Comparison with full numerical solution of MCT-ITT

q grid approximations, which must be done for the full numerics. However, anticipating the discussion on Fig. 4.4,c^{( ˙γ)} = 1.83 seems even to be slightly too large and therefore the value of 3 from Ref. [41] must be regarded as an even worse approximation.

Figure4.2shows a calculation of the anisotropic amplitudeGˆq^afrom Eq. (4.7) in 2d and 3d. The amplitudes of both dimensions show a quantitatively similar behaviour for q6 12.6, while for largerqthe 3d amplitudes are clearly smaller. This shows, that amplitudes, like for the mentioned final-decay-regime oscillations in Sec.3.2.1, are smaller in 3d, which implies that particles circumvent each others more easily in 3d. The qualitative behavior is similar, which supports the claim that the essential physics of a three dimensional system under shear is already described within a two dimensional shear plane.

ææ

Fig. 4.2:Anisotropic amplitudeGˆq^a of the density correlatorΦq(t), Eq.(4.1), in 2d(blue discs) and 3d(red squares). The symbols are connected with straight lines. The 2d data possesses an error of±10%depending on the method of numerical derivation used for theS_q^MHNC.

4.3 Comparison with full numerical solution of MCT-ITT

This section validates the newly developed anisotropic contribution to the β master equation under shear by comparison to the full numerical solution of MCT under shear in 2d and 3d. The full numerics in 2d was done by Matthias Krüger, based on his works [23,50]; see also Sec. 5.2and Ref. [42] for his according rheological calculations.

With all theβparameters from Tab.4.1and anisotropic amplitudesGˆq^afrom Fig.4.2at hand, a comparison ofG(t) + ˆGq^aγt˙ with(Φq(t)−f_q^c)/hqfrom the 2d full numerics can be made and is presented in Fig.4.3for exemplary wavevectors at the first and secondS^MHNC_q peaks. The asymmetric term in theβanalysis shows a good qualitative and quantitative correspondence within the β scaling regime. Especially the quadrupolar nature of the

4.3 Comparison with full numerical solution of MCT-ITT

small strain regime, i.e. the elastic regime, can be verified in the full numerics via this comparison. As theβ analysis requiresγt˙ ≪1, faster decaying correlators match better than slower decaying ones. With growingγt, the symmetries in the SF distortion change,˙ see Sec. 3.4, and also Φq(t) loses the quadrupolar qxqy symmetry of the β correlators.

As mentioned above, c^{( ˙γ)} = 4.67 is overestimated compared to c^{( ˙γ)} = 3.36 of Ref. [23], however,G(t)decays only slightly too fast. Forε= 10⁻³, small deviations in the plateau values f_q occur. This is due to the fact that G(t) is a perturbation of order √

ε, cf.

Eq. (B.11), and higher order contributions inεalready show their influence.

q=6.6 numerical results (Φq(t)−f_q^c)/hq (dashed lines), but for different wavevector moduli q (see panels). Green color labels the q orientation of qx = qy, red labels qy = 0 and grey qx = 0 (they separate in the full numerics for increasingγt), and˙ magentalabels qx =−qy. The sets {A, B, C, D} correspond toγ˙ = 10{−9,−9,−7,−3}, and ε={10⁻³,0,−10⁻³,−10⁻³}, respectively.

The C curves show an isotropic fluid decay of Φq(t), with Wi ≪ 1. The critical quiescent correlatorΦq(t)is shown assolid blue line. Both panels share the same y-axis label.

A complete comparison of anisotropic amplitudesGˆ^aq extrapolated from the 2d calcula-tions with the actualβ analysis forq 612.8is shifted to App.B.4, because of numerical subtleties, which makes the comparison rather technical.

Figure4.4shows a comparison of the anisotropicβ correlator with the 3d full numerics of Chap. 3 in the same manner as Fig. 4.3does for 2d. The wavevector moduli q = 6.6 and q = 7.4 are chosen. For the former |Gˆq^a| is maximal in 3d, just like in 2d, but with inverted sign. The anisotropic deviations to the isotropicβ analysis are most pronounced for those q moduli, but less pronounced (and less interesting than in 2d) for larger q, cf. Fig. 4.2. The q_x, q_y plane is illustrated, because every other inclination angle yields just a prefactor of sin²(ϑ_q), Eq. (4.7). The qualitative and quantitative agreement is also in 3d very high, especially the change in sign of Gˆq^a around the SF peak of q = 7is the same in β analysis and full numerics. The quadrupolar q_xq_y symmetry of the small strain regime is evident in β correlators and 3d full numerics. According to Sec. 3.4, the full numerics becomes non-quadrupolar for increasing strain, while the β-correlator

4.4 Summary

anisotropies remain quadrupolar. An overestimation of c^{( ˙γ)}= 1.83 can be verified, even if this value ofc^{( ˙γ)} is smaller than in 2d. This justifies thatc^{( ˙γ),i}≈3from Ref. [41] is too large. Besides the largec^{( ˙γ)}, thoseβ correlator modes which increase in value due to the qxqy symmetry, decay much too slow compared to the full numerics, because they leave the range of theγt˙ ≪1regime. The smallerc^{( ˙γ)}(compared to 2d) causes then a too slow decays ofG(t). Also in 3d, small deviations inεdecrease already the agreement with the full numerics noticeably, becauseG(t) is a perturbation of order√

ε, cf. Eq. (B.11).

q=6.6 A

B

10² 10⁴ 10⁶

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

t(d²/D0) G(t)+ˆGa q˙γt,(Φq(t)−f

c q)/hq q=7.4

C

E

D

10² 10⁴ 10⁶ 10⁸

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2

t(d²/D0)

Fig. 4.4:Both panels show 3d β correlators G(t) + ˆGq^aγt˙ (solid lines) in comparison to the full numerical results (Φq(t)−f_q^c)/hq of Chap. 3 (dashed lines), but for different wavevector moduli q, see panels. For allq holdsqz = 0(qx, qy plane). Green color labels theq orientation of qx =qy, red labels qy = 0, grey labelsqx = 0, and magentalabels qx =−qy (note that red and greyseparate in the full numerics for increasing γt). The sets˙ {A, B, C, D, E} correspond to, P e0 = 10{−6,−7,−6,−3,−9}, and ε = {0,−10⁻³,10⁻³,−10⁻³,0}, respectively. The critical quiescent correlatorΦq(t)is shown assolid blue line. Both panels share the same y-axis label.

4.4 Summary

In this chapter, an anisotropic contribution to the intermediate small strain regime of mode coupling theory (MCT) under simple shear was derived theoretically and approved in comparison to the full numerical solution of MCT in two and three dimensions (2d and 3d). The anisotropic term is of quadrupolar symmetry and cannot be neglected (like in Refs. [41, 55]), especially around the first peak region of the structure factor. It was verified for the full numerics that the initial quadrupolar symmetry in the shear induced decay decreases with time, cf. Sec. 3.4. A tendency of the 3d correlators to decay at larger strains than in 2d was observed. Besides this, taking the third spacial dimension into account does not add new qualitative differences to the intermediate time regime, in which the β analysis is valid; the 3d correlators inherit the quadrupolar symmetry from the two dimensional shear-flow, shear-gradient plane.