Nonlinear glassy rheology

COCIS-00938; No of Pages 12 Current Opinion in Colloid & Interface Science xxx (2014) xxx–xxx

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Nonlinear glassy rheology Thomas Voigtmann ⁎ Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany Department of Physics, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany

a r t i c l e

i n f o

Article history: Received 15 October 2014 Received in revised form 31 October 2014 Accepted 1 November 2014 Available online xxxx Keywords: Rheology Glass transition Amorphous solids Plastic deformation and flow

a b s t r a c t Dense colloidal suspensions and related glass-forming fluids show pronounced nonlinear (non-Newtonian) flow effects, since slow internal relaxation competes with the imposed external driving. The microscopic processes governing the macroscopic material behavior are nonlocal both in time and space, encoding the divergence of viscosity in the fluid, and the rigidity of the amorphous solid that forms at the glass transition. Current theoretical approaches starting from either extreme, the homogeneous fluid flow and the spatially heterogeneous amorphous solid, are reviewed. © 2014 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-SA license (http://creativecommons.org/licenses/by-nc-sa/4.0/).

1. Introduction The dynamics of dense fluids – crystallization and other phase transitions aside – is governed by slow collective structural relaxation processes. Their time scale τ can be many orders of magnitude larger than the time scale τ0 associated with free-particle motion, and so large that fluid-like relaxation of density fluctuations ceases for all practical purposes. The system then turns from a liquid to an amorphous solid, the glass. Rheology – the study of flow and deformation of matter under external forces, traditionally on macroscopic scales – of dense fluids is a broad, growing research area [1,2]. It includes soft-matter colloidal suspensions, polymer melts, gels, foams, and granular matter, but also hard-matter bulk metallic glasses (BMG) [3]. Applications range from the handling of paints, coatings, pulp suspensions, concrete, mud and lava flows, the production and sensatory perception of food, biophysical matter [4], to the development of high-performance alloys. Slow thermally driven dynamics causes visco-elasticity (as noted initially by Maxwell): on observation times t ≪ τ, the liquid structure appears essentially frozen, with (ideally elastic) solid-like response. Only for t ≫ τ, fluid-like viscous dissipation emerges. Imposed flow of rate γ interrupts the equilibrium relaxation on a time scale τ γ ∼1= γ . The relevant dimensionless group is the Péclet (or Weissenberg) number Pe ¼γ τ , not the bare Péclet number Pe0 ¼γ τ0 . If Pe ≫ 1, nonlinear-response phenomena render the fluid's rheology quite different from that of a simple Newtonian fluid — it becomes non-Newtonian.

⁎ Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany. Tel.: +49 2203 601 3846. E-mail addresses: [email protected], [email protected].

In the extreme case, τ → ∞, arbitrarily slow flow is a highly nonlinear phenomenon related to the yielding of the amorphous solid (viscoplastic flow of a yield-stress material). The regime Pe ≫ 1 ≫ Pe0 close to the glass transition is that of nonlinear glassy rheology. Here the microscopic processes that govern macroscopic flow and deformation, share both fluid-like and amorphous-solid-like aspects. This is reflected in different theoretical approaches (Fig. 1): microscopic theory of soft materials (with moduli from Pa to kPa) starts from homogeneous flow and nonlinear effects that arise due to temporal-history effects. Approaches rooted in plasticity theory (for hard materials with moduli in the GPa range, and increasingly transferred to soft materials) take the frozen amorphous solid as a reference state and emphasize aspects of mesoscale heterogeneities and their long-range elasticity-mediated spatial correlations. The challenge remains to merge these approaches to describe the fully spatially and temporally nonlocal rheology on a wide range of scales. Recent reviews by Brader [5] and Barrat and Lemaître [6] address colloidal-fluid respectively amorphous-solid rheology. I aim to provide a bird's eye view of both realms, and possible connections between them. This also concerns the connection between the glassy rheology of soft and hard materials. Colloidal systems allow to access the microscopic dynamics under flow by combining macroscopic stress/strain measurements with particle-resolved experiments (confocal imaging “rheoscopy”) [7–10], or scattering techniques [11,12]. We focus here on macroscopic rheology in dense colloidal (hard-sphere-like) and related amorphous systems, leaving aside micro-rheology and a large body of literature dealing with complex fluids, time-dependent (thixotropic) rheology, the rheology of polymers or granular matter, and many more (see, e.g., Refs. [13–17]). To keep the review within reasonable bounds, a rather subjective choice of topics (and references) has been made.

http://dx.doi.org/10.1016/j.cocis.2014.11.001 1359-0294/© 2014 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-SA license (http://creativecommons.org/licenses/by-nc-sa/4.0/).

Please cite this article as: Voigtmann T, Nonlinear glassy rheology, Curr Opin Colloid Interface Sci (2014), http://dx.doi.org/10.1016/ j.cocis.2014.11.001

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T. Voigtmann / Current Opinion in Colloid & Interface Science xxx (2014) xxx–xxx

Fig. 1. Schematic regimes of glassy rheology in terms of temperature (or inverse density) and length scale addressed, from microscopic dynamics to continuum mechanics. The homogeneous flow regime is treated in microscopic approaches such as mode-coupling theory (MCT), while for the amorphous solid, descriptions in terms of shear transformation zones (STZ), local plastic events, and their mesoscale elasto-plastic coupling are relevant.

2. The rigidity transition: glass vs. jamming The dynamical shear modulus G(t) and its Fourier transform via the frequency-dependent storage and loss moduli, display the Maxwelltype intermediate-time rigidity as a distinct signature of glassy dynamics [18]. In linear-response, G(t) stays close to a plateau G∞ for τ0 ≪ t ≪ τ, before decaying to zero. In the idealized glass (τ = ∞), limt → ∞G(t) = G∞ sets a finite low-frequency modulus as the system becomes nonergodic. The resulting jump in the low-frequency shear modulus from zero to G∞ is predicted by mode-coupling theory of the glass transition (MCT) [19] (caused by dynamical arrest of density fluctuations), and by the replica theory of structural glasses [20,21] (as a broken continuous symmetry of shifting one replica of the system with respect to the others). The thermal energy density sets the scale, G∞ = O(20 kT/d3), in hardsphere-like glasses (particle diameter d; thermal energy kT). The glass transition marks the divergence of the linear response (zero-shear) viscosity η. For hard-sphere suspensions with packing fraction φ = (π/6)ϱd3 (number density ϱ), this occurs at φc ≈ 0.58 [22], rather than at the random-close packing limit φrcp ≈ 0.64. The emergence of rigidity at the jamming transition connected with the latter (probed in granular matter or non-Brownian suspensions of deformable particles so that φ N φrcp can be reached) is profoundly different: here, the shear modulus rises as a power law from zero, interpreted as critical behavior from the percolation of a force network and the notion of marginal mechanical stability of the packing [23–26]. In the amorphous solid, one defines elastic moduli as derivatives of the free energy with respect to displacements around some reference position. Yet, the microscopic definition of a displacement field is not straightforward. Using suitably time-averaged reference positions, the jump in G(∞) at the glass transition re-emerges from such an elasticity-theory approach [27]. To fully reconcile the inherently dynamic and static perspectives in the viscoelastic liquid respectively amorphous solid case, remains a topic of debate [28]. Non-affine displacements are important in amorphous solids. They significantly reduce the elastic moduli compared to the affine Born

contribution familiar from crystals [28–31] and depend on the quench procedure used to arrive at the amorphous solid [32]. Continuum elasticity and isotropy break down below a length scale where elastic heterogeneities emerge [33,34]. Mesoscopic elastic moduli Clocal, defined by σlocal = σlocal + Clocal ⋅ ϵlocal (with the local-strain vector ϵlocal in 0 Voigt notation), follow broad distributions linked to specific vibrational properties of the glass [35,36]. They are much broader than in crystals; nano-indentation atomic-force acoustic microscopy (AFAM) measurements on metallic glasses support this [37]. The term σlocal is non-zero 0 in general [34]; the related statistics of stresses of inherent structures has been linked to G∞ [38]. 3. Steady-state rheology: flow curves The difference between glass and jamming transitions is borne out by the nonlinear flow curves, i.e., the maps of shear stress σ as a function of an applied constant shear rate γ under steady-state conditions (Fig. 2). Experiments and simulations close to the colloidal glass transition [39,41], and experiments on bulk metallic glass formers and silicate melts all show a similar signature for Pe0 ≪ 1. A linear-response regime, σ∝ γ implying a constant (Newtonian) viscosity, is found for Pe ≪ 1; this regime shifts to lower shear rates upon approaching the glass transition. In the glass (upper curve in the left panel of Fig. 2), a plateau prevails in the σ –versus– γ curve down to the lowest accessible shear rate. In theoretical idealization, limγ→0 σ γ ¼ σ y N0 marks a dynamical yield stress. Any small imposed flow shear-melts the glass, but at an energydensity cost of at least σy. Reminiscent of the behavior of the shear modulus, the value σ(0) of the flow curve jumps discontinuously from zero in the liquid to a finite value σcy N 0 at the glass transition. In fact, one finds σy/G∞ ≈ 1/10 for the data shown in Fig. 2. Thus, σy = O(kT/d3) for hard spheres; the effect of strong (short-range) attractions has more recently been discussed [42–45]. The yield-stress plateau implies shear thinning, η ¼ σ= γ ∼1= γ as the predominant non-Newtonian flow effect. Although suggestive, shear thinning is not related to “laning” of particles in the flow direction (which only appears for Pe0 ≈ 1 [46,47]).



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Fig. 2. Flow curves (steady-state shear stress σ as function of applied shear rate γ). Left: for a hard-sphere-like colloidal suspension (filled symbols; in units of thermal energy density kT/

d3 = kT/(2R3H) and the bare Péclet number Pe0 ¼γ τ 0 ), for increasing effective packing fraction (bottom to top). Lines are obtained using mode-coupling theory (ITT-MCT); open symbols demonstrate that an estimate from linear-response rheology fails to describe the flow curves. From Ref. [39]. Right: in the limit τ → ∞, for an athermal elasto-plastic model (symbols; the line is a fit using a phenomenological Hershel–Bulkley power law), from Ref. [40].

Flow curves for athermal granular materials close to jamming look superficially the same, tempting to treat jamming and glass transition on the same footing [48]. It has recently become clear that this view is too simplistic [49–51]. Close to the glass transition on the liquid side, the flow curves indicate the finite yield stress at the transition by an intermediate-shear-rate plateau. Asymptotic scaling laws hence involve only time scales through Pe, but keep the stress scale fixed [49]. Close to jamming, scaling laws reminiscent of critical phenomena allow to collapse the flow curves by scaling both Pe and σ [52,53]. This implies the flow curve directly at the jamming transition to follow a power x law, σ ∼γ , and hence σcy = 0 at the jamming transition. Ikeda, Berthier, and Sollich [50,51] classify the difference between glassy and jamming flow curves by their relevant energy scales: thermal fluctuations govern σy,g = O(kT/d3) at the glass transition, but the interaction strength U from particle overlaps sets σy,j = O(U/d3) ≫ σy,g at jamming, on a time scale τj = (kT/U)τ0 ≪ τ0. Glassy rheology falls in the regime Pe0 ≪ 1, jamming physics emerges for Pe j ¼γ τ j ∼1. While hard-sphere colloidal suspensions usually probe glassy rheology, foams are in the jamming regime; oil-in-water emulsions present an interesting cross-over case [51] that deserves closer investigation in the future.

4. Theoretical descriptions: fluid vs. amorphous solid Continuum rheology addresses the Eulerian stress tensor σ appearing in the Navier–Stokes equations. Linking it to the dynamics in terms of constitutive equations starting from microscopic theory, taking into account the principles of continuum mechanics, is a hard task in general. The generics of the glass transition and disorder, reviewed in Refs. [19,54,55], offer starting points for the development of constitutive equations rooted in fundamental statistical mechanics.

non-Markovian time-nonlocal slow relaxation of density fluctuations (describing visco-elasticity and an idealized dynamical glass transition as τ → ∞), which the latter tends to decorrelate (causing shear thinning). Due to its approximations, ITT-MCT is best suited for Pe0 ≪ 1 (but arbitrary Pe). Generalized nonlinear Green-Kubo relations, derived within ITT, are approximated in MCT through flow-dependent dynamical two-point correlation functions ϕ! ðt; t 0 Þ, formed with a density fluctuation to k ! wave vector k at time t that has, under flow advection, evolved from ! one to k ðt; t 0 Þ at t′. The expression for the stress tensor reads Z σðt Þ ¼

t

dt −∞

0

Z

! 0 2 dk V ! ½−∂t0 Btt 0 ϕ! t; t ; ½B 32π3 k k ðt;t 0 Þ

ð1Þ

(starting in stress-free equilibrium at t → − ∞). The geometry of the flow enters through a (tensor-valued) functional V of the Finger tensor Btt 0 , a rotational-invariant measure of the deformation between t′ and t. For the density correlation function (not denoting the dependence on the deformation history for notational simplicity), h

i t −1 0 ′′ ′′ 0 ′′ 0 Γ kðt;t0 Þ ∂t þ 1 ϕk t; t þ ∫t 0 dt mk t; t ; t ∂t 00 ϕk t ; t ¼ 0;

ð2Þ

motion on the time scale τ0. The with Γ kðt;t 0 Þ related to short-time ′′ 0 memory kernel mk t; t ; t represents the history-dependent friction and is the point where flow history enters in Eq. (2), rendering Eq.(1) nonlinear in the flow rate. MCT approximates it again in terms of ϕk t; t 0 , addressing in particular length scales of typical inter-particle separations. The coefficients of Eqs. (1) and (2) can be fixed by the equilibrium static structure factor Sk. The growing memory with increasing coupling strength drives a bifurcation transition in Eq. (2) identified as the ideal dynamic glass transition. Flow supresses this memory. One finds density correlations to decay from an intermediate nonergodicity plateau at the time scale τ without flow, and at τγ ∼1= γ for Pe ≫ 1, resulting in a discontinuous appearance of a dynamical yield stress at the glass transition. The relaxation ∼τγ causes a dynamical yield stress and is confirmed by computer simulation [62, 63] and (with apparently systematic deviations) by confocalmicroscopy experiments [7]. Schematic simplifications of ITT-MCT [64] replace Eq. (1) by σ ðt Þ ¼ t ∫−∞ dt 0 γ ðt 0 ÞG t; t 0 ; γ (for the case of simple shear), relating the nonlinear dynamical shear modulus G(t, t′) to a correlator ϕ(t, t′)2 (the restriction of Eq. (2) to a single dominant mode in wave-vector space). Schematic models allow to fit the main aspects of nonlinear glassy rheology close to the glass transition [65] (lines in Fig. 2), and treat various forms of time-dependent shear [66–68] and geometries [64,69] (rationalizing the von Mises criterion for three-dimensional

4.1. Mode coupling theory (ITT-MCT)

The mode-coupling theory of the glass transition provides a generic bifurcation scenario encoding the discontinuous jump of G(∞). Fuchs and Cates [56,57] combined MCT with the integration through transients (ITT) framework to deal with nonlinear rheology of colloidal suspensions, starting from the evolution equation for the many-body nonequilibrium distribution function (the Smoluchowski equation). Later extensions deal with arbitrary time-dependent incompressible and homogeneous flow [58–60] or non-colloidal systems [61]. ITT-MCT encodes the trapping of particles in cages formed by the neighbors (transiently in the fluid, permanently in the ideal glass), and flow-induced advection of fluctuations. The former leads to a


4


yielding [70]). The broken time-translational invariance in G(t, t′) under time-dependent flow is crucial for some features of nonlinear glassy rheology [67]. The full ITT-MCT for hard spheres [45] quantitatively calculates stresses and structural anisotropies, although it overestimates them. The radial and angular dependence of g ðrÞ (or the q-dependence of the Fourier transform, the distorted static structure factor) after startup or cessation of flow reveals spatially resolved information on the flowinduced structural dynamics [11,46,71,72]. For Pe0 ≪ 1, dense suspensions remain rather isotropic. This is reflected also in single-particle fluctuations (corrected for the affine drift). In multi-particle quantities, such as correlations of displacement vectors across space (which are special four-point correlation functions), the effect of shear is more strongly felt [73]. The recent shift of emphasis from two-point to four-point correlations (moving beyond MCT), reflects a similar discussion in the dynamics of quiescent glass formers. 4.2. Fluidity models t

The schematic form of the Green–Kubo relation, σ ðt Þ ¼ ∫−∞ γ ðt 0 Þ G t; t 0 ; γ dt 0 , together with the (crude) approximation of an exponentially decaying dynamical shear modulus (on a time scale τM), yields the Maxwell model for viscoelasticity,

∂t σ þ ð1=τM Þσ ¼ G∞ γ :

ð3Þ

This model links the linear-response structural relaxation time (τM ≈ τ) to the viscosity, η = G∞τ. A simple nonlinear generalization −1 to steady shear, τ −1 þ γ =γc (with a characteristic strain scale M ¼τ γc), captures shear thinning and the dynamic yield stress qualitatively. The ad-hoc extension to nonsteady flow is not obvious. If τM ≡ τM γ ðt Þ , memory decorrelates as a function of accumulated strain. An integral version of Eq. (3) breaking this dependence on strain only, captures creep phenomena [74]. Fluidity models treat f(t) = 1/τM(t), termed fluidity, as a time- and space-dependent state variable, postulating a dynamical equation for this variable. A generic form is ∂t f ¼ −λð f Þ þ μ f ; σ; γ þ D∇2 f with positive nonlinear functions λ and μ and a diffusive term added in spatially nonlocal models. A simple choice akin to the one used to describe long-time creep in granular matter is [75]

appropriate units of energy), between which the system changes by activated hopping dynamics. Between hops, each mesoscopic trap is strained affinely under an imposed deformation, γ local ¼γ . The local stress is given elastically, σlocal = Gγlocal, and the macroscopic stress is the average over all local ones. The locally stored elastic energy lowers the barriers, so that the hopping probability ∝ exp[−(E − (1/2)σ2local)/x] (leading to shear thinning). Each hop acts as a local yield event, resetting the local strain, γlocal = 0, and selecting a new trap at random. The temperature-like parameter x is a main control parameter in SGR. For x N 2, the system has the rheological signature of a fluid. For x b 1, the system is in a continuously aging glassy state. There is a peculiar intermediate regime 1 b x b 2 in SGR where the response is that of a x−1 power-law fluid, σ ∼γ . At x = 1, a yield stress continuously develops (as the SGR does not feature intermediate-time rigidity). For this latter reason, the SGR is not truly a model for nonlinear glassy rheology. However, it has great success in explaining many generic features of the nonlinear rheology of a broad class of soft materials. SGR specifically addresses structural aging for x b 1, which makes it a useful tool to discuss the shear-induced rejuvenation of many gels, foams, or emulsions. Since it incorporates the mean-field notion of the coupling of local yield events, x is called a noise temperature. This implies that a coupling between x and the flow should be present. It has been modeled as a relaxation–diffusion dynamics [82], ẋ(y) = − (x(y) − xss)/τx + D∂ 2y x(y) + S(y) (assuming spatial variation along the y coordinate). Here, S(y) describes the flow-induced generation of noise as the result of plastic events. It is possible to interpret x as a thermodynamically consistent configurational temperature (of some slow mesoscopic degrees of freedom), identifying it with the effective temperature χ introduced in the theory of shear transformation zones (see below) [83]. In similar vein, SGR has been recently recast as a consistent mesoscopic nonequilibrium thermodynamic model [84].

2

∂t f ¼ −af þ rσ γ ;

ð4Þ

where a and r are model parameters. With an appropriate differential equation, fluidity models describe the internal restructuring in thixotropic materials. To address spatially non-local dynamics in channel flow, Goyon et al. [76,77] used the diffusive term to describe relaxation towards the homogeneous fluidity, interpreting D ∼ ξ2 as a cooperativity length scale [78,79]. The velocity profile for channel flow can then be evaluated and yields the typical “plug-flow” shape for yield-stress fluids (where the fluid in the center of the channel is transported along essentially unsheared). Fluidity models offer a convenient playground to test ideas related to non-local dynamics, since they are easy to implement and offer a natural physical explanation of the terms appearing in their equations, while their form is reasonably close to what one expects from more microscopic approaches. 4.3. Soft glassy rheology (SGR) An influential model of the nonlinear rheology of soft nonergodic materials (not necessarily glasses) is the soft glassy rheology (SGR) model by Sollich, Cates, and coworkers [80,81] (extending a trap model by Bouchaud). One assumes a free-energy landscape of many mesoscopic traps with an exponential barrier distribution, p(E) ∝ exp[− E] (in

4.4. Shear transformation zones (STZ) In the limit where thermal fluctuations are weak, non-affine deformations in amorphous solids occur through individual localized (in time and space) events termed shear transformation zones (STZ). This mirrors the (historically related) observation of local bubble rearrangements (T1 events) in foams [16]. The current STZ theory (save early precursors [6]) goes back to Falk and Langer [85] (reviewed in Refs. [86,87]). STZ are envisaged as twostate objects (aligned along or against the flow) existing in some density n±, and global yielding the collective effect of many local STZ. The balance of STZ is determined by an effective disorder temperature χ, leading to a formulation of the theory in terms of thermodynamics of two subsystems separating fast and slow degrees of freedom (this is how glassy dynamics enters) [88–91]. An internal state change of STZ causes a local strain rate, so that the overall plastic strain rate (denoting simple shear for simplicity) is given pl by γ ∝hRþ nþ −R− n− i (where angular brackets are a suitable average over STZ properties) with transition rates R± that depend on state parameters and the stress. The material stops flowing if either activity is exhausted (n+ = n− = 0) or the different STZ polarizations balance (n+/n− = R −/R +). The STZ densities evolve according to a master equation

n ¼ −R n þ R∓ n∓ þ X :

ð5Þ

The source term X models flow-dependent creation and destruction of STZ due to thermal and mechanical noise. The steady-state density of STZ is given by the Boltzmann factor, ∝ exp[−1/χ], where χ is a (dimensionless) configurational temperapl ture. The plastic strain rate can then be written as, γ ∞ expl½−1=χ f ðsÞ, where f(s) is the (tensorial) function of the deviatoric stress. A dynamical equation for χ expresses relaxation towards equilibrium



and the counter-action of plastic work driving the system away from equilibrium, 0 1 dχ χ e pl ∇2 χ: @ A þ Dγ ¼ αχ f ðsÞexp½−1=χ 1− pl dγ χ ss 1=γ

ð6Þ

Here, χss is the (shear-rate dependent) steady-state configurational pl temperature, taken to depend on 1=γ (the time scale typical for shearaccelerated dynamics). Modeling this term empirically to incorporate the change in dynamics around the glass transition, the STZ theory reproduces the linear viscoelastic moduli [92] (although the fit parameters can be discussed critically [39]) and typical flow curves of BMG around Pe ≈ 1 with a dynamical yield stress [88]. The effective-temperature concept is very appealing. Effective temperatures Teff derived from deviations of fluctuation–dissipation ratios by a (roughly) constant term have been used to describe sheared glassy and granular systems [93,94]. But the analogy between thermal fluctuations and those induced by mechanical driving may be questioned [40]. Evidence for STZ (of about 5 to 10 particles in diameter) comes from atomistic simulations and more recently also in direct imaging of colloidal glasses [8] (Fig. 3). The appearance of STZ ties in well with the concept of structural heterogeneities widely discussed in quiescent glasses [96,97]. The STZ loci correlate with the local elastic moduli and soft modes of the amorphous structure [98–101].

4.5. Elasto-plastic models Localized plastic events cause, by appeal to continuum elasticity theory, long-ranged stress- and strain-fields in the surrounding amorphous solid. Elasto-plastic models focus on this long-ranged coupling of plastically rearranging zones. The calculation of the response of an ideal elastic medium to a change in shape of some localized region is attributed to Eshelby. The total strain ϵ = ϵel + ϵpl is decomposed into an elastic part, and a localized plastic contribution. Assuming linear elasticity outside the plastic zone, σ = 2G∞εel, mechanical equilibrium of the elastic solution and incompressibility determine the displacement field u that de fines ε ¼ ∇u þ ∇uT =2, h i pl 0 pl 0 u ¼ −2G∞ ∫dr Q r; r ∇ ε r :

ð7Þ

0 In an infinite homogeneous medium, the Green function Q r; r is 0 given by the Oseen tensor O r−r , and a Fourier transform yields upl ðqÞ

5

¼ 2G∞ iOðqÞ ϵ pl;T q, with OðqÞ ¼ 1−qq=q2 = G∞ q2 . If one assumes the plastic strain field to carry the symmetry of the imposed simple shear flow, there results the elastic propagator for the shear stress field, ð1Þ 0 0 pl 0 σ xy ðrÞ ¼ 2G∞ ∫dr G r−r ϵxy r :

ð8Þ

The dominant angular variation of GðrÞ is that of a quadrupole [G(r, ϑ) = (1/πr 2 )cos(4ϑ) in 2D] featuring an algebraic decay, GðrÞ 1=r dim in dim = 2 and dim = 3 dimensions. The propagator Gðr; r 0 Þ is a central quantity in elasto-plastic models, describing how one plastic event can trigger others, leading to spatially correlated collective events and avalanches in the macroscopically yielded regime [8,102,103]. Newer models incorporate a balance of time scales for the transition of local regimes from elastic to yielded, and back again [103,104]. The effect of plastic events is best seen in zero-temperature vanishingly slow flow, probed by athermal quasistatic simulations (AQS). They proceed by alternating small strain steps with an energy-minimization technique, and hence access the regime τ pl ≪τγ ≪τ, where τpl is the time scale intrinsic to an individual plastic event. Plastic events in AQS correspond to the transition from one minimum in the energy landscape to the next, leaving the signature quadrupolar pattern in the nonaffine displacement field, and discernible stress drops even at very small strains. Avalanches of plastic events in 2D at finite shear rate, as identified through maps of a coarse-grained nonaffine displacement field (Fig. 3), and their scaling with shear rate and system size have been discussed by Lemaître and coworkers [95]. They suggest a scaling of av−1=dim alanche size as γ at low enough shear rates. A related scaling of shear-induced diffusivity, D ∼ L, with the linear dimension L of the system is taken as an indicator for the occurrence of avalanches [105,106]. The rate of avalanche events is found to obey power-law scaling that depends on the microscopic dynamics [107,108]. (For a related discussion, see Ref. [109].) At high shear rates, no strong size dependence is expected as plastic events are triggered more frequently through the driving than through elastic correlations. The fate of long-ranged correlations at finite temperature and finite deformation rate is currently debated. For a certain window of shear rates and temperatures, thermal fluctuations were found to simply lower the strains at which plastic events occur [106,110]. Some studies found already small finite temperatures to suppress the long-range organization into avalanches [111] and the signature quadrupolar symmetry [112]. The spatial and also temporal correlation between plastic events is discussed through a quantity reminiscent of the elastic propagator

z (µm)

20 15 10 20

25

y (µm)

30

Fig. 3. (Left) shear-transformation zone (STZ) in confocal microscopy of a colloidal glass. Colors are a measure of local strain, dashed lines indicate regions of positive and negative strain surrounding the STZ. Reprinted from Ref. [8]. (Right) Map of nonaffine displacements from a 2D 2D T = 0 simulation under steady shear for a strain interval Δγ = 0.20, showing the appearance of correlated avalanches. Reprinted from Ref. [95].


6


0 G r; r . One starts from the commonly used measure of nonaffinity, subtracting from the local particle displacements that generated by a best-fit locally affine deformation [85]. The least-square deviation δ2 is used to characterize the plasticity (and other measures are broadly consistent [114]). The correlation of two plastic events at, re 0 0 spectively, ðr; t Þ and r ; t ¼ ðr þ Δr; t þ Δt Þ, is measured by the plastic correlator ED EE E E DD DD 0 0 2 2 0 2 2 − δt ðr Þ δt 0 ðr Þ C2 r; r ; t; t ¼ δt ðr Þδt 0 r r t

r

r t

ð9Þ

(suitably normalized to unity). It shows strong positive correlations in streamwise and crosswise directions, in agreement with the quadrupolar elastic coupling (see Fig. 4). Nicolas et al. [113] proposed a lattice model accounting for local yield events coupled by elastic propagators that captures (but somewhat exaggerates) the relevant signatures found in simulation. The spatial decay of C2 ðat Δt ¼ 0Þ has been studied extensively in confocal microscopy and simulation [73,97,114–117]. In the regime of shear-dominated displacements, the typical quadrupolar signature is found, together with a power-law decay in Δr. When thermal noise dominates, the correlations become rather isotropic (as the linearresponse expectation for Pe ≪ 1 suggests) and shorter-ranged. The temporal persistence of C2 decreases with increasing Pe, in agreement with the finding that shear heterogeneities induced by elasto-plastic coupling are more dominant at low shear rates [113]. The time for unfolding of correlated arrangements of plastic events (avalanches) is set by the propagation of shear- and stress-waves [117,118]. Studying a spatial autocorrelation of the accumulated strain over a time lag Δt, Chattoraj and Lemaître [119] find Eshelby-type correlations to persist in this quantity even in the Newtonian flow regime. The significance of this finding is still under debate [116]. 5. The dynamics of stresses We now turn to some phenomena that highlight the temporalhistory effects of stresses in nonlinear glassy rheology. 5.1. Startup transients: stress overshoots The transient behavior after sudden startup of flow probes the dynamics of the system en route to its nonequilibrium steady state. Starting from equilibrium, the stress–strain curve σ(γ) displays an intermediate overshoot. This is known from metallic glasses under tensile strain [3] and, more recently, colloidal suspensions [72,74,120]. It is a genuine nonlinear-response phenomenon. The height σmax of the overshoot depends on Pe and, deep in the glass, on sample age [72,74,121]. σmax is sometimes referred to as a yield stress (for materials that break before entering plastic flow). A connection to transient shear

banding is sometimes invoked [82] but this does not appear to be a prerequisite [71]. The stress overshoot reveals the dynamics of flow-induced cage breaking: cages transiently store more elastic energy than they support under flow. Overshoots occur at strains γc of a few to ten percent, compatible with mean cage sizes, and are accompanied by transiently superdiffusive non-affine mean-squared displacements [71]. ITT-MCT rationalizes them by the decorrelation of the coupling of stresses to density fluctuations [45,120]. Similar cage-breaking signatures are seen under applied stress [74] and in constant-velocity microrheology [122]. The shear-history dependence of cage breaking is evident in the socalled Bauschinger effect: upon flow reversal from the steady state, the stress overshoot vanishes, while it remains intact upon reversal after small strains and reappears after a quiescent waiting time [68,123, 124]. The effect can be understood from the loss of temporal memory after accumulated strains of γc [68]. The broken symmetry of the pre-strained initial state is indicated by a nonvanishing second derivative of stress versus strain [123]. This quantity has been qualitatively linked to the imbalance between positively and negatively aligned STZ. Within STZ theory, the existence of such an internal degree of freedom is crucial to understand stress-overshoot transients [88].

5.2. Flow cessation: residual stresses The temporal-history effects of nonlinear glassy rheology are strikingly felt in the appearance of remnant stresses long after the removal of past flow. Stresses then decay, ultimately on a time scale associated with τ. In the ideal glass, ITT-MCT predicts limt→∞ σ t; γ ¼ σ ∞ γ N0, a true persistent residual stress, in agreement with experiments on colloidal glasses and computer simulation [125,126] (see Fig. 5). Jammed soft-particle suspensions display similar residual stresses [127]. The value of σ∞ depends on the past flow rate. Residual stresses are a consequence of the non-trivial time dependence emerging from microscopic theory in constitutive equations such as Eq. (1). Simpler standard-rheology models miss this and predict, e.g., stresses to ultimately vanish after reversing step strains, although plastic rearrangements cause irrecoverable stress decay [67] (also in oscillatory shear). The dependence of residual stresses on the past flow rate is striking because it is a genuine nonlinear-response effect incompatible with the standard linear-response Onsager hypothesis [126]. Residual stresses produced (or even deliberately imposed) by either mechanical, chemical, or thermal processing history play an important role in determining macroscopic material properties, not only in soft-matter glasses, but also in safety glass and smartphone covers and various other applications [128], and to control plasticity in metallic glasses [129] or the rupture of thin polymer films [130]. The emergence of complex spatial patterns of residual stresses in bulk glasses still needs to be understood theoretically.

! Fig. 4. Spatial correlations of plastic events, measured through the plastic correlator C 2 Δ r ; Δt , Eq. (9), for Pe0 = 10−4 from MD simulation, for increasing time lag Δt (left to right). The quadrupolar symmetry induced by the Eshelby-type stress- and strain fields is clearly seen. From Ref. [113].



(a)

(c)

7

of a critical amplitude connected to critical-point-like phenomena has recently been discussed as a mechanism for yielding in low-temperature BMG and colloidal glasses [147–149] (although its relevance when thermal fluctuations dominate can be debated [141]). 6.2. Stress-controlled flow: creep The appearance of a yield stress causes rich phenomenology when driving flow through applied stress. While applied rate-controlled flow eventually fluidizes the system in all spatial directions [131], one expects a transition from a deformed solid at low applied stress (certainly at σ b σy), to plastic flow at large stress (for σ N σc above some static yield stress σc). Below σy, no steady-state flow is possible, and the resulting γ ðt Þ displays a long transient time dependence. It may even decay to zero only algebraically (in particular close to σy), causing long-lasting “anomalous” deformation referred to as creep. Several temporal regimes for creep have been identified in hard-sphere-like colloidal glasses [74] (see Fig. 6), in striking resemblance to what is discussed in hard solids. After some initial (essentially elastic) transient, power-law creep, γ ðt Þ∼t −x with, 0 b x b 1, is observed. In fact, x ≈ 0.6 has been suggested early on in (non-amorphous) metal wires by Andrade, where it is explained in terms of dislocation dynamics. The wide-spread appearance of Andrade creep also in various amorphous systems [150–152] points to a more generic mechanism that is, however, not fully understood yet. At later times, logarithmic creep, γ ðt Þ∼1=t, first empirically found in hard matter is reported in several soft-matter systems [74,75,150]. Again, a microscopic understanding is still incomplete. Flow remains homogeneous in the Andrade regime. For σ N σc, the eventual fluidization of the system (i.e., the material failure from a hard-solid perspective) appears as a rather rapid, sample-age dependent process triggered on a time scale that diverges with (σ − σc) [74,153]. During this process, which is connected with the stress overshoot in rate-controlled startup flow, heterogeneities may become important. But detailed studies of local dynamical properties during creep have only started recently [152,154,155].

(b)

(d)

Fig. 5. Stress decay σ(t), normalized by the steady-state value σss, as a function of scaled

time γ t after cessation of past shear flow of rate γ , from (a) MD simulation, (b) ITTMCT, (c) hard-sphere and (d) thermosensitive PS-PNIPAM suspensions. Non-decaying curves indicate persistent residual stresses in the glass (same state with different preshear rates). Reprinted from Ref. [125].

6. Dynamical yielding

As noted above, the application of a steady deformation rate causes the amorphous solid to yield and enter a plastically flowing state, where it is fluidized in all spatial directions [131]. The application of stresses (instead of the deformation rate) or non-steady perturbations can probe the nonequilibrium transition from a solid-like deformed state to the fluid-like yielded state in different ways. 6.1. Oscillatory shear (LAOS) Large-amplitude oscillatory shear (LAOS) provides systematic access to non-steady-state properties in nonlinear rheology. One considers (shear) strains γ(t) = γ0sin(ωt), as a function of amplitude γ0 and frequency ω (also in its microrheological counterpart [132]). Several methods of analysis exist and provide demanding tests of constitutive models [133], prominently in terms of higher harmonics in Fouriertransform rheology. Glassy LAOS rheology identifies a yield strain γy compatible with the stress-overshoot position [134]. In systems with short-ranged attraction, LAOS reveals two-step yielding (first of particles, then of clusters of particles) [135–138]. A comprehensive discussion of nonlinear-glassy-rheology aspects of LAOS in ITT-MCT, experiment, and simulation on hard-sphere systems has been given [66,139]. As the strain amplitude γ0 is increased, the response evolves from that of an elastic solid (with no or few higher harmonics) to that of an alternatingly yielded and elastic material (implying a large contribution from rather high-order harmonics). Accumulated plasticity may lead to eventual yielding even at (arbitrarily) small γ0, after a large number of cycles. This has been seen in a colloidal polycrystal [140] and close to jamming [141]. In this light, the true steady-state under oscillatory strain-driven flow may always be yielded [142]. Within STZ, this emerges since plastic events are always present. The time to reach the steady state is practically infinite [143], since a sufficiently small number of local plastic events are still effectively reversible [144]. One may in fact need to distinguish strain-driven from force-driven oscillatory shear here [145]; the latter may allow the system to remain intact as a solid even when the former, strictly speaking, does not. There still remains a rather sharp crossover at some critical strain amplitude γc0, where the transition to the irreversibly yielded steady state is rapid and the nature of the dynamics changes [146]. The concept

(a)

(b)

Fig. 6. Deformation γ(t) and shear rate γ following application of a constant stress σ to a hard-sphere-like colloidal glass. (i)–(v) mark several regimes identified in the response, including (ii) power-law creep, (iv) cage yielding, and (v) logarithmic creep. Reprinted from Ref. [74].


8


At least mean-field models suggest that the static yield stress σc (not to be confounded with the stress-overshoot value σmax that is sometimes taken as a yield stress, but is no intrinsic material property) should equal the dynamic one σy. 7. Shear thickening Shear thickening, a dramatic increase in viscosity upon increasing the mechanical driving (famously in cornstarch-in-water suspensions), is often seen in the regime of large Pe0 [156]. There, details of the particle interactions become important. We briefly mention some recent discussion of discontinuous shear thickening (a rapid increase of σ above a typical threshold) that are related to glassy rheology and a backcoupling of stresses to the dynamics. Traditional explanations of shear thickening include clusters induced by HI, or dilatancy (the tendency of densely packed materials to expand under flow due to hard-core constraints in the particle interactions) followed by stress-induced jamming [46,156]. Recently, the role of frictional contacts that form when applied stress overcomes the hydrodynamic lubrication forces between hard particles was studied [48,157–160]. Observing that the jamming transition may shift with interparticle friction and that the latter is changed by applied stress, Wyart and Cates [160] construct S-shaped flow curves that lead, in the extreme case to discontinuous shear thickening. The notion of a stress-backcoupling to the dynamics has also been incorporated into a schematic ITT-MCT model that captures aspects of discontinuous shear thickening qualitatively [161,162]. This model assumes Brownian motion to be still important, which is not always the case in experiments on shear thickening. 8. Flow localization: shear banding A hallmark of the spatial nonlocalities that govern nonlinear glassy rheology deep in the glass, is the localization of strain in narrow regions. Such shear bands are in fact the main failure mode of deformed amorphous solids and in particular BMG [163,164]. Shear banding can arise from flow curves that are non-monotonic for γ →0 (since regions with dσ =d γ b 0 are mechanically unstable); this is usually found in thixotropic soft-matter systems [165,166]. In systems with monotonic flow curves, transient shear bands may still arise [167]. Coexistence between arrested and flowing regions is captured in the mesoscale models discussed above by adding spatial variation and diffusive dynamics to the internal state parameters (effective temperature, fluidity) [76,82,77]. Introducing a local restructuring time for the relaxation of yielded regions back to elastic (τel), nonmonotonic flow curves result if

τel ≫ τpl, the time scale for the transition from elastic to yielded. This favors the formation of shear bands, where the long-range elasto-plastic interactions are crucial [103] (Fig. 7), and lead to preferential collective alignment of plastic events. Shear localization is generally attributed to strain-induced softening of local plastic regions, that thus become more susceptible to further flow. In colloidal suspensions, flow-induced particle migration is discussed as a further feedback mechanism, elaborated as flowconcentration coupling by Besseling et al. [9]. Essentially, fluctuations towards larger density locally create lower shear rates, causing further migration of particles to this region, due to a flow-induced (osmotic) pressure that increases with increasing γ. If this accumulation is stronger than diffusive spreading of particles and stress redistribution, a pos itive feedback arises if ∂γ Π∂ϕ σ = ∂ϕ Π∂γ σ N1. This criterion could be verified in experiment [9] and simulation [168]. The importance of strain-localized flow, and possibly also the mechanisms relevant for it, depend on the regime addressed — in terms of shear rates and proximity to the glass transition.

9. Conclusions and outlook Nonlinear glassy rheology encompasses two regimes that mirror the transition from fluid to solid. The flow of dense colloidal suspensions and glass-forming liquids is governed by pronounced temporal memory effects. In the (deeply quenched) glass, however, the plastic deformation of the amorphous solid is predominantly understood in terms of spatially heterogeneous dynamics. A particularly intriguing route of current and future development is connected to the question how far these different approaches can be extended to cover overlapping regimes of applicability. This would need to result in a fully (spatially and temporally) nonlocal description of flow and deformation of disordered systems. For the currently homogeneous, non-Markovian description of nonlinear glassy rheology close to the fluid–glass transition, based on ITT-MCT and the temporal memory integral in Eq. (1), this means to allow for spatial variations in the flow and to incorporate the fluctuations that lead to the emergence of such inhomogeneities. From the point of view of elasto-plastic models, temporal effects in the interaction of spatially localized correlated plastic events, as described by the spatial integral in Eq. (8), need to be added, dropping the separation of time scales of individual plastic rearrangements, τpl, and that set by the flow, τ γ ∼1= γ. A closely related question concerns the fate of localized plastic rearrangements in the flowing state where thermal fluctuations are strong [110,119]. A suitable measure identifying such events (in some statistically-averaged notion) in a model-inspecific way will need to be found. This may relate to the identification of dynamical

Fig. 7. Local number of plastic events in a given strain interval, from an elasto-plastic model with large relaxation time of yielded regions, coupled with a long-range elastic Green's function (resulting in a shear band, left), and an ad-hoc short-range propagator (right). Reprinted from Ref. [103].



heterogeneities close to the quiescent glass transition, and their fate under flow [97,105,155]. The study of nonlinear-response phenomena then becomes a versatile tool to address fundamental questions about the underlying glass transition, allowing to disentangle various microscopic processes whose interaction is not obvious from the linear response alone. Further insight on the microscopic mechanisms of yielding can be provided by investigating the aspects of different flow geometries (for example extensional flow, or combining steady shear flow in one direction with oscillatory flow) [64,69,70]. The combination of different applied fields (for example, mechanical with magnetic fields to address magnetostriction) allows to study new nonequilibrium state transitions that arise from the introduction of a further control parameter [169,170], and provides interesting links to phenomena of other driven nonlinear dynamical systems (e.g., Barkhausen or crackling noise [171]). Also, experiments with locally resolved response to local perturbation, such as nano- (or meso-) indentation measurements applied both to BMG and colloidal systems [172], or actively driven micro-rheology understood as a tool on its own (rather than as a surrogate to macrorheology as is often the case), will play an increasing role in disentangling the microscopic processes behind nonlinear glassy rheology. Microscopic theory needs to address the common, but also distinguishing features of soft colloidal rheology and the plasticity of hard amorphous materials, most notably bulk metallic glasses. Some generic features such as the nonlinear flow curves indicating shear thinning, or the appearance of stress overshoots, most likely have the same physical origin in both material classes. But BMG are typically highly brittle (a major obstacle for their application, as it makes them prone to catastrophic failure), while soft materials are ductile. These features are linked to the interparticle interactions (and the resulting properties of STZ, say) [173,174] and need further investigation. A multi-scale description building on microscopic and mesoscopic theories of nonlinear glassy rheology, combined with macroscopic fluid dynamics, will allow to address the coupling of structural timeand length-scales to hydrodynamic ones and macroscopic heterogeneities emerging from the flow (such as those seen in residual stresses). This requires combining numerical tools for the Navier–Stokes equations (including lattice-Boltzmann, LB, or finite-element and finitevolume methods, FEM/FVM) with microscopic simulation or theory [175], and to incorporate, in particular, integral constitutive equations and the treatment of nearly unyielded regions [176] that still involve some plastic activity. Plastic events and their surface effects, e.g., in pressure-driven channel flow cause deviations from the scaling laws expected from continuum mechanics [76–78,104,177]. An example for the coupling of structural and hydrodynamic time scales is the propagation of elastic waves triggered by plastic events, studied in a LB-based model of glassy rheology [117,118]. (The combination of integral CE with classical fluid mechanics also provides microscopic predictions for various regimes of rod climbing [178].) Hence, bridging the scales from microscopic dynamics and mesoscopic plasticity to macroscopic flow kinematics will provide an interesting bridge from statistical physics to materials science applications. Having the applications of rheology (mentioned in the introduction) in mind, the theoretical knowledge about the dynamical processes in nonlinear glassy rheology that has emerged in the past few years, can help in improving the so-far empirical constitutive equations used in macroscale modeling. Such a linking back of fundamental physics to chemical engineering will be a further step to predictive materials design of process-history dependent, noncrystalline materials. Acknowledgments I thank M. Fuchs and J. Baschnagel for helpful comments, and the participants of the 570.WE-Heraeus-Seminar “Nonlinear Response in Nonequilibrium Liquids and Complex Systems” for their engaged discussions.

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