Dynamical fluctuation effects in glassy colloidal suspensions

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1 Current Opinion in Colloid & Interface Science 12 (2007) Dynamical fluctuation effects in glassy colloidal suspensions Kenneth S. Schweizer Department of Materials Science and Frederick Seitz Materials Research Laboratory University of Illinois, Urbana, IL, 61801, USA Received 28 July 2007; accepted 30 July 2007 Available online 9 August 2007 Abstract Fundamental understanding of heterogeneous dynamics in concentrated glassy hard sphere fluids and colloidal suspensions, even at the single particle level, requires major theoretical advances. Recent simulations and confocal microscopy experiments suggest strong nongaussian dynamical fluctuation effects and activated transport emerge well before an apparent kinetic glass transition is reached. New theoretical approaches that can predict the observable signatures of intermittent large amplitude motions and the associated fluctuation phenomena are discussed. Comparisons are made with experiments, computer simulations, and prior theory for average dynamical properties Elsevier Ltd. All rights reserved. Keywords: Colloidal suspensions; Hard spheres; Glassy dynamics; Caging; Nongaussian fluctuations; Dynamic heterogeneity; Activated barrier hopping 1. Introduction Hard sphere fluids have long served as the basic reference system for understanding the consequences of entropy and excluded volume in thermal liquids. Suspensions of hard sphere colloids are their real world surrogates and play an important role in modern colloid and materials science. The structure, equilibrium phase behavior, and dynamics of hard spheres in the normal fluid state (defined as volume fractions below the onset of crystallization, ϕ 0.494) are well understood [1]. A simple dynamical description of the latter built on binary collisions weakly perturbed by structural packing constraints ( caging )is largely adequate [2]. However, at higher volume fractions, which is the analog of the supercooled regime for thermal liquids, particle motions slow down dramatically, viscoelasticity emerges, and a disordered solid, a glass, forms on the experimental time scale [1,3,4]. Ideal mode coupling theory (MCT) [3,5 ] is the first microscopic description of glassy hard sphere dynamics and describes well many average dynamical [3,6,7 ] and mechanical [8] properties as probed by traditional ensemble-averaged methods. Recent advances in the imaging of colloidal suspensions based on confocal microscopy enables new dynamical fluctuation processes to Tel.: ; fax: address: kschweiz@uiuc.edu. be quantified, particularly at the single particle level where good statistical sampling can be achieved [9 ]. The introduction of short range attractions results in novel dynamical arrest phenomena (gelation), and multiple aspects of the average consequences of physical bond formation at high volume fractions have been successfully predicted by ideal MCT [10 ]. One might think the problem of glassy dynamics in quiescent hard sphere fluids and suspensions is solved. I believe this is not true for the reasons that form the subject of this article. Most fundamentally, a host of dynamical fluctuation phenomena that explicitly probe the nongaussian nature of particle motions, of widely recognized importance in supercooled thermal liquids [11], are not properly described by MCT. This article discusses recent theoretical attempts to treat such effects for hard spheres focused on the conceptually simplest single particle level, which is the aspect most well established in colloid experiments and fluid simulations. Such elementary dynamical fluctuation phenomena have received little quantitative theoretical analysis. Only approaches that can predict (or have the potential to) transport properties and time correlation functions, including the nongaussian aspects, are discussed. Determining the correctness of theories requires significant input from both simulation and experiment, and recent progress on these fronts is also summarized in an integrated manner. Many aspects of the experimental phenomenology of supercooled liquids can be rationalized based on vastly different /$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi: /j.cocis

2 298 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) physical pictures. This thorny issue of degeneracy of explanation is potentially severe for suspensions composed of colloids of diameter σ 300 nm 2 μm. The reason is that the Brownian time scale, defined as s 0 ufr 2 =k B T ¼ r 2 =D 0 (where D 0 is the dilute solution Stokes Einstein diffusion constant), is typically s, roughly orders of magnitude larger the analogous time scale for atomic or molecular liquids. Since experiments generally measure dynamics only up to ,000 s, the degree of slowing down beyond the normal dense fluid region is limited to a glassy precursor regime and quantitative aspects are necessarily important. For example, the shear viscosity increases by only a factor of 100 as the volume fraction increases from 0.5 to the laboratory vitrification value of [12 ], which is far below random close packing (RCP) at ϕ RCP There also exist multiple real world complications, usually ignored in theoretical and/or simulation studies, which should be kept in mind. (1) Long range hydrodynamic interactions are the primary difference between hard sphere fluids and suspensions. Simulations that neglect them based on Newtonian, Brownian (no inertia or momentum conservation), stochastic, and even Monte Carlo dynamics all agree on the basic one and two body aspects of glassy precursor dynamics [13,14,15 17 ]. (2) Polydispersity renders difficult the accurate determination of volume fraction, increases ϕ RCP, and accelerates dynamics. (3) Gravity and sedimentation modifies the kinetic vitrification process [18,19]. (4) Deviations from the hard sphere potential exist arising from grafted molecular layers, residual charge [9 ] and/or particle deformability [20 ]. (5) Physical aging can occur associated with intrinsic dynamical time scales exceeding laboratory measurement time scales [9,19]. In Section 2 the concept of average versus fluctuation dynamical properties is discussed. Section 3 summarizes recent theoretical work, which is contrasted with computer simulations and experiments in Sections 4 and 5, respectively. The article concludes in Section 6 with a brief summary and future outlook. 2. Average and fluctuation dynamical properties and phenomena Ensemble-averaged dynamical properties fall into two broad classes : scalars and time-dependent functions. Rheological properties, except for the viscosity, will not be discussed [8]. An average property is nonzero for a strictly gaussian dynamical process. Examples are the self-diffusion constant D, the shear viscosity η, the particle mean square displacement, MSDðtÞuhð Y r ðtþ Y r ð0þþ 2 i, the single particle (incoherent) dynamic structure factor at a wavevector q, F s ðq; tþuhe i Y qð Y rðtþ Y rð0þþ i, and the normalized collective dynamic structure factor,fðq; tþusðq; tþ=sðqþ. The latter describes the time autocorrelation of total density fluctuations, dqðqþ, on a length scale 2π/q, where S(q) is the equilibrium structure factor. Length scale dependent relaxation times, τ inc (q) andτ(q), can be defined from F s (q,t) and F(q,t), respectively. The collective structural relaxation time is usually identified with the cage scale peak of S(q) at q = q 2π/σ which quantifies short range order; its single Fig. 1. Schematic of some average dynamical properties. The main panel shows the time-dependent dynamic structure factors for various volume fractions and cage scale wavevectors. A two-step decay, corresponding to a fast β and slow α process separated by a quasi-plateau, emerges at high volume fractions. The insets show: (i) log log sketch of the time-dependent mean square displacement where the regime of maximally nonfickian diffusion is characterized by an effective exponent, and (ii) log-linear plot of the rapid growth of scalar properties (viscosity, inverse diffusion constant, collective and single particle alpha relaxation times) which change functional forms (and need not all be identical) as the normal fluid regime is exited. particle analog, τ, is defined as F s ðq ; s Þue 1. Fig. 1 indicates the qualitative behavior of some average properties: (i) the emergence of a q-dependent, 2-step decay at high volume fractions corresponding to a fast and local in cage β-process and a slow cage escape α-process [3,4,5 ], (ii) strong volume fraction dependence of scalars, and (iii) intermediate time anomalous (nonfickian) diffusion, MSDðtÞ~t 4ð/Þ, characterized by the smallest, less than unity exponent, Δ [21 ]. Fluctuation properties are either exactly zero if the dynamics is a gaussian process or exhibit qualitative deviations from gaussian behavior. The relevant phenomena are schematically illustrated in Fig. 2 and include the following. (i) Decoupling of diffusive and relaxation processes as encoded in a volume fraction dependence of the product of the self-diffusion constant and shear viscosity, Dη, or its purely single particle analog Dτ. Based on a gaussian or Stokes Einstein perspective these decoupling factors should not depend on volume fraction. (ii) The traditional nongaussian parameter (NGP), a 2 ðtþuf3hr 4 ðtþi=5hr 2 ðtþi 2 g 1, is zero for a gaussian process and is a measure of dynamic heterogeneity on the late β/early α time scale when the MSDbbσ 2. Its maximum value and corresponding time scale grow in a distinctive manner with volume fraction. (iii) A recently proposed new NGP, defined as g 2 ðtþuð1=3þhr 2 ðtþih1=r 2 ðtþi 1 [27 ], preferentially weights deviations from gaussian dynamics on the slowest α -relaxation time scale. The asymmetric shape of this function, and the larger amplitude and much stronger volume fraction dependence of its characteristic time scale, contrast sharply with the more well studied NGP. (iv) For a Fickian process F s ðq; tþye q2 Dt. Nongaussian effects can be quantified at long times via a length-scale dependent diffusion constant as F s ðq; tþye q2 DðqÞt, or relaxation time as F s ðq; tþue t=sðqþ. A characteristic crossover length scale for attainment of Fickian diffusion, n D ~1=q D, corresponds to the

3 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) condition R(q) q 2 Dτ(q) 1 forqbq D. (v) The inverse Fourier transform of F s (q,t) is the van Hove function, G s (r;t), which is the probability distribution for particle displacements of absolute magnitude r at a time t. The closely related dimensionless function P(r;t) r 3 G s (r;t) weights more strongly large amplitude motions [27 ]. For a Fickian process G s (r;t) is gaussian and fully specified by the MSD. However, if the dynamics is strongly intermittent it becomes drastically nongaussian at intermediate times (e.g., exponential tails emerge), and P(r;t) attains a bimodal form signaling the emergence of distinct fast and slow populations. Dynamic heterogeneity is most fundamentally probed via multi-point time correlation functions and susceptibilities [28,29 ]. The single particle nongaussian aspects discussed above are consequences of space-time mobility fluctuations. A wavevector dependent 4-point function, χ 4 (q,t), or its real space analog, G 4 (r;t), describes the correlated dynamics of two different particles at two different times [29 ]. Equivalently, they quantify the spatial fluctuations of standard 2-body correlations such as the single particle dynamic self-correlation function. The spatially integrated G 4 (r;t) is a dynamic susceptibility, χ 4 (t). Simulations [13,28] have motivated microscopy analysis of colloid trajectories in terms of correlated string motions and mobility clustering [24 ]. Experimental investigation and microscopic calculation of multipoint dynamical objects is very challenging, and this topic is largely beyond the scope of the present article. Perhaps the most basic question with regards to fluctuation effects is whether particle trajectories are relatively smooth characteristic of a gaussian (like) process, or intermittent with large amplitude displacements characteristic of a highly non- Fickian process. The presumption of gaussian type motion underlies MCT [22,23 ] which does not account for ergodicityrestoring activated events thereby resulting in an ideal kinetic glass transition well below RCP. Both fluid simulations and suspension experiments find that at sufficiently high volume fractions (or low enough temperatures in thermal liquids) dynamics qualitatively changes to solid-like whence relaxation and flow is associated with rare stochastic jumps or hops on the particle radius length scale [13,14,15,20,24,25,26 ].These intermittent motions restore ergodicity and destroy the ideal glass transition. The single particle nongaussian effects have been observed in simulations of hard-sphere-like fluids (Newtonian and Brownian) and atomic thermal liquids, and in many cases in colloidal experiments, in the dynamical precursor regime. In my opinion a thorough understanding of activated hopping and nongaussian effects at the single particle level is a critical theoretical task. Progress for hard spheres will provide a valuable foundation for treating thermal liquids and other complex fluids. 3. Theories There exist two classes of dynamical theories of glassy hard sphere fluids and suspensions which qualitatively differ by whether direct interparticle forces or solvent-mediated long range hydrodynamic interactions are assumed to be dominant. Two fundamentally distinct subclasses exist in the former category corresponding to whether they predict, or do not predict, an ideal kinetic arrest transition below RCP. Fig. 2. Schematic of some fluctuation dynamical properties. Top panel shows the evolution with volume fraction of the classic and new nongaussian parameters (NGP), α 2 (t) and γ 2 (t), respectively. The latter has a larger amplitude, more asymmetric shape with a sharp long time decay, and peaks at a longer and more volume fraction dependent time scale than the classic NGP. The upper inset sketches in a log-linear format how the time scale of the maximum amplitudes vary relative to the mean single particle alpha relaxation time as a function of volume fraction. The lower inset indicates in a linear-linear format the decoupling of the volume fraction dependence (failure of the Stokes Einstein relation) of the self-diffusion constant and viscosity or alpha relaxation time. The lower panel illustrates the time evolution of the dimensionless probability distribution function P(r;t) as a function of logarithmic particle displacement at a high volume fraction. It changes from a unimodal form at early times, to a bifurcated function at intermediate times indicative of the presence of fast and slow subpopulations, to the diffusive Fickian form at long times. The inset shows the volume fraction dependent transition of the wavevector (length scale) dependent quantity RðqÞuq 2 Ds inc ðqþ from a constant Fickian behavior to a nongaussian, q-dependent form; the characteristic inverse length for crossover to Fickian dynamics, q c, is indicated Ideal mode coupling theory and recent extensions The pioneering and most widely applied approach is the wavevector space based ideal MCT built on the idea that all aspects of glassy dynamics are controlled by collective density fluctuations on the cage scale [3,5 ]. The theory approximately relates structure (via S(q)), forces and collective density fluctuations via a nonlocal in time equation-of-motion for S(q,t) that encodes a nonlinear feedback mechanism : dfðq; tþ dt þ q2 D s Fðq; tþþ SðqÞ Z t 0 dt V Mðq; t t V Þ dfðq; t V Þ dt V ¼ 0 ð1þ where F(q,t) S(q,t)/S(q) and D s is a short time diffusion constant. Many particle caging is contained in the memory

4 300 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) function, M, which is a spatially-resolved generalized stress autocorrelation function. This complex object is approximated as a nonlinear functional of F(q,t), fully quantified by S(q), via a gaussian closure (dynamical mean field) approximation [3,22,23 ]. Specifically, the three and four point time correlation functions that enter the memory function are expressed ( factorized ) as products of S(q,t). Single particle dynamics is then self-consistently determined using the collective dynamic structure factor as input [30]. Eq. (1) is a closed, self-consistent equation for S(q,t) and has a rich mathematical structure [3,5 ]. Its solution exhibits a type of bifurcation singularity corresponding to a literal kinetic arrest ( ideal glass formation). For hard spheres the singularity occurs at a critical volume fraction ϕ c based on using Percus Yevick theory for S(q). In applications to experiment and simulation ϕ c is treated as a fit parameter via the specification of the separation parameter ε (ϕ c ϕ)/ϕ c.as the kinetic singularity is approached a two-step decay of F(q,t) and F s (q,t) emerges involving fast (β) and slow (α) processes which become increasingly separated by a quasi-plateau as ε 0 +. MCT makes many detailed and intriguing predictions including intricate inter-relationships between the β-process and the early stages of the α-relaxation [3,4,6 ]. All slow dynamics are slaved to collective density fluctuations, and all terminal time scales and transport coefficients diverge in an identical critical power law manner. For example, τ τ(q) η D 1 (ϕ c ϕ) v, where for hard spheres ν 2.6 and fits to suspension experiments typically yield ϕ c Dynamic light scattering studies [4,6 ] of S(q,t) for qσ 4 8 and volume fractions up can be consistently interpreted by MCT. A major difficulty is the prediction of a divergence well below RCP which is an artefact of the MCT factorization approximation due to its neglect of highly nongaussian activated processes. Perhaps surprisingly, strong signatures of the latter are observed in simulations [13,14,15, 16,27,31,32] and confocal experiments [9,20,24, 25,26 ] in precisely the volume fraction (or temperature for liquids [13,33]) regime where MCT can describe many average properties well. This represents a major puzzle. The emergence of activated processes and nongaussian fluctuation effects are believed by many, including the present author, to be intimately related. The gaussian nature of MCT implies it predicts little or no dynamical fluctuation effects. For example, small NGP's of a maximum amplitude which saturates at 0.3 as ε 0 [30,33,34 ], tiny degrees of decoupling ( 10 20% level) [34,35 ], no bifurcation of the particle displacement distribution P(r;t) or exponential tails of the van Hove function [34,36 ], and nearly Fickian behavior of single particle relaxation times τ inc (q) q 2 [34,35 ]. There have been several recent fundamentally-based efforts to improve MCT while retaining the ideal glass scenario. The first, fully microscopic attempt delayed the memory function gaussian factorization approximation by one level in a continued fraction spirit [37 ]. The primary consequence is a significant increase of the predicted ϕ c, but it appears all qualitative features of the original MCT remain the same. A field theoretic formulation of MCT has been developed [38,39 ] largely within a so-called schematic model framework [3 ] where all q-dependence and microscopic connections to S(q) are sacrificed for mathematical and computational simplicity. At the schematic level there are no uniquely specified formulas for the memory function and empirical parameters enter. The technical advance is the approximate retention of longer wavelength critical-like fluctuations in the spirit of harmonic-like corrections to the traditional mean field ideal MCT. This new inhomogeneous MCT suggests a more spatially-extended view of cages and predicts (weakly) diverging correlated cluster length scales as the ideal glass transition is approached [38 ]. Since ideal MCT is a mean field theory a diverging length scale cannot be rigorously defined. However, a diverging 4-point dynamical susceptibility is predicted even within the mean field approximation which is taken as the signature of a diverging correlation length. Inhomogeneous MCT makes novel predictions for multi-point functions which allows identification of such a growing dynamical length scale, number of correlated particles, and quantification of the q-dependent amplitude of mobility fluctuations, as the ideal glass transition is approached. Decoupling of the form Dη ε v associated with a distribution of relaxation times is also predicted [39 ]. However, the applicable volume fraction range, quantitative magnitude, and degree of exponent universality is unclear. Activated barrier hopping physics is not addressed, and no microscopic quantitative results for the hard sphere nongaussian dynamical fluctuation effects of primary concern in this article have been obtained Beyond MCT approaches and activated processes The original attempts to include ergodicity restoring events in MCT has many variants [3,5 ] all based on some kind of low order nonlinear coupling of density fluctuations to kinetic degrees of freedom (currents). These extended-mct approaches have phenomenological elements that strongly restrict their predictive power. Recently it has been argued they are qualitatively incorrect [40,41 ] since the kinetic couplings do not describe activated hopping processes and are negligible corrections which do not cutoff the ideal glass transition. The most recent variant includes a defect slow mode and average (not fluctuation) properties of hard sphere fluids have been computed [42]. Several distinct direct force based theories that do not have MCT-like kinetic singularities have been recently proposed. They all acknowledge the usefulness of ideal MCT but do not employ coupling to density currents as an ergodicity-restoring mechanism. Only one of these approaches [43,44 ] is implemented at a microscopic level, employs no adjustable (fit) parameters, has no kinetic singularities below RCP, has quantitatively computed average and fluctuation properties of hard sphere fluids, and has been confronted in a detailed manner with computer simulations and colloid experiments. For this reason it is discussed first and in considerable depth. The nonlinear Langevin equation (NLE) approach [43,44 ] is formulated in real space and builds on early insights concerning

5 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) connections between MCT and a density functional description of a glass as an aperiodic solid [45]. It adopts the more limited goal of describing only single particle dynamics and quantifies caging constraints via S(q). These simplifications render tractable a microscopic and predictive treatment of barriers and activated transport. The theory has been derived using time-dependent density functional and local equilibrium ideas at a lightly coarsegrained dynamical variable, not traditional ensemble-averaged, level. Its relationship to Kramers' theory and heavily coarsegrained descriptions of the slow dynamics of a nonconserved order parameter have been established [44 ]. The essence of the NLE theory is a stochastic, nonlinear, time local equation-ofmotion for the (scalar) displacement of a tagged particle from its initial location, r(t). In the overdamped limit (inertia neglected) one has : f s ArðtÞ At ¼ AF eff ðrðtþþ þ df ðtþ ArðtÞ Here, ζ s is a short time friction constant, δf(t)the corresponding white noise random force, and F eff (r) is a nonequilibrium free energy which contains competing contributions that favor delocalization (fluid state) and localization due to (ϕ-dependent) caging (see Fig. 3). At long times Eq. (1) is replaced by a simple 3- dimensional linear Langevin equation with a renormalized friction constant that incorporates the barrier escape process [44 ]. The simplified (so-called naïve [43,45]) MCT nonergodicity transition, which occurs at ϕ c 0.432, is replaced by a smooth dynamical crossover. Specifically, at ϕ c the nonequilibrium free energy changes from a monotonically decaying form with its attendant smooth fluid-like (Brownian) particle trajectories, to a new activated dynamics form where F eff (r) is characterized by a transient localization well and entropic barrier. Physically, particles become localized for long periods of time until rare noise-driven fluctuations trigger diffusive transport over the barrier which is identified with the alpha relaxation process. Naïve MCT is recovered if a dynamical gaussian approximation for the ensemble-averaged MSD is made, or the noise in Eq. (2) is dropped. A similar in spirit, but extended, approach that treats some aspects of two particle dynamics and confined hard spheres has been very recently developed [46]. The NLE theory, in conjunction with a Green Kubo based treatment of transport coefficients, describes well the mean alpha relaxation time, self-diffusion constant and shear viscosity data of equilibrated colloidal suspensions up to the highest volume fractions studied, ϕ= [43,47 ]. The physical picture is activated transport over low or modest entropic barriers (F B 1 7 k B T), which is also the origin of the strong nongaussian effects. The quantitative predictions are likely sensitive to the theoretical and model simplifications, but these issues have not been systematically investigated. Despite its nonsingular nature below RCP, over the experimentally studied volume fraction range the quasi-analytic scalar property calculations can be empirically fit by adjusting the location of the divergence in the MCT critical power law, or the essential singularity of the free volume or entropy crisis (Adam Gibbs) models [43,47 ]. ð2þ Fig. 3. Nonequilibrium free energy (units of the thermal energy) of the nonlinear Langevin equation theory as a function of dimensionless particle displacement [43 ]. The top curve is monotonically decaying and is for a volume fraction of 0.4 which is below the dynamic crossover or naïve MCT ideal glass transition. The other three curves correspond to higher volume fractions (ϕ=0.50,0.54,0.58) where entropic barriers exist. The inset is a cartoon of trajectories below, and well above, the dynamical crossover corresponding to continuous random-walk-like motion, and intermittent or abrupt hopping type transport, respectively. Single particle Brownian trajectory simulation has been employed to fully solve Eq. (2) over the range ϕ= [48,49 ]. A host of inter-related and strong non-mct dynamical fluctuation effects are predicted due to the intermittent nature of particle trajectories and the attendant broad distribution of hopping times. The nongaussian effects are of a purely dynamic origin, not induced by static structural fluctuations nor a probabilistic nonequilibrium free energy. Highlights include the following. (1) The nonfickian exponent Δ decreases roughly linearly above ϕ 0.5 (where F B k B T). If the calculations above ϕ = 0.57 are empirically extrapolated (as often done in analyzing measurements or simulations) the exponent appears to vanish at ϕ c This could be interpreted as signaling an ideal glass transition, and the theoretically deduced value of ϕ c agrees with experimental analysis [6,7 ] and MCT-inspired fits of the NLE theory scalar property predictions [43,47 ]. However, this ideal glass state is not physical within the nonlinear Langevin equation theory since Δ remains nonzero below RCP. (2) Decoupling of Dτ emerges for ϕn0.5 and grows to a factor of 15 at ϕ=0.57. For the highest ϕ range a fractional Stokes Einstein law emerges, D (1/τ ) (3) The length scale dependence of decoupling is dramatic and is well described by a jump diffusion model [50 ] corresponding to R(q) q 2 Dτ(q) 1+(qξ D /2π) 2, where the Fickian crossover length scale ξ D increases linearly from 1 to 3 particle diameters for ϕ= The crossover p length is related to the decoupling factor as n D ~ ffiffiffiffiffiffiffiffi Ds T, a result also analytically derived using a general statistical mechanical analysis [50 ] and deduced from kinetically constrained models [51 ]. Specifically,

6 302 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) ξ D grows as a remarkably weak function of the mean relaxation time which can be represented as either a logarithmic form, or as ξ D (τ ) in the high ϕ regime [50 ]. (4) All the volume fraction and wavevector (qσ ) dependences of X(q) τ(q)d/(τ(q)d) 0, where (τ(q)d) 0 is the normal fluid value, collapse onto a dynamicalscaling-like form X=1+c(qξ D ) v, where c is a constant and v =1.7±0.2. (5) The NGP maximum amplitude (at a time t α )is 0.3 at the naïve MCT transition, in agreement with its full ideal MCT analog [30]. However, with increasing volume fraction it grows strongly attaining much larger values, e.g. 8atϕ=0.55. (6) The alternative NGP peaks at t γ 2τ and is significantly larger than the classic NGP. Nevertheless, a (untested) prediction is that it is proportional to its classic analog thereby suggesting a deep connection between nongaussian effects on the late β and final α time scales. The ratio of peak time scales, t γ /t α, grows strongly with volume fraction, from 1 in the normal fluid regime to 100 at ϕ This emphasizes the different space time regimes of nongaussian dynamics probed by these functions. (7) As the barrier increases the real space probability distribution P(r;t) undergoes a transition from a gaussian form in the normal fluid regime to an increasingly bimodal form at intermediate times. The degree of bimodality is a maximum at a time very close to τ. The signature of large amplitude hopping motion in G s (r;t) is an exponentially decaying tail with an amplitude and range that grows with increasing volume fraction. The NLE theory also makes several inaccurate predictions [49 ]. For example, the 2-step decay aspect of F s (q,t) is too weak suggesting that the theory tends to quantitatively overestimate transient localization. The temporal form of the alpha relaxation is only slightly perturbed from an exponential decay with no clear wavevector trend of the apparent stretching exponent. However, overall the NLE approach seems rather consistent with ideal MCT for many average properties in the experimentally accessible volume fraction regime while simultaneously predicting enormously larger nongaussian dynamical effects. In this sense it may provide insight to the puzzle of why ideal MCT works well for average properties but not nongaussian fluctuation effects. The other recently proposed beyond MCT theories [40,52 ] are in early stages of development, and are presently at a schematic model level and/or have not been quantitatively applied to hard spheres for fluctuation quantities [40,52,53,54 ].A fully nonperturbative approach has been proposed within a time correlation function and schematic model framework [52 ]. Many particle dynamic correlation effects through infinite order are approximately summed thereby formally accounting for ergodicity restoring events. Most of the β-process and intermediate time predictions of ideal MCT survive, including time and wavevector scalings, thereby demonstrating a degree of robustness of the MCT factorization approximation. However, the critical power law divergence of all long time relaxation and transport processes are destroyed and replaced by a continuous but exponential growth. Although mathematically complex, this is a fundamental approach that holds much promise if it is worked out at a microscopic level. An alternative schematic model avoids the MCT singularity by employing an ad hoc (but physically motivated by hopping physics arguments) unconventional closure approximation for relating the memory function to S(q,t) [40 ]. A qualitative discussion of its relationship to colloid DLS experiments, and how some features of MCT may survive the destruction of the singularity, has been presented. The random first order phase transition theory has been combined [53,54 ] with ideal MCT. A crossover to transport based on an entropic droplet activated dynamics picture cuts off the ideal glass singularity, and calculations of S(q,t) at a schematic model level have been performed [54 ]. An older dynamic density functional approach based on an equilibrium free energy includes activated hopping at a somewhat abstract collective density field level [55]. Its numerical complexity has limited the results obtained to date. Additional theoretical approaches exist [51,54,56 58] which either treat hard spheres but do not compute dynamics [56], or have not been quantitatively applied to hard sphere fluids [51,54,57]. A mixed mode coupling and density functional approach has addressed some aspects of tagged particle motion, but the results appear similar to ideal MCT [58] Long range hydrodynamic interactions A qualitatively different approach argues that long range hydrodynamic interactions (HI), not direct interparticle forces, play the critical role for glassy colloid dynamics [59]. A fundamental distinction between fluids and suspensions is thus emphasized, and the assumption long range HI are screened is not made. A kinetic glass transition is predicted as defined by a vanishing self-diffusion constant, D (ϕ 0 ϕ) 2 where ϕ 0 = A simplified version of the theory has been proposed, and arguments advanced for the destruction of the ideal glass transition due to polydispersity effects [60,61]. Recent quantitative applications have been largely focused on the mean square displacement, M 2 (t). Based on an assumed Gaussian form for F s (q,t), the MSD obeys the equation-ofmotion: dm 2 (t)/dt=6d+6(d s D)e λ(ϕ)m 2(t), where D(D s ) is the volume fraction dependent long (short) time diffusion constant and λ(ϕ) is an adjustable (fit) function. All nongaussian effects are assumed to be small [60 ]. Quantitative calculations and fits agree well with experiments and simulations, although not obviously better than a recent ideal MCT analysis of colloidal MSD data [62]. 4. Computer simulations Testing theories which ignore long range HI can, in principle, be most directly made by comparison with simulations. Many studies have been recently performed for moderately polydisperse spheres that interact via a steep hard-sphere-like repulsion [31,35,36,63 65 ]. As mentioned in the

7 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) Introduction, a variety of dynamical laws are employed with small differences found for the slow dynamics aspects, a statement also true for the popular thermal Lennard Jones binary mixture model (LJBM) [15 17 ]. Inertia and momentum conservation are not fundamentally relevant to glassy dynamics, consistent with a recent theoretical analysis [41 ]. Quantitative testing of theories are sensitive to simulation details, although the stronger limitation may be that only 3 4 orders of magnitude of slow relaxation are probed. Landscape inspired analyses [32,66, 67,68 ] find hopping transport emerges at an onset temperature (T 0 NT c )orvolumefraction(ϕ 0 bϕ c )wellbefore the empirically-deduced MCT transition. Successful interpretation of diffusion and relaxation simulation data in terms of the phenomenological activated trap model support this view [32,66,67 ]. Associated with the crossover is the emergence of all signatures of single particle heterogeneous dynamics Hard-sphere-like simulations Half a dozen research groups [31,35,36,63 65 ] have recently carried out simulations of polydisperse hard-sphere-like (PHS) systems in volume fraction ranges corresponding to the glassy suspension regime. Almost all find large NGP's, mobility bifurcation, and strong decoupling which rapidly grows with increasing volume fraction. These features are in contrast with ideal MCT but are in good qualitative, and reasonable quantitative, accord with the nonlinear Langevin equation approach [48,49 ]. Large diffusion-relaxation decoupling effects, attributed to hopping-driven intermittent motion and mobility bifurcation, have been documented via the ratio Dτ /(Dτ ) 0, where (Dτ ) 0 is the normal fluid value [31 ]. This ratio begins to significantly deviate from unity at ϕ 0.56 and rapidly increases by over an order of magnitude ( 16) at ϕ = Mobility bifurcation has been vividly demonstrated at ϕ = 0.55 for a nearly hard-spherelike system [36 ]. Specifically, two well separated peaks of the probability distribution function P(r;t) emerge corresponding to fast and slow subpopulations. The most recent simulation found diffusion viscosity decoupling starts at ϕ 0.54 and is a factor of 20 at ϕ [65 ]. The self-diffusion constant decreases with volume fraction by an amount consistent with suspension experiments which challenges arguments [59,61] that long range HI are dominant. Simulations find large nongaussian parameters of maximum amplitude 3 5 for ϕ [36,64].An excellent linear correlation between (t α α 2 (t α )) 1/4 and ϕ has been discovered [31 ], which is consistent with prior suggestions [69] and quantitatively predicted by the NLE theory [48 ]. The consensus of recent PHS simulation studies is that strong single particle nongaussian fluctuation effects exist at the high volume fractions relevant to glassy suspension experiments. One comprehensive study did find significantly smaller nongaussian effects and good agreement with ideal MCT for many average properties [35 ]. The alpha relaxation time follows a τ inc (q) q 2 Fickian law for the relatively high wavevectors studied (qσ 4). Nongaussian effects emerge at the highest volume fractions with some indication of mobility bifurcation and modest (factor 3 4) decoupling Thermal binary mixture simulations Many LJBM simulation studies have been recently performed based on various dynamical laws [16,17,32 34,67,70,71]. Within the MCT interpretive framework, the separation parameter is ε T (T T c )/T c,wheret c is an empirically extracted ideal glass transition temperature. By comparing PHS and LJBM simulations based on ε T and ε (ϕ c ϕ)/ϕ c a consistent overall picture seems to emerge. The collective alpha relaxation time, its single particle analog, and stress relaxation time all behave very similarly, suggesting a common underlying relaxation process [33]. Strong decoupling of the temperature dependences of the single particle relaxation time and self-diffusion constant occurs, D 1/τ z and z 0.75 [33,70], withdeviationsfromstokes Einstein behavior of roughly an order of magnitude at T c. The maximum NGP amplitude grows strongly with cooling below the onset temperature achieving values of 3 5 at or near T c, and further increases to 10 at 0.9T c [71]. These values are an order of magnitude or more larger than their normal fluid and MCT maximum analogs. Calculations of the new NGP, γ 2 (t), reveal a peak amplitude at t γ 2τ that is 3 4 times larger than its classic NGP analog [27 ]. The time t γ strongly separates from its classic NGP analog with cooling (factor at T c ). At lower temperatures subpopulations emerge as indicated by a bimodal P(r;t) that is maximally bifurcated at roughly t γ, but are nonexistent (unimodal) on the classic maximum NGP time scale [27,70]. A dynamic correlation length, ξ, has been extracted over a wide range of temperatures and wavevectors [72 ] and grows from 0.4 to 3 particle diameters as temperature is lowered from T 0 to T c. A universal form for the space time nongaussian behavior emerges given by R(q) q 2 Dτ(q); 1+c (qξ) x,wherec is a constant and x 1.7. Although established for the thermal LJBM model, essentially identical dynamic scaling behavior is predicted by the hard sphere NLE theory [48 ]. 5. Colloid suspension experiments 5.1. Transport coefficients and incoherent dynamic structure factor The self-diffusion constant has been measured using confocal microscopy [20 ] and incoherent DLS [21 ] up to ϕ where D/D 0 decreases to The latter value corresponds to a mobility reduction of 3 4 orders of magnitude in the glassy regime (ϕn0.5). The high volume fraction diffusion data is accurately predicted by the NLE theory [47 ]. It can also be well fit via adjustment of ϕ c based on MCT [21 ], HI theory [20 ] and the free volume model [47 ]. An analysis of shear viscosity measurements [12 ] up to ϕ=0.562 reaches similar conclusions. Early DLS studies found negligible decoupling in the normal fluid region [73]. However, this issue has not been definitively addressed in the glassy regime since measurements of selfdiffusion and viscosity by the same group using the same materials do not exist. Arguments have been advanced for significant decoupling in suspensions based on frequencydependent measurements [74], but do not seem definitive for the question of transport coefficient decoupling.

8 304 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) By employing a clever binary mixture formulation the incoherent dynamic structure factor has been studied [7 ].The volume fraction dependences of the collective and single particle alpha relaxation times, and D, were found to be identical to within experimental uncertainties. This implies negligible decoupling and gaussian behavior up to ϕ Moreover, the volume fraction dependence is well fit by the MCTcritical power law with an exponent very close to that a priori predicted. A simple Gaussian F s (q,t)=exp( q 2 MSD(t)/6) was measured corresponding to τ(q) q 2 The nonfickian aspects were recently quantified via the maximally anomalous diffusion exponent, i.e. MSDðtÞ~t 4ð/Þ [21 ]. The effective exponent varies smoothly with volume fraction as Δ 0.5 at ϕ=0.494, 1/3 at ϕ=0.545,and b 0.1 at ϕ=0.57, which is consistent with the HI [60,61], MCT [62] and NLE approaches [49 ]. The overall implication of this experimental study is that nongaussian fluctuation effects are tiny, in contrast to simulations and some theories, and the confocal microscopy experiments now discussed Confocal microscopy Recent visualization of colloid trajectories at high volume fractions find dramatic nongaussian behavior characterized by strong intermittency and hopping type motions on the (sub) particle length scale [9,20,24,25,26 ]. Significant residual motion beyond the putative glass transition has been reported, and the suggestion it represents an ultra-slow activated alpha relaxation was advanced [20 ]. The nongaussian parameters are large, with amplitudes in the range of 2 5 for ϕ [20, ,75 ], in reasonable accord with simulations and the nonlinear Langevin equation theory [48 ]. The microgel experiments [20 ] find the NGP amplitude increases strongly from b 0.5 at ϕ=0.48, to 5atϕ=0.56. The time scale for the NGP maximum, t α, occurs near the end of the quasi-plateau region of the MSD, i.e. the late β/early α process stage [26 ], and grows weakly with volume fraction. Confocal measurements of the MSD and nonfickian exponent Δ for microgels [20 ] and polymethylmethacrylate colloids [26 ] are qualitatively similar to each other and the DLS experiment [21 ]. One measurement of the new NGP, γ 2 (t), has been reported at ϕ=0.6[75 ]. In agreement with simulation [27 ] and the NLE theory [48 ], its shape is qualitatively different than the classic NGP, the peak occurs at a much larger time scale (factor 20), and its amplitude is significantly larger. Measurements of the van Hove distribution reveal strong nongaussian behavior [20,24,25,26 ], including the existence of fast and slow subpopulations as observed in simulations [13,34,36 ] and predicted by the NLE theory [48 ]. Confocal microscopy has also been utilized to estimate a correlation length, ξ, describing cooperative particle displacements [76 ]. A relatively weak increase of ξ from roughly 1.5 to 3 particle diameters was observed as the volume fraction increases from 0.46 to Curiously, the behavior of this dynamical correlation length appears to be qualitatively similar to the Fickian crossover length scale determined from the thermal LJBM simulations [72 ] and the NLE theory analysis of F s (q,t) for hard spheres [48 ]. 6. Conclusions and outlook In the absence of long range hydrodynamic interactions numerous simulations and the microscopic nonlinear Langevin equation theory [48,49 ] clearly find that strong single particle nongaussian dynamical fluctuations effects and activated hopping transport emerge well below the empirical MCT transition volume fraction deduced from fitting. The question can be raised as to whether many body hydrodynamic interactions could somehow quench nongaussian fluctuation effects in suspensions and mitigate activated processes. The fact that simulations (and theory) which neglect long range HI can reproduce the volume fraction dependences of the experimental viscosity and self-diffusion constant, the confocal microscopy observations, and the harmony between Newtonian, Brownian and stochastic simulations for single particle average and fluctuation properties, argue against such a speculation. New experiments that quantitatively establish the behavior of the classic and new nongaussian parameter (including their possible inter-relationships), decoupling of various scalar and wavevector dependent quantities, the real space displacement probability distribution function, and F s (q,t) over a wide q- range, all as systematic functions of volume fraction, are highly desirable. The HI [59] and new generalized MCT based theories [37,40,52,54 ] need to be quantitatively and microscopically implemented to establish their predictions for hard sphere nongaussian fluctuation effects. After this article was completed a new simulation analysis of computer models and experiments (including hard sphere suspensions), and a generic phenomenological model for the van Hove function, appeared [77 ].It emphasizes the universal nature of particle displacements, coexistence of slow and fast particles, intermittent hopping, and the emergence of a nongaussian exponential tail of G s (r;t) well below (above) the empirically-deduced ideal glass transition volume fraction (temperature). Ideal MCT is a documented success for many average dynamical properties of hard sphere suspensions in the observable precursor regime [3,4,5 ]. However, its predictions are closely tied to the underlying ideal glass singularity. Hence, there remains the subtle issue of whether a theory built on a kinetic divergence (well below RCP) that is destroyed by fluctuations can fully describe average dynamics based on an avoided singularity mechanism. A primary theoretical goal for the near future should be a full synthesis of MCT with recent efforts to go beyond it in order to achieve a quantitative and microscopic description of average and fluctuation aspects of single particle and collective pair dynamics. This advance will better elucidate the connections between ergodicity-restoring barrier hopping motions and dynamic heterogeneity phenomena. Growing theoretical and experimental interest in the nonlinear rheology of hard sphere suspensions may also shed more light on the role of activated processes. Emerging experimental [78,79,80] and simulation studies [29 ] of multi-point dynamical correlations, ultra-slow relaxations and physical aging of glassy fluids and suspensions will also provide new insights concerning the space time aspects of mobility fluctuations that will deepen understanding and further

9 K.S. Schweizer / Current Opinion in Colloid & Interface Science 12 (2007) challenge theory. However, experimental measurement of 4- point dynamical correlation functions is very difficult. Moreover, a conceptual complication is they depend on both the thermodynamic ensemble and the type of dynamics [29 ], and hence fundamental differences between colloidal suspensions and thermal liquids is anticipated. These difficulties have motivated the study of 3-point dynamical correlation functions which are experimentally accessible and can be related to the 4- point susceptibilities via a rigorous inequality [29,78 ]. Whether dynamic heterogeneity effects in hard sphere fluids and nonthermal granular materials are related is also a fascinating question, especially in light of the recent discovery of surprising connections between them [77,81]. Finally, the fluctuation phenomena discussed in this article are even richer for colloids, nanoparticles and proteins that interact via short range attractions and form gels and attractive glasses [10 ]. New structural mechanisms for dynamic heterogeneity emerge, and very large single particle nongaussian effects have been observed in both experiment [79,82,83 ] and simulation [36,84 ]. Qualitative differences of the multi-point dynamical susceptibilities of hard and sticky spheres have begun to be documented [85]. Such dramatically heterogeneous dynamics, including the presumably coupled activated processes of physical bond breakage and cage escape, remain essentially unaddressed by theory and represent a major challenge for the future. Acknowledgements I acknowledge support for my work in the area of glassy colloidal systems from the National Science Foundation via DMR and the U.S. Department of Energy, Division of Materials Sciences via DEFG02-91ER I am grateful to G. Yatsenko, K. 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