Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition
Giovanni Brambilla, Djamel El Masri, Matteo Pierno, Ludovic Berthier, Luca Cipelletti, George Petekidis, Andrew B. Schofield
aa r X i v : . [ c ond - m a t . s o f t ] A p r Probing the equilibrium dynamics of colloidal hard spheres abovethe mode-coupling glass transition
G. Brambilla, D. El Masri, M. Pierno, L. Berthier, and L. Cipelletti
Laboratoire des Collo¨ıdes, Verres et Nanomat´eriaux, UMR 5587,Universit´e Montpellier II and CNRS, 34095 Montpellier, France
G. Petekidis
IESL-FORTH and Department of Material Science and Technology,University of Crete, GR-711 10 Heraklion, Greece
A. B. Schofield
The School of Physics and Astronomy,Edinburgh University, Mayfield Road,Edinburgh, EH9 3JZ, United Kingdom (Dated: May 29, 2018)
Abstract
We use dynamic light scattering and computer simulations to study equilibrium dynamics anddynamic heterogeneity in concentrated suspensions of colloidal hard spheres. Our study covers anunprecedented density range and spans seven decades in structural relaxation time, τ α , includingequilibrium measurements above ϕ c , the location of the glass transition deduced from fitting ourdata to mode-coupling theory. Instead of falling out of equilibrium, the system remains ergodicabove ϕ c and enters a new dynamical regime where τ α increases with a functional form that was notanticipated by previous experiments, while the amplitude of dynamic heterogeneity grows slowerthan a power law with τ α , as found in molecular glass-formers close to the glass transition. PACS numbers: 05.10.-a, 05.20.Jj, 64.70.P- ϕ , in analogy to the glass transition of molecular liquids [4] and the jam-ming transition of grains [2]. However, the nature of the colloidal glass transition, its preciselocation, the functional form of the structural relaxation time divergence, and the connectionbetween slow dynamics and kinetic heterogeneities remain largely open issues [5, 6].For hard spheres at thermal equilibrium, several distinct glass transition scenarios havebeen described. In the first, the viscosity or, equivalently, the timescale for structuralrelaxation, τ α ( ϕ ), diverges algebraically: τ α ( ϕ ) ∼ ( ϕ c − ϕ ) − γ . (1)This is predicted [7] by mode coupling theory (MCT), and supported by the largest set oflight scattering data to date [5]. Packing fractions ϕ c ≈ . − .
59 are the most often quotedvalues for the location of the ‘colloidal glass transition’. It is widely believed that a trulynon-ergodic state is obtained at larger ϕ [1, 3, 5]. Within MCT, the amplitude of dynamicheterogeneity quantified by multi-point correlation functions also diverges algebraically. Inparticular, the four-point dynamic susceptibility should diverge as [8]: χ ∼ ( ϕ c − ϕ ) − ∼ τ /γα , a prediction that has not been tested experimentally.Several alternative scenarios [9, 10, 11] suggest a stronger divergence: τ α ( ϕ ) = τ ∞ exp (cid:20) A ( ϕ − ϕ ) δ (cid:21) . (2)Equation (2) with δ = 1 is frequently used to account for viscosity data [6] because itresembles the Vogel-Fulcher-Tammann (VFT) form used to fit the viscosity of molecularglass-formers [4], with temperature replaced by ϕ . Moreover, it is theoretically expectedon the basis of free volume arguments [9], which lead to the identification ϕ ≡ ϕ rcp , therandom close packing fraction where osmotic pressure diverges. Kinetic arrest must occurat ϕ rcp (possibly with δ = 2 [10]), because all particles block each other at that density [10,12, 13]. Entropy-based theories and replica calculations [11] predict instead a divergence of2 α at an ideal glass transition at ϕ < ϕ rcp , where the configurational entropy vanishes butthe pressure is still finite. Here, the connection to dynamical properties is made throughnucleation arguments [14] yielding Eq. (2), with δ not necessarily equal to unity [15]. In thiscontext, the amplitude of dynamic heterogeneity should increase only moderately, typicallylogarithmically slowly in τ α [16].In molecular glass-formers where dynamical slowing down can be followed over as manyas 15 decades, the transition from an MCT regime, Eq. (1), to an activated one, Eq. (2),has been experimentally demonstrated [4]. For colloidal hard spheres, the situation remainscontroversial, because dynamic data are available over a much smaller range [1, 5, 6, 17],typically five decades or less. Crucially, equilibrium measurements were reported only for ϕ < ϕ c , leaving unknown the precise nature and location of the divergence. Theoreticalclaims exist that the cutoff mechanism suppressing the MCT divergence in molecular systemsis inefficient in colloids due to the Brownian nature of the microscopic dynamics, suggestingthat MCT could be virtually exact [18]. This viewpoint is challenged by more recent MCTcalculations [19], and by computer studies of simple model systems where MCT transitionsare avoided both for stochastic and Newtonian dynamics [20, 21].Here, we settle several of the above issues by studying the equilibrium dynamics of col-loidal hard spheres using dynamic light scattering and computer simulations. By extendingprevious data by at least two orders of magnitude in τ α , we establish that the volume frac-tion dependence of both τ α and χ follows MCT predictions only in a restricted densityrange below our fitted ϕ c ≈ .
59. Unlike previous studies, we provide equilibrium measure-ments above ϕ c , thereby proving unambiguously that in our sample the algebraic divergenceat ϕ c is absent. Instead, a new regime is entered at larger ϕ , where the dynamics is welldescribed by Eq. (2) with δ ≈ ϕ much larger than ϕ c . The amplitude of kineticheterogeneities then grows slower than a power law with τ α , as in molecular glasses close tothe glass transition.Dynamic light scattering (DLS) experiments are performed in the range 0 . < ϕ < . σ =260 nm, stabilized by a thin layer of grafted poly-(12-hydroxy stearic acid) (PHSA). Thesize polydispersity, about 10%, is large enough to prevent crystallization on a timescale ofat least several months. The particles are suspended in a mixture of cis-decalin and tetralinthat almost perfectly matches their average refractive index, allowing the dynamics to be3robed by DLS. Additionally, a careful analysis of the combined effects of optical and sizepolydispersity shows that we probe essentially the self-part of the intermediate scatteringfunction [22]: F s ( q, t ) = * N N X j =1 e i q · ( r j ( t ) − r j (0)) + . (3)Here, r j ( t ) is the position of particle j at time t , q is the scattering vector ( qσ = 6 . F s ( q, t ). We carefully check equilibration by following the evolution ofthe dynamics after initialization, until F s stops changing over a time window of at least10 τ α . Samples are prepared by dilution, starting from a very concentrated batch obtainedby centrifugation. All volume fractions relative to that of the initial batch are obtained witha relative accuracy of 10 − , using an analytical balance and literature values for particle andsolvent densities [17]. Relative volume fractions are converted to absolute ones by comparingthe experimental ϕ dependence of the short-time self-diffusion coefficient measured by DLSto two sets of theoretical calculations [25] at low density, ϕ ≤ .
2. For less polydispersesamples, this calibration method yields ϕ values compatible with those obtained by mappingthe experimental freezing fraction to ϕ f = 0 .
494 [26]. The remaining uncertainty on theabsolute ϕ is about 5%, because [25] contains two slightly different predictions. To ease thecomparison with the simulations, we set the absolute ϕ , within this uncertainty range, sothat our experimental and numerical τ α closely overlap for ϕ > . σ and 1 . σ , known to efficiently prevent crystallization. We workin a three dimensional space with periodic boundary conditions, and mainly use N = 10 particles. No noticeable finite size effects were found in runs with N = 8 · particlesperformed for selected state points. In an elementary move, a particle is chosen at randomand assigned a random displacement drawn within a cubic box of linear size 0 . σ centeredaround the origin. The move is accepted if the hard sphere constraint remains satisfied.One Monte Carlo step corresponds to N such attempts. The dynamics is characterized bythe self-intermediate scattering function, Eq. (3), measured for qσ = 6 .
1, close to the firstdiffraction peak.Representative F s ( q, t ) obtained by DLS are plotted in Fig. 1, showing that the relaxation4 (cid:0)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7) (cid:8)(cid:9)(cid:10)(cid:11) (cid:12)(cid:13)(cid:14) (cid:15)(cid:16)(cid:17) (cid:18)(cid:19)(cid:20) (cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" FIG. 1: (Color online) Time dependence of the self-intermediate scattering function F s ( q, t ) in DLSexperiments at qσ = 6 . ϕ c ≈ . is fast and monoexponential at low ϕ , while a two-step decay is observed when increasing ϕ ,reflecting the increasingly caged motion of particles in dense suspensions [3]. We measurethe structural relaxation time by fitting the final decay of F s to a stretched exponential, F s ( q, t ) = B exp[ − ( t/τ α ) β ].Figure 2a shows τ α ( ϕ ) for both experiments and simulations. Time units are adjustedto maximize the overlap ( ≈ ϕ . Our experimentaldata are well fitted by Eq. (1) in the range 0 . < ϕ ≤ . ϕ c = 0 . ± . γ = 2 . ± .
1. For the slightly less polydisperse sample of Ref. [5], a similar power lawbehavior with γ ≈ . ϕ c = 0 . − .
595 was reported, the two quoted values of ϕ c stemming from experimental uncertainty in the volume fraction determination. However,our measurements for the largest densities strongly deviate from the MCT fit. Attemptsto include points at ϕ > .
59 in the MCT fit yield unphysically large values of γ . Similardeviations are found in our simulations, showing that hydrodynamic interactions play littlerole in experiments performed at large ϕ , although they probably explain the discrepancywith simulations at low volume fraction, see Fig. 2a. Therefore, our results unambiguouslydemonstrate that the mode-coupling singularity is absent in our hard sphere colloidal system,as is also found in molecular glass-formers [4].What is the fate of the fluid phase above ϕ c ? Figures 2a and 2c show that the increaseof τ α at high ϕ is extremely well described by an exponential divergence, Eq. (2). Wefind that the data can be fitted well using the conventional form with δ = 1, yielding5 .0 0.2 0.4 0.6-4-2024 -3 -2 -1 l og ( t a / t ) j a) l og ( t a / t ) j t a / t (1- j / j c ) -1 b) l og ( t a / t ¥ ) A / (j - j ) c) FIG. 2: (Color online) a) Relaxation timescale τ α for hard spheres in experiments (black circles)and simulations (open triangles), respectively in units of τ = 1 sec and τ = 7 · MC steps. Thered dashed line is a power law fit, Eq. (1), with ϕ c = 0 .
590 (vertical dotted line) and γ = 2 . ± . ϕ = 0 .
637 and δ = 2. The zoomin the inset shows that the MCT singularity is absent. b) Same data plotted against 1 / (1 − ϕ/ϕ c ).A straight line with slope γ is obtained in an MCT regime covering almost 3 decades in τ α . c)Data for ϕ > .
41 plotted using reduced variables with ϕ = 0 .
637 and 0 .
641 for experiments andsimulations, respectively. ϕ ( δ = 1) ≈ . ± . δ is allowed to depart from unity.The optimal value, robust for both experimental and numerical data, is δ = 2 . ± .
2, whichyields our best estimate for the location of the dynamic glass transition: ϕ ≈ . ± . ϕ ≈ . ± .
002 (simulations). Figure 2c shows the linear dependenceof log τ α on ( ϕ − ϕ ) − , demonstrating the exponential nature of the dynamic singularity.The behavior of dynamical heterogeneity provides additional evidence of a crossover froma restricted MCT regime to an ‘activated’ type of dynamics. Using methods detailed in [27,28], we study the evolution of the three-point dynamic susceptibility defined by: χ ϕ ( q, t ) ≡ ∂F s ( q, t ) /∂ϕ . This linear response function is directly connected to a four-point dynamicsusceptibility: χ ( q, t ) = N h δF s ( q, t ) i , where δF s ( q, t ) denotes the fluctuating part of the6elf-intermediate function; χ is a powerful tool to quantify dynamic heterogeneity in glass-formers [16], because it represents the average number of molecules whose dynamics arecorrelated. In hard spheres, the following relation holds [27]: χ ( q, t ) = χ ( q, t ) | ϕ + ρk B T κ T ( ϕχ ϕ ( q, t )) , (4)where ρ is the number density, κ T the isothermal compressibility (measured in simulations,taken from the Carnahan-Starling equation of state in experiments), and χ ( q, t ) | ϕ denotesthe value taken by χ ( q, t ) in a system where density is strictly fixed. Only the second con-tribution to χ ( q, t ) in (4) can be accessed experimentally, but both terms can be determinedin simulations. We obtain χ ϕ ( q, t ) by applying the chain rule to the fitted F s ( B, τ α , β ) [28],where the ϕ dependence of B, τ α , β is fitted by smooth polynomials. Our results are in-dependent of the choice of fitting functions, and consistent with that obtained from finitedifferences between data at nearby ϕ , when available. Figure 3 shows the peak of dynamicalsusceptibilities as a function of ϕ . First, we numerically establish in Fig. 3a that the termcomprising χ ϕ is the main contribution to χ when ϕ > .
52, implying that a good estimateof χ can be obtained using three-point functions in hard spheres, as surmised in [27], andestablished for molecular glass-formers in [8]. For both simulations and experiments, theMCT prediction for the algebraic divergence of χ ( q, t ) only holds over a limited densityrange. Indeed, when plotted against τ α , χ eventually grows slower than a power law, asfound for the size of dynamically correlated regions in molecular glasses close to the glasstransition [28]–a hallmark of activated dynamics [16].Our results establish the existence of a non-trivial, exponential divergence of τ α ( ϕ ) at acritical volume fraction ϕ ≈ .
637 much above the putative ‘colloidal glass transition’ at ϕ c ≈ .
59. It is natural to ask whether ϕ and ϕ rcp coincide. This is a difficult questionbecause ϕ rcp can always be shifted to a larger value by trading order and packing [12]. Forthe binary mixture studied here, the onset of jamming has been located at ϕ J = 0 .
648 [13].Furthermore, for 10 % polydispersity, the estimate ϕ rcp ≈ .
67 was obtained in numericalwork [29], well above ϕ . Finally, we have employed Monte Carlo simulations to producedisordered hard sphere configurations with finite pressure above ϕ by a very fast compres-sion of fluid configurations used to produce the equilibrium data in Fig. 1 (open symbols).These results support the possibility that ϕ < ϕ rcp , implying a fundamental difference [30]between the glass [11] and jamming [13] transitions in hard spheres.7 tuv wxyz {|}~ (cid:127)(cid:128)(cid:129)(cid:130) (cid:131)(cid:132)(cid:133)(cid:134) (cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:146)(cid:147)(cid:148)(cid:149)(cid:150)(cid:151)(cid:152) (cid:153)(cid:154)(cid:155)(cid:156) (cid:157)(cid:158)(cid:159)(cid:160) ¡¢£⁄ ¥ƒ§ ¤'“ «‹› fifl(cid:176) –†‡·(cid:181)¶•‚„”»…‰(cid:190)¿(cid:192)`´ˆ˜¯˘˙¨(cid:201)˚¸(cid:204)˝˛ˇ—(cid:209)(cid:210) r (cid:211)(cid:212)(cid:213) k (cid:214) j (cid:215) c j (cid:216)(cid:217)(cid:218)(cid:219)(cid:220)(cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)ŁØŒº(cid:236)(cid:237)(cid:238)(cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı(cid:246) t a (cid:247)łøœß c (cid:252) j r (cid:253)(cid:254)(cid:255) k (cid:1) j (cid:2) c (cid:3)(cid:0) FIG. 3: (Color online) Peak of dynamic susceptibilites, Eq. (4), measured in a) simulations andb) experiments. In a) both contributions to χ are compared, validating χ ϕ as a valuable toolto quantify dynamic heterogeneity in hard spheres. The predicted MCT algrebraic divergence(red dashed line) holds over a small density range. The inset shows that the size of dynamicheterogeneities grows slower than a power law at large τ α , as found in molecular glass-formers. In conclusion, we report a set of dynamic data for a well-known colloidal hard sphere sys-tem covering an unprecedented dynamic range of equilibrium relaxation timescale. Whilethe onset of dynamical slowing can be described by an MCT divergence at a critical volumefraction ϕ c , upon further compression a crossover from an algebraic to an exponential diver-gence at a much larger volume fraction ϕ is observed, accompanied by a similar crossoverfor the growth of dynamical correlations. Our results show that the apparent singularity at ϕ c does not correspond to a genuine ‘colloidal glass transition’, suggesting that the MCTtransition is generally avoided in colloidal materials, just as in molecular glass-formers.This work was supported by the Joint Theory Institute at the University of Chicago,the European MCRTN “Arrested matter” (MRTN-CT-2003-504712), the NoE “SoftComp”(NMP3-CT-2004-502235), Region Languedoc-Roussillon and ANR “DynHet” and CNES8rants. L.C. acknowledges support from the IUF. [1] P. N. Pusey, and W. van Megen, Nature , 595 (1986).[2] Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales , Eds:A . J. Liu and S. R. Nagel (Taylor and Francis, New York, 2001).[3] P. N. Pusey, and W. van Megen, Phys. Rev. Lett. , 2083 (1987).[4] P. G. Debenedetti, and F. H. Stillinger, Nature , 259 (2001)[5] W. van Megen et al. , Phys. Rev. E , 6073 (1998).[6] Z. Cheng, J. Zhu, P. M. Chaikin, S. E. Phan, and W. B. Russel, Phys. Rev. E , 041405(2002).[7] W. G¨otze, J. Phys.: Condens. Matter , A1 (1999).[8] L. Berthier, et al. , J. Chem. Phys. , 184503 (2007); J. Chem. Phys. , 184504 (2007).[9] M. H. Cohen and D. Turnbull, J. Chem. Phys. , 1164 (1959).[10] K. S. Schweizer, J. Chem. Phys. , 164506 (2007).[11] M. Cardenas, S. Franz, and G. Parisi, J. Chem. Phys. , 1726 (1999); G. Parisi and F.Zamponi, J. Chem. Phys. , 144501 (2005).[12] S. Torquato, T. M. Truskett, and P. G. Debenedetti, Phys. Rev. Lett. , 2064 (2000).[13] C. S. O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel, Phys. Rev. Lett. , 1045 (1989).[15] J.-P. Bouchaud and G. Biroli, J. Chem. Phys. , 7347 (2004).[16] C. Toninelli et al. , Phys. Rev. E , 041505 (2005).[17] S.-E. Phan et al. , Phys. Rev. E , 6633 (1996).[18] S. P. Das and G. F. Mazenko, Phys. Rev. A , 2265 (1986).[19] M. E. Cates and S. Ramaswamy, Phys. Rev. Lett. , 135701 (2006); A. Andreanov, G. Biroli,and A. Lef`evre, J. Stat. Mech. P07008 (2006)[20] G. Szamel and E. Flenner, Europhys. Lett. , 779 (2004).[21] L. Berthier and W. Kob, J. Phys.: Condens. Matter , 205130 (2007).[22] P. N. Pusey, H. M. Fijnaut, and A. Vrij, J. Chem Phys. , 4270 (1982); P. N. Pusey, J. Phys.A , 119 (1978).[23] B. J. Berne, and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).
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