a peak concomitant with a crossover between G′ andG′′ as the system ages and undergoes arrest, permitting anovel observation of the structural relaxation “in reverse.”Our results indicate that this is a characteristic feature ofstructural arrest in these systems. The shape of the losspeak is strongly dependent on the applied stress, with a

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A. Negi 1 C. Osuji1

significant narrowing displayed at higher stresses.Our system is an aqueous suspension of Laponite which

consists of disc-like clay particles of 25 nm diameter and1 nm thickness. “Wigner” glasses are formed at low ionicstrength in dilute conditions due to long range electro-static repulsion between the particles [11]. The systemdisplays glassy dynamics, as shown by previous reports[10, 12, 13]. Samples are prepared by mixing LaponiteXLG (Southern Clay Products) into ultra-pure water, ad-justed to pH 9.5 to ensure chemical stability of the parti-cles [14]. The suspensions are mixed for 20 minutes andthen allowed to develop quiescently for several days, defin-ing a well controlled and reproducible initial state. Thesystem is thus in the full aging regime, as opposed tothe short time “gelation” regime, the delineation betweenwhich has been described in the literature [15]. Measure-ments are conducted on an MCR 301 rheometer (Anton-Paar) mounted on a air table for mechanical noise isola-tion. Evaporation of water from the sample is successfullysuppressed without perturbing the system via applicationof a thin film of mineral oil at the sample edge. In allmeasurements, samples are first subjected to a rejuvenat-ing shear at γ̇=100 s−1 for 100 s. This eliminates any flowhistory and provides a reproducible initial state in whichthe viscosity is well defined within ±5%. The sample isthen brought to rest and allowed to sit without externalperturbation for 1 second before the start of subsequentmeasurements. We monitor the dynamic modulus via ap-plication of a small probe stress oscillation σp(t) atop afixed mean-field stress σm applied to the sample. Theoverall stress is σ(t) = σm + σp0

sin(ωt) where the ap-plied frequency ω=10 rad/s. Probe stresses were variedfrom 0.2 to 1 Pa. In each case, the response of the sys-tem was verified to be linear in σp0

, confirming that theexperiment samples the linear properties that exist in thepresence of the mean-field stress σm. Parallel superposi-tion rheology as applied here has been successfully usedto study the dynamics of a variety of complex fluids undershear. It is to be noted that the interpretation of resultsfrom these flows may be complicated by coupling betweenthe stationary and oscillatory components and a shear ratedependence of the perturbation spectrum [16]. This lim-its strict quantitative interpretation of data, but not theutility in providing a general description of the stress orshear rate dependence of dynamics [17]. For this reason,we assess our results here only qualitatively, as is commonpractice [18,19]. For observations of the viscosity bifurca-tion, following the initialization described, the sample issubjected to a fixed stress and the time dependent shearrate developed in response is observed. The sample is re-initialized and the stress is iteratively increased to cover asuitable range.We examine the frequency dependent and non-linear

dynamics of the system via frequency and strain sweeps.Samples are allowed to sit quiescently for 30 minutes fol-lowing the rejuvenating shear at γ̇=100 s−1. This dura-tion is large relative to the sweep time so the system does

not age considerably during the actual measurement. Thefrequency sweep is typical of this class of materials, witha wide regime where both the storage and loss moduliscale weakly with frequency. In this case, G′

∼ ω0.01 andG′′

∼ ω−0.2, Figure 1a, for ϕ=3.5 wt.%. In the strainsweep, there is a linear viscoelastic regime of constantG′,G′′ extending up to γ ≈ 10%. This is followed by asteady decrease in the storage modulus and a peak in theloss modulus on increasing strain as the system becomesnon-linear, Figure 1b. Deep into the non-linear regime,both moduli follow a power-law dependence on γ with theratio between the exponents approaching 2 with increasingstrain, indicative of strong fluidization of the system. Thesystem displays physical aging with the complex modulusscaling as G∗

∼ t0.2, after cessation of the rejuvenatingshear, inset Figure 1a.

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Fig. 1: Strain and frequency sweeps recorded at ω=10 rad/s.and γ=2%, respectively, for ϕ=3.5 wt.%. Inset: Aging of thecomplex modulus with time after cessation of the rejuvenatingflow, ω=10 rad/s.

Application of a fixed stress σm after sample rejuvena-tion produces an initial shear rate which then evolves intime as the sample ages and its viscosity changes. Forstresses larger than some critical stress, σm > σc, a steadyshear rate was achieved, indicating that physical agingduring the measurement window was suppressed. Appli-cation of smaller stresses σm < σc resulted in a time-dependent decrease of the shear rate during the course ofthe measurement. The data show an initial regime of slow

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Stress activated dynamics during structural arrest

power-law decay followed by a terminal regime where γ̇(t)appears to diverge towards zero as the viscosity increasesrapidly, Figure 2.

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Fig. 2: Evolution of the shear rate for stresses less than thebifurcation stress σc for ϕ=2.75, 3.0 and 3.5 wt.%, (top tobottom).

The dynamical arrest that occurs in the presence of

these sub-critical mean field stresses is due to the ag-ing of the system resulting in eventual vitrification. Onecan think of vitrification as a structural relaxation con-ducted “in reverse.” In an equilibrium system such as amodel Maxwellian material, the structural relaxation istraversed by progressively decreasing frequency. In a non-equilibrium system, observation at a fixed frequency asa function of time exposes the underlying frequency de-pendence as the system ages with the passage of time.The near linear dependence of the characteristic time onsample age in many glassy materials which enables time-elapsed time universal scaling and the width of the glassypower law regime where G′

∼ ωx with 0 < x < 1, both un-derline this point. One can expect then that fluidizationof a glassy system by an applied stress σ < σc will reveala dynamical signature akin to that of the structural relax-ation as the material ages and enters the arrested state.

This is precisely what is observed. As shown in Figure3, there is a crossover betweenG′ and G′′ coincident with apeak in G′′ at a characteristic time τ(σm) during the agingof the system under moderate mean-field stresses. Similardata have been observed by Ovarlez et al. in their studyof a thixotropic bentonite suspension where composition,stress and temperature dependence could be rescaled togenerate a master curve of the aging response of the sys-tem. [20] The data for the current system are particularlywell defined and show striking parallels with the proto-typical frequency dependence found in soft systems wherethere is a transition via a peaked loss modulus from theterminal regime where G′

∼ ω2, G′′∼ ω to a slow ar-

rested state with weakly increasing G′ and decreasing G′′

[21]. The linearity of the response to the probe stress isconfirmed by the stress independence of the modulus, asshown in Figure 4.

Notably, τ appears to vary exponentially with 1/σm.The data have been fit to Equation 1, as shown in Figure5, reflecting the slight curvature observed.

τ ∼ exp [−E/(σm − σ0)] (1)

In the context of stress as an effective temperature,the timescale for diffusive particle motion should scaleas τd ∼ exp(E/σm). Correspondingly, the timescale forparticle arrest should scale inversely, as τ−1

d . The param-eter σ0 accounts for the fact that by necessity the arresttimescale diverges above a critical stress beyond which ag-ing of the system will not bring it to arrest. This stressshould be identical to the bifurcation stress, σc discussedearlier. We observe good correspondence between σd fromthe fits of Figure 5 and estimates of σc taken from thetime-dependent shear rate experiments. Both E and σ0

exhibit an apparent power law dependence on the volumefraction, scaling with an exponent ≈ 4, although the com-position range is quite limited, inset Figure 5. More sig-nificantly, however, E scales directly proportional to σ0,indicating that the critical bifurcation stress can be inter-preted in terms of an energetic barrier to particle motion.

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A. Negi 1 C. Osuji1

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Fig. 3: Time evolution of storage (solid symbols) and loss mod-uli (open symbols), under different applied stresses for ϕ=2.75,3.0 and 3.5 wt.%, (top to bottom) measured with probe stressesof 0.2, 1 and 1 Pa.

This is in agreement with observations which note closecorrespondence between bifurcation stresses and the yieldstress in other pasty materials [22].

The loss peak that develops as the sample ages and ar-rests exhibits a marked dependence on σm, broadeningsignificantly as the stress is decreased, or conversely, nar-rowing at higher stresses. A Lorentzian function was used

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0.2 Pa, 5 Pa 0.4 Pa, 5 Pa 0.7 Pa, 5 Pa 1 Pa, 5 Pa 0.2 Pa, 10 Pa 0.4 Pa, 10 Pa 0.7 Pa, 10 Pa 1 Pa, 10 Pa

Fig. 4: Time evolution of storage (solid symbols) and loss mod-uli (open symbols), under different probe stresses σp0

, for twodifferent mean field stresses (5, 10 Pa) for ϕ=2.75 wt.%.

to provide an empirical estimate of the peak width. It wasfound to decrease markedly from approximately 3 decadesfor σm=13 Pa to less than 1.4 decades for σm=28 Pa., forϕ = 3.5 wt.% This narrowing can be understood as dueto a narrowing of the distribution of lossy modes in theglass, as similarly observed in glassy polymers subjectedto external stresses which likewise activate segmental mo-tion the system [23]. This variation in the shape of theloss peak precludes re-scaling into a universal form. As aresult, the data do not display time-stress superposition,just as time-temperature superposition is lost in equilib-rium systems on approaching Tg where the width of theloss modes also become temperature dependent, broaden-ing with decreasing T [24, 25].

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Fig. 5: Dependence of the crossover time τ on the meanfield stress σm. The lines are fits to the exponential formτ = τ0 exp [−E/(σm − σ0)]. Inset: Dependence of fit parame-ters E and σ0 on composition.

Coussot et al. have advanced a model which describes a

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Stress activated dynamics during structural arrest

time dependent viscosity resulting from a competition be-tween aging (structuring) and rejuvenation in the presenceof an applied stress [26]. This phenomenological approachdescribes the system in terms of a canonical structural pa-rameter λ which grows with a characteristic time, but alsois degraded by the action of shear, with system dependentconstants of proportionality and a specific viscosity thatis a simple power-law function of λ. Our system displaysthe hallmark behaviors of the model such as an asymp-totic characteristic time for the onset of structural arrestat small stresses, an exponential dependence of this timeat intermediate stresses and a fast transient leading to asteady state viscosity at high applied stresses. Our resultshere provide, for the first time, a detailed look at the dy-namics of the system in the low stress regime where arrestoccurs and shows that the features of this data are broadlyconsistent with observations made during viscosity bifur-cation experiments.Recent work has shown that the competition between

aging and rejuvenation in soft glassy materials is depen-dent on the magnitude of stress or strain imposed [9], in-cluding the observation of an over-aging regime [27]. Here,we see that for our repulsive glass, the flow arrest timeincreases exponentially, and eventually diverges with in-creasing stress. We do not observe an overaging regimewithin which the flow arrest time decreases with increas-ing stress. The reasons for this remain unclear. Onepossibility is simply that the stresses used in the currentwork are too large for the system under study to elicit anoveraging response. For the range of stresses consideredhere, the elastic modulus of the colloidal glass was inde-pendent of the stress applied during structural arrest. Asshown in Figure 3, within the experimental uncertainty,the system asymptotes towards a single curve, display-ing power-law aging of the elastic modulus at long times.This is in contrast to data and a conceptual model de-rived from bentonite suspensions where both the elasticmodulus and the yield strain were seen to increase withthe magnitude of the stress applied during the liquid-solidtransition [28]. More careful measurements, particularlyout to longer times, are required to conclusively test thismodel in the present system, but it provides a useful start-ing point for the consideration of the effect of stress on themechanical properties of the colloidal glass after structuralarrest.The dynamic measurements demonstrate that the aging

of the system and resulting arrest give rise to a temporalresponse that parallels that of the frequency dependenceobserved in non-aging systems on traversing the structuralrelaxation time. The peak in the time dependent loss mod-ulus is a dynamical signature of the structural arrest dueto aging. Analogous to the manner in which stress acti-vates particle dynamics in equilibrium systems, it is shownhere that it slows aging, delaying the onset of structuralarrest in this out of equilibrium system. The dependenceof the arrest time on stress is strikingly similar to that ofthe relaxation time in thermally activated fragile glass for-

mers on approaching the glass transition, highlighting theactivating role of stress in structural glasses and furthervalidating the concept of stress as an effective tempera-ture. To the best of our knowledge, this represents the firstsystematic characterization of stress mediated structuralarrest in a colloidal glass due to physical aging. Theseresults may spur progress on quantitative numerical andexperimental investigation of the role of stress in the agingdynamics of soft glassy materials.

∗ ∗ ∗

The authors thank P. Coussot and G. Ovarlez for in-sightful discussions and gratefully acknowledge NSF fund-ing under CBET-0828905.

REFERENCES

[1] Thirumalai D. and Mountain R. D., Phys. Rev. E , 47(1993) 479.

[2] Wyss H. M., Miyazaki K., Mattsson J., Hu Z., Re-

ichman D. R. and Weitz D. A., Phys. Rev. Lett. , 98(2007) 238303.

[3] Haxton T. K. and Liu A. J., Phys. Rev. Lett. , 99 (2007)195701.

[4] Ilg P. and Barrat J. L., Europhys. Lett. , 79 (2007)26001.

[5] Pine D. J., Gollub J. P., Brady J. F. and Leshansky

A. M., Nature , 438 (2005) 997.[6] Schapery R. A., Polym. Eng. Sci. , 9 (1969) 295.[7] Da Cruz F., Chevoir F., Bonn D. and Coussot P.,

Phys. Rev. E , 66 (2002) 051305.[8] Coussot P., Soft Matter , 3 (2007) 528.[9] Cloitre M., Borrega R. and Leibler L., Phys. Rev.

Lett. , 85 (2000) 4819.[10] Knaebel A., Bellour M., Munch J., Viasnoff V.,

Lequeux F. and Harden J., Europhys. Lett. , 52 (2000)73.

[11] Bonn D., Tanaka H., Wegdam G., Kellay H. andMeunier J., Europhys. Lett. , 45 (1998) 52.

[12] Bonn D., Tanase S., Abou B., Tanaka H. and Meu-

nier J., Phys. Rev. Lett. , 89 (2002) 015701.[13] Joshi Y. M. and Reddy G. R. K., Phys. Rev. E. , 77

(2008) 021501.[14] Mourchid A. and Levitz P., Phys. Rev. E , 57 (1998)

R4887.[15] Tanaka H., Jabbari-Farouji S., Meunier J. and

Bonn D., Phys. Rev. E. , 71 (2005) 21402.[16] Vermant J., Walker L., Moldenaers P. and Mewis

J., J. Non-Newtonian Fluid Mech. , 79 (1998) 173.[17] Dhont J. K. G. and Wagner N. J., Phys. Rev. E , 63

(2001) 021406.[18] Mewis J., Kaffashi B., Vermant J. and Butera R.,

Macromolecules , 34 (2001) 1376.[19] Anderson V., Pearson J. and Sherwood J., J. Rheol.

, 50 (2006) 771.[20] Ovarlez G. and Coussot P., Physical Review E , 76

(2007) 11406.[21] Fielding S. M., Sollich P. and Cates M. E., J. Rheol.

, 44 (2000) 323.

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A. Negi 1 C. Osuji1

[22] Coussot P., Nguyen Q., Huynh H. and Bonn D.,Journal of Rheology , 46 (2002) 573.

[23] Lee H.-N., Paeng K., Swallen S. F. and Ediger

M. D., Science , 323 (2009) 231.[24] Olsen N. B., Christensen T. and Dyre J. C., Phys.

Rev. Lett. , 86 (2001) 1271.[25] Oh J., Kim C., Kim H. and Hwang Y., Physica A , 315

(2002) 308.[26] Coussot P., Nguyen Q. D., Huynh H. T. and Bonn

D., Phys. Rev. Lett. , 88 (2002) 175501.[27] Viasnoff V. and Lequeux F., Phys. Rev. Lett. , 89

(2002) 065701.[28] Ovarlez G. and Chateau X., Physical Review E , 77

(2008) 61403.

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