FDT Violation Quantitative – Numbers for the FDR

1 +^Nq_|˙γ|a^(z=0)

q

∂

∂tC_q(t) γt˙ =O(1). (4.47) The long time FDR

Xq(t→ ∞) = 1 +N_q(z= 0)

|γ˙|aq

(4.48) is time independent and also independent of shear rate for ˙γ → 0. It is hence non-analytic in the glass as pointed out before. As we will see later, Nq(z = 0) is negative in MCT approximations and the FDR is smaller than unity. All these findings are in agreement with the simulation results in Ref. [1].

In Sec. 4.5.5, we present a schematic model which allows to calculate the convolution in Eq. (4.32) and compare the results to the Markov approximation in Eq. (4.45). We will see that Eq. (4.45) is quite accurate but there are slight differences to the full convolution.

4.5.4 FDT Violation Quantitative – Numbers for the FDR

We will now perform MCT approximations for the memory functions nq(t) andmq(t).

While the approximation for the latter is possible on ‘standard routes’ [60], the function n_q(t) in (4.36) is more involved. It contains two evolution operators, one for the corre-lation timetand one for the transient times, which entered through the ITT approach.

It furthermore contains derivatives with respect to s, as is shown by rewriting (4.36) via the following identity, which holds for general functions f({y_i, z_i}) andg({r_i}). It is important to note that without identifying theses-derivatives, the correct ˙γ-dependence of ∆χq(t) would not be achieved¹,

X

i

(∂_i+F_i)·∂f^∗

∂r_i g= 1 2

h

Ω^†f^∗g−f^∗Ω^†g+ (Ω^†f^∗)gi

. (4.49)

1Remembering our arguments in Sec.4.5.2, the reader realizes that finding the correct ˙γ-dependence was one of the major tasks in this approach.

4.5 Mode Coupling Approach

This leads to three contributions fornq(t),

n_q(t) = n⁽¹⁾_q (t) +n⁽²⁾_q (t) +n⁽³⁾_q (t),

A large benefit from Eq. (4.49) is that the s-integration in n⁽¹⁾_q could be done directly.

n⁽¹⁾_q will turn out to be the dominating term and will be connected to the waiting time derivative from Chap.2 in Sec. 4.6. We can now approximate the formally exact expressions for nq in (4.50) and mq (4.37) by projection onto densities. Again, this physical approximation amounts to assuming that these are the only slow variables, sufficient to describe the relaxation of the local structure in the glassy regime. The special form of the functions n^(2,3)q makes it necessary to use also the triple density projector. The detailed derivation of the expressions below can be found in appendix B. We show that the functionsn⁽²⁾_q andn⁽³⁾_q have a term in common with different sign, which cancels, see Eq. (B.12). ¯n⁽²⁾q and ¯n⁽³⁾q denote the functions without these terms.

We want to stress again that in the following expressions for ¯n⁽²⁾q and ¯n⁽³⁾q , derivatives with respect tosshow up due to the SOs in the exact expressions,

n⁽¹⁾q (t) = γS˙ _q²

4 Fluctuation Dissipation Relations under Steady Shear

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

0 5 10 15 20

nq(1,2,3) (z=0)

q

n_q⁽¹⁾

−n_q⁽²⁾

−n_q⁽³⁾ Σ_i n_q⁽ⁱ⁾

Figure 4.2: The three memory functions for q = qe_z. Shown are the time-integrated functions,nq(z= 0) =R^∞

0 dt^′nq(t^′).

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1

0 5 10 15 20

n q(1,2,3) (z=0)

q

n_q⁽¹⁾

−n_q⁽²⁾

−n_q⁽³⁾ Σ_i n_q⁽ⁱ⁾ -1/2 f_q 0.05 h_q

Figure 4.3: The three memory functions for q = qey. Shown are the time-integrated functions,nq(z= 0) =R^∞

0 dt^′nq(t^′). According to our arguments in Sec.4.6.1,n⁽¹⁾q (z= 0) should coincide with −1/2fq. This would lead toXq = ¹₂ in Fig.4.4. For the dashed line see the discussion in Sec.4.6.2.

4.5 Mode Coupling Approach correlation function connected to the structure factor via the Ornstein-Zernicke equation S_k = 1/(1−nc_k) [29]. The memory function mq is to lowest order in ˙γ approximated by

mq(t) = S_q 2nq⁴

Z _∞

0

dt

Z d³k

(2π)³S_k(−t)−qS_k(−t)V_qk⁽²⁾V_qk(−t)⁽²⁾ Φ_k(−t)(t)Φ_k(−t)−q(t). (4.52) From the expressions in (4.51) one can now see the earlier proposed properties, Eq. (4.42).

The functionnq can schematically be written nq(t) =aγ˙γ tf˙ (t) +b

Z _∞

0

dsγ˙γs g(s)˙ h(t). (4.53) The first term in (4.53) corresponds to n⁽¹⁾q , the second term to n^(2,3)q . f(t), g(t), h(t) are functions of t/τ in the liquid and ˙γt in the glass. Eq. (4.42) follows. The fact that the terms in (4.53) start linearly with t and srespectively comes because V⁽²⁾ in Eq. (4.51) is symmetric ink_x,V⁽²⁾ =V⁽²⁾(k_x²), and becauseV⁽¹⁾ at timet= 0 (ors= 0) is anti-symmetric in k_x, V⁽¹⁾(−k_x) = −V⁽¹⁾(k_x), and the property n⁽¹⁾_q (t → 0) ∼ γ˙²t follows after integration overd³k. The linear increase with time follows for example from k_y(t) =k_y−γtk˙ _x. We see thatn⁽¹⁾q has a very different form compared ton^(2,3)q , but the Laplace transforms atz= 0 are similar.

For the numerical evaluation of Eqs. (4.51) and (4.52), the transient correlator Φq(t) and the static structure factors S_q and S^{( ˙}_q^γ) are needed. As a purely technical simpli-fication, we use the isotropic approximation [27], which reads for long times in glassy states

Φ_q(t)≈Φ_q(t) =f_qe^{−chqfq|γ|t˙} , (4.54) with the non ergodicity parameterf_q and the amplitude h_q. Eq. (4.54) is in accordance with the expansion near the critical plateau, see Sec.1.6.2. The parametercis related to the parameters in theβ-equation, c=p

c^{( ˙γ)}/(λ−1/2), we usec= 3 [27]. For the static equilibrium structure factor, we use the Percus-Yevick closure [29], and approximate S_q=Sq^{( ˙γ)}, which holds astonishingly well at small shear rates [1], although the structure is nonanalytic [61]. In the limit of small shear rates, the contribution of the short time decay of the correlators to the above expressions vanishes. The above expressions were evaluated using spherical coordinates with gridk_max = 50, ∆k= 0.05 and ∆θ= ∆φ= π/40 or smaller. The time grid in botht andswas ˙γt= 2^i/4/10¹⁰, starting from i= 90 corresponding to ˙γt≈6 10⁻⁴.

Figs.4.2and4.3show the Laplace transforms of the memory functions,n^(1,2,3)q (z= 0), for thez- and they-direction respectively, at the critical packing fraction. It can be seen that the sum of all three functions is negative, this will lead to an FDR smaller than unity. Also, the sum is very nearly equal ton⁽¹⁾q , sincen^(2,3)q are smaller and in addition to this partially cancel each other. We do not show the function m_q, which is with Eqs. (4.54) and (4.39) given by

mq(z= 0) = f_q²

ch_q|γ˙|. (4.55)

4 Fluctuation Dissipation Relations under Steady Shear

0.6 0.7 0.8 0.9 1

0 5 10 15 20 25 30

Xq(t→∞)

q

z-direction y-direction

Figure 4.4: Long time FDR as function of wavevectorqfor density fluctuations in z- and y-direction in the limit of small shear rates. The packing fraction is the critical one,φ=φc.

We verified this by comparison to the result of Eq. (4.52). Now we can compute the value of the long time FDR in Eq. (4.48), which is given with (4.54) and (4.40),

X_q(ε≥0,γ˙ →0) = lim

˙

γ→01 + f_qn_q(z= 0)

|γ˙|c h_qm_q(z= 0). (4.56) This is shown in Fig. 4.4for the parameters given above and n_q(z= 0) ≈n⁽¹⁾q (z= 0).

The FDR ranges between zero and unity for all wavevectors in accordance with Ref. [1].

Its dependence on wavevector is stronger than found in Ref. [1], where a q-independent FDR was proposed. Given the complexity of the involved functions, the results are still satisfying. In Sec.4.6, we will show that the findings of Chap.2lead to aq-independent FDR in first approximation and to q-dependent corrections.

Im Dokument Properties of Non-Equilibrium States : Dense Colloidal Suspensions under Steady Shearing (Seite 70-74)