Statics

qi = 2π

L ·ni, i∈ {x, y} (3.24) where the brackets h...i denote averaging over simulation runs and all q values with the same absolute value |q|, L the simulation box size and r_i(t) the position of particle i at time t.

For all coherent correlators shown about 160 independent systems were prepared on a cubic lattice. As the system starts from a cubic lattice, and is then inflated to the binary system with the desired packing fraction, it is necessary to wait for the system to relax before mean-ingful stationary averages can be taken. Equilibration was performed with Newtonian dynamics (without imposing the Brownian time step) for a the time interval of 10⁵ in units ofd_s/p

hv_i²i, i ∈ {x, y}. Correlation functions in Brownian dynamics were measured in a time window of 5·10³ time steps (in units of d²_s/D0).

3.2.1. Statics

The first point we want to address are the non-ergodicity parameters. A common description of the shape of the α relaxation is in terms of stretched exponential Kohlrausch laws,

Φ^αβ_q (t) = A^αβ_q exph

−(t/τ_q^αβ)^βqαβi

, (3.25)

with the stretching exponent β_q^αβ, a relaxation time scale τ_q^αβ and the amplitude A^αβ_q . For structural relaxation in equilibrium system β_q^αβ < 1 is required. The α-master function from MCT (equation(3.23)) is different from the Kohlrausch form, however the theory predicts that for large wave numbers, the two functional forms become identical, and β_q^αβ → b [66]. The relaxation time τ_q^αβ is connected by a q-dependent and a density dependent prefactor to the scaling timet^′_σ in MCT (equation(3.13)). The Kohlrausch amplitudeA^αβ_q provides an estimate for the MCT non-ergodicity parameter (F_q^c)^αβ. Since the α process starts below this plateau value, |A^αβ_q | ≤ |(F_q^c)^αβ| should hold. However, in practice, the separation of the α process from the β relaxation is not clear enough to fulfill this prediction.

Kohlrausch fits are hindered by some subtle problems: Lacking a clear separation of the α process, the fit parameters enclose a dependence on the fit range. A priori, it is unclear how to choose the optimal fit range, as for very long times one expects the relaxation to become (non stretched) exponential again, and for short times, deviations originating from the β

re-laxation. Thus the fit range was fixed, so that the parameters only exhibit the weakest (the region were they are almost constant) dependence on the boundaries. This procedure leads to tD0/d²_s ∈ [15; 5000] for δ = 5/7 and ϕ = 0.79 and with the various xs. For δ = 1/3 and

Figure 3.5.: Left: Normalized critical non-ergodicity parameters of the simulated collective den-sity correlatorsΦ^bb_q (t) of the big particles. Right: Normalized critical non-ergodicity parameters of the simulated collective density correlatorsΦ^ss_q (t) of the small particles. All non-ergodicity pa-rameters were extracted from Kohlrausch fits for the size ratioδ = 5/7and the packing fraction ϕ= 0.79. For small and big particles the non-ergodicity parameters increase upon increasingxs. Number concentrations of the small particles are as labeled in the legend. The solid red lines give the MCT results calculated with a simulated structure factor input at ϕ^c_{M CT} = 0.6920and xs= 0.5.

Figure 3.5 shows approximate values for the normalized critical non-ergodicity parameters (F_q^c)^bb/(S_q^c)^bb for the big (left panel) and small discs (right panel) at δ = 5/7 and differ-ent values for xs, extracted from the simulation data via the Kohlrausch fitting method just discussed. In both panels the MCT critical non-ergodicity parameters forxs = 0.5are included.

The MCT calculations were performed by David Hajnal with simulated structure factors of exactly the same system as input, yielding a critical packing fraction of ϕ^c_{M CT} = 0.6920. On a qualitative level, the MCT results are in good agreement with the simulation results concerning the overall q-dependence. For the big particles, the relation |A^αα_q | ≤ |(F_q^c)^αα| is well fulfilled for all qd_s except for some outliers. The same holds for the small particles but for qd_s .5the Kohlrausch fit yields larger estimations for the non-ergodicity parameters. Both particle sizes show an increase of the amplitudes extracted from simulation results upon increasingxs, which is on a qualitative level in agreement with the MCT results [43, 44], see also effect (iii) from above.

0

Figure 3.6.: Left: Normalized critical non-ergodicity parameters of the simulated collective den-sity correlatorsΦ^bb_q (t) of the big particles. Right: Normalized critical non-ergodicity parameters of the simulated collective density correlatorsΦ^ss_q (t) of the small particles. All non-ergodicity pa-rameters were extracted from Kohlrausch fits for the size ratioδ = 1/3and the packing fraction ϕ = 0.81. A pronounced increase of the non-ergodicity parameters upon increasing x_s can be seen. Number concentrations of the small particles are as labeled in the legend. The solid black lines give the MCT results calculated with a simulated structure factor input atϕ^c_{M CT} = 0.6991.

Now, we have a look at the system with radius ratio δ = 1/3. Figure 3.6 contains the ap-proximate values for the normalized critical non-ergodicity parameters(F_q^c)^bb/(S_q^c)^bb for the big discs (left panel) and for the small discs (right panel) and different values forx_s, extracted from the simulation correlators with the Kohlrausch fitting procedure. Additionally in both panels the critical non-ergodicity parameters calculated with MCT were included for the system with xs = 0.5.

Again the MCT calculations were performed by David Hajnal with simulated structure factors of exactly the same system as input, giving a critical packing fraction of ϕ^c_{M CT} = 0.6991. For the present value ofδ, we observe that the MCT calculations yield systematically smaller values for the non-ergodicity parameters, compared to the estimation from the simulation results. The underestimation of the non-ergodicity parameters may be attributed to the underestimation of ϕ^c: MCT predicts arrest at lower densities, but the non-ergodicity parameters may increase with density as the denser glass is stiffer with respect to density fluctuations. However, sim-ulation results in both panels indicate a systematic increase in the non-ergodicity parameters upon increasing x_s which is, as expected, more strongly pronounced than for the caseδ = 5/7.

Again on a qualitative level an agreement with MCT predictions, which was called effect (iii) above [43, 44], can be found.

Let us investigate the critical amplitude (following equation(3.9)) as a further interesting static

Figure 3.7.: Left: Critical amplitudes obtained by equation (3.26) for the big and small particles withq0d= 1.9. The data were extracted from the collective correlators atϕ= 0.79,δ = 5/7and x_s = 0.5. Right: Critical amplitudes for δ= 1/3and x_s = 0.5 obtained by equation (3.26) for the big and small particles withq₀d= 1.8. Solid black and red lines depict MCT results obtained with the same structure factors as in figure 3.5and figure3.6respectively.

In order to determine the critical amplitudes from the simulation correlators, one can define the function [67]: with tj chosen in the β-scaling regime which in our case is determined to tj ∈ [0.93; 16.245]

forδ = 5/7and tj ∈[2.41; 67.87] forδ = 1/3. The last equality follows from equation (3.15) and thus allows us to extract the critical amplitudes (H_q^c)^αβ up to a factor (H_q^c₀)^αβ.

In the left panel of figure 3.7 the values for the normalized critical amplitudes (H_q^c)^αα/(S_q^c)^αα are shown. They are estimated from the simulation correlators with the method described for both the big and the small discs atδ = 5/7 and xs = 0.5.

Corresponding results for δ= 1/3are shown in the right panel of figure 3.7.

While, beside the numerical uncertainty at low qd_s, the simulation data for δ = 1/3 are in a good agreement with the corresponding results from MCT, for the case δ = 5/7 we observe larger discrepancies for all qds. Again for both MCT calculations simulated structure factors of the considered system were used as input.