Structure of the systems - Searching for the glass

5.4 Searching for the glass

6.2.2 Structure of the systems

For the computation of radial distribution functions𝑔(𝑟)and partial structure factors𝑆(𝑞)in binary sys-tems the same techniques were used as for the monodisperse syssys-tems described above (cf. Section6.1.3).

Of special interest are the samples NIRM6A/B/C which have an increasing volume fractionΦ and dy-namically go from supercooled to rather solid. In Figure6.19𝑔(𝑟)and𝑆(𝑞)for those systems is displayed.

Length units of the mean interparticle distance𝑟_𝑚 are used to normalize the shift of the structure peaks due to the increasing particle density. Small particles have the label A and big particles the label B, so that AA denotes the correlation among small particles, AB the correlation between small and big and BB the correlation among big particles. Although the MSD curves of the three samples in Figure6.18 reveal that the systems are dynamically quite different, the structural changes looks very small. While one can clearly see an increase of the peak heights of the pair distribution function𝑔(𝑟), especially for

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Figure 6.19: Partial pair distribution functions𝑔(𝑟)(left panel) and partial structure factors𝑆(𝑞)(right panel) for three samples going from supercooled (NIRM6A) to rather solid (NIRM6C). A is the label for small particles and B the label for big particles. For both quantities units of the mean interparticle distance𝑟_𝑚are used so that differences in the particle density𝑛do not lead to a shift of the peaks.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Figure 6.20: Partial pair distribution functions𝑔(𝑟)and partial structure factors𝑆(𝑞)for the two very different binary mixtures NIRM9C and RPMIX5A. structure factor𝑆_tot =𝑆_𝐴𝐴+ 2𝑆_𝐴𝐵+𝑆_𝐵𝐵corresponding to the partial functions shown in the two fig-ures above.

6.2 Binary systems the principal peak, the increase is very small (almost invisible) for the peaks of𝑆(𝑞). This was already observed for the monodisperse systems (cf. Fig.6.4). The reason could be that the peaks of𝑔(𝑟)are more sensitive to the local structure, while the peaks in𝑆(𝑞)rather illustrate characteristic particle distances in the system.

In Figure6.20we see a comparison of two very different samples. All NIRM samples (including NIRM9C) consist of small and big particles with a diameter of𝜎_𝐴 = 1.0µm and𝜎_𝐵 = 2.0µm while RPMIX5A has larger particles with𝜎_𝐴 = 1.8µm and𝜎_𝐵 = 2.6µm. Since the size ratio𝜎_𝐴∕𝜎_𝐵 is closer to 1 for RPMIX5A, the principal peaks of the partial pair distribution functions (next nearest neighbour peaks) are closer together than for NIRM9C. However, a view on the partial structure factors (right panel in Fig. 6.20) reveals no qualitative differences, there is only a shift of the peak positions. A compari-son to𝑆(𝑞)for NIRM6 (right panel in Fig. 6.19) shows that the peaks for𝑆_𝐵𝐵(𝑞)(big-big) are lower for NIRM9C and RPMIX5A and the peaks for 𝑆_𝐴𝐴(𝑞) (small-small) are higher. Here the reason is clearly given by the different mixing ratios: NIRM6 has54 % big particles while RPMIX5A has42 % and NIRM9C only35 %. More big particles lead to higher peaks in𝑆_𝐵𝐵(𝑞)and analogously more small particles to higher peaks in𝑆_𝐴𝐴(𝑞).

Altogether the qualitative differences between the partial structure factors of all systems are small. An interesting remark here is that there is no significant sign for a pre-peak in𝑆_𝐴𝐴(𝑞). This means that the samples do not exhibit clusters of small particles.

Finally Figure6.21displays the total pair distribution function𝑔_tot(𝑟)and the total structure factor𝑆_tot(𝑞), where all particles in the sample are included ignoring their type. In contrast to the results for the sim-ulated low density and highly charged binary system presented in Chapter2.4.2(see Figure2.24), the structure of the systems shown here is not well comparable to a monodisperse system. The nearest neigh-bour peaks in the partial pair distribution functions are more narrow and too far from each other, so that they do not add up to a single big peak in𝑔_tot(𝑟). This results in quite small principal peaks in𝑆_tot(𝑞). Due to its size ratio being closer to1, RPMIX5A has the highest peak, while NIRM9C has the lowest one as a result of its more asymmetric size ratio (𝜎_𝐴∕𝜎_𝐵 = 0.5) and mixing ratio (𝑥_small = 0.65).

6.2.3 Comparison to MCT

It follows a very similar comparison to what was shown before for the monodisperse systems in6.1.4.

Once more, Th. Voigtmann’s MCTSolver was used to numerically solve the dynamic MCT equations with the experimentally determined (partial) structure factors as input. The procedure is very similar to the monodisperse case. An additional input is the mixing ratio𝑥=𝑁_𝐴∕(𝑁_𝐴+𝑁_𝐵). Furthermore, one needs to determine two short time diffusion coefficients𝐷^(𝐴)₀ and𝐷₀^(𝐵)for the two particle species. While 𝑥is measured within the particle detection, a fit is required for the two diffusion coefficients. As before, this is done by adjusting𝐷₀so that the MSD computed with the theory coincides with the experimental value for a short lag time𝑡near the transition to the plateau. The ratio of the diffusion constants is kept constant as𝐷^(𝐴)₀ /𝐷^(𝐵)₀ =𝜎_𝐵∕𝜎_𝐴, according to the proportionality𝐷₀∝ 1∕𝜎(Stokes-Einstein relation).

Localization length

As for the monodisperse systems the localization length𝑟_𝑠is determined from the mean squared displace-ment (MSD) at the lag time𝑡= 100s (cf. Equ.6.1). At this time all MSD curves have either reached the plateau or they are already in the regime of the𝛼-relaxation (see Figure6.18).

0.02 0.04 0.06 0.08 0.10 0.12 0.14

MCT

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m ea su re d

small big NIRM6A NIRM6B NIRM6C NIRM9A NIRM9C1 NIRM9C RPMIX5A

Figure 6.22:Comparison of localization lengths𝑟_𝑠derived from the MSD at𝑡= 100s directly measured in the experiments (𝑦-axis) and indirectly determined by solving the dynamic MCT equations with ex-perimentally determined structure factors as input (𝑥-axis). Results for small and big particles are shown together. The value for the small particles in NIRM6A is not shown (not measurable because the parti-cles moved too fast for a reliable tracking). Lengths are given in units of the mean interparticle distance 𝑟_𝑚. Error bars reflect a10 %error for MCT values, a5 %error for the experimental determination of the MSD of big particles and a10 %error for the experimental MSD of small particles.

Figure6.22 is a scatter plot comparing𝑟_𝑠 directly measured in the experiment to𝑟_𝑠 computed within MCT employing the measured structure factors presented in the paragraph above. Within the error bars the agreement between theory and experiment is very good. Remember that no fitting parameter is used in this comparison. The adjusted short time diffusion constants do not play a role for the height of the plateaus and hence have no effect on the MCT localization length, except for the two supercooled systems NIRM6A and NIRM6B, where𝑡= 100s is in the𝛼-relaxation regime. That the agreement is a lot better than for the monodisperse systems (cf. Figure6.5) is less surprising with the knowledge that the image quality of the measurements was quite good (see. e.g. Figure3.20).

In the discussion about monodisperse systems it was mentioned that MCT is rather expected to underes-timate localization lengths. Such an underestimation was not observable for monodisperse systems and now for binary systems the agreement is so good that one cannot speak of over- or underestimation at all. However, one should remember that positioning errors (although small) should still have their an influence on the partial structure factors. These errors lead to a decrease of the structure peaks and hence to higher theoretical localization lengths within MCT. But a reliable estimation and correction of the positioning errors is not easy. Their influence on the MSD is an increase compared to the true values for both experimental and MCT outcomes. Tests on rigid samples (3.3.2) indicate that the error in the experimental MSD is less than10 %. Still the influence on MCT results could be somewhat bigger due to its high sensitivity on the structure factor input. But only if the positioning errors would really lead to a considerably bigger error in the MCT outcome, the agreement between theory and experiment would be destroyed. For now, one can conclude that there is a surprisingly good agreement between measured and computed localization lengths.

6.2 Binary systems

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MS D

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small big

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NIRM6A NIRM6B NIRM6C

Figure 6.23:Mean squared displacement for small and big particles from measurements (diamonds and squares) and computed via MCT (dashed lines and solid lines) using structure factors determined from the measured particle positions. In the left panel we see curves for two systems deeper in the glassy phase, in the right panel systems close to the glass transition line are presented. In the left panel three measurements are combined for each curve, 2D measurements for short lag times, 3D measurements for longer lag times. For NIRM6A there is no experimental result for the small particles since they were too fast for reliable tracking.

Mean squared displacement

For a more detailed comparison of theory and experiment one can have a look at the mean squared displacements (MSDs) in Figure6.23. Except for the small particles in sample NIRM9A short time evolutions and plateau heights are in very good agreement. Similar as in the monodisperse case the discrepancy of about20 %in the plateau heights for the small particles of NIRM9A could have its reason in a somewhat worse quality of the microscope images, which has a larger effect on the positions of the small particles.

The relaxation seen in the experimental curves for large lag times𝑡 >5000s is certainly explainable by unavoidable fluctuations in the experiment, like for example temperature fluctuations leading to small particle drifts (<1µm in 10 hours). So the samples NIRM9A, NIRM9C and NIRM6C can be considered as being solids in the glass phase. But for NIRM6A and NIRM6B the relaxation sets in earlier so that one could interpret it as the𝛼-relaxation in supercooled liquids. The theoretical curves support this picture.

Indeed MCT results show that the ideal glass transition is crossed going from NIRM6A (Φ = 35 %) to NIRM6B (Φ = 36 %). In the experiment one would rather see the transition between NIRM6B and NIRM6C (Φ = 39 %). Still it is a remarkable success of the theory that, only using structural information, it predicts the dynamic transition point at almost the same position as experimentally measured.

Density correlators

Again, the comparison of experiment to MCT is finalized with a study of the (self-) intermediate scatter-ing functions𝐹(𝑞, 𝑡)and𝐹_𝑠(𝑞, 𝑡). The focus lies on sample NIRM9C, which is quite deep in the glassy phase.

In Figure6.24we see correlators for a scattering vector𝑞 close to the main peak of the total structure factor𝑆_tot(𝑞). The agreement between theory and experiment is very good for the self-part𝐹_𝑠(𝑞) but lacks accuracy for the coherent partial intermediate scattering function𝐹(𝑞), especially for the correlation between small and other small particles. But even there the relative deviation is only10-15 %.

10 ³10 ²10 ¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶

Figure 6.24: Sample NIRM9C: Comparison of partial intermediate scattering functions𝐹(𝑞, 𝑡)(left panel) and self-intermediate scattering functions𝐹_𝑠(𝑞, 𝑡)(right panel); once directly measured and com-puted from the particle trajectories (diamonds) and once calculated via MCT from measured structure factor input (lines). Scattering vector𝑞= 7.0∕𝑟_𝑚is chosen close to the main peak of𝑆_tot(𝑞).

0 5 10 15 20 25 30 function of𝑞at the constant lag time𝑡= 100s (a value corresponding to the plateau region of the MSD).

0 5 10 15 20 25 30

Figure 6.26: Comparison of the total intermediate scattering function𝐹_tot(𝑞, 𝑡) = (𝐹_𝐴𝐴+ 2𝐹_𝐴𝐵 + 𝐹_𝐵𝐵)∕(𝑆_𝐴𝐴 + 2𝑆_𝐴𝐵+𝑆_𝐵𝐵)once directly measured and computed from the particle trajectories (di-amonds) and once calculated via MCT (lines). The gray line shows a typical critical non-ergodicity parameter (NEP) for a monodisperse system computed within MCT using MPB-RMSA for the structure factor (system parametersΦ = 0.2 %, ̃𝑘= 8, ̃𝛾= 2500).

6.2 Binary systems The comparison proceeds with the consideration of the IMSFs as functions𝑞 for the constant lag time 𝑡= 100s in Figure6.25. That lag time was chosen for the determination of the localization length from the MSD. Again, the coincidence between theoretical and experimental curves is very good, in this case for both, self-part (right panel) and the normal (coherent) IMSF (left panel).

Finally in Figure6.26shows to the total IMSF𝐹_tot(𝑞, 𝑡), again for the fixed lag time𝑡 = 100s. For this late lag time one can interpret𝐹_tot(𝑞, 𝑡 = const.) as the total non-ergodicity parameter 𝑓_tot(𝑞) (NEP) describing the remaining (frozen in) density correlations in the solid. A very good coincidence between 𝑓_tot(𝑞) and NEPs of monodisperse systems was already observed for the simulated binary mixtures in Chapter2.4.2, considered directly at the glass transition (critical NEPs) (cf. Figure2.25). Also in the experimental case, for the sample NIRM6B close to the glass transition, one can observe a close re-semblance of𝐹_tot(𝑞, 𝑡 = 100 s)and the critical NEP of a (typical) monodisperse system. This supports the statement in Chapter2.4.2, that the physics in well mixed binary systems is very similar to that in monodisperse systems. However, the fact that crystallization is avoided makes glassy binary systems better accessible and also allows one to study well-aged glasses.

6.2.4 Glassy dynamics, dynamical heterogeneity

Mobile and less mobile regions

A good way to look at the heterogeneous dynamics of glassy samples are plots of the particle trajectories color coded with the individual localization length𝑟_𝑠as defined in Equation6.3.

Of special interest is the comparison of supercooled systems with systems in the glassy phase. This is done in Figure6.27with the two samples NIRM6B and NIRM6C. The sample NIRM9C shown in Figure6.28is even deeper in the glass. The outcome is very similar to what was observed before for the monodisperse samples RP44_D4 and RP44_B4 (see Figure6.11). Going deeper into the glassy phase the regions with mobile particles shrink to smaller sizes. Here for the binary samples they are smaller, their size is often only about2-3mean interparticle distances𝑟_𝑚.

Note that there is no dynamical separation, mobile regions mostly include both, small and big particles.

But it is clear that the individual mobility of small particles is almost twice as high, they contribute to a much higher degree in the “fast” regions than the big particles. The role of small particles is even more important for NIRM9C, where the mixing ratio is𝑁_small∕𝑁_big = 2 ∶ 1, compared to NIRM6, where it is almost1 ∶ 1. The shape of the “fast” regions is less compact than it was the case for monodisperse systems. They look more filament-like than sphere-like. One should remark that in NIRM9C the volume occupied by small particles is only1∕5of the volume occupied by big particles. For NIRM6 this ratio is even smaller with1∕10. So despite their contribution to the total volume being much smaller, they dominate the fast dynamics. One can interpret their role as that of a lubricant in a mechanical system.

Their presence prevents the system from getting stuck.

Time evolution of mobile regions

Another question is how mobile regions evolve with time. Are they fixed to their locations in the sample?

Do they move? Again a look at particle trajectory plots helps to get an impression. In Figure6.28we see a comparison of the trajectories of the same sample at the same location but with a time difference of one day between the starting points of the trajectories. The time intervalΔ𝑡covered by those trajectories corresponds to the onset of the𝛼relaxation, in the sense that the MSD starts to leave the plateau value at the corresponding lag time.

One observes that most of the mobile regions are still there after one day. Some of them changed their shape by shrinking or growing, for instance near the upper center and near the right border. One bigger region somewhat downwards and slightly right of the center almost disappeared completely. Another one in the upper left is has not changed at all. One can conclude that on the timescale of days one can indeed observe changes in the mobility landscape. But these changes are rather small, at least for a well relaxed glassy sample like the one shown in Figure 6.28, which remained untouched for more than a week. Furthermore, mobile regions cannot be described as moving, they rather behave like laser light speckles that appear and disappear. This behaviour was also reported in glassy binary colloidal systems in two dimensions [80].

Figure 6.27:Dynamical heterogeneity of NIRM6B and NIRM6C made visible in plots of the trajectories color coded with the individual localization length𝑟_𝑠(cf. Equ.6.3). In both panels red color bar applies to small particles, blue one to big particles. Time length of the trajectories is4000s; time interval between snapshots is20s and100s, respectively. Thickness of the visible box in𝑧direction is3mean interparticle distances𝑟_𝑚. Trajectories are projected onto the𝑥-𝑦plane. Deeper in the glass (NIRM6C, right panel) the size of mobile regions is substantially smaller.

Figure 6.28: Dynamical heterogeneity in two measurements of the binary sample NIRM9C. We see the same part of the sample at different waiting times𝑡_𝑤after the sample was shaken and put under the microscope. Color codes and color bars have the same meaning as in Figure6.27above. Time length of the trajectories is10000s; time interval between snapshots is50s. Thickness of the visible box in𝑧 direction is5mean interparticle distances𝑟_𝑚. Most of the mobile regions are still there after one day.

6.2 Binary systems Comparison to a hard sphere binary system

Lynch et al. [103] investigated a binary system that is very similar to our sample NIRM9C. They had par-ticles with a size ratio𝜎_𝐴∕𝜎_𝐵 = 1.1µm∕2.4µm in a mixture with a volume fraction ratioΦ_𝐴∕Φ_𝐵 = 1∕6. Using hard spheres the total volume fraction of their system was quite large withΦ = Φ_𝐴+ Φ_𝐵 = 62 %. For NIRM9C the size ratio is𝜎_𝐴∕𝜎_𝐵 = 1.0µm∕2.0µm and the ratio of volume fractions isΦ_𝐴∕Φ_𝐵 = 1∕5. The difference of their hard sphere system to the charged system presented here is that the total volume fraction in NIRM9C is less than half, withΦ = 28 %. So except charges and density they are rather similar.

Applying Equation6.1to the MSDs measured by Lynch et al. [103], one finds that the overall localization lengths in their system are 𝑟_𝑠,𝐴 = 0.18µm= 0.10𝑟_𝑚 and𝑟_𝑠,𝐵 = 0.10µm= 0.06𝑟_𝑚 (with the mean interparticle distance 𝑟_𝑚 = 𝑛^−1∕3). For NIRM9C one obtains𝑟_𝑠,𝐴 = 0.14µm = 0.07𝑟_𝑚 and𝑟_𝑠,𝐵 = 0.08µm= 0.04𝑟_𝑚. So despite of the lower density the localization lengths in the charged system are lower. Thus, one can conclude that NIRM9C is deeper in the glassy phase compared to the sample investigated in [103].

With𝑟_𝑠,𝐴= 0.12𝑟_𝑚and𝑟_𝑠,𝐵 = 0.07𝑟_𝑚the localization lengths in the sample NIRM6B are actually more similar to that hard sphere mixture. But then, NIRM6B is also the sample that can be interpreted as the one closest to the glass transition. Note however that it has a smaller volume fraction ratio (1∕10) and a higher total volume fraction (Φ = 39 %) than NIRM9C. But since NIRM6B consists of the same small and big particles the size ratio is the same, so that the comparison to [103] is reasonable.

For the ratio between localization lengths for small and big particles one obtains𝑟_𝑠,𝐴∕𝑟_𝑠,𝐵≈ 1.7, for both NIRM9C and NIRM6B. In fact, one can find the same ratio in the hard sphere system studied by Lynch et al. [103]. This makes clear that there is no substantial difference between the mobilities in glassy hard sphere and glassy charged spheres systems; at least not in the intermediate time scale corresponding to the plateau region of the MSDs.

For the individual localization lengths𝑟_𝑠(Equ.6.3) the picture is very similar. Lynch et al. [103] show a slice of their sample where they colored those particles with the30 %largest particle displacementsΔ⃗𝑟 within a time of 10 minutes. They found clusters of mobile particles looking very similar to the mobile regions presented in Figure6.27and6.28. Even the size of a few mean interparticle distances is the same as in the glassy systems NIRM6C and NIRM9C.

Van-Hove functions

A phenomenon that is always observed together with dynamical heterogeneities is the non-Gaussian behaviour of dynamical distribution functions like the van Hove function. Recalling the definition of its self-part𝐺_𝑠(𝑟, 𝑡)(see Chapter1.2.2) we remember that it measures the probability to find a particle at the distance𝑟after a fixed lag time𝑡, if this particle was at the origin for𝑡= 0. For particles doing free Brownian motion, e.g. colloids in the liquid phase, one obtains a perfect Gaussian distribution in𝑟with a growing width as a function of time. Another simple theoretical model that also shows such behaviour is that of a trapped particle doing Brownian motion in a harmonic potential. A discussion on that model is given in AppendixB.

The trajectory already revealed that the glassy systems discussed here show heterogeneous dynamics.

“Fast”particles in mobile regions do not seem to be trapped in a potential. Their cages, built by the nearest neighbours, are not stable. On the other hand particles in “slow”, less mobile regions can be regarded as trapped in a potential. Up to second order in distance𝑟from the minimum, any potential well

Im Dokument Glass transition in charged colloidal suspensions (Seite 171-187)