After typically several weeks of treatment, the sample properties are sucient for data acquisition. Then, images are processed and coordinates are extracted as described in section 3.2.2. The coordinates are tracked over time in situ to obtain the trajectories of each particle. These coordinates are stored on the hard disc of the controlling computer in data blocks each storing 1000snapshots. Every data block consists of four columns:
x-coordinate y-coordinate recording time ± particle label
The particle label additionally contains the information of the species. It is negative for small and positive for big particles.
3.4.1 'Multiple τ ' time steps
Data sets have to be stored over many days when investigating the sample at strong interaction due to the long relaxation times. Observing typically 3000 particles in the eld of view, this would exceed storage volume if the time intervalτ between snapshots was constant, e.g. τ = 1sec. Furthermore, the quantities of interest, e.g. mean square displacements, are usually analyzed on a logarithmic time scale, and therefore it is not necessary to have the same high repetition rate of snapshots for the whole time of acquisition.
Therefore, the time stepτ between particle snapshots is doubled every1000 snapshots typically starting with τ = 0.5sec. However, this requires to track all particles in situ because the time gaps between the stored snapshots become too long to track particles after data acquisition. In this way the size of data sets is limited to a few hundred megabytes, which is still convenient for the subsequent data evaluation on conventional personal computers.
3.4.2 Drift compensation
In the ideal case a at and equilibrated sample shows no net drift of the particles in the eld of view. However, at low Γ some small net drift is often inevitable causing loss of particle information at the edges of the eld of view. To compensate a possible drift, the actuators of the microscope optics mount (see Figure 3.2) are compensating the average particle displacement during data acquisition. The drift is calculated and averaged from the trajectories of all particles in the eld of view, and then x- and y-velocities of the actuators are separately adjusted by a feedback loop.
The total trajectory of the compensating actuators and therefore of the whole optics
Figure 3.15: Motor drift compensation for x- and y-direction is plotted left and the resid-ual overall drift in the sample is plotted right. The starting points (green) at the graphs origin and the ending points (red) are highlighted. The interaction strength wasΓ = 527 and the sample had a relative concentration of ξ = 29%. The recording time was 49 hours. The dark disc for comparison has the diameter of a big particle (d= 4.5µm).
mount is shown in Figure 3.15 (left). The displacement during two days is compara-ble to the size of a big particle drawn in the same graph for comparison. This shows how quiescent the monolayer is for high interaction strengths Γ. At lower interaction strengths, the net drift can be larger up to≈10µm/hour but is precisely compensated.
The accuracy of the compensation is demonstrated in the right graph of Figure 3.15.
The residual average displacement of all measured particles is plotted, and it only ex-tends over a few microns. This residual drift of all measured particles is subtracted before data evaluation.
The drift compensation and additionally the subtraction of the residual drift after measurement is essential for the investigation of the system dynamics. Average dis-placements at high interaction strengthsΓcan be much smaller than the drift (compare plateau height in mean square displacements in chapter 4 with average drift). Slight drift deviations strongly matter.
By applying the drift compensation it is assumed that the average displacement orig-inating from the system is zero in the eld of view, and only perturbations are com-pensated. However, it cannot be excluded that inherent information of the system are obscured, e.g. long wavelength uctuations as expected for 2D systems (Mermin-Wagner-Hohenberg theorem [21, 22]).
Glassy dynamics
In this chapter the dynamical properties of the 2D binary mixture of dipoles are discussed, and it will be demonstrated that it is a good model system for a strong glass former in two dimensions. The results of the comparison with Mode Coupling Theory (MCT) and experimental data are based on the publication [26].
The evolution of the relaxation in a supercooled liquid can be typically divided in three parts:
1. For short delay times particles can move freely in their cage of nearest neighbors, and therefore the mean square displacements increases linearly in time as the motion is diusive. This behavior is independent from temperature.
2. At intermediate timescales the system dynamics slows down (β-relaxation) and the particles are trapped in their cage of nearest neighbors. The mean square dis-placement or the intermediate scattering function respectively develop a plateau.
This plateau becomes more pronounced as the system temperature is lowered.
3. Finally, for long delay times the α-relaxation causes the mean square displace-ments to increase. The onset of the α-relaxation is shifted towards later times with decreasing system temperature.
The dynamics in the transition regimes between these three regions appear to be universal for most glasses [4]. The intermediate scattering function shows power law behavior in the onset of both, the β-relaxation and the α-relaxation. Furthermore, it obeys the time-temperature superposition principle, i.e. the curves of theα−relaxation collapse on a single master curve if scaled in time. These experimental observations are well described by MCT (see section 2.2).
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4.1 Comparison with Mode Coupling Theory (MCT)
The experimental system truly behaves glassy and not only in the qualitative way as just described. This is demonstrated in Figure 4.1 by a comparison of mean square displacements and static structure factors obtained from MCT and experiment respec-tively. The solid curves are obtained from static structure factors of monodisperse hard discs (see section 2.2), and the data points are calculated from the coordinates of the big particles of the experimental system. The normalization of time and area is obtained from the data set for Γ = 527 and is used for all comparisons with MCT. Therefore, the only parameter left for adjustment is the packing fraction of the discs, and that is mapped to the interaction strength Γ for best overlap of the curves. This mapping is shown in the lowest graph. Good agreement is found over many decades in time for all types of particles1. The data for Γ = 423 (top graphs) is compared to an MCT curve with a packing fraction close to but below the glass transition (² = φcφ−φc = −10−2).
φc is the critical packing fraction, where MCT shows a dynamical bifurcation for the model glass former of hard discs (see section 2.2). In that graph, the α-relaxation is clearly visible. For the higher interaction strength Γ = 527 (middle graphs) a curve in the glassy state at ² = +10−7 was found to correspond best to the experimental data.
The mean square displacement of the MCT curve is bounded and the system is not ergodic anymore. The obtained ts are remarkably good considering the simplication of mapping the binary dipolar system onto monodisperse hard discs. The position and height of the primary peak in the hard disc structure factor closely agrees with the position of the experimental partial structure factor. Further, the eective diameter of the hard discs in the comparison at Γ = 527 is Ref f ≈ 20.5µm, which is close to the experimental value of the average particle distance lA= 21.8µm. It is thus concluded that MCT captures the ratio of localization length to particle distance.
From this comparison, a critical interaction strength Γc for the glass transition is ex-pected in the range of 400 <Γc<550. However, the approach to the glass transition as seen in the mapping of Γto ² is very slow.
1Here, only the comparison of the big species is shown, but the other curves correspond beside a vertical shift when plotted in a logarithmic scale (see section 4.2).
Figure 4.1: Mean square displacements and structure factors of experimental data from 2D binary dipoles (big particles only, ξ = 30 %) is compared with 'Mode Coupling Theory' (MCT) for hard discs. The data in the upper graphs(Γ = 423) is compared to a MCT curve with ²=−10−2 very close but below the glass transition.
The data in the middle graphs (Γ = 527) is mapped to ² = +10−7, a critical curve in the glassy state. The packing fraction of hard discs is manually chosen such that best agreement for the dynamics is found. This mapping of Γ to ² is shown in the lowest graph.
The dynamics correspond over several decades in time and the rst peaks of the structure factors agree remarkably well. At Γ = 527, the unit of time is tted as τBD = 1/470sec, and the unit of length is tted as 420 µm. These values were used for all curve ts.