Comparison to MCT

Using the structure factors𝑆(𝑞)as input it is now indeed possible to compute dynamic quantities with the MCT equations introduced before in section1.3.2. The ability to do this is a considerable improve-ment to earlier experiimprove-ments, where the structure factors always had to computed in the framework of an integral equation approximation (like MPB-RMSA). In earlier works, the only link to experimentally de-termined structure factors were fits of theoretical curves to the data. Beck et al. [25] did this for charged monodisperse systems using the rescaled mean spherical approximation (RMSA) for their structure fac-tors. This was necessary since their light scattering data was restricted to the region around the principal peak. Nevertheless, they obtained a quite good agreement between theoretical and experimental results

6.1 Monodisperse systems for dynamic quantities. For a binary mixture of repulsive particles in 2D, Bayer et al. [49] used a mod-ified Percus-Yevick closure relation designed for hard disks for the computation of 𝑆(𝑞). Theoretical structure factors were fitted to the experimental data and then used as input to the MCT. They also found an accordance of theory and experiment in a comparison of experimental and theoretical mean squared displacements. So here, for the results presented in the following, no detour was made via a theory for 𝑆(𝑞). The structural data measured in the experiment was directly used for calculations of the dynamics using MCT.

As for the purely theoretical calculations presented in Chapter2, again Th. Voigtmann’s MCTSolver was employed to solve the MCT equations numerically. Technically, the calculations were done on an equispaced grid with 200𝑞 vectors, the largest𝑞 being 10 times larger than the position of the principal peak of𝑆(𝑞). Note that a higher resolution in𝑞did not change the results.

Localization length

A direct test for mode coupling theory is to check whether the predicted localization lengths𝑟_𝑠 for the glassy systems are comparable to those in the experiment. The results in Figure6.3 already exhibit a correlation of𝑟_𝑠 with the separation from the theoretical MCT transition line𝜖_𝛾, computed from fits to MPB-RMSA structure factors. In the following the detour via a structure factor theory is avoided and the experimental structure factors are directly as input to the MCT equations.

As discussed in Chapter2, from solving the full dynamic MCT equations for the coherent correlator 𝜙(𝑞, 𝑡)one goes further to solve for the tagged correlator𝜙_𝑠(𝑞, 𝑡)and finally gets the mean squared dis-placement (MSD) via the low-𝑞limit of𝜙_𝑠(𝑞, 𝑡). This was done to compute the MSD using the measured 𝑆(𝑞)and the particle density𝑛. In order to adjust the time scale, a third input is necessary, which is the short-time diffusion coefficient𝐷₀. As a first guess one can derive it via the Stokes-Einstein relation for free particles (Equ.1.21). For the systems at hand this always implied that the plateau was reached earlier in the theoretical MSDs compared to the measured MSDs. Therefore𝐷₀was reduced by factor of about 0.1so that the theoretical curves reach the plateau at a similar lag time. However, this “fitting” was not done for each measurement, the same𝐷₀is used for all systems with the same particle size.

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

MCT

_rs

0.02 0.03 0.04 0.05 0.06 0.07 0.08

m ea su re d

rs

FL4_A4 FL4_A5 FL4_B2 FL13_A3 FL13_A5 FL13_B5 RP44_B3 RP44_B4 RP44_C3 RP44_C4 RP44_C5 RP44_D4

Figure 6.5: Comparison of localization lengths𝑟_𝑠(derived from the MSD at𝑡= 100s) directly mea-sured in the experiments (𝑦-axis) and indirectly determined by solving the dynamic MCT equations with experimentally determined structure factors as input (𝑥-axis). Lengths are given in units of the mean interparticle distance𝑟_𝑚. Error bars reflect a10 %error for MCT values (see text for a discussion) and a 5 %error for the experimental determination of the MSD (cf.3.4.4).

For the comparison of theoretically computed localization lengths𝑟_𝑠 with experimental ones, the defi-nition in Equation6.1was used to obtain the localization length from the MSD at the specific lag time 𝑡= 100s. The result is plotted in Figure6.5. At first, one can see that only four systems are close to the gray line where𝑟_𝑠 is the same for experiment and MCT. But one has to keep in mind that qualitatively very different systems with different particle sizes are shown together in a single plot.

A closer look at the different conditions in the measurements can explain why the localization lengths computed within MCT are larger than the measured ones for most of the systems. As discussed before in the presentation of the measured MSD curves (see Fig. 6.2), some of the systems might still be in a relaxation process. For instance, FL4_A4 and FL4_A5 are measurements on the same system, but they show qualitatively different MSD curves. While FL4_A4 bends off the plateau region very early, FL4_A5 bends off almost 2 decades later. In such non-relaxed systems it is clear that the structural peaks of𝑆(𝑞) are less developed and henceforth MCT predicts the system to be softer so that the computed localization length𝑟_𝑠is larger.

Another issue is the different quality of the confocal microscopy images. The comparison in Figure6.6 shows that the quality of the images for RP44D is a lot better than that for RP44C. It is obvious that the positioning error in RP44C is expected to be larger, which leads to smaller peaks in the structure factor and finally the prediction of a larger𝑟_𝑠 within MCT. In Figure 6.5one can see that the three different samples RP44_C1 to RP44_C3 still lie on a line parallel to the grey line, where experiment and MCT give the same localization lengths. This means that the trend going from smaller to larger lengths is the same in both, theory and experiment.

The particle size plays a big role, too. Since the particles used in FL13 are smaller (only1.7µm compared to2.6µm for RP44) the error in the positioning is again larger, leading to a larger predicted localization length 𝑟_𝑠. Due to the high sensitivity of the theory on 𝑆(𝑞) it is clear that the positioning error has a stronger influence on the MCT result than on the experimental result for𝑟_𝑠.

Earlier works on hard spheres have reported that MCT predicts lower values for the volume fractions at the glass transition (e.g. [99]) compared to experiments (e.g. [18]). Therefore, one speaks of MCT underestimating the transition point, i.e. it is shifted towards weaker couplings𝛾 or lower densities𝑛. That’s why one would rather expect that the localization length computed within MCT is smaller. How-ever, in those works theoretical structure factors were used without any fit to experimental data. Here the measured particle positions are used directly for the calculation of𝑆(𝑞). Taking this into account, the different trend to getting the same or even a larger localization length than in the experiment is not contradicting the results in earlier works.

Figure 6.6:Comparison of the image quality for different samples. The main reason for the degradation of the quality is the bleaching of the fluorescent dye. Another one is the particle size: The positioning of smaller particles suffers from a lower accuracy.

6.1 Monodisperse systems

10 ¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

lag time

t

[s]

10 ³ 10 ² 10 ¹ 10⁰

MS D

r2

(

t

)/

r2 m

FL4_B2 RP44_B3 RP44_B4 RP44_D4

10 ¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

lag time

t

[s]

10 ³ 10 ² 10 ¹ 10⁰

RP44_C3 RP44_C4 RP44_C5

FL13_A3 FL13_A4_2D

Figure 6.7:Mean squared displacement from measurements (diamonds) and computed via MCT (lines) using𝑆(𝑞)determined from the measured particle positions. In the left panel we see curves with a good coincidence, the right panel consists of those with a rather bad coincidence. Note that also in the right panel experimental and theoretical curves show the same trends. The different short-time behaviour for 𝑡 <10𝑠is a consequence of the different particle sizes in the considered systems.

10⁰ 10¹ 10² 10³ 10⁴ 10⁵

lag time

_t

[s]

10 ³ 10 ² 10 ¹ 10⁰

MS D

r2

(

t

)/

r2 m

+ + RP44_D4

RP44_D4,

_MCT,n±1%

RP44_B4

RP44_B4,

_MCT,n±1%

Figure 6.8: Comparison of MSDs computed via MCT (lines) and directly measured data (diamonds).

Cyan and orange lines illustrate the deviations upon a change of the particle density𝑛by±1 %.

Mean squared displacement - MCT error estimation

Another possible test is to examine the agreement of measured and MCT calculated mean squared dis-placements (MSDs). In Figure6.7the MSDs for several of the measurements are shown. In the left panel we see those samples with a good coincidence between experiment and theory, in the right panel some of the samples where MCT largely overestimates the MSD and consequently also the localization length 𝑟_𝑠 (see Fig.6.5). It is interesting to see that for RP44_D4, the system that is expected to be supercooled due to its much higher𝑟_𝑠, experimental and theoretical curves are very close. Even the𝛼-relaxation takes place at about the same lag time. Remember that the only fitting involved is the adjustment of the short dime diffusion𝐷₀. The same𝐷₀is used for all RP44 samples and is fixed in a way that the𝛽-decay at short times fits to the measured MSDs around𝑡 = 25s (only possible for RP44_B/D, the plateau is too different for the other RP44 samples). No fitting was done to catch the time scale of the𝛼-relaxation.

As pointed out above, all the systems show a depart from the plateau at large lag times𝑡that could be interpreted as some kind of𝛼-relaxation. But since all these systems are not “well-aged” and actually do crystallize hours or days after the measurement, a further relaxation is not contrary to expectation.

In Figure6.8one can see how a change of±1 %in the particle density𝑛affects the MSD curves resulting from MCT. An error of 1 % in the measurement of𝑛 is a realistic assumption in for the experiments presented here. In order to obtain this result, the value of𝑛(used in the vertex for the memory kernel, cf.

Equation1.41) was changed and the structure factor was adapted using the proportionality(𝑆(𝑞) − 1) ∝𝑛 (see Equ.1.14). At𝑡= 100s, the lag time at which𝑟_𝑠is determined, the deviation in the MSD is almost 10 %. Using equation6.1, one can thus derive that the localization length𝑟_𝑠changes relatively by∓10 % upon a change of𝑛by only±1 %. This illustrates the strong sensitivity of MCT on𝑆(𝑞)and𝑛.

Density correlators

The comparisons of MCT and experiment are finalized with a comparison of measured and computed in-coherent and in-coherent density correlators, often named (self-) intermediate scattering functions (IMSFs) 𝐹_𝑠(𝑞, 𝑡)and𝐹(𝑞, 𝑡). Their direct computation from the particle trajectories is discussed in the appendix (C.5andC.4). In Figure6.9both functions are shown with their dependence on the lag time𝑡 for the two samples RP44_B4 and RP44_D4 that already showed a good agreement between theory-experiment in their MSDs. For both scattering vectors𝑞𝑟_𝑚 = 7(close to the principal structure factor peak) and 𝑞𝑟_𝑚 = 18 (near the third peak) one can see that MCT and experiment agree quite well, except for the decay in the curves for RP44_B4 at times𝑡 >1000s.

The agreement of the IMSF as a function𝑞for a constant delay time𝑡= 100s is even more convincing.

This is illustrated in Figure6.10. Within MCT the critical non-ergodicity parameter𝑓_𝑐(𝑞)is the remainder of the coherent density correlator 𝐹(𝑞, 𝑡) at 𝑡 → ∞ for a system directly at the glass transition line (cf.2.1.3). Therefore, it makes sense to compare a theoretical𝑓_𝑐(𝑞), computed using MPB-RMSA for the structure factor, to the experimental results. This is done with the gray line in Figure6.10. It is rather surprising that this line is always below the other lines. With RP44_B4 (red line) being a system in the glassy phase and RP44_D4 (blue line) being a supercooled system one would expect the gray line to lie in between. But this fits to the picture that MCT together with a theoretical structure factor underestimates the transition point. An underestimation of𝑓_𝑐(𝑞)is a reasonable consequence.

10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶

lag time t [s]

0.0 0.2 0.4 0.6 0.8 1.0

im sf F ( q, t ) ,F

s

( q, t )

q

·

r

m

= 7

10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶

lag time t [s]

0.0 0.2 0.4 0.6 0.8 1.0

q

·

r

m

= 18

RP44_D4, self RP44_D4, full RP44_B4, self RP44_B4, full

Figure 6.9:Comparison of intermediate scattering function𝐹(𝑞, 𝑡)and self-intermediate scattering func-tion𝐹_𝑠(𝑞, 𝑡)once directly measured and computed from the particle trajectories (diamonds) and once calculated within MCT (lines). Left panel shows curves for the scattering vector𝑞𝑟_𝑚 = 7close to the principal peak of𝑆(𝑞), right panel for𝑞𝑟_𝑚= 18in the vicinity of the third peak.

6.1 Monodisperse systems

Figure 6.10: Comparison of intermediate scattering function𝐹(𝑞, 𝑡)(left panel) and self-intermediate scattering function 𝐹_𝑠(𝑞, 𝑡)(right panel) as a function of𝑞 for the constant lag time𝑡 = 100s (in the plateau region of the MSD). Diamonds correspond to experimental data, solid lines are computed within MCT. The gray line shows a typical critical non-ergodicity parameter𝐹(𝑞, 𝑡 → ∞)computed within MCT using MPB-RMSA for the structure factor (system parametersΦ = 0.2 %, ̃𝑘= 8, ̃𝛾= 2500).

Im Dokument Glass transition in charged colloidal suspensions (Seite 158-163)