Dynamics and Time Scales of Higher-Order Correlations in Supercooled Colloidal Systems

The dynamics and time scales of higher-order correlations are studied in supercooled colloidal systems. A combination of X-ray photon correlation spectroscopy (XPCS) and X-ray cross-correlation analysis (XCCA) shows the typical slowing of the dynamics of a hard sphere system when approaching the glass transition. The time scales of higher-order correlations are probed using a novel time correlation function gC, tracking the time evolution of cross-correlation function C. With an increasing volume fraction, the ratio of relaxation times of gC to the standard individual particle relaxation time obtained by XPCS increases from ∼0.4 to ∼0.9. While a value of ∼0.5 is expected for free diffusion, the increasing values suggest that the local orders within the sample are becoming more long-lived for larger volume fractions. Furthermore, the dynamics of local order is more heterogeneous than the individual particle dynamics. These results indicate that not only the presence but also the lifetime of locally favored structures increases close to the glass transition.

For a total of 100 particles, Brownian motion was simulated by moving each particle in random directions. The step size of the movement follows a Gaussian distribution and was chosen so that a decorrelation of the g 2 function can be observed within 1000 steps. The path of one particle over 1000 steps is shown in Figure S1. Figure S1: Path of a simulated particle over 1000 steps.
Both, the normalized g 2 and the normalized g C functions at various q-values were calculated for this system and are shown in Figure S2. The q-values were defined by assuming a wavelength of λ = 1.5, a distance between the sample and detector of d = 5m and a detector pixel size of 50 × 50 µm 2 . A q-map was then calculated via q = 4π λ sin(arctan( r d )/2), where r is the distance from the center. The g C functions decorrelate from 1 to 0 over a time ∆t and have a very similar shape when compared to the g 2 functions. For a given value of q the g C functions always decorrelate faster than the g 2 functions. Eq.
[2] of the main manuscript was fitted to both the g 2 and the g C functions and their relaxations times τ were determined.
The quotient of the relaxation times τ g C /τ g 2 is shown in Figure S3 as a function of q. For most q, τ g C /τ g 2 is around 0.5. At q = 0.05 nm -1 , τ g C /τ g 2 decreases significantly, with a much larger errorbar. This is due to low statistics at this q, since it corresponds to the first minimum of the intensity due to the single particle form factor. This shows, that if the sample shows only diffusion and no spatial correlations, g C (∆t) decrease twice as fast as g 2 (∆t), τ g C = 0.5τ g 2 . Note that this S-2 Figure S2: Intermediate scattering functions | f (q, ∆t)| 2 and normalized g C (q, ∆t) functions for various values of q. The corresponding fits are shown as solid lines.
S-3 result agrees with the fact, that for g C squares of the intensity I are correlated whereas the intensity I is directly correlated for g 2 . Figure S3: Ratio τ g C /τ g 2 for the simulated particles.
As an example of a non diffusive system, a system showing a square symmetry that rotates around its center was simulated. The g 2 and g C functions for this example are shown in Figure S4 as dashed lines. In this case, the g2 function shows the typical oscillatory shape that has been reported in many studies (e.g. on flowing liquids and soft matter under (oscillatory) shear). As such a rotation does not alter the higher-order correlation and thus C(∆), its time correlation is constant, indicating a stable structural correlation. In a second step, diffusive motion was added to the individual particles while the rotation of the whole structure was still present. This means, that the higher-order correlation vanishes with time, as can be seen by the drop of g c with increasing rotation angle in Figure S4.

Additional analysis of the dynamics
The determined relaxation times τ are shown as a function of q in Figure S5. For all samples τ decreases with increasing q. For volume fractions φ < 0.45 a slowing down of the dynamics at q nn = 0.046 nm -1 is observed, which is the position of the first structure factor peak and is known as the de Gennes narrowing. Functions of type τ = aq p were fitted to the measured values and S-4  Figure S4: Correlation functions g 2 and g C for a simulated square rotating around its center without diffusive motion of the individual particles (dashed lines) and with diffusive motion of the individual particles (solid lines).
are shown as solid lines in Figure S5. For HS2, the functions were only fitted to the values of q < 0.04 nm, that did not show the de Gennes narrowing. Figure S5: Relaxation times τ as a function of q for HS1 (circles) and HS2 (triangles) at various volume fractions.
The KWW exponents γ, determined by fitting Eq. 2 to the measured g 2 , are shown in Figure S6, along with the exponents p of the q-dependence of the relaxation time. Together, the KWW exponent γ and the exponent p specify the dynamics of a system. We find three dynamic domains: (i) for φ < 0.45 the dynamics characterize a subdiffusive liquid (p ≈ −2 and γ < 1), (ii) for 0.45 < φ ≲ 0.55 a transition towards ballistic dynamics (−1.5 < p < −1 and γ ≈ 1) is ob-S-5 served, and (iii) for φ > 0.55 stress-dominated dynamics are found as in many (colloidal) glasses (γ >= 1.5 and −1.5 < p < −1). The boundaries of these domains, however, are not clear-cut.
volume fraction exponent exponent p Figure S6: Exponents γ and p as a function of the volume fraction φ for HS1 (circles) and HS2 (triangles). The grey, dashed lines indicate the dynamical regions that we found.
In order to check for non-isotropic dynamics that are found for stress-dominated dynamics, g 2 were calculated as a function of the azimuth angle ω. The relaxation times as a function of ω are shown in Fig S7 for  Based on the model introduced in S1 we used the function D 0 /D(q nn ) = D exp a · φ 1+b with three fit parameters D, a and b to fit our data to the model of long-lived neighbours. Note that this model relies on parameters that could be measured experimentally in the microscopy experiment (e.g. number of neighbours, g(r), spring constant k) but cannot be obtained in our scattering experiment on smaller particles. Therefore we cannot further disentangle our fitting parameter a. The fits to the data are shown in Figure S8 with the black line representing the fit to the VFT model.
In Fit 2 all three parameters are fitted, while Fit 3 sets D = 1 as expected in the dilute limit. We calculated two-time correlation functions to check our samples for dynamical heterogeneity. S2-S5 They are shown in Fig. S9 for the two highest volume fractions and for the g 2 and the g C functions. The two-time correlation functions show stable dynamics with only small modulations.
From the two-time correlations we determined the variance χ which is shown in Fig.S10 for the g 2 functions. The order of magnitude of χ is 10 −4 , which is low compared to other heterogeneous systems where variances of at least 10 −3 to 10 −1 are reported. S6-S10 Due to low statistics χ could not be calculated for the g C functions. Overall, this showed that the degree of dynamical heterogeneity is small. S-7 Figure S8: Relative diffusivity D 0 /D(q nn ) as a function of the volume fraction φ for HS1 (circles) and HS2 (triangles). The lines represent fits to different models.