Separation from the glass transition, mean squared displacement, localization

charac-terized by their parameters volume fractionΦ, effective charge𝑍_{ef f}and Debye screening length𝜅⁻¹. Us-ing the latter two one can compute the dimensionless couplUs-ing constant̃𝛾and the dimensionless screening parameter ̃𝑘of the systems as defined in Equations2.1,1.2, and1.3. In Chapter2the ideal glass transi-tion lines𝑔𝑎𝑚𝑚𝑎(̃𝑘)̃ were determined within the framework of MCT using structure factors computed in the modified Poisson-Boltzmann rescaled mean spherical approximation (MPB-MRSA). The positions of the monodisperse glassy samples in the theoretical MCT phase diagram are shown in Figure2.1.

Note that a higher number of effective charges𝑍_{ef f} leads to higher value of̃𝛾and a big screening length 𝜅⁻¹results in a small screening parameter̃𝑘. Since the dimensionless parameters̃𝛾and̃𝑘are normalized by the mean interparticle distance𝑟_𝑚 = 𝑛^−1∕3 (cf. Equ.2.1) the influence of the particle density𝑛, or respectively the volume fraction Φ, on the position of the glass transition line is quite small. This is reflected by the fact that the two MCT transition lines in Figure6.1forΦ = 5 %andΦ = 48 %are very close to each other. Interestingly, all measured systems are predicted to be below the theoretical glass transition. However, the error bars for ̃𝛾and̃𝑘are quite large since the effective charge and the screening lengths were derived from fitting simultaneously theoretical structure factors and pair distribution func-tions computed within the MPB-RMSA theory (cf. Chapter1.2.4). As mentioned before the quality of those fits is rather limited (see Fig.5.15).

4 5 6 7 8 9 10

k

10²

10³ 10⁴ 10⁵

glass

liquid

FL4_A, _=40%

FL4_B, _=35%

FL13_A3, _=21.5%

FL13_A5, =24%

FL13_B5, =23%

RP44_B3, _=19%

RP44_B4, _=19.2%

RP44_C3, _=22.3%

RP44_C4, _=22.0%

RP44_C5, =21.5%

RP44_D4, =19.5%

MCT transition =5%

MCT transition _=48%

Figure 6.1:Points corresponding to the monodisperse glassy systems listed in Table5.3in the parameter space of coupling constant̃𝛾and screening parameter̃𝑘(using length units of the mean interparticle dis-tance𝑟_𝑚). Volume fractionsΦof the systems are given in the legend. The error bars denote the estimated relative error of10% in the determination of̃𝛾and̃𝑘from simultaneous fits of the structure factor and the pair distribution function. The solid lines are theoretical MCT transition lines from Chapter2.

6.1 Monodisperse systems

10 ² 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_A4 FL4_A5 FL4_B2 FL13_A3 FL13_A4_2D FL13_A5 FL13_B5 RP44_B3 RP44_B4 RP44_C3 RP44_C4 RP44_C5 RP44_D4 Lindemann 7%

free diffusion, t

Figure 6.2: MSDs in units of the mean interparticle distance𝑟_𝑚of the monodisperse glassy samples listed in Table5.3and Figure6.1. The finite exposure time problem is avoided by the usage of 2D-recomputed𝑥and𝑦positions as described in Chapter3.4. Light red horizontal line corresponds to the value where the particles on average have moved7 %of the mean interparticle distance in one of the three directions (calculated from solving√

⟨𝑟²(𝑡)⟩∕6∕𝑟_𝑚= 7% for⟨𝑟²(𝑡)⟩).

Knowing that the theory predicts all the systems to be liquid, the next step is to look at their dynamics described by the mean squared displacement (see Figure6.2). At first glance one realizes that the 3D measurements cannot cover the initial (𝛽-) relaxation at short lag times because the time interval of several seconds between two 3D snapshots is too long. Only one 2D measurement (FL13_A4_2D) indicates in which time regime the𝛽-relaxation takes place. But for the decision if a system is liquid or solid the𝛽 relaxation is not relevant. One can see that all the systems show an increase of the MSD towards the largest lag times, similar to an𝛼-relaxation. However, this does not imply that really all the measurements were done in the supercooled regime. This also becomes clear from the fact that some of the curves clearly show superdiffusive behaviour (steeper than∝𝑡), ruling out the usual𝛼-relaxation scenario.

An aging sample will always show some kind of relaxation dynamics. As mentioned before, measure-ments that showed inhomogeneous collective drifts were excluded for further evaluation, for example those with a gradient from top to bottom (see Figure5.10). But it is clear that such a test by eye might not resolve smaller drifts or inhomogeneities that are still visible in the MSD. A further problem is that although several thousand particles contribute in the computation of the MSD, the observed system size might be too small to see a well ensemble-averaged MSD. Later in this chapter follows a discussion on the dynamical heterogeneity of the systems, which supports this statement.

The horizontal line in Figure6.2corresponds to an average displacement of the particles by7 %of the mean interparticle distance. Theoretical predictions for the localization lengths at the MCT glass transi-tion (see Figure2.9) are also around7−8 %of the mean interparticle distance. According to a Lindemann-like criterion, systems that cross the gray line in Figure6.2could be classified as melted or liquid systems.

However, obviously a liquid system should not show a long plateau region in the MSD curve that is below this “Lindemann line”. One rather expects a flatter part that is actually crossing the line and is followed by a steeper𝛼-type growth∝ 𝑡for late𝑡(this is shown for example in Figure2.11). Only the magenta (RP44_D4) and the dark blue line (FL4_A4) show this behaviour in Figure6.2. This leads to the con-clusion that the other measurements are more or less in the glassy phase and their growth of the MSD at large𝑡is the result of relaxation processes in the glass or experimental perturbations and not because the systems are actually supercooled. An evidence for this idea can be seen in consecutive measurements on the same system (normal and dashed lines with the same color, in Figure6.2). For instance the𝛼-decay

for FL4_A4 comes a lot earlier than that for FL4_A5. Both were measurements on the same system, the second one started several hours later. One can conclude that with a longer waiting time the sample has relaxed and the𝛼-decay goes to later times.

With this in mind it is clear that many of the measurements on monodisperse systems are actually not done on samples that are relaxed well-enough; they are not well-aged. However, as mentioned before, all the monodisperse samples showed crystallization after a too long waiting time. Therefore the selection of measurements is basically a compromise between a too short waiting time, resulting in a badly relaxed system, or a too long waiting time, resulting in a crystalline system. Nevertheless, it is possible to relate the localization lengths𝑟_𝑠, seen in the measurements, to the separation𝜖_𝛾 = (𝛾−𝛾_𝑐)∕𝛾_𝑐from the MCT glass transition as determined from the phase diagram in Figure6.1. The localization length is determined from the MSD as:

𝑟_𝑠 =

√⟨Δ𝑟²(𝑡)⟩∕6 (6.1)

Here a delay time of𝑡= 100s was chosen, which is a time where all the MSD curves in Figure6.2are in the plateau region. The resulting diagram is shown in Figure 6.3. Note that despite the large error bars there is a clear trend to a decrease of the localization length coming closer to the MCT-predicted transition line at𝜖_𝛾 = 0. This means the theory indeed has indeed some predictive power. Actually one would expect systems with𝑟_𝑠 <7 − 8 %to be solid according to the Lindemann criterion, which means they should have a separation 𝜖_𝛾 > 0to the glass transition line. Instead, all samples are predicted to be below the line (𝜖_𝛾 < 0). Given this, one might conclude that the coupling parameters ̃𝛾 predicted at the MCT transition line are too high, the transition actually takes place for lower ̃𝛾 values. However, it has been observed that the fits of experimental data to theoretical MPB-RMSA structure factors and pair distributions are not very reliable (cf. Figure 5.15), which leads to large error in the estimation of the system parameters and thus also the separation𝜖_𝛾. Therefore it makes sense that this is not a problem of MCT but rather one of MPB-RMSA.

Ignoring any MCT results, it makes sense to declare RP44_D4 as the only supercooled system. FL4_A4 relaxes to FL4_A5 which then shows a considerably lower localization length𝑟_𝑠. The MSD curves of all other systems have clearly larger plateaus and most notably, their localization length shown in Figure6.3 is clearly below the Lindemann criterion, even if a relatively small value of7 %of the mean interparticle distance is chosen for the threshold.

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3

separation

0.02 0.03 0.04 0.05 0.06 0.07 0.08

loc ali za tio n len gt h

rs

/

rm

FL4_A4

FL4_A5 FL4_B4 FL13_A3 FL13_A5 FL13_B5 RP44_B3 RP44_B4 RP44_C3 RP44_C4 RP44_C5 RP44_D4

Figure 6.3: Localization length𝑟_𝑠in units of the mean interparticle distance𝑟_𝑚, derived from the MSD at time𝑡= 100s, plotted against the relative separation𝜖_𝛾from the MCT glass transition line (forΦ = 30 %), which is computed from the data presented in Figure6.1. Error bars correspond to the5 %error in the determination of the MSD (cf.3.4.4) and to the10% error in the determination of𝜖_𝛾from̃𝛾and̃𝑘.

6.1 Monodisperse systems 6.1.3 Structure of glassy systems

Since the structure factor 𝑆(𝑞) is the main input to the MCT equations some effort was put into its computation. In confocal microscopy the limited number𝑁 of observed particles does not allow for a precision as high as it is possible in light scattering experiments, where𝑁 ∼ 10⁹. However, light scattering experiments are always limited to a small range of scattering vectors𝑞, usually somewhere around the principal peak. Comparisons to MCT thus require a theory like MPB-RMSA to compute𝑆(𝑞) for larger𝑞. Such limitations do not exist when the particle positions are available, e.g. from confocal microscopy data. Limitations in the accuracy are set by the number of particles. The lowest possible𝑞 vector is given by the size of the observed box𝐿as𝑞_min =𝜋∕𝐿. With several thousand particles and a reasonable box size one can determine𝑆(𝑞)with a precision good enough for usage as input to MCT.

In the appendicesC.1andC.2the computation of𝑔(𝑟)and𝑆(𝑞)from particle particle positions and the influence of positioning errors is discussed in detail. Errors in the particle positions lead to a broadening of the peaks in the pair distribution function𝑔(𝑟). Their influence on𝑆(𝑞)is mainly a multiplication with a function𝑓(𝑞) < 1leading to a simple shrinking of the structure peaks without any broadening. With estimated errors below50nm ≃ 1 %𝑟_𝑚 (see3.3.3)𝑓(𝑞)is very close to1. Therefore no corrections for 𝑆(𝑞)were applied due to positioning errors. Another reason is that one can only determine a positioning error in rigid samples, which means that there is no possibility to reliably compute𝑓(𝑞) for actually measures samples.

In solid or almost solid systems the finite exposure time problem, which is discussed in Chapter3.4, has an opposite effect compared to positioning errors. The particles mostly stay close to a potential minimum and since they move during the image capture they seem less mobile, so that in the end each particle seems to be more tightly bound to its minimum. For𝑆(𝑞)this leads to higher peaks compared to a measurement with a very short exposure time. As described in3.4.2, an idea is to use𝑥and𝑦coordinates computed directly from the corresponding 2D slice for each particle, which reduces the effective exposure time to time for a single 2D slice. In𝑧direction this is not possible since several consecutive 2D slices are needed to obtain a reliable𝑧coordinate. Hence for the𝑧coordinate, the error due to the finite exposure time was determined and corrected for with the method described in Section3.4.5. In the common case this resulted in a shrinkage of the principal peak of𝑆(𝑞)of about2 %. It is important to point out that these corrections (2D recomputation for𝑥and𝑦, correction in𝑧) were done because the structure factors are

0.5 1.0 1.5 2.0 2.5 3.0

r / r

m

0 1 2 3 4

g ( r )

increasingly glassy

FL13_B5 FL13_A5 RP44_B4 RP44_D4

0 5 10 15 20

q r

m

0 1 2 3 4

S ( q )

Figure 6.4:Pair distribution function𝑔(𝑟)and structure factor𝑆(𝑞)for four systems with different sep-arations from the glass transition. Both are shown in units of the mean interparticle distance𝑟_𝑚. The localization length (according to the MSD, cf. Fig.6.2) of the listed systems decreases from RP44_D4 to FL13_B5.

later used as input to the mode coupling theory. The MCT equations are very sensitive to small changes of𝑆(𝑞)and therefore any small correction can have a tremendous effect.

A comparison of the results for four different samples is presented Figure 6.4. As discussed above RP44_D4 is definitely a supercooled system, the other samples should be more or less deep in the glassy phase. It is interesting to see that within a small error (< 5 %) the structure factors of all four systems or more or less equal. With about 4.0 the peak height of the principal peak of𝑆(𝑞) is in a range where MCT predicts the glass transition for systems with a moderate screening length of about 𝜅⁻¹ =𝑘⁻¹𝑟_𝑚 = 0.1𝑟_𝑚, where𝑟_𝑚is the mean interparticle distance (see Figures2.7and2.8). As expected from those purely theoretical results, the value𝑆(𝑞_max) = 4.0is considerably higher than the Hansen-Verlet criterion (𝑆(𝑞_max= 2.85) for crystallization. And also as expected from Hansen and Verlet’s work, all these systems do crystallize within several days (see5.4.1).

In the pair distribution function, which is less noisy due to the better statistics, one can finally see some differences between the four systems. Most obviously, the peak heights differ in a range between3.25for the rather supercooled system RP44_D4 and4.25for more localized system RP44_B4. But according to the lower MSD plateaus one would expect the peak heights of FL13_A5 and FL13_B5 to be even higher, the systems should be deeper in the glassy phase. That this is not the case could be due to the different particle sizes (2.6µm for RP44 and1.7µm for FL13) and screening parameters ̃𝑘(see Fig.6.1) which could cause the transition to occur for lower peaks. Another explanation could be the larger positioning errors for FL13 particles due to their smaller size. Nevertheless, the trend of a higher principal peak for the more localized system FL13_B5 is still correct.

In both𝑔(𝑟)and𝑆(𝑞)one can observe a shoulder or even a split in the second peak which is not seen in theoretical structure factors according to MPB-RMSA. Since MPB-RMSA is actually designed for the liquid phase, it is clear that it cannot describe all the structural details. In the MC simulations presented in Chapter2.4.1such a split was found as well (cf. Fig.2.20) and other experiments on charged colloids [23, 94] also reported it. An interesting comparison can be done with the work of Sirota et al. [23]. They were the first to investigate the structure of dense charged colloidal systems and they find a similar principal peak height𝑆(𝑞_max) = 4for their typical glassy samples. They detected the split in the second peak of𝑔(𝑟) and mentioned that it fits very well to the structure seen in a random close packing (rcp) of spheres. The split also resembles well the structure of metallic glasses [23,97]. With a common neighbour analysis (CNA) Clarke and Jónsson [98] could attribute this split to certain distances𝑟appearing in polytetrahedral structures of randomly close packed hard spheres. Sirota et al. [23] also observed another small peak for their charged spheres, just in between the first peak and the second (split) peak. They could attribute the corresponding distance to a local face centered cubic (fcc) structure, so that it could well be the detection of small fcc crystallites. The fact that there is no such small peak for the systems presented here is an indication that the methods to exclude crystalline or polycrystalline samples have worked very well.

6.1.4 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

Another issue is the different quality of the confocal microscopy images. The comparison in Figure6.6

Im Dokument Glass transition in charged colloidal suspensions (Seite 154-0)