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 intoρq(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 Φq= 1 for all times.
In a flow field of simple shear, Eq. (1.1), the forward advected wavevector reads q(t) =q−qxγt˙ ey, (2.25) with unit vector iny direction ey. 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).
2.2 Integration through transients
2.2.1 Generalized Green-Kubo relation
ITT uses the following operator identity to replace non-equilibrium averagesh.itby equi-librium averages h.i
eنt= 1 + Z t
0
dt′eΩ†t′Ω†. (2.26) It must be assumed that the system is in equilibrium at some timet0 = 0. TheΩacting on Ψe is the adjoint ofΩ†, hence it follows from partial integration
DeΩ†t′Ω†. . .E (2.2)(2.3)
= D
κ:σˆeΩ†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 functionsSqf g(t)
f(t) =hfq=0it/V =hf0i/V + 1 V
Z t 0
dt′D
κ:σˆeΩ†t′f0E
(2.28) Sqf g(t) =
fq∗gq
N+ 1 N
Z t 0
dt′D
κ:σˆeΩ†t′fq∗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 Φq(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 eΩ†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 Ψe 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 ρkρp together with the Gaussian approximation of MCT yields the crucial quantity for this work: an approximation for the transient stress tensorσ(t).
2.2 Integration through transients
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 eΩ†t → QeQΩ†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 ρkρp 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
|ρkρpihρ∗kρ∗p|
N2SkSp , with ρ∗kρ∗p|ρkρp
≈ hρ∗k|ρki ρ∗p|ρp
=N2SkSp,
(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 2eQΩ†t′QP2Q∆f0E
≈ 1 V
Z t 0
dt′X
k>p
Vk(κ−t′)p(−t′)Vkpf
N2SkSp Φk(−t′)Φp(−t′), (2.33)
2.2 Integration through transients
where the following definition were used:
Vk(κ−t)p(−t) ≡
κ:σQρˆ k(−t)ρp(−t)
=−Nκ:k(−t)⊗p(−t)Sk(′
−t)
k(−t)δk,−p (2.34) Vkpf ≡
ρ∗kρ∗pQ∆f0
⇒ Vkpσ =−Nk⊗pSk′
k δk,−p (2.35) Dρ∗k′ρ∗p′eQΩ†tQρkρp
E≈N2Sk(−t)Sp(−t)Φk(−t)Φp(−t)δk(−t),k′δp(−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 Vkpκ and Vkpf or Vkpσ see Ref. [57]. The SF derivative with respect to the index is denoted by Sk′ ≡ dSk/dk. Equation (2.33) holds also for structure functions, when replacing f by fq∗gk (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)·ET(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) =eκt. 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 dDk 4(2π)D
∂t′(k·B(t′)·k) k⊗k kk(−t′)
Sk′S′k(−t′)
Sk2 Φ2k(−t′)(t′), (2.39)
2.2 Integration through transients
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 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 tensorσxy, viz. 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 Without wavevector advection, gxy(t,γ˙ = 0) recovers the quiescent MCT expression for the stress autocorrelation function [51]. For an ideal elastic solid, the modulusgxy would be constant and stress and accumulated strainγt˙ would be proportional. If gxy(t) does not depend on γ˙ and decays on an intrinsic time scale τα, the finite time integral over gxy(t) is the long time viscosityη0xy 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) = Sqρρ(t), Eqs. (2.6) and (2.14). The vertex Vkpρρ for the SF reads [53] (δk,−p from Eq. (2.34) already evaluated)
Vkpρρ =
2.2 Integration through transients
where the first term yields an anisotropic contribution to the SF distortion and the second term an isotropic one. The isotropic term is problematic and not yet well understood, which is possibly connected to the fact that time-independent mean values of fluctuations automatically drop out of the time evolution of Eq. (2.28), but could be re-evoked by projection on P2. In the steady state, the isotropic term implies a wrong asymptotic behaviour of the SF for largeq, viz.Sq→∞( ˙γt→ ∞) 91. Only self correlations should survive in this limit and henceSq→∞( ˙γt→ ∞)→! 1; find a closer discussion in Ref. [53].
A calculation and illustration of a similar isotropic term is discussed in Ref. [64].
In the actual context, only the anisotropic term in Eq. (2.44) is regarded for sim-ple shear flow. The (transient) anisotropic SF distortion δSq(t) ≡ Sq(t)−Sq reads (Eqs. (2.34) and (2.44) in Eq. (2.33))
δSq(t) = ˙γ Z t
0
dt′[∂t′(k·B(t′)·k)]
q(−t′) Sq(′ −t′)Φ2q(−t′)(t′) or (2.45) δSq(t) = ˙γ
Z t 0
dt′qxqy(−t′)
q(−t′) S′q(−t′)Φ2q(−t′)(t′), (2.46) where the last line holds for simple shear, Eq. (1.1), and is evaluated numerically in Sec.3.4.
This can be reformulated as time integral over the derivative of the equilibrium SFSq, weighted by the time dependence of the transient density correlator Φq(t). The latter captures the memory stored in the system of the distortion ofSq at the earlier times.
δSq(t,γ˙) = Z t
0
dt′ ∂Sq(−t′)
∂t′ Φ2q(−t′)(t′). (2.47) This equation can be decomposed into to terms, the advected derivation of Sq and the weights Φ2q(−t′)(t′). Because the former term depends just on accumulated strain γt˙ it decays on a strain scale γ∗∗, cf. Sec. 5.1 and Refs. [12, 42]. In an ideal elastic material it isΦq(t) = 1, therefore Ref. [65] calls thisanelasticdecay, because the affine distortion annihilates structural correlations at long times, i.e.Sqanel(t→ ∞,γ)˙ →1. This behaviour is elastic (but nonlinear) in that sense that reversing the flow at some intermediate time reverses the effect. IfΦq(t)decays shear induced afterτqor afterταin the fluid phase, the memory is lost and the structural change is irreversible. The deformation is then plastic.
In consequence, Eq. (2.47) leads to either anelastic or plastic structure deformation, with possible different strain/time scales arising from the competition of structural memory, encoded in the Φq(t), and affine motion, encoded in the advected vector q(t). Using Eqs. (2.46) and (2.47) in (2.41) yields
σxy(t) =n
Z dDk 2(2π)D
kxky
k c′kδSk(t,γ˙), (2.48) where the direct correlation function ck (given through the Ornstein-Zernike equation Sq = 1/(1 −ncq)) plays the role of an effective potential. This direct connection of structural distortion and shear stress will yield very illuminating insights in Secs. 3.4