Local fourfold order

As the system does not exhibit any long-range orientational order in the binary case, the local fourfold order is now investigated. The explicit fraction of SQ unit cells found in the local order is discussed rst. To compare this order for dierent parameters, three selection criteria for the occurrence of square unit cells are introduced:

1. Selection of congurations with 1 small particle surrounded by 4 big particles 2. Fourfold local bond order parameter Ψ of congurations selected in 1)

3. Bond length deviation parameter b of congurations selected in 1)

In the rst criterion it is checked whether the four closest particles of a small particle are big ones. The fourfold bond order parameter of these congurations for the second criteria is dened as

where θ_{N N} are the angles of the four bonds between the small bead in the cen-ter and the surrounding big particles relative to a xed reference. The bond order parameter p=√

Ψ^∗Ψ is a measure for the local fourfold symmetry. For perfect right angles this parameter is p= 1. All other congurations have lower values p >0. To characterize a SQ square, this criterion alone is not sucient since other congu-rations, like e.g. a rhombus, have right angles too. Therefore, in the third criteria the bond length deviation parameterb is introduced as

b= 1

with the four bond lengthsl_{N N} and their average lengthl. For a perfect rectangle (not only squares) this quantity becomes zero. All three criteria combined can be used as a local fourfold order parameter.

The graphs in Figure 5.15 show how sensitive these parameters depend on the interaction parameter Γ. A strong and continuous increase of order is found in both criteria. The graphs Figure 5.15A and B show the distribution of both parameters b andpfor two extreme values ofΓ. For the uid case (Γ = 5) the local order is very low as the distribution ofbis broad and centered around a value of about15%. At Γ = 662 the distributions ofb andpare strongly peaked reecting the high local order. Average values of b and p saturate for increasing Γ as shown in Figure 5.15C and D. These

Figure 5.15: Graph A: Histogram of bond length deviations averaged over the four bonds of quadrangular congurations (1 small bead in center surrounded by 4 big beads) are shown for two dierent values of Γ.

Graph B: Histogram of local bond order parameters for two values of Γ calculated only for quadrangular congurations. Local fourfold order and average bond length are strongly enhanced with increasing Γ.

Graph C: The average deviation of bond lengths in SQ congurations (4 big, 1 small) is shown versus Γ.

Graph D: The average bond order parameter p=√

Ψ^∗Ψof SQ congurations is plotted versus Γ. The dashed lines indicate the saturation values b= 5.7% and p= 0.92. The symbols (open squares, lled circles) dier because two dierent samples were analyzed.

Both samples have relative concentrations of ξ ≈45%.

saturation values are used as selection thresholds to determine whether a quadrangular conguration is labeled as square unit cell.

In the top panels of Figure 5.16 these selected SQ congurations are highlighted in particular snapshots for two dierent interaction strengths. The graph of Figure 5.16 shows the fraction of SQ unit cells normalized by the number of small particles (the maximum number of possible SQ congurations). A continuous increase in the fraction of SQ unit cells is observed indicating the tendency for crystallization. The measurements for two dierent samples agree well forΓ<220but signicantly deviate for larger values of Γ, although both samples had comparable relative concentrations.

Two reasons may explain this: i) The sample with less order forΓ>220(lled circles) happened to have a slight total drift while the other (open squares) was absolute quiescent. A small drift causes a non negligible shear in the sample which may have lead to a reduction in local order. ii) The samples have dierent preparation histories, one is cooled down very slowly (open squares), whereas the other (lled circles) was cooled down rapidly and the eld of view was varied. Therefore the coincidence at low Γand the deviation at highΓcan be explained by the non-ergodicity of the system at these strongly supercooled states on the timescale of measurement. Whereas for lower Γ the fraction of unit cells seems to be an equilibrium quantity, for higher Γ it strongly depends on the history and individual composition of the sample.

The strong occurrence of local order reveals that the structure of this glass former is directly related to crystallinity. Therefore, it is assumed that local crystallinity plays an important role for the nature of the glass transition, at least in this particular system.

Figure 5.16: Top: Snapshots for Γ = 51 (left) and Γ = 520 (right). The highlighted particles have SQ unit cell conguration. The criteria of whether a quadrangular con-guration is considered as a SQ unit cell is b < 5.7% and p = √

Ψ^∗Ψ > 0.92, the saturation values taken from Figures 5.15C and D.

Bottom: Fraction of SQ unit cells versus Γ. The symbols (open squares, lled circles) dier because two dierent samples were analyzed (bothξ ≈45%). For low Γboth sam-ples exhibit same amounts of SQ order. For high Γ, where the system is non-ergodic, the individual local particle distribution and sample history plays a role. A continuous increase in local order is observed for decreasing system temperature.

Chapter 6

Local Dynamics

In the previous chapters the average dynamics (chapter 4) and the average structure were analyzed (chapter 5) revealing the glassy character of the system. Furthermore, the local structure was investigated (chapter 5) exhibiting two characteristic ordering phenomena, partial clustering and local crystalline order. To complete the picture, this last chapter is concerned with the locally resolved dynamics of the glass forming system. It will be shown that the distribution of fast and slow moving particles is not homogeneous but highly correlated over large areas. These dynamical heterogeneities show that the qualitative relaxation mechanism is not just a local phenomenon. Fur-thermore, it is strongly dependent on the interaction strength Γ.

It is assumed that dynamical heterogeneities are directly related to the solidication process of glass formers in general as reviewed in [86].

6.1 Dynamical Heterogeneity

To investigate the local dynamics, two quantities are introduced that are plotted spa-tially resolved. They are used complementary to describe the same scenario.

Displacement-elds The starting- and end-position of all trajectories for both species is connected after a given delay time τ. The starting point is marked with a bold dot that labels the particle species by its color (big: red, small:

blue). In displacement-elds the correlated dynamic is exhibited on an absolute length scale, and spacial and directional correlation in dynamics are easily seen.

Compared to trajectory plots they appear less confusing for longer delay times as the information is reduced to starting- and end position. A disadvantage of the absolute length scale is that the displacements are very small for high interaction strengths, and correlated dynamics are almost not visible. However, 'hopping' processes are still easily observed.

98

Integrated squared displacement-elds (ISD-elds) They are used to investi-gate correlated dynamics on a normalized length scale, i.e. the individual dynam-ics of single particles in comparison to all the others in the eld of view. The ISD is dened for each particle i as

c_i = ˜c

over all integrated squared displacements of all N particles j in the eld of view.

The delay time is denoted as τ, the color code of the average ISD as ˜c, and the individual squared displacement for each particle i as ∆r²_i(τ). The ISD are plotted spatially resolved and color-coded with a dynamic range from 0−255 (black: slow, red: average, white: fast). Particle species are not distinguished in the calculation of ISD but they are plotted with dierent cover disc sizes.

The average color code was chosen c˜= 180and ISD higher than 255 are plotted with color code 255 (cuto). The integral is used to weight the displacement at each time equally. Particles that have large displacements but end up in the vicinity of the starting point still have large values of ISD characterizing them as mobile particles. The normalized scale has the advantage compared to displacement elds that information on local dynamical correlation can be still observed although the absolute displacement might be small. However, directional correlations are obscured in contrast to the displacement-elds.

In the following the dynamic spatial correlation of three measurements will be discussed for a common delay timeτ ≈1800 sec. They have dierent interaction strengthsΓbut comparable relative concentrations ξ ≈45%. The delay time was chosen such that it lies well inside the plateau of the mean square displacements for a strong supercooled sample (see Figure 4.3, section 4.3) so that mean square displacements dier consid-erably in this time interval. The displacement- and ISD-elds are plotted for three dierent values of the interaction strength Γ in the Figures 6.1, 6.2 and 6.3 respec-tively.

Figure 6.1: Top: Displacement-eld for a time interval τ = 1800sec at Γ = 5 and ξ = 41% for the whole eld of view 1158 ×865 µm². Small (big) particles are highlighted blue (red). Bottom: Integrated square displacements (ISD) are plotted (col-orcode: 0−255 black-red-white) for the same conguration. Particle species are indi-cated with dierent disc sizes. Both plots show uncorrelated dynamics of both particle species.

Figure 6.2: Top: Displacement-eld for a time interval τ = 1840sec at Γ = 110 and ξ = 48%for the whole eld of view 1158×865µm². Small (big) particles are high-lighted blue (red). Bottom: Integrated square displacements (ISD) are plotted (color-code: 0−255 black-red-white) for the same conguration. Particle species are indicated with dierent disc sizes. Both plots show strong dynamical heterogeneities.

Figure 6.3: Top: Displacement-eld for a time interval τ = 1800sec at Γ = 394 and ξ = 45% for the whole eld of view 1158 ×865 µm². Small (big) particles are highlighted blue (red). Bottom: Integrated square displacements (ISD) are plotted (col-orcode: 0−255 black-red-white) for the same conguration. Particle species are in-dicated with dierent disc sizes. Only small absolute displacements are seen in the displacement-eld but in the ISD-eld strong dynamical heterogeneities are visible.

Weak interaction: Figure 6.1 shows the dynamics for a very low interaction strength Γ = 5. The displacement-elds show that particles can move over dis-tances larger than a typical inter-particle distance in the given time interval of τ = 1800 sec. The displacements of the small particles are usually larger than that of the big ones, and all particles move in arbitrary directions mostly inde-pendent from their direct neighbors. The ISD-eld shows that the dynamics are almost uncorrelated. Black (slow) and white discs (fast) are often found in close vicinity. Dark discs are mostly identied as big particles and white discs as small particles showing the dynamical separation between species due to their Stokes friction as discussed in section 4.2.

Intermediate interaction: Figure 6.2 shows the dynamics for an intermediate in-teraction strength Γ = 110. The scenario now exhibits highly correlated and heterogenous dynamics: percolating areas spanning the whole eld of view are seen in the displacement-eld with large sections moving cooperatively in the same direction. Other parts in between stay quiescent over the observed time in-terval. Furthermore, the fast areas exhibit a nite lateral width not only strings of moving particles. In the ISD-eld these fast areas are found at the same places as in the displacement-eld. There, gradients over a few inter-particle distances are identied from fast to slow regions in lateral direction to the motion: white (fast) areas are usually surrounded by orange (average velocity) areas, followed then by red and black (slow). The contrast from slow to fast areas is quit strong and covers the full dynamical range of this analysis (0< c <255).

For this interaction strength the color coding cannot be assigned anymore to a cer-tain particle species like in the weak interaction regime. Here, fast and slow par-ticles are found of both species. The interaction strengthΓis now aboveΓ_C = 60, the value for crystallization of a one-component system of big particles [18, 19], and it is also well aboveΓ≈50where a signicant amount of local square struc-ture starts to evolve (see Figure 5.16, section 5.2.3). Indeed, the quiescent areas (black) in this example can often be identied with the stable crystallite struc-tures discussed in 5.2, hexagonal areas of big particles, square crystallites and combinations of both.

Strong interaction Figure 6.3 shows the dynamics for a strong interaction strength Γ = 394. Again, the scenario is changing. At this high interaction strength the displacement eld is hardly visible in most regions. In some areas a cooperative displacement is found but the eld exhibits less curvature than in the example described before in the intermediate regime (Figure 6.2). In some places single 'hopping' processes are identied where a single small particle has a strong dis-placement of a typical inter-particle distance l_A. The surrounding particles do not signicantly move.

The ISD-eld still shows strongly correlated and extended dynamic regions. In contrast to the intermediate regime, there is no obvious correlation between local crystalline structure and the dynamic heterogeneity. For example in the lower

left corner a large crystallite of square order is found (see displacement-eld) and the particles of this region are quite mobile compared to the other regions in the eld of view (see ISD eld).

It is suspected that the main contribution to the dynamical heterogeneities in this interaction strength regime originates from reversible elastic deformations and not from plastic ow as in the intermediateΓ-regime. Here, the delay time τ is well inside the strongly developed plateau of the mean square displacement (see Figure 4.3, section 4.3) and the whole sample is regarded as a homogenously elastic monolayer, at least on the timescale of this particular delay time τ.