Time dependent flows in arrested states

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Time dependent flows in arrested states

Dissertation

zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften

an der Universität Konstanz im Fachbereich Physik, Lehrstuhl Prof. Dr. Matthias Fuchs,

vorgelegt von Christian Peter Amann

Tag der mündlichen Prüfung: 18.12.2013

1. Referent: Prof. Dr. Matthias Fuchs

2. Referent: Prof. Dr. Georg Maret

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Non sunt multiplicanda entia sine necessitate.

John Ponce (1603 – 1661)

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Danksagung

Ich möchte mich bei folgenden Personen dafür bedanken, dass sie zum Gelingen dieser Arbeit und meiner Ausbildung zum Wissenschaftler beigetragen haben:

• Prof. Dr. Matthias Fuchs für die Gelegenheit, an seinem Lehrstuhl eine Disserta- tion anzufertigen, der zu jedem Zeitpunkt bereit war, aufkommende Fragen mit seinem großen Fachwissen zu beantworten und immer auch Verständnis für die wiederkehrenden Probleme des Alltags hatte,

• Prof. Dr. Georg Maret für die freundliche Übernahme des Zweitgutachtens und Prüfungsvorsitzes,

• Jun.-Prof. Dr. Fabian Pauly als drittem mündlichem Prüfer,

• Frau Marianne Griesser für die unkomplizierte und freundliche Unterstützung in allen bürokratischen Belangen,

• allen (ehemaligen) Angehörigen des Lehrstuhls Fuchs und der Gruppe Voigtmann, die mit wissenschaftlichem Rat (vor allem Dr. Thomas Voigtmann), oder beim Korrekturlesen dieser Arbeit (Dr. Christian Harrer, Fabian Frahsa und Sebastian Fritschi) geholfen haben, oder moralische Unterstützung in Form lustiger Gruppen- abende gaben,

• all authors of Ref. [15], especially Dr. Dmitry Denisov and Dr. Peter Schall for kindly providing unpublished measurement data,

• all co-authors of Refs. [13,42,43],

• meiner lieben Sarah, die mich gerade in der letzten, schwierigsten Phase dieser Arbeit begleitet hat,

• meiner Familie und ganz besonders meinerMutter, deren Unterstützung unverzicht- bar für jede meiner Durststrecken ist.

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1 Introduction

1.1 Preamble

What do golf clubs and dispersion paint have in common? The former consist of a metallic glass (at least sometimes [1]), the latter of solvently dispersed colloids, which are a thousand times bigger than metal constituents. Also the energy scales are completely different, metals are literally ‘hard’, paint is ‘soft’ (matter). They seem harder to com- pare than apples and oranges, which are at least both round. However, the answer might be equally simple. Both, metallic atoms and colloids are spherical objects and, in the actual context, densely and amorphously packed. Due to the dense packing, hydrodynamic or atomic interactions, Brownian motion or ballistic motion might take place on short time scales, while the motion of the amorphous structure evolves on a macroscopic time scale, which diverges under approach of a glass transition point (in density or temperature). There is a theory for this structural arrest, the mode coupling theory (MCT) and its rheological features are the issue of this work. One generic field of its application are colloidal dispersions. The question, if the underlying, structural concepts are more universal, i.e. the introductory question, will be addressed in Chap.5.

The following chapters will deal with a three dimensional investigation of the macroscopic, mechanic response of dense colloidal dispersions to homogeneous strain flow. In general, this is a non-trivial task, because structural relaxation times compete with the timescale of the affine motion, which is induced by the outside perturbation. A nonlinear behavior emerges in consequence, which is of big interest in material science [2, 3].

Prominent examples are the stress overshoot in a stress-strain curve and shear thinning of the viscosity of (shear molten) dense liquids. However, MCT allows also for concep- tual investigations, like on fluctuation-dissipation theorems or effective non-equilibrium temperatures [4,5]. Here, nonlinear response to start-up strain is the tool to investigate macroscopic response, like shear stresses or structure factor (SF) distortions, but also microscopic features, like mean square displacements. These nonlinear investigations go on experimental side beyond colloidal dispersions and are investigated also for metallic glasses near their melting temperature [6,7].

Starting from Smoluchowski dynamics, MCT uses a projection on particle-density fluctuation modes to provide a closed set of equations of motion (EQM) for the intermediate scattering function or transient density correlator Φq(t), with wavevector mode q. It describes the slow time evolution of a colloidal system’s structure. In consequence, a critical packing fraction φ_c can be calculated, at which density fluctuations do not completely relax anymore. The interpretation is that particles are stuck in a cage of a certain mean size. Experiments discovered in hard-sphere systems a glass transition point φ_g at which the relaxation time of Φ_q(t) diverges [8]. It lies somewhat above φ_c; Fig. 1.1

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1.1 Preamble

gives an illustration. Further projections on density fluctuations allow to approximate

RCP HCP

Fig. 1.1:Sketch of the glassy arrest as function of packing fractionφin a hard-sphere dispersion (in 3d). MCT predicts a critical packing fraction of φc = 0.516, the measured glass transition takes place atφg≃0.575[8]. The closest amorphous (random) packing RCP lies atφRCP≃0.64.

Above, only crystalline configurations are possible with hexagonal close packingφHCP= 0.74as upper bound.

ensemble averages of observables by integrating through the transient dynamics (ITT).

Generic examples are the macroscopic stress tensorσas response to strain flow or the SF distortion, which means the evolution of the systems structure. The strength of the ITT formalism is that it provides non-equilibrium, transient (non-steady), non-linear observables. The meaning of this is best illustrated at thestress overshoot, cf. Fig. 3.9. After start-up of simple shear, the stress σ responds linearly to the shear strain γ, which is called elastic regime. The linear theory of elasticity deals with the full tensorial nature of this subject [9]. When the strain reaches about 100%, a stationary flow with constant stress is reached (steady state). Hydrodynamics, especially the Navier-Stokes equation [10], deals with this regime, when structural relaxation times can be neglected. In the intermediate, transient regime, the stress might be higher than in the steady state, which is typical for shear molten solids or viscoelastic fluids. This nonlinear behavior is called stress overshoot. MCT-ITT provides a description, starting from a microscopic picture, of the elastic-, transient-, and viscous-response regime and their material parameters (elastic moduli, viscosity, etc.).

Colloidal dispersions bear within this considerations a very intriguing feature, viz.

their constituents are comparably big, but still microscopic. They are subject to an observable Brownian short-time dynamics at room temperature and can even be tracked directly with confocal microscopy [11–13]. But also light scattering [14] and X-ray scattering provide insights into the structural changes, which has been already combined with rheological measurements in real time [15]. Lots of experience exists on fabricating glass forming colloidal dispersions and investigating them under shear flow, which gave rise to numerous comparisons of MCT-ITT to experiments [11, 16–18] and simulations [13,19–23]. This work continues these comparisons, because nothing better can happen to a theory than a permanent discovery and refinement of its strengths and weaknesses in describing real physical systems.

Colloidal experiments and hard-sphere simulations use polydisperse [24] or bidisperse [19–21] mixtures to prevent a shear or ageing induced crystallization into a periodic structure. A second branch in Fig.1.1, which is not shown, would describe how the fluid phase turns into a crystalline phase upon increasing the packing fraction above φlrp≃0.52. A

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1.1 Preamble

MCT for polydisperse system has been developed already [25], but generalizing it via ITT to describe strain flow is yet numerically too challenging. A crystallization however of the system is unwanted, because equilibrium MCT and MCT-ITT make use of translationally invariance in Fourier space to simplify equations drastically. Heterogeneities are thus not dealt with by MCT-ITT especially not shear banding, which is assumed to be another reason for stress overshoots [26–29]. The shear-transformation-zone theory (STZ) claims to describe this phenomenon, based on dynamical heterogeneities [30], which are claimed to be observed in experiments already [31]. MCT leaves this flank open, but there are attempts to regard MCT as a mean field theory [32,33] (of possibly heterogeneous system realizations). However, MCT has the advantage of being not phenomenological (which STZ is) and close to the glass transition point, while STZ often deals with small temperatures deep in glass phase [34,35].

The deformation-symmetry considerations of STZ, which date back to older flow defect theories [36,37] using Eshelby’s continuum theory [38], could bring both theories closer.

It will be illustrated in this thesis that the elastic straining regime goes hand in hand with quadrupolar structural distortions. MCT-ITT will reveal that a decrease of this quadrupolar symmetry around the first SF peak produces the stress overshoot. The change in structural symmetry from the elastic to the steady state, viz. from quadrupolar to hexadecapolar, is one of the important results of this work. Thus, even without inhomogeneous flow, MCT-ITT is able to produce stress overshoots as consequence of a change of symmetry in structural distortions.

Another (here minor) topic is the coupling of rheological response (transient viscosity) to particle mean square displacements (MSD). Based on earlier findings [12], it will be further elaborated that the stress overshoot and super-diffusive motion are closely connected, which was found in simulations [12, 21, 39] and experiments [12, 13]. In metallic glasses, it is also already possible to connect structural decay with single particle diffusion [40].

1.1.1 Structure of this work

Some introductory remarks about the generic physical system behind this work and MCT-ITT conclude this introduction.

Chapter 2 gives a theoretical overview over MCT-ITT. A most recent formulation is shown to illustrate the tensorial character of the theory in time-independent, homogeneous, but otherwise universal start-up flow.

In Chap. 3, the fully anisotropic, three dimensional results of MCT-ITT are shown and discussed. At first, some details about the numerical implementation are made, followed by a discussion of the transient density correlator. Shear and normal stresses are presented and discussed, as well as the SF distortion in 3d. A close connection between the symmetries of the SF distortion and the transient stress regime are identified.

Comparisons with experiments and a MD simulation proof the qualitative agreement with MCT’s correlators [11,19] and SF distortions [15]. In conclusion, a heuristic argument for the Lindemann criterion is derived from symmetry considerations of the MCT equations of stress and SF distortion.

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1.2 Dense colloidal dispersions

Chapter4 derives and illustrates an anisotropic contribution to the MCT-ITT β ana- lysis [41]. It is compared and approved by the microscopic calculations of Chap. 3 and the coherent 2d calculations behind Ref. [23]. The anisotropic correction is quadrupolar and cannot be neglected, especially not around the first SF peak.

Chapter 5 finally deals with the schematic F₁₂^{( ˙}^γ)model of MCT. It is generalized to describe nonlinear, transient stress-strain behavior in a universal flow geometry. A validation with 2d microscopic MCT follows together with a comparison to colloid experiments in shear flow [42]. To test the universality, compressional experiments on metallic glasses are compared to theF₁₂^{( ˙}^γ)model [43], revealing a universal behavior of the structural decay process under strain of hard-sphere like, amorphous systems. This process is described by MCT. Making use of a generalized Stokes-Einstein relation reveals the close connection of the stress overshoot and super-diffusive particle motion. A comparison to MD simulations validates this feature of MCT and as well depicts its limits.

A summary closes every chapter (but the theory Chap. 2), together with some ideas for a future development or refinement of the underlying theory. The thesis finishes with a conclusion in German language, followed by a list of figures, a list of tables, and a list of the used literature.

Some numerical details, formulas, or technical proofs are shifted to the Appendix.

1.2 Dense colloidal dispersions

Some introductory remarks about the physical system, which is generic for this work, can be made before going into the details of each theoretic section.

1.2.1 Glass forming hard-sphere systems in strain flow

The system under consideration consists of N spherical particles with diameter d dispersed in a volume V. It is homogeneous, i.e. translationally invariant, and thus amorphous. Along with the deduction of the theory, the thermodynamic limit is taken and only the particle density n = N/V remains finite, while V and N diverge. As most important quantity, the packing fraction φ is frequently used, it describes the fraction of space occupied by the particles. In 3d, it is φ = n πd³/6 and in 2d φ = n πd²/4.

Often, also here, particles exhibit anexcluded volume interaction, i.e. their pair potential diverges if their centers move closer than d and is otherwise zero. Such particles are denotedhard spheres.

The hard-sphere particles constitute the glass forming system, which is described under strain flow by MCT-ITT. They are dispersed in a fluid medium, which is itself not subject to the theoretical description, but provides three concepts. First, it has a temperature T and gives rise to a Brownian motion of the particles with diffusion constant D₀ (on short time scales). Second, it is incompressible and thus gives rise to the argument that the hard-sphere system itself is incompressible. Third, it imprints a homogeneous strain field via a Stokes drag on each particle and allows to formulate a theory of dense dispersions under strain. Besides this, the solvent’s physical properties are not taken into

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consideration. Its viscosity is neglected, as well as possible hydrodynamic interactions between the particles. That means that the particle momenta are not conserved by the solvent and play no role in the collective dynamics. The underlying assumption is that for a dense system at the glass transition, the influence of the particle momenta averages out on time scales of (glassy) structural relaxation and density fluctuations in space and time are solely sufficient to describe the evolution of the system. The particle density is thus assumed to be the relevant slow variable, while particle momenta are assumed to decay fast and to be irrelevant.

The particleiis subject to interparticle potential forcesF_i=−∂_iU({r_i})and a Stokes force induced by a homogeneous time-independent strain flow of the solvent, viz. κ·r_i. The strain-rate tensor κis incompressible, i.e. Trκ= 0. It reads for simple-shear flow

κ≡γ˙sex⊗ey =





0 γ˙_s 0 0 0 0 0 0 0



, (1.1)

which is the most important case throughout this work. The e_α denote unit vectors in α direction. Uniaxial compressional flow of the formκ= ˙γu(1−3ex⊗ex)/2will play a role in Chap. 5. At this point it is setγ˙ ≡γ˙_s andγ˙_u ∝γ˙ to control every homogeneous strain field with one parameter, cf. App.A.4.3. Throughout this work, the shear rate γ˙ is set

˙

γ >0 ∀ t>t₀ and γ˙ ≡0 ∀ t < t₀, (1.2) which is denoted the idealized start-up of strain process. Without loss of generality, the start-up timet₀ ≡0.

A recently very successful experimental model system to illustrate and proof MCT-ITT are thermosensitive core-shell particles immersed in water (abbreviated PNiPAM system, see Sec. 5.3). Figure 1.2 shows a microscopic snapshot together with an illustration of simple-shear flow in a colloidal dispersion. The packing fraction can be tuned by temperature via the swelling of the crosslinked network around the particle cores. The elasticity of the network is high enough to regard the particles still as hard spheres [17].

The microscopic equation of motion for the dispersed particles is the Langevin equation for overdamped motion [44] and reads

m∂²ri

∂t² =−λ ∂ri

∂t +κ·ri

+Fi+ξ_i≈0, (1.3)

with friction constant λ and a white noise ξ_i, i.e. with zero mean and uncorrelated

fluctuations D

ξ^α_i(t)ξ_j^β(t^′)E

= 2λ k_BT δ_i,jδ_α,βδ(t−t^′), (1.4) where α, β indicate directions and deltas are Kronecker/Dirac deltas.

The Fluctuation-Dissipation theorem known as the Einstein relation D0 = kBT /λ holds. It is assumed to be valid for short enough times for single particles in the collective dynamics. In the overdamped dynamics of a micro- or nanoscopic dispersion,

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γ=∆x/y

Fig. 1.2:Left panel: thermosensitive core-shell particles immersed in water as model system for a colloidal hard-sphere glass; the picture is taken from Ref. [17]. See Refs. [16–18] for a detailed description of their temperature controlled radius and packing fraction. The right panelshows a sketch of simple-shear flow imprinted on a colloidal dispersion by the solvent, with resulting shear stressσxy on the colloids.

the coefficient m of the inertial force (i.e. the particle mass) is much smaller than the other force coefficients. Hence, the left side of Eq. (1.3) can be set to zero. As r_i is now a stochastic variable gouverned by a first order equation of motion in time, the corresponding time evolution equation of its probability density function Ψ({r_i}, t) is the Smoluchowski equation [45].

1.2.2 Vitrification dynamics and mode coupling theory

MCT focuses on the dramatic slowing down of the structural dynamics when approach- ing a critical glass-transition packing fraction φ_c. Below it, structural correlations relax to zero with a structuralα relaxation timeτ_α, which is orders of magnitude bigger than the Brownian diffusion time τ_B of a dilute suspension. Above φ_c, density fluctuation correlations remain positive even for long times, i.e. the structure is frozen, which de- fines an ideal glass. In a hard-sphere dispersion, φ is the relevant control parameter, because temperature only influences the short-time scale of Brownian motion via the Stokes-Einstein relation. If the particles however interact with a finite potential force, an equivalent consideration for the critical temperature Tc must be added.

MCT-ITT describes structural relaxation under a (here) time-independent strain rate

˙

γ (for t > t₀). The effect of strain becomes relevant when 1/γ˙ approaches τ_α. For smaller strain rates, the structure behaves fluid like with a long-time viscosityη₀. When the Weissenberg number (or dressed Péclet number) Wi≡ γτ˙ _α is bigger than 1, strain thinning due to a strain-induced relaxation time τ_q ∝1/γ˙ occurs. This also happens in the glass, which causes density correlations to decay to zero, called shear melting. In the glass and for γ˙ → 0 a dynamic yield stress σ⁰ = σ( ˙γ → 0) > 0 can be observed,

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which indicates an infinite viscosity. If the strain rate is too high, the bare Péclet number Pe₀= ˙γ d²/D₀ gets bigger than 1 and a structural slowing down is prevented. The particle dynamics is then not well described by MCT’s only input quantity, the equilibrium structure factorSq. Thus MCT-ITT should be used for Pe0 ≪1 and Wi&1.

MCT-ITT implicitly assumes to be in an infinitely aged state, i.e. prior to straining the state of the system is unique and the start-up time of strain t₀ is arbitrary. A dependence of the transient regime, i.e. of the stress overshoot on ageing, like observed in experiments [46, 47] and simulations [48], can thus not be examined by MCT-ITT.

However, MCT-ITT is capable of describing residual stress after switch-off of an applied shear rate [49], but which is not the issue of this thesis.

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2 Microscopic mode coupling theory:

integration through transients

This chapter gives an overview of mode coupling theory with integration through transients (MCT-ITT). A closed set of integro-differential equations is derived, which make it possible to calculate non-equilibrium expectation values of macroscopic observables like the stress tensor or (transient) structure factor (SF) distortions. Calculations of microscopic observables, like single-particle mean square displacements (MSD) [23, 50], will appear in this work only as side topic, Secs.5.1.2 and 5.5.

Mode coupling theory has been developed by Wolfgang Götze, Lennart Sjögren, and co-workers to describe, starting from microscopic dynamics, the glass transition of glass- forming liquids in the equilibrium/quiescent state, e.g. for colloidal hard-sphere dispersions. A detailed review of their work is Ref. [51]. The ITT approach was developed by Matthias Fuchs, Michael E. Cates, and co-workers to apply the principles of MCT in a (strongly) out-of-equilibrium environment of strain flow [52,53] and more generally use them for macro- and microrheology [54]. This includes the description of a nonlinear rheological behaviour after start-up of a constant shear process, which is the constituting case of this work. Section 2.3 will conclude with a closed set of equations of motion (EQM) for the transient density correlator Φ_q(t), which is the heart of MCT. To derive this closure approximations is a lengthy task and, even when well explained, challenging to recalculate step by step. It is not the aim of this work to proof the plausibility of this approximations, but rather to illustrate their implications and perform the technical implementation of the resulting MCT equations in three dimensions (3d). Therefore it will only be sketched in the following sections, how to arrive at the desired EQM. A detailed description of the necessary steps and their motivations can be found in Ref. [51]

(quiescent MCT), in Refs. [50,53,55] (MCT under shear), and a most recent formulation for time-dependent, universal (but homogeneous) strain geometry in Ref. [56].

The notation of this chapter is closer to Ref. [56], because a universal strain geometry is regarded in Chap.5, with compressional flow in Sec.5.4. Also MCT-ITT is illustrated in its most recent formulation. However, most of this work deals with a time-independent, simple-shear geometry and many equations can be simplified and adapted to this special case, which is reviewed in Refs. [53, 57]. It is thus a contribution of this chapter to illustrate the MCT-ITT equations from Ref. [56] in the desired simplified form.

For the theoretical generalization and experimental validation of a schematic MCT (theF₁₂^{( ˙}^γ)model) see Chap. 5.

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2.1 Smoluchowski dynamics in strain flow

A Brownian dynamics simulation of colloidal particles typically uses the Langevin equation (1.3) as microscopic starting point [22, 58]. As MCT is a statistical theory of macro- and microscopic response functions, the Smoluchowski equation for the probability density function (PDF) Ψ({r_i}, t) of the system in phase space is the fundamental microscopic equation. Because the particle momenta decay instantly in the overdamped dynamics, the phase space variable Γ reduces to the set of all particle trajectories {r_i} (Γ→ {r_i}).

2.1.1 Probability density function The Smoluchowski equation reads [44]

∂_tΨ(Γ, t) = Ω Ψ(Γ, t), with Smoluchowski operator Ω =X

i

∂_i·

D₀∂_i− D₀

k_BTF_i−κ(t)·r_i

. (2.1)

Smoluchowski and Langevin equations (with white noise) are equivalent descriptions of an overdamped system [45].

Throughout this work (exceptions noted explicitly), it is chosen d = D₀ = k_BT = 1 as normalization for the appearing length scales (d), time scales (d²/D₀), and energy scales (kBT). This renders the numerical values of Pe0 = ˙γ d²/D0 and γ˙ equal in the actual context and their notion becomes commutable. As already mentioned, a time- independent, simple-shear flow geometry is regarded, so the aspect of time-dependence (of κ(t)) can be dropped for simplicity. It is reduced to the instantaneous start-up process, Eq. (1.2); see Ref. [56] for a time-dependent MCT-ITT.

The Smoluchowski operator Ω can be separated into an equilibrium term Ω_e and a non-equilibrium term δΩ according to

Ω = Ω_e+δΩ, with Ω_e=X

i

∂_i·(∂_i−F_i), δΩ =−X

i

∂_i(κ·r_i), and Ω_eΨ_e(Γ) = 0.

(2.2)

The familiar Boltzmann distribution Ψ_e(Γ) ∝ exp (−U({r_i})) solves the Smoluchowski equation in equilibrium with potential U({ri}). Thus, for the non-equilibrium term it holds

δΩΨ_e(Γ) =κ:σˆΨ_e(Γ), (2.3) with microscopic-stress-tensor elements σˆ_αβ =δ_α,β−P

ir^α_iF_i^β, Eq. (2.16), and a:b≡ P

αβa_αβb_αβ. Because of incompressibility (Trκ = 0), the term with δ_α,β vanishes and

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only the potential part of the stress tensor (viz. the second term of σˆ_αβ) remains of interest.

BecauseδΩis time-independent, a steady stateΨ_s(Γ), withΩΨ_s(Γ) = 0will be reached after a long enough relaxation time [45], denoted flow regime.

The formal transient solution of Eq. (2.1) is

Ψ(Γ, t) =e^ΩtΨ_e(Γ), (2.4)

assumed that Ψ(t) = Ψ_e for t < 0. All statistical observables could in principle be calculated withΨ(t), but as Eq. (2.4) is in general unsolvable for a many-particle system, the operator exp(Ωt) must be approximated appropriately, e.g. in dense dispersions by MCT.

2.1.2 Observables and correlation functions

Time-dependent expectation valuesf(t)of phase-space observablesf(Γ)at timet, which are themselves not explicitly time dependent, are calculated with the PDF Ψ(Γ, t) via

f(t)≡ hfit≡ Z

dΓ Ψ(Γ, t)f(Γ) = Z

dΓf(Γ)e^Ω(Γ)tΨ_e(Γ). (2.5)

Indexed bracketsh.i_tdenote an averaging with Ψ(Γ, t), while h.i denotes an equilibrium average using Ψ_e(Γ).

The definition of an equal-times correlation off with another observable greads hf^∗git≡

Z

dΓf^∗(Γ)g(Γ)e^Ω(Γ)tΨ_e(Γ) (2.6) and is in principle just a special case of Eq. (2.5). Iff_q andg_q are corresponding Fourier modes of f and g, a structure function is defined as Sq^{f g} ≡

f_q^∗gq

t

N. The complex conjugate f^∗ is used to give the expectation value the properties of a scalar product between f and g.

A two-times correlatorC^{f g} off measured att^′andgmeasured at timet > t^′is defined as

C^{f g}(t^′, t)≡ hf^∗git^′,t ≡ Z

dΓdΓ^′P(Γ, t|Γ^′, t^′)Ψ(Γ^′, t^′)f^∗(Γ^′)g(Γ), (2.7) with conditional probabilityP(Γ, t|Γ^′, t^′)of the system being in stateΓattafter evolving from state Γ^′ at t^′. Time evolution via Smoluchowski dynamics yields P(Γ, t|Γ^′, t^′) = exp (Ω(Γ)(t−t^′))δ(Γ−Γ^′), which yields after partial integration

hf^∗git^′,t= Z

dΓg(Γ)e^Ω(Γ)(t⁻^t^′⁾Ψ(Γ, t^′)f^∗(Γ). (2.8) The substitution of Ψ(Γ, t^′) by exp(Ωt^′)Ψ_e is a non-trivial step, because the operator exp(Ω(t−t^′))acts also onf^∗(Γ). In the actual context, it is sufficient to set t^′ =t₀ = 0, thus it isΨ(t^′) = Ψ_e and the two-times correlator becomes atransient correlatorC^{f g}(t).

This gives rise to asingle-time MCT. See Refs. [50,59] for a two-times MCT, which is

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necessary for time-dependent κ(t) [56] or waiting time analyses, wheretw ≡t^′−t0 >0.

Setting t=t^′ yields the equal-times correlator of Eq. (2.5).

From partial integration in Eq. (2.8) follows (with Trκ= 0) hf^∗gi0,t=D

f^∗e^Ω^†^tgE

, with (2.9)

Ω^†=X

i

∂_i·(∂_i+F_i+κ(t)·r_i). (2.10) The right hand side of Eq. (2.9) is an equilibrium average with the adjoint operator Ω^† ofΩ. Analogously to Eq. (2.2) it is defined

Ω^†_e≡X

i

∂_i·(∂_i+F_i) and δΩ^†≡X

i

∂_i·κ·r_i ^Trκ=0= X

i

r_i·κ^T ·∂_i.

(2.11)

If the equilibrium average is regarded as scalar product off andg, the adjoint equilibrium term Ω^†_e is Hermitian with respect to this average, i.e. D

f^∗Ω^†_egE

=D

gΩ^†_ef^∗E . The non-equilibrium term δΩ^† is not Hermitian in this sense.

Like Eq. (2.9), Eq. (2.5) can be integrated partially, yielding hfi_t=D

e^Ω^†^tfE

. (2.12)

In principle this partial integration corresponds to a switch from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The problem of calculating a non- equilibrium PDF Ψ(t) is replaced by the problem of evolving the observables in time.

However, equilibrium averaging is a fundamental feature of the ITT formalism.

Fluctuation modes

To exploit the translationally invariance of the system, fluctuation modes fq of the observables in Fourier space are regarded. They are given as

f_q=X

i

X_i^f(Γ)e^iq^·^rⁱ, (2.13)

where X_i^f(Γ) relates the fluctuation mode to the particles and is set by the observable.

A possibleq dependence ofX_i^f(Γ) can be neglected in the actual context. Afluctuation mode could be defined more precisely asδfq ≡fq−hfqi. However, a constant observable average hf_qi is both, unaffected by the ITT formalism and it does not affect the ITT formalism itself, as will be shown in Sec. 2.2.2. Therefore only observable fluctuations are of interest and Eq. (2.13) is denoted as such. Some observables are of distinguished interest in this work, they are presented in the following.

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Transient density correlation

The most important observable in MCT-ITT is the density fluctuation ρq = P

ie^iq^·^rⁱ, which is received in setting X_i^ρ = 1. The time dependent structure factor (SF) is the density autocorrelation function

S_q(t)≡ ρ^∗_qρ_q

t/N. (2.14)

In equilibrium, the index-vector q simplifies to a scalar index q, because the SF S_q is isotropic in equilibrium. It turns out that this equilibrium quantity is, for a fixed mean particle density n, the only external input needed (besides the Brownian time scaleD0) for the MCT-ITT formalism, which makes it clearly a microscopic, non-phenomenological theory. The value q = 0 is unimportant, one verifies that S₀ = N diverges in the thermodynamic limit, but remains constant in time. The limit q → 0 however yields as particle-number-fluctuation correlator the isothermal compressibility χ_T via S_q_→₀ = N/V k_BT χ_T [60].

The heart of MCT is the transient density correlator Φq(t), which is defined as the normalized two-times autocorrelation function C^ρ^q^,ρ^q(t)(0, t),

Φ_q(t)≡ 1 N S_q

ρ^∗_q(0)ρ_q(t)(t)

0,t. (2.15)

For a generalization C^ρ^q^,ρ^q(t)(t^′, t), with t^′ 6= 0 see Refs. [56, 59]. A strain advected wavevector is denoted by q(t); find in Sec. 2.1.3 its definition and explanation.

Stress tensor

The fluctuations of the microscopic stress tensor [61] are gained in setting (X_i^σ)_αβ =δ_α,β+1

2 X

j(6=i)

(r_i^α−r_j^α)du(|r_i−r_j|)

dr^β_i , (2.16)

withu(|r_i−r_j|)being the pair potential between particles iandj. Substituting(X_i^σ)_αβ into Eq. (2.13), using Newton’s third law and the symmetry of the stress tensor, and regarding the zero wavevector value yields the microscopic stress tensor

ˆ

σ_αβ(q= 0) =δ_α,β−X

i

r_i^βF_i^α. (2.17)

The second term is denotedpotential part of the stress tensor, because it stems from the particle interactions, while the particle motion causes a momentum flux in the system, which gives rise to a pressure term, viz. the δ_α,β. As the regarded system is incompressible, the kinetic termδ_α,β vanishes in MCT-ITT, cf. Eq. (2.3) with Trκ= 0.

This stress observable has two important roles. First, it governs the time evolution of the PDF via Eq. (2.3) and second, its expectation value (when divided by V) is the macroscopic stress tensor of the system

σ(t) = hσ(qˆ = 0)it/V, (2.18) which can be measured in rheological experiments.

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2.1.3 Translationally invariance and wavevector advection

The system is translationally invariant, hence Ψ(t) must possess this invariance. How- ever, the non-equilibrium part δΩ of the Smoluchowski operator, Eq. (2.3), is not invariant under adding a constant vector a to each trajectory r_i. Therefore a term A≡ −P

i∂_i·(κ·a) arises in the Smoluchowski operator, Eq. (2.1). In a universal strain geometry, A and Ω do not commute, but for simple shear, Eq. 1.1, they do, be- causeκ·κ= 0. Nonetheless, in making use of the homogeneity ofκ, the sumP

i∂_i can always be brought to the right hand side of each term involvingAof the series expansion ofe^Ω+A[56]. ApplyingP

i∂_ionΨ_ein Eq. (2.4) yields a termP

iF_i= 0, which vanishes due to Newton’s third law. In consequence, Ψ(t) is translationally invariant, because all terms with A vanish. This invariance must also hold, when switching from Ωto the adjointΩ^†.

An arbitrary translation of all r_i by a adds an adjoint operator A^† =P

i∂_i·(κ·a) to Ω^†. Now it becomes clear, why observable modes in Fourier space allow to make use of the translationally invariance of the system. Translationally invariance for a single observable fluctuation yields the following condition, cf. Eqs. (2.12), (2.13), and Ref. [56], e^Ω^†^(Γ^′^)tf_q(Γ^′)=^! e^i¯^q(t)^·^ae^Ω^†^(Γ)tf_q(Γ), with (2.19)

¯

q(t)≡q·e^κt, (2.20)

where Γ^′=b{r_i+a}and Γ=b{r_i}. For a translationally invariant observable, it must hold X_i^f(Γ^′) = X_i^f(Γ), thus P

j∂_jX_i^f(Γ) = 0 (all non-zero-order Taylor coefficients must vanish). The emerging constant phase factor q(t)¯ ·a must vanish for arbitrary a after averaging with Ψ_e. Hence, only forq= 0 can it beD

e^Ω^†^tf_qE

>0, i.e.

hf_qi_t=δ_q,0hf_qi_t. (2.21) Moreover is hf₀it/V = f(t). The volume V emerges, because it is f₀ = R

d^Dr f(r) by definition of the Fourier transform.

For structure functions follows the condition of diagonality, which reads De^Ω^†^(Γ^′^)tf_k^∗gq

E _!

=e^i(¯^q(t)⁻^¯^k(t))^·^aD

e^Ω^†^(Γ)tf_k^∗gq

E

⇒ hf_k^∗gqi_t=^! δ_q,khf_k^∗gqi_t. (2.22) It follows from Eq. (2.9) for the two-times correlator that

Df_k^∗e^Ω^†^(Γ^′^)tgq

E _!

=eⁱ⁽⁻^k+^q(t))^¯ ^·^aD

f_k^∗e^Ω^†^(Γ)tgq

E

⇒ hf_k^∗gqi_0,t=^! δ_q(t),k_¯ hf_k^∗gqi_0,t. (2.23) Rephrasing the above condition of k=^! q(t)¯ in words reads: The wavevector of a mode q at later time t is correlated to a mode k=q(t)¯ at earlier time t^′ (= 0). The q mode

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2.2 Integration through transients

is advected to the strain flow κ, thus q(t)¯ is denoted reverse advected wavevector. In consequence

q(t)≡q·e⁻^κt (2.24)

is denotedforward advected wavevector. The notation ofq(t)andq(t)¯ is exactly inverted compared to Ref. [56]. In this way it fits better to the notation of Ref. [53] and other works on simple, time-independent shear, which is also important here.

With Eq. (2.23) and (2.24), theq(t)in Eq. (2.15) can be validated: theΦq(t)is a two- times density autocorrelator, which measures the decay of density fluctuations in time. A density modeρq(0)has evolved intoρ_q(t)(t)at timet >0because of flow advection. The autocorrelator definition (2.15) accounts for that. If no decorrelation due to deformation or stochastic fluctuations took place, it would remain Φ_q= 1 for all times.

In a flow field of simple shear, Eq. (1.1), the forward advected wavevector reads q(t) =q−q_xγt˙ e_y, (2.25) with unit vector iny direction e_y. Figure 2.1illustrates this case.

Fig. 2.1:Illustration of shear advection taken from Ref. [53]. An initial wavevectorq(0) =qex

is advected by shear intoq(t>0) = (1,−γt,˙ 0). The wavelengthλx= 2π/qxremains constant, while λy = 2π/qy shrinks reciprocal to γt. For all˙ t, q(t) is orthogonal to planes of constant fluctuation amplitudes and their wavelength decreases with increasingq(t) =qp

1 + ( ˙γt)². Thus Brownian motion can more easily smear out advected fluctuation modes.

2.2 Integration through transients

The integration-through-transients (ITT) formalism provides a method to calculate the complete time evolution of a system under homogeneous strain flow. This is a very powerful tool, because calculating a non-equilibrium PDF Ψ(t) is usually possible only in some special cases or for the steady state. There is however the necessary condition of an incompressible strain flow (Trκ= 0).

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2.2.1 Generalized Green-Kubo relation

ITT uses the following operator identity to replace non-equilibrium averagesh.i_tby equilibrium averages h.i

e^Ω^†^t= 1 + Z t

0

dt^′e^Ω^†^t^′Ω^†. (2.26) It must be assumed that the system is in equilibrium at some timet₀ = 0. TheΩacting on Ψ_e is the adjoint ofΩ^†, hence it follows from partial integration

De^Ω^†^t^′Ω^†. . .E ^(2.2)_(2.3)

= D

κ:σˆe^Ω^†^t^′. . .E

. (2.27)

Using this identity together with Eq. (2.9), (2.12), (2.21), and (2.22) yields for observable expectation valuesf(t) and structure functionsSq^{f g}(t)

f(t) =hf_q=0it/V =hf₀i/V + 1 V

Z t 0

dt^′D

κ:σˆe^Ω^†^t^′f₀E

(2.28) S_q^{f g}(t) =

f_q^∗gq

N+ 1 N

Z t 0

dt^′D

κ:σˆe^Ω^†^t^′f_q^∗gq

E. (2.29)

A generalization of the above scheme to two-times correlations is not necessary, because in the actual context only transient correlators are regarded. The time evolution of the transient density correlator Φ_q(t), Eq. (2.15), as a closed set of equations of motion is depicted in Sec.2.3. It is the fundamental derivation of MCT-ITT.

Equation (2.28) is a generalized Green-Kubo relation, which can be seen, when using the microscopic stress tensor σˆ itself as observable f. All deviatoric terms vanish in equilibrium. A macroscopic deviatoric stress tensor σ evolves due to a conjugated field, which is the strain tensor κ. The transport coefficient, viz. the proportionality factor between observable and conjugated field, is a non-equilibrium autocorrelation (accounted for with e^Ω^†^t) between fluctuations of the microscopic version of the observable itself, viz. σ. This Green-Kubo relation is denotedˆ generalized, because the underlying non- equilibrium averaging allows for nonlinear stress-strain relationships [62]. The fact that equilibrium averaging is also used for an ideal glass, which is non-ergodic, is a subtlety addressed again in Sec. 2.3.2. An assumption must be made that a Boltzmann-like PDF Ψ_e holds in a well relaxed glass state.

2.2.2 Projection on density modes

It is assumed, Sec.1.2.1, that in an overdamped colloidal system (without hydrodynamic interactions) the particle density fluctuationsρq are therelevant slow variables, i.e. they describe already all structural processes and all interesting quantities can be projected on them. The following steps show that the ITT time evolution can be projected into a space orthogonal to density fluctuations. A projection on density pairs ρ_kρp together with the Gaussian approximation of MCT yields the crucial quantity for this work: an approximation for the transient stress tensorσ(t).

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Because equilibrium averages are regarded as scalar products, it is possible to adopt the familiar bra-ket notation, viz. hf^∗gi ≡ hf^∗|gi, and define projection operators like in the matrix mechanics of quantum theory via

P ≡X

q

|ρqihρ^∗_q| N Sq

, Q≡1−P, and P+Q= 1. (2.30) They are idempotent and orthogonal.

The ITT time evolution automatically erases linear projections on density, which can be proven by inserting P into Eq. (2.28) and using

Dκ:σeˆ ^Ω^†^tρq

E= 0. (2.31)

The above equation holds forq6= 0, because of translationally invariance, cf. Eq. (2.21).

It also holds forq= 0, because of the incompressibility condition Trκ= 0(for isotropic pressure terms inσ) and because all deviatoric stresses must vanish in equilibrium. Oneˆ can use as well Eq. (2.27) and thatρ0 =N is a conserved particle number, Eq. (2.13), i.e.

∂_tρ0 = Ω^†ρ0 = 0. In consequence, it is also proven that projections on zero eigenvalues of the adjoint Smoluchowski operator (viz. conserved quantities) lead not to divergences of the ITT time evolutions, which follow from Eq. (2.27). Hence, a projection e^Ω^†^t → Qe^QΩ^†^tQQdoes not change the ITT time evolution.

In the following, the crucial mode coupling approximation is shown. The ITT time evolution, which lives in a space orthogonal to linear density modes, is projected on pairs of density modes ρ_kρ_p assuming that this captures the relevant slow dynamics of the system. Higher density fluctuation correlations (i.e. higher stochastic moments) either vanish or reduce to quadratic ones. This is denoted Gaussian approximation, because it assumes the density fluctuations to be Gaussian stochastic variables. The according projection operator P₂ is defined as

P₂ ≡X

k>p

|ρ_kρpihρ^∗_kρ^∗_p|

N²S_kS_p , with ρ^∗_kρ^∗_p|ρ_kρ_p

≈ hρ^∗_k|ρ_ki ρ^∗_p|ρ_p

=N²S_kS_p,

(2.32)

where a Gaussian approximation is used in the second line.

The projection P₂ is used in Eq. (2.28) to obtain f(t)− hf₀i/V ≈ 1

V Z t

0

dt^′D

κ:σQPˆ ₂e^QΩ^†^t^′^QP₂Q∆f₀E

≈ 1 V

Z t 0

dt^′X

k>p

V_k(^κ₋_t′)p(−t^′)V_kp^f

N²S_kS_p Φ_k(₋_t^′₎Φ_p(₋_t^′₎, (2.33)

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where the following definition were used:

V_k(^κ₋_t)p(₋_t) ≡

κ:σQρˆ _k(₋_t)ρ_p(₋_t)

=−Nκ:k(−t)⊗p(−t)S_k(^′

−t)

k(−t)δ_k,−p (2.34) V_kp^f ≡

ρ^∗_kρ^∗_pQ∆f₀

⇒ V_kp^σ =−Nk⊗pS_k^′

k δ_k,−p (2.35) Dρ^∗_k′ρ^∗_p′e^QΩ^†^tQρkρp

E≈N²S_k(₋_t)S_p(₋_t)Φ_k(₋_t)Φ_p(₋_t)δ_k(₋_t),k^′δ_p(₋_t),p^′ (2.36) The last line is the central approximation of quiescent MCT and is also used in MCT- ITT; cf. Ref. [51] and also Ref. [63] for colloidal fluids. For a more detailed calculation of the vertices V_kp^κ and V_kp^f or V_kp^σ see Ref. [57]. The SF derivative with respect to the index is denoted by S_k^′ ≡ dS_k/dk. Equation (2.33) holds also for structure functions, when replacing f by f_q^∗g_k (and V by N, cf. Eqs. (2.28) and (2.29)). The fluctuation of f₀ is redefined as ∆f₀ ≡f₀− hf₀i. Even though constant terms hf₀i are erased by the exact ITT time evolution, Eq. (2.28), the approximate projection on P₂ could restore them. This causes a problematic term in the time evolution of the structure factorSq(t), cf. Eq. (2.44). Fortunately there is no such impact on the stress tensor σ(t), because hσi = 0holds in equilibrium (the pressure term is neglected). Note that there is a sign error (and also a N missing) in the vertices in Ref. [56], which fortunately cancels.

2.2.3 Stress tensor in MCT-ITT

Before showing the MCT-ITT approximation of the stress tensor, the Finger tensorB, also called left Cauchy-Green tensor, is introduced; it is defined via

B(t)≡E(t)·E^T(t), with (2.37)

∂lnE(t)/∂t=κ. (2.38)

The deformation gradient tensorE(t, t^′)≡∂r(t)/∂r(t^′)transforms a trajectory at earlier timet^′ to one at later timet, which is induced by the strain rate κ(t). Within this work, κ is homogeneous and time independent, alsot^′ =t0 = 0, thus r(t) = E(t)·r. Taking the time derivative and then the spacial gradient, with chain rule applied, of the last expression yields ∂_tE(t) =κ·E and thus E(t) =e^κt. The wavevector advection from Eq. (2.24) can now be expressed via q(t) =¯ q·E(t) and q(t) = q·E⁻¹(t). The Finger tensor will play an important role in the actual generalization of the schematicF₁₂^{( ˙}^γ)model in Sec. 5.1. It plays also a role in illustrating the principle of material objectivity [56], which states that a time dependent rotation of the observed body (or the observer) leaves a constitutive relation between strain and stress tensor invariant, like Eq. (2.39) (field and conjugate observable). This must be true, when all inertial effects, caused by the rotation, can be neglected, e.g. in an overdamped system.

The following approximation for the stress tensorσ(t)can be derived from Eq. (2.33)- (2.36) (withhσˆ₀i= 0 in equilibrium) as

σ(t) = Z t

0

dt^′

Z d^Dk 4(2π)^D

∂_t^′(k·B(t^′)·k) k⊗k kk(−t^′)

S_k^′S^′_k(₋_t′)

S_k² Φ²_k(₋_t′)(t^′), (2.39)

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where the Finger tensorB is used. At this point, a typical switch from aksummation to a kintegration has been performed, which corresponds to applying the thermodynamic limit, i.e. N, V → ∞, while keeping the densityn=N/V fixed. In terms of wavevector summations this means

X

k

−→ V (2π)^D

Z

d^Dk, with dimension D, (2.40) because summation and integration volume must be kept equivalent.

For simple shear, the form of Ref. [53] can be restored for the shear-stress element of the stress tensorσ_xy, viz.

σ_xy(t) = ˙γ Z t

0

dt^′

Z d^Dk 2(2π)^D

k_x²ky(−t^′)ky

kk(−t^′)

S_k^′S_k(^′ ₋_t′)

S_k² Φ²_k(₋_t′)(t^′), (2.41) which is the final approximation used for implementation in Sec.3.3.

Equation (2.41) can be used to derive a constitutive equation [57] for the shear stress of the form

σ_xy(t) = Z t

0

dt^′γ(t˙ ^′)g_xy(t−t^′,[ ˙γ]) ^(1.2)−→ γ˙ Z t

0

dt^′g_xy(t^′,γ),˙ (2.42) with ageneralized shear modulus

g_xy(t,γ) =˙ 1 2

Z d^Dk (2π)^D

"

k²_xk_y(−t)k_y kk(−t)

S_k^′S^′_k(₋_t) S_k²

#

Φ²_k(₋_t)(t). (2.43) Without wavevector advection, gxy(t,γ˙ = 0) recovers the quiescent MCT expression for the stress autocorrelation function [51]. For an ideal elastic solid, the modulusg_xy would be constant and stress and accumulated strainγt˙ would be proportional. If g_xy(t) does not depend on γ˙ and decays on an intrinsic time scale τα, the finite time integral over g_xy(t) is the long time viscosityη⁰_xy of a Newtonian fluid, stress and shear rate are then proportional. This is referred to be the Maxwell model of linear response. Viscoelastic media exhibit a non-linear behaviour inγ˙, because of aγ˙ functionality ofgxy(t,[ ˙γ]). MCT can obviously provide a microscopic description of such viscoelasticity. Equation (2.43) will proof to be useful in Sec.3.3and yields the basis for a more general, strain-universal consideration within schematic MCT in Sec. 5.1.

2.2.4 Structure factor in MCT-ITT

It will become clear in Sec. 3.4 that structural changes and rheological quantities are most closely connected. To look at the evolution of a medium’s structure under shear means to investigate the change of the structure factor Sq(t) = Sq^ρρ(t), Eqs. (2.6) and (2.14). The vertex V_kp^ρρ for the SF reads [53] (δ_k,−p from Eq. (2.34) already evaluated)

V_kp^ρρ =

ρ^∗_kρ_kQ∆(ρ^∗_qρ_q)

/N = 2N S_q²δ_q,k−S₀

S_k+n∂S_k

∂n

∂_n(nS_q), (2.44)

Time dependent flows in arrested states