The glass transition of a selected mixture

In this section, we select one of the systems (δ = 5/7, xs = 0.5) from the preceding sections and perform a more profound MCT analysis with the aim to check some asymptotic results and to determine the ideal glass transition point within MCT. Having gathered all that information about our system we will have laid the foundation to drive the system away from equilibrium by applying external forces in chapter 5.

An approach to testing the the factorization theorem given by equation (3.15) is to consider the function [69]

X_q^αβ(t) = Φ^αβ_q (t)−Φ^αβ_q (t^′)

Φ^αβq (t^′)−Φ^αβq (t^′′) (3.27) with fixed timest^′ < t^′′to be chosen appropriately from theβ-scaling regime. This is in analogy to equation (3.26). Then equation (3.15) predicts

X_q^αβ(t) = G(t)−G(t^′)

G(t^′)−G(t^′′) +O(|σ|) (3.28) not to be dependent on wave number, to leading order in the separation parameter σ. Hence, superimposing the functions X_q^αβ(t) for different qds, one should be able to fix the two times t^′ and t^′′ uniquely such that a time window appears where all X_q^αβ(t) collapse. An advantage of this procedure is that the critical amplitude drops out and does not need to be fitted.

Figure 3.10 shows the simulation results forX_q^αα(t)at ϕ = 0.79, δ = 5/7, x_s= 0.5, t^′ = 1.5, and t^′′ = 17.9 for different wave numbers qds. Indeed, within the numerical accuracy of our simulations, for bothα=band α=s the data for differentqdscollapse onto each other within

a time window of about two decades, similar to previous results for binary Lennard-Jones mix-tures in three dimensions from Gleim et al. [69] and multi-component analyses of polydisperse quasi hard spheres of Weysser et al. [31].

-4 -2 0 2 4

10^-2 10^-1 10⁰ 10¹ 10² 10³ Xss q (t)

tD₀/d_s² -4

-2 0 2 4

Xbb q (t)

t’ t’’

qd=3.06 qd=5.86 qd=7.76 qd=10.56 qd=15.86

Figure 3.10.: Functions X_q^αα(t) calculated from equation (3.27) at ϕ = 0.79, δ = 5/7, and x_s = 0.5 by fixingt^′ = 1.5 andt^′′ = 17.9 for the big (upper panel) and the small (lower panel) particles. The MCT factorization theorem is validated by observing the data collapse for different qd_s in a time window spanning [t^′;t^′′]. Arrows indicate both timest^′ andt^′′.

A more sensitive test of MCT asymptotics is the so-called ordering rule: As in the next-to-leading order corrections to the factorization theorem the same q dependent correction ampli-tudes appear, the deviations before the collapse regime must be in the same direction as after the collapse window. Hence, correlators entering the collapse region in a certain order when numbered from top to bottom should leave the collapse window in exactly that ordering [56]. In a nutshell: The ordering from top to bottom is preserved. Figure 3.10confirms this prediction.

Now, we test the validity of theα-scaling law given by equations (3.23) and (3.22). According to these equations plotting the correlators as a function of t/t^′_σ should make the data collapse for long times on a master curve on approaching ϕ →ϕ^c from the liquid.

To determine a relaxation time τ_ϕ with τ_ϕ/τ^∗ ∝ |σ|^−γ from the simulation the averaged nor-malized correlators Φq(t) ≡ P

αβΦ^αβ_q (t)/P

αβS_q^αβ at qds ≈ 6 (at the structure factor peak of P

αβS_q^αβ) with ϕ < 0.79 have been shifted to coincide in the final decay with the highest

packing fraction ϕ = 0.79. The highest packing fraction thereby defines τϕ=0.79/τ^∗ = 1 and is our best approximation for theα-master function. Checking that τϕ/τ^∗ is independent onq we can validate the α-scaling. We have chosen qds ≈ 6 at the structure factor peak, as the strength of theα-process is maximal and a separation from theβ-process can be achieved. The right panel of 3.11 shows the exemplary result of the scaling for two different wave numbers.

The data clearly exhibit the two-step relaxation pattern of glass-forming liquids with increasingly stretched plateaus upon increasing the packing fraction. The shifted correlators approach an α-master curve with the highest densities collapsing over almost three decades in time.

0

Figure 3.11.: Left: Normalized total collective correlators from simulations for δ = 5/7,xs = 0.5, andqd_s= 6.06. The solid lines show corresponding MCT results where the packing fractions have been mapped according to equation (3.29). The inset shows the α-relaxation timescales and the fitted power law τϕ/τ^∗ = A|ϕ−ϕ^c_sim|^−γ with γ = 2.4969 (fixed from MCT), A = 1.63303·10⁻⁶, and ϕ^c_sim = 0.79481. Right: Rescaled correlators for ϕ < 0.79 to collapse on oneα-master function at long times. The rescale times are independent onqd_s.

After checking a few asymptotic results full numerical MCT calculations will be presented. As input simulated static structure factors were used for MCT to calculate the critical packing fraction ϕ^c_{M CT} ∼= 0.6920 and the exponent γ = 2.4969. The obvious mismatch in ϕ^c neces-sitates a comparison at equal separation from the transition point. That entails matching the separation parameter σ which is not easy to obtain from the simulation, but a peculiarity of this special system helps us to circumvent the problem.

In the left panel of figure 3.11 the simulation results, for the normalized total collective corre-lators Φ_q(t) are shown at δ = 5/7, x_s = 0.5, and qd = 6.1 for different packing fractions ϕ within the liquid regime close to vitrification.

The inset of the left panel of figure 3.11 shows the results for τϕ/τ^∗ obtained from the

shift-ing process seen in the right panel of figure 3.11. The straight line shows the result from a power-law fit τϕ ∼ |ϕϕ −ϕ^c_sim|^−γ with fixed γ = 2.4969 which yields the extrapolated value ϕ^c_sim ∼= 0.79481 for the critical packing fraction which is approximately 15% larger than the value predicted by MCT. For|ϕ−ϕ^c_sim|.0.01the ϕ-dependence ofτ is excellently described by the MCT exponent γ.

Although MCT underestimates the critical packing fraction, it nevertheless describes very well the ϕ-dependence of the α-relaxation process in the liquid regime close to the glass transition.

Using this information, a quantitative comparison of time-dependent correlation functions from MCT to those from the simulations can be done. For this purpose one has to take into account that MCT overestimates glass formation. It is well known that this results in predicting the glass transition at a too low critical packing fraction ϕ^c. Hence, the relevant parameter when comparing MCT and simulation results is the separation parameter σ (see equation (3.12)) which depends linearly on the distance parameter via σ=Cε. As the constant C is evaluated at the critical packing fraction with the corresponding structure factor, it is reasonable to assume that MCT does not yield the same prefactor as the simulation. For instance Flenner and Szamel found that both prefactors differ in a three dimensions binary Lennard Jones mixture [70]. In order to construct a mapping of the packing fractions ϕsim used in the simulation onto some appropriate ϕM CT to be used for the corresponding MCT calculations, we postulate that the separation parameters for both systems must be equal. This leads us to the ansatz

εM CT = (Csim/CM CT)εsim ≡A εsim (3.29) with some appropriately chosen constant A, which in this special case can be found empirically as A∼= 1. With this, all input parameters for the MCT equations are uniquely determined.

The solid lines in the right panel figure 3.11 represent MCT results for the normalized total collective correlation functions corresponding to the simulation data represented by symbols.

Corresponding results for the normalized partial correlators Φ^αα_q (t)/S_q^αα are shown in the two panels of the second row of figure 3.12. It can be observed that for the chosen wave number MCT tends to underestimate the correlation functions in the transient time regime t ∼= 2.5· 10⁻³ and overestimates the plateau values at times within theβ-scaling regime. Beside these quantitative deviations, MCT describes very well the qualitative t and ϕ dependences of the simulated correlation functions for ϕ ≥ 0.78. Note especially that in this parameter regime MCT describes the final part of theα-relaxation process also on the quantitative level correctly.

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Figure 3.12.:Normalized collective correlators from the simulations forδ = 5/7,x_s= 0.5,qd_s= 4.57 (top), qds = 6.07 (middle) and qds = 9.3 (bottom). The solid lines show corresponding MCT results where the packing fractions have been mapped according to equation (3.29) which is also used in figure 3.11. The left column shows the small particles’ normalized correlator Φ^ss_q (t)/S_q^ss while the right column shows the big particles’ correlator Φ^bb_q (t)/S_q^bb. A connection between overestimating/underestimating the plateaus, and overestimating/underestimating the α-relaxation is most obvious for the top and bottom panel.

Of course, the grade of quantitative compliance of the simulation and MCT results is also dependent on the wave number q selected and the different particles composing the mixture.

This is demonstrated in figure3.12 forΦ^αα_q (t)/S_q^αα atqds = 4.6(upper panels), atqds = 6.06 (middle panels) and atqds = 9.3(lower panels). The right column shows the correlators for the big, the left column the ones for small particles. For instance, MCT strongly overestimates the plateau values and also the α-relaxation times for Φ^bb_q (t)/S_q^bb at bothqds = 4.6 andqds = 9.3.

ForΦ^ss_q (t)/S_q^ssatqds = 4.6, on the other hand, MCT underestimates both the plateaus and the α-relaxation times. WhereasΦ^ss_q (t)/S_q^ss atqds = 9.3MCT slightly overestimates the plateaus, but excellently describes theα-relaxation processes.

The middle panels of figure 3.12 give a more detailed picture of the left panel of figure 3.11.

Both, small and big particles’α-relaxation time is captured well by MCT, with a slight overesti-mation for the small particle ones’. For both particle types the plateau values are overestimated by a few percent. All these data indicate a connection between the overestimation (underesti-mation) of the plateaus and the overestimation (underesti(underesti-mation) of the α-relaxation times by MCT as the relaxation time in-/decreases on in-/decreasing the plateau value.

0.3 0.5 0.7 0.9

0 2 4 6 8 10 12 14

βαα q

qd_s

α=bα=s

10⁴ 10⁵ 10⁶ 10⁷

0 2 4 6 8 10 12 14

ταα q

qd_s

α=bα=s

Figure 3.13.: Left: Kohlrauschβ_q^αα for the big (α=b) and the small (α=s) particles. Right:

Kohlrausch τ_q^αα for the big (α = b) and the small (α = s) particles. Both extracted from the correlators at ϕ = 0.79. The MCT von Schweidler exponent b = 0.5571 is included as dashed black line.

For completeness the remaining Kohlrausch parameters are shown in figure 3.13. In the left panel the Kohlrauschβ_q^ααstretching exponent for the big (α=b) and the small (α=s) particles is shown. For large q the stretching exponent converges to β_q^αα →0.6±0.05. Unfortunately noisy data prevents a more precise determination of the high q limit. Nevertheless the high q limit is in good accordance with MCT’s von Schweidler exponent, equation (3.20), calculated to b= 0.571 with the simulated structure factors.

The τ_q^αα relaxation times in the right panel show that the Kohlrausch fits presented before

are reliable, if one accepts a few outliers at low q (which is also the case for the stretching exponents). They smoothly vary, showing an overall decrease from the highest relaxation times for low q to the lowest for high q in accordance to what is known for three dimensional systems [31].