Mean square displacements h∆r2(τ)i are shown in Figure 4.3 for two samples with dierent relative concentrationsξ. The left graphs result from a sample withξ= 45 % and the sample of the right graphs from ξ = 29 %. Particle species were distinguished (top and middle graphs) and not distinguished (bottom graphs). The data has sucient statistics on timescales ranging from 0.1sec up to several days. The data sets for shortest times were recorded separately without in situ data processing. This allows to measure delay times down to the maximum camera frame rate of 10Hz. Note, that the shown data has sucient statistics at the lowest delay timeτ = 0.1sec, where average displacements of less than200nmfor the big particles are clearly resolved (with lA≈23µm). For low interaction strengths the curves do not signicantly deviate from free diusion. Only a slight bend is found between the short- and long-time regime.
For increasing interaction strength Γ a pronounced plateau develops, and the onset of the α-relaxation is shifted towards longer delay times. As the time and length scales are plotted logarithmic, this demonstrates the drastic slowdown of the dynamics. The curves forΓ = 459andΓ = 527have their plateau stretched up to105 secwhich is more than a day. This means that the particles are conned to their average positions for this long time. Note, that the plateau is far below the value ofh∆r2i= 1corresponding to an inter-particle distance lA. Thus, the system can be truly regarded as arrested on the timescale of days.
A further nding from this data is the high magnitude ofΓneeded to achieve dynamical arrest in the binary system. The value of Γ has to be much higher than in a one-component sample, where a phase transition into a hexagonal crystal atΓ = 60already leads to an arrest2 [18]. The addition of small particles suppresses this dynamic arrest as seen for example from the data atΓ = 110exhibiting almost free diusion of the big particles (black curves). To reach arrest in the binary system, the interaction strength Γ has to be chosen higher by an order of magnitude.
2Although the denition ofΓused here is not the same as in the one-component system, the com-parison of the magnitudes is possible. According to the dierent denitions, they do not signicantly dier (<16%). For details see section 2.1.2.
Figure 4.3: Mean square displacements h∆r2(τ)i are shown for dierent interaction strengths Γ and two dierent relative concentrations ξ. The h∆r2(τ)i are plotted sepa-rately for the dierent particle species A and B. Graphs from top to bottom: A, B, A and B.
Left graphs: ξ = 45 %, values of Γ correspond to all graphs in the left column. Small discontinuities at τ ≈ 30sec are visible because two dierent measurements for each curve were joined. The short-time data was recorded with the highest possible camera frame rate of 10Hz for ≈30sec without 'in situ' data processing. The long-time data sets were recorded directly afterwards.
Right graphs: ξ = 29 %, values of Γ correspond to all graphs in the right column.
In the following, the question is further addressed whether a distinguished value Γc can be determined marking the glass transition, a value analogous to the critical packing fraction φd=3c = 0.58 in a 3D hard sphere system [6] or φd=2c = 0.697 for 2D hard discs [26].
Therefore, the stretching index ν is introduced, a quantity that suciently marks the correct critical packing fraction in a 3D colloidal hard sphere system [36]. The stretching index ν is dened as the exponent of h∆r2(τ)i ∝ τν in the point of inection in the mean square displacement, i.e. the slope of the h∆r2(τ)i curve in a logarithmic plot at this time (for more details see appendix B.7). Theoretically, the index ν is 1 for free diusion and 0 for dynamical arrest. For all kinds of slowed down dynamics, ν has a value in between. This index ν is plotted versus interaction strength Γ in Figure 4.4. The mean square displacements of two dierent samples with dierent relative concentrations ξ were used (shown in Figure 4.3). A linear t is chosen to extrapolate the stretching index to Γ = 0 and ν = 0. Two results are obvious: Firstly, the index ν is only decaying linearly with Γ, not comparable to a 3D hard sphere colloidal system where a very steep decay at φd=3c = 0.58is observed [36].
Secondly, it is found that the dynamics is dependent on the relative concentration ξ. The stretching index decays faster for the sample with lower relative concentration (ξ = 29 %) and is extrapolated to ν = 0 at Γc ≈ 300. The sample with ξ = 45 % extrapolates to Γc ≈ 450. Again, it is concluded that the presence of small particles uidizes the system.
The extrapolation toΓ = 0 does not lead to ν= 1 as expected for free diusion but to a lower value of ν = 0.8. An explanation might be that for low interaction strengths Γ additional repulsive interaction become relevant, mediated via hydrodynamic interaction or close encounters of the hard colloidal spheres3.
The information from the stretching index ν was obtained from intermediate timescales, from the plateaus of the mean square displacements. Now, the long-time relaxation is used to investigate the glassy dynamics, in particular the long-time diusion constantDα. From mean square displacements it is not easy to extrapolate to a linear diusive regime for strong interactions because the onset of the relaxation has just started. Another methods to extract the long-time diusion constants is tting an exponential decay T(q, τ) = exp[−q2Dατ] to the end parts of the self-intermediate scattering function FS(q, τ) as described in appendix B.7. There, the long-time relaxation is stretched over a wider range compared to the plot of the mean square displacement simplifying the extrapolation of the data. This is shown in the upper graphs of Figure 4.5 where the self-intermediate scattering function is plotted for two dierent samples with relative concentrations ξ = 45% and ξ = 29%. The Gaussian approximation is used for the scattering angle q = π, which is assumed to be suciently small for the approximation. The diusion constants are manually adjusted to describe the end part of the data where the dynamics is expected to
3Although 'hydrodynamic interaction may enhance self diusion' (K. Zahn [49]) in the system at hand, it still can be expected that it slows down dynamics compared to freely diusing particles not interacting at all (ν= 1).
Figure 4.4: The stretching index ν of mean square displacements is plotted versus interaction strength Γ for two dierent relative concentrations ξ. Particle species were not distinguished. The data points deviate from linear behavior only for the highΓ data points (cross symbols). A linear t is applied to the bold data points to extrapolate to stretching index ν = 0 (arrest of dynamics at Γ≈300 and Γ ≈450 respectively). The extrapolation to Γ = 0 results in ν ≈ 0.8, a value below ν = 1, which is expected for free diusion.
become diusive. For high values of Γ this is only possible by extrapolation of the data. A typical feature of glass formers is reected in the scaling of the long-time relaxation behavior: the shape of the ISF curve is identical beside a shift in time for all values of Γ. It obeys the time-temperature superposition [4] and not only for the long-time diusive regime but already at intermediate times when the curves depart from their plateau. This will be discussed in the following section 4.4. Here, this property is used to justify the extrapolation.
The lower graph of Figure 4.5 shows the inverse of the diusion constant Dα versus Γ on a logarithmic scale. The inverse diusion constant can be regarded as the system viscosity η(Γ) and is increased by several orders of magnitude in the measured range of Γ. This drastic increase is characteristic for glass formers. The increase of 1/Dα is described best by an exponential function corresponding to an Arrhenius-like behavior η(Γ)∝exp[A·Γ] as found for activated processes with an activation barrier A [76]. This is characteristic for strong glass formers and was already observed in [40, 41]. A Vogel-Fulcher-like divergence [76] characteristic for fragile glass formers is not observed.
Figure 4.5: Top: Intermediate scattering functions in 'Gaussian approximation' for dif-ferent interaction strengthsΓand dierent relative concentrations (left: ξ = 45%, right:
ξ = 29%). Particle species are not distinguished. Exponential decays are extrapolated from the end part of the curves where the dynamics becomes increasingly diusive for longer delay times τ. Bottom: The inverse of the long-time diusion coecient Dα as obtained from the extrapolations of the upper graphs is plotted versus Γfor both relative concentrations. An increase of relative concentration ξ enhances relaxation dynamics.
In strong glass formers the divergence of viscosity is less steep compared to fragile glass formers (see Figure 1.1 in chapter 1). This corresponds to the observa-tion from the stretching index (Figure 4.4) and the mapping parameters of the MCT comparison (Figure 4.1): the 2D system at hand is approaching the dynamical arrest slowly and not at a sudden compared to other glass formers as e.g. a 3D hard sphere glass [6], which can be regarded as a fragile glass former [77]. The reason for this is partially suspected in the softness of the dipolar potential. In a system with a hard sphere interaction potential, the potential barrier of a caged particle can suddenly di-verge to innity simply because the gap between two neighboring particles is too small for another particle to pass through. In a soft potential this barrier is always nite and increases continuously. Furthermore, it is unknown what role dimensionality plays in the relaxation scenario. Long wavelength density uctuations that are expected in 2D but not in 3D systems, might additionally enhance the relaxation dynamics [21, 22].
The relative concentration ξ has a strong inuence on the samples dynamics. The vis-cosity of the sample with relative concentration ξ = 29 % is shifted by a factor of ≈ 3 compared to the sample with ξ= 45 %. As found already in the stretching indexν, the increase of the amount of small particles signicantly speeds up relaxation dynamics.
In a molecular dynamics simulation study for binary mixtures of hard spheres an anal-ogous dependence of the long-time relaxation dynamics on the relative concentration of small particles was found [78]: for large size disparity (analogous to the high asym-metry of magnetic moments in the dipolar system at hand) a speedup of the long-time relaxation was found. For a small size disparity one nds the opposite behavior, a slowdown.