Binary mixtures

The best way to avoid crystallization is to increase the polydispersity in the particle systems. By using mixtures of two particle sizes this can be done in a very controlled way, also allowing a simple enough description with theories like MCT. As discussed above, the usage of small and big particles coloured with different dyes proved to be impossible, due to unexpected changes in the particle interactions (see section5.3.2). Therefore only particles were mixed together, that stem from the same chemical synthesis procedure and have the same labelling dye. For the detection and positioning of the particles in the confocal microscopy images, a new algorithm is introduced in full detail in Chapter3. With this software it is actually an advantage to have the particles coloured with the same dye, since both particle species are on the same image. There is no time lag which would bias the temporal correlation of the dynamics of small and big particles.

0 10 20 30 40 50 60 70 0.06

0.08 0.10 0.12 0.14 0.16 0.18

density n [µm−3]

x, y, z [µm]

big small

Figure 5.16: Illustrating the homogeneous distribution of small and big particles in a glassy binary sample of NI2 (1.0µm) and NI4 (2.0µm) NIR particles with a mixing ratio𝑥_small = 0.67at a volume fraction of 20%. Left panel: Density profiles in all three directions (bin size is 2µm). Right panel: A 3D snapshot of the observed volume rendered using positions and particle sizes determined by the particle detection software.

Figure 5.17: Slices at𝑧= 0through the 3D pair distribution functions𝑔(⃗𝑟)of the same binary mixture as presented in Figure5.16. None of the three partial distribution functions shows signs of a crystalline or otherwise anisotropic structure.

First promising binary samples were made as a combination of NI2 (1.0µm) and NI4 (2.0µm) particles (see Table5.1. Both particle species were distributed homogeneously in the whole sample. There was no sign of phase separation or clustering (see Figure5.16). Furthermore, no anisotropies nor any crystal-like symmetries can be observed in the three-dimensional partial pair distribution functions𝑔_𝛼𝛽(⃗𝑟)(see Figure5.17).

It is even possible fit the experimentally determined radial distribution functions𝑔_𝛼𝛽(𝑟)and structure fac-tors𝑆_𝛼𝛽(𝑞)to those from Monte Carlo simulations (see Figure5.18). Those simulations were performed with the methods described in Chapter2.4.1with a system size of 2000 particles and allowing an equili-bration time of 5000 MC cycles. The effective charges𝑍^(𝑎)

ef f and𝑍^(𝑎)

ef f and the screening length𝜅⁻¹were adapted for the best possible agreement of theoretical curves with the experimental data. The quality of the fits is remarkably good, having in mind that the Debye-Hückel theory describes the system with three relatively simple pair potentials for the interactions of the particles. Together with the particle densities, the potentials were the only input to the MC simulations.

5.4 Searching for the glass

Figure 5.18: Fitting experimentally determined partial pair distribution functions and structure factors with theoretical curves from Monte Carlo simulations. In the left panel “aa” (small-small) and “ab”

(small-big) are lifted by 2 and 1 for a better view on the curves. The quality of the fit is remarkable good.

0 10000 20000 30000 40000 50000 60000

time t [sec]

Figure 5.19:Evolution of the number density𝑛of small and big particles in a mixture of RP45(1.8µm) and RP44(2.6µm) particles. Within a typical measurement time the density of the small particles in-creased while that of the big ones dein-creased (red curves). One single sample at a somewhat higher volume fraction showed a much slower evolution of the number densities with no clear trend during a measurement time of50 000seconds (blue curves).

In order to facilitate the particle detection, attempts were made with mixtures of bigger particles. Another intention was to investigate higher effective charges. To this end, particles were loaded with as much dye as possible with the procedure described above. But it turned out that mixtures of RP44 (2.6µm) and RP45 (1.8µm) Rhodamine particles were not suitable for reliable measurements. Even if they came from the same synthesis process, the density mismatch between the two species posed a big problem. The density matching procedure was performed for each particle size separately. But after the mixed sample had been under the microscope for several hours, one could observe that the number density of small particles increased, while that of the big particles decreased (see Figure5.19). With relative changes of about 5-10% in 12 hours, the timescale for this to happen was very slow. After evaluating the data it was even observed that smaller particles came into the observed volume from above, while big particles left it upwards. It seems that for mixtures of these bigger particles the density mismatch between small particles and big particles is higher compared to the systems with smaller particles (NI2 and NI4). Gravitational drifts lead to a de-mixing of the two particle species. Only one single sample at a higher volume fraction (see Figure5.19) was relatively stable with little fluctuations during a 12 hour long-time measurement.

However, after three days one could again observe larger changes in the densities that rendered the sample useless for further measurements.

Abandoning all samples having density mismatch problems or otherwise inhomogeneous collective drifts, one is left with a relatively small list of binary mixtures showing glassy dynamics, which is presented in Table5.4. The values for𝑍_{ef f}, and𝜅⁻¹were determined from the tedious work of fitting the parameters

samples particles Φ 𝑛∕µm⁻³ 𝑥_𝑎 𝑍^(𝑎)

ef f 𝑍^(𝑏)

ef f 𝜅⁻¹ 𝑁 # msmts

NIRM6A 35% 0.119 0.46 180 580 0.30µm 450 1 in 3D

NIRM6B 36% 0.123 0.46 180 580 0.30µm 1450 1 in 3D

NIRM6C NI2 39% 0.133 0.46 180 580 0.29µm 1500 1 in 3D

NIRM9A and 24% 0.112 0.64 180 580 0.48µm 500-8000 5 in 3D1 in 2D

NIRM9C1 NI4 28% 0.128 0.65 150 580 0.47µm 1300 2 in 3D

NIRM9C 28% 0.132 0.65 150 580 0.47µm 600-4000 26 in 3D,11 in 2D

RPMIX5A +RP45RP44 16% 0.028 0.58 240 600 0.78µm 800 3 in 3D

Table 5.4:Measurements of binary samples with supercooled or glassy dynamics (plateau in the MSD).

Effective charge numbers𝑍_{ef f}and screening length𝜅⁻¹are fit parameters from fitting with curves from MC simulations. Volume fractionΦ, number density𝑛and the fraction of small particles𝑥_𝑎=𝑁_𝑎∕(𝑁_𝑎+ 𝑁_𝑏)were determined from the particle positions found by the detection software. “# msmts” means the number of measurements done with that sample,𝑁denotes the average number of observed particles.

carrying out many Monte Carlo simulations (see section 2.4.1). The sample NIRM9C was stable for a very long time and could therefore be measured many times in the course of several months. Note that NIRM9C1 is an initial version of that sample with a slightly lower particle density; NIRM9C is the final version.

Relation between charge number and size of the colloids

From Table5.4one can derive the ratios of the effective charges of small and big particles𝑍_{ef f}^(𝑎)∕𝑍_{ef f}^(𝑏). Always using particles in the same suspension, one obtains the numbers0.31,0.26and0.4. They are very similar to the ratios of the corresponding squared diameters𝜎²_𝑎∕𝜎_𝑏² = 0.25,0.25and0.48(see Table5.4).

Consequently, for the suspensions at hand, the number of surface charges is about proportional to the surface area of the particles. This is in contrast to the results of Schöpe [62, pages 110,119, 123] who found that the effective charge of PS particles doubles when the diameter is doubled: Particles with a diameter of85nm were seen to have an effective charge number of530and larger ones with a diameter of156nm had the number945.

This discrepancy could be explained by counterion condensation. Schöpe’s PS particles are ten times smaller in diameter but have a similar or even a higher number of effective charges. The dissociating surface groups on the particles, that are the reason for the electrical charges, have a much higher density in those PS systems. Therefore many of the surface groups are not able to dissociate. The electric potential is so strong so that the corresponding counterions do not leave the surface (counterion condensation [39]).

With this explanation it seems quite normal that only for bigger particles with fewer charged surface groups the intuitive rule𝑍_{ef f} ∝𝜎²is fulfilled.

5.5 Conclusion

5.5 Conclusion

In this chapter the experimental setup and the samples used for the investigation of charged colloidal systems with confocal microscopy were introduced. It was demonstrated that the biggest challenge is the production of reliable samples. Particle interactions are very sensitive on the ion concentration in the solvent, which makes it difficult to reproduce systems with exactly the same properties once again. Both, the ion number densities in the solvent CHB/decaline and the charge numbers on the PMMA particles are very small compared to other systems (e.g. PS particles in water). This results in a high sensitivity on small (wanted or unwanted) additions of charges to the system.

It turned out that suspensions of monodisperse particles always show a tendency to crystallization. Mea-surements on glassy monodisperse systems are limited to suspensions in which the crystallization is so frustrated that it does not start immediately in all parts of the sample. In binary systems crystallization can be inhibited completely. They proved to be well suited for long time measurements to investigate glassy behaviour.

It was shown that the characterization of samples in terms of the two most important system parameters, effective charge𝑍_{ef f} and Debye screening length𝜅−1, is possible with the use of Monte Carlo simula-tions. To this end, experimentally determined structure factors𝑆(𝑞) and pair distribution factions𝑔(𝑟) are reproduced theoretically by a proper adjustment of those parameters in the simulations.

Chapter 6

Confocal microscopy: Results and discussion

After all the preliminary work on many of the subtleties hidden in particle detection, tracking and the production of glassy (but non-crystalline) samples this chapter is dedicated to the results of the confocal microscopy experiments. The focus of this chapter lies on two subjects. One is the comparison of the results to mode coupling theory (MCT). The other is a characterization of the heterogeneous dynamics found in glassy systems. Of special interest are the correlations between local structure and dynamics.

Furthermore, it will be investigated how the dynamics changes with the growing age of a sample, the so-called aging process.

Monodisperse systems and binary mixtures are discussed in two separate sections of this chapter. It turns out that there are many similarities which will also be elaborated here.

Contents

6.1 Monodisperse systems . . . 143