α-master curves

As mentioned in section 5.4, MCT-ITT gives plausible reasons to assume that correlation func-tion approach a master curve for small shear rates and states close to the glass transifunc-tion. More precisely said: In states with Pe0 ≪ 1 and Pe≫ 1, these master curves depend on time only via accumulated strain γt.˙

Figure 5.13 shows the transient incoherent correlators from figure 5.9 rescaled by γt. The left˙ panel shows theqy = 0 and the right panel theqx =qy direction. Although the correlators are as close asε =−6·10⁻³ to the glass transition, they do not collapse on rescaling them byγt.˙ Unfortunately Pe0 ≪1and Pe≫1can’t be fulfilled, asτ_α is too small, and thus the structural relaxation still to fast, to find the scaling for the shear induced relaxation.

0

Figure 5.13.: Left: Transient incoherent density correlators for the q_x = 0 direction. Right:

Transient incoherent density correlators for theq_x=q_ydirection. Both panels show the correlators in the liquid ϕ = 0.79 rescaled by γt, for the absolute wave vector˙ |q|d = 6.06 (black) and

|q|d= 16.22 (blue). The shear rates are from left to right Pe₀ = 2·10⁻⁴, 6·10⁻⁴, 2·10⁻³, 6·10⁻³, 2·10⁻² and 6·10⁻². The correlators don’t collapse on the master function, as the structural relaxation obviates Pe≫1and Pe0 ≪1at the same time.

Going over the glassy system with ϕ = 0.81 in figure 5.14, demonstrates that the simulation curves indeed approach a master function for the system for this density. Here Pe≫ 1 always holds asτα is infinite or large compared to the window accessible in the simulation.

0

Figure 5.14.: Left: Transient incoherent density correlators for the q_x = 0 direction. Right:

Transient incoherent density correlators for theq_x=q_ydirection. Both panels show the correlators in the glass ϕ = 0.81 rescaled by γt, for the absolute wave vector˙ |q|d = 6.06 (black) and

|q|d= 16.22 (blue). The shear rates are from left to right Pe₀ = 2·10⁻⁴, 6·10⁻⁴, 2·10⁻³, 6·10⁻³,2·10⁻² and 6·10⁻². For Pe₀ . 6·10⁻³ the correlators approach a master function.

The red dashed line shows examples of the compressed exponential fits. The fit parameters are the same in left and right panel.

Comparing the properties of the master curves with MCT-ITT results directly, it is important to note that in both theory and simulation, only parts of the final relaxation process, depending on the direction, can be well fitted by a compressed exponential defined as in equation(5.49). The direction qx =qy and qx =−qy can be well fitted in the simulation, and thus these directions can be used for a comparison of the compression exponents. On the other hand in MCT-ITT the qx = 0 direction is suitable for a fit and will thus be used for the comparison. As already pointed out in the simulation, figure 5.9 and 5.11, the curves are almost isotropic and well described by a compressed exponential up to γt˙ = 0.1. For γt >˙ 0.1 the qx = 0 and qy = 0 develop a rather stretched exponential foot, while the qx =qy and the qx =−qy direction can still be fitted by the compressed exponential. Due to the isotropy the results for the exponent and the relaxation times are almost the same for q_x =q_y and the q_x =−q_y direction.

In the left panel of figure 5.15, a comparison of the relaxation timescale obtained from this fitting procedures between MCT-ITT and simulation results is shown.

10^-2

Figure 5.15.: Left: The relaxation time scaleτ_q^{( ˙γ)} of the master function from the simulation at ϕ= 0.81 for all wave vectors. MCT-ITT results from [24] are included. MCT-ITT overestimates the simulation by a factor of about 70 to 5 for small q and for large q respectively. Right: The Kohlrausch compressing exponent β_q of the master function at ϕ = 0.81 for all wave vectors.

MCT-ITT results from [24] are included.

For the simulations, the fit was done with the smallest shear rate (Pe0 = 2·10⁻⁴) accessible, which will be the ’master function’. The figure shows that, while the overall shape of the time scale as function of q is the same in the theory and simulation, the theory overestimates the relaxation time by about a factor of 70 for the smallest q. For large q, the agreement is much better as there is roughly a factor of5difference. Forq= 6.6, the difference is roughly a factor

20, i.e. 4 times larger than the deviation for the initial decay from the plateau, as discussed for figure 5.11. This additional factor of 4 can hence be attributed to the underestimation of compressed exponentials in theory.

This argument can also be observed in the right panel of figure 5.15, where the stretching exponent (in our case rather a compressing exponent) for the master functions are compared.

As discussed before, the fact that the exponent is larger than unity can be interpreted as a signature of non-stationarity, as it seems to only appear for transient quantities [16,24]. While MCT-ITT correctly captures this nontrivial feature on a qualitative basis, the exponent appears much larger in the simulations. We additionally see that the exponent in the simulations has a maximum as function of q, which can again be understood as an effect of the stress over-shoot: For large q, the functions have relaxed to zero before the overshoot sets in (compare the timescales in the left panel of figure 5.15), hence they do not feel the effect of overshoot and they are less compressed. As mentioned above, this effect seems not to be captured by the theory, as the exponents increase steadily with q. This effect might also explain the approach of theory and simulation for large q as seen in both panels of figure 5.15. It should also be emphasized that the directionqx = 0 has for most cases the least steep curves, for other direc-tions, exponents as large as nearly 2 can be observed in theory.

Finally, the anisotropy for different wave vectors will be discussed. Figure 5.16 shows the master curve obtained at ϕ = 0.81 and Pe0 = 2·10⁻⁴ shifted by different timescales γ˙ ∈ {10⁰,10⁻², 10⁻⁴, 10⁻⁶}. The correlators for the qy = 0 and qx = 0 direction develop a foot in their final decay phase. For very low q, this foot takes up about 40% of the relaxation from the plateau, while for higher qthe foot only governs the last20% of the final relaxation. These approximate values can be found by comparing the plateau height with the point, where the correlators start to split in the final decay. Unlike in the theory, seems this effect not to vanish for highq and to become rather constant, saturating at about 15−20% of the plateau height.

This is in contrast to the MCT-ITT curves shown in [24]. Here the correlators also develop an angle dependent foot at long times, and the shape of the curves is very different from a stretched exponential. Unlike in the simulation is the foot more pronounced for small wave vectors, but almost vanishes for larger wave vectors. Also, the oscillation below zero seen in figure 5.16 for the final decay of the qx = qy and qx = −qy directions doesn’t appear in the MCT-ITT curves.

0 0.2 0.4 0.6 0.8 1

10^-8 10^-5 10^-2 10¹ 10⁴

Φqs (t,t0)

t^·γ q_y=0

q_x=0 q_x=q_y q_x=-q_y

Figure 5.16.: Final decay of the transient incoherent correlators at ϕ = 0.81 in the glass for all the directions qx = 0, qy = 0, qx = qy and qx = −qy for wave vectors with |q|d = 2.99,

|q|d= 6.63, |q|d = 8.96 and |q|d = 12.6 from right to left. All correlators are at the lowest shear rate accessible Pe₀= 2·10⁻⁴ but shifted by a factor (γ˙ = 10⁰,10⁻²,10⁻⁴ and10⁻⁶ from right to left) to make them fall apart.