Glassy dynamics, dynamical heterogeneity

In the beginning of this chapter it turned out that all the monodisperse glassy systems show some kind of relaxation that results in a rise of the MSD curves for large lag times (cf. Fig.6.2). In the following we take a closer look at the particle trajectories trying to search for spatial or structural correlations.

Individual localization lengths and particle trajectories

In order to get a measure for the mobility of a single particle during a specified time intervalΔ𝑡one can use the radius of gyration of its trajectory defined by:

𝑅²_𝑔 = 1 and⃗𝑟_meanis the mean position during that time. This quantity was already introduced by Widmer-Cooper et al. [100] as the “propensity for motion” of a particle. Within this work a slightly modified version of this quantity is used, making comparisons to other lengths in the system more easy. From𝑅_𝑔 one can derive anindividual localization lengthfor each particle as:

𝑟_𝑠 =

√

𝑅²_𝑔∕𝑑 with dimension 𝑑 = 3 (6.3)

Note that the symbol𝑟_𝑠is reused here since this definition gives values that are very close to the overall localization length that was defined before in Equation6.1, where the MSD is evaluated at a given lag timeΔ𝑡. The value of𝑟_𝑠is comparable to the radius of a sphere that contains most of the trajectory⃗𝑟(𝑡) during the chosen time interval given by the time intervalΔ𝑡. For a system in the glassy phase, where particles are trapped in the cages of their nearest neighbours, the exact choice ofΔ𝑡is not critical as long as the time interval is larger than the time scale of the𝛽-relaxation. Individual localization lengths can then be interpreted as individual cage sizes.

Figure 6.11: Dynamical heterogeneity of RP44_D4 (rather supercooled, left panel) and RP44_B4 (deeper in the glass, right panel) made visible in plots of the trajectories color coded with the individ-ual localization length𝑟_𝑠(see text). The time length of the trajectories is5000s; time intervals between snapshots are25s. In𝑧direction the thickness of the box shown here is8mean interparticle distances𝑟_𝑚. Trajectories are projected onto the𝑥-𝑦plane. Deeper in the glass not only the localization lengths are smaller but also the size of regions with less localized/more mobile particles and their number is smaller.

Figure 6.12:Dynamical heterogeneity of FL13_A5 (left panel) and FL13_B5 (right panel), increasingly deeper in the glass. Like in Figure6.11the color code indicates the individual localization length𝑟_𝑠of each trajectory. The time length of the trajectories is again5000s; time intervals between snapshots are 20s. The thickness of the box shown here is only4mean interparticle distance𝑟_𝑚. Interestingly, the size of the mobile regions (high𝑟_𝑠) is not shrinking although on average the localization length is clearly smaller for the system deeper in the glassy phase (the trajectories form smaller spots).

Mobile and less mobile regions

In Figures6.11and6.12we see trajectories of the same four qualitatively different systems as discussed before in Figure6.4. With the sequence RP44_D4, RP44_B4, FL13_A5, FL13_B5 the systems are listed in descending order according to their overall localization lengths. Not to overcrowd the picture, only trajectories of particles within a box of a certain thickness in 𝑧-direction are plotted, projecting them onto the𝑥-𝑦plane. The individual localization length𝑟_𝑠is used as the color code. Since it is clear that𝑟_𝑠 grows with the length of the time intervalΔ𝑡(similar to the lag time for the MSD),Δ𝑡 = 5000s is used in all four trajectory plots to allow for a meaningful comparison.

6.1 Monodisperse systems With these pictures it becomes even more reasonable to identify RP44_D4 as a supercooled system and to say that the other ones more likely belong to the glassy phase. One can see that large regions in RP44_D4 consist of mobile particles with localization lengths above10 %of the mean interparticle distance. For sample RP44_B4, which has the same volume fraction (Φ = 19 %) but a larger screening length, this is not the case. One can only see relatively few small regions with particles that are more mobile. Going further to systems deeper in the glass (FL13_A5 and FL13_B5) there are still regions of higher mobility, but altogether the localization lengths are a lot smaller (note the different scales of the color bars in Figures6.11and6.12). Interestingly, there seems to be no further trend for the size of the mobile regions.

They do not become smaller for systems deeper in the glass and stay at sizes of up to 5 mean interparticle distances𝑟_𝑚 in diameter.

Apparently, for supercooled systems close to the glass transition the mobile regions are larger than in the glassy phase. But once in the glass, these regions do not become smaller when going over to stiffer systems with lower overall mobilities. This picture was already found in earlier works on hard spheres.

Weeks et al. [69] reported that the average number of particles contributing to a fast cluster grows when going from the liquid to more and more supercooled systems. In the glassy phase the size of fast regions is again much smaller, but does not grow or shrink when going deeper into the glass.

Mobility and local density

Naturally the question arises what characterizes the mobile and the less mobile regions. A first obvious test is to check whether the local particle density plays a role. For this purpose one has to compute the Voronoi decomposition of the particle positions. This results in one polyhedron for each particle, the Voronoi cell, that contains all the points being closer to that particle than to any other particle. The volume of the Voronoi cell is a measure for the local density at that particle’s position. In dimensionless length units𝑟_𝑚 (mean interparticle distance) the average Voronoi volume𝑉_vol is exactly1. An upward deviation𝑉_vol>1points to a lower local particle density and a downward deviation𝑉_vol<1to a higher one.

0.00 0.05 0.10 0.15 0.20

r

s

/ ^r

^m

0.6 0.8 1.0 1.2 1.4

V

vor

/ r

3 m

RP44_D4

0.00 0.05 0.10 0.15 0.20

r

s

/ ^r

^m

0.6 0.8 1.0 1.2 1.4

V

vor

/ r

3 m

RP44_B4

10⁰ 10¹

fre qu en cy

10⁰ 10¹

fre qu en cy

Figure 6.13:Correlation between the individual localization length𝑟_𝑠(Equ.6.3) and the mean Voronoi volume (1∕𝑉_vor is the local density). The measurement time used to compute𝑟_𝑠and the time average of the Voronoi volume isΔ𝑡 = 10000s. Length unit is the mean interparticle distance𝑟_𝑚. Each of the ∼ 3000particles gives one count in the 2D histogram. Left panel shows the supercooled system RP44_D4, right panel the rather glassy system RP44_B4. To the left of the red vertical lines are the20 % least mobile (“slowest”) particles and to the right of the green vertical lines are the20 %most mobile (“fastest”) particles.

In Figure6.13we see 2D histograms for two different samples counting individual localization lengths 𝑟_𝑠and average𝑉_volvalues from∼ 3000particles during a measurement time of10000seconds. There is no clear sign of a correlation between local density1∕𝑉_vol and mobility𝑟_𝑠. High mobility values occur independently of the local density and not only in regions with with a low density, as one might have expected. As anticipated, the system deeper in the glassy phase (RP44_B4, right panel) shows a more narrow distribution with smaller𝑟_𝑠values. However, the distribution of Voronoi volumes is very similar for both samples.

Mobility and local structure

As discussed above, it turned out that there is no distinct sign of a correlation between local density and mobility of the particles. But one can still look at the structural properties of mobile and less mobile particles. The idea used here is to compute the radial distribution function (RDF) with a restriction to the20 %“slowest” and to the20 %“fastest” particles defined by the red and green vertical cutoff lines in Figure6.13. More precisely, the correlation of the “fastest” particles with all particles is computed by:

𝑔_fast(𝑟) = 𝑥²_f𝑔_f,f(𝑟) + 𝑥_f(1 −𝑥_f)𝑔_f,nf(𝑟) 𝑥²

f +𝑥_f(1 −𝑥_f) (6.4)

Here𝑥_f is the number fraction of the “fastest” particles (here𝑥_f = 0.2),𝑔_f,f is the partial RDF for the correlations between a “fast” and another “fast” particle and𝑔_f,nf is the correlation between a “fast” and any other “not fast” particle. The denominator ensures a correct normalization so that𝑔_fast(𝑟→∞) = 1. The definition of𝑔_slowfor the “slowest” particles is analogous to that of𝑔_fast.

The results for the same two systems as discussed in the previous paragraph is presented in Figure6.14 together with the complete RDF including all particles (grey line). For the interpretation of𝑔_slow and 𝑔_fast one should first recall that𝑛𝑔(𝑟) d𝑉 is the probability to find a particle in the volumed𝑉 at distance 𝑟if there is a particle at the origin. Therefore, with the definition in Equation 6.4, 𝑛𝑔_fast(𝑟) d𝑉 is the probability to find any particle in the volume d𝑉 at the distance 𝑟if there is a “fast” particles at the

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Figure 6.14:Partial radial distribution functions𝑔_slow(𝑟)(red) and𝑔_fast(𝑟)(green) giving the correlation of the20 %least mobile (“slow”) and20 %most mobile (“fast”) particles with all particles (cf. Equ.6.4).

The grey line shows the normal RDF including all particles. This allows for a comparison of the structure in the neighbourhood of “slow” and “fast” particles.

Measurements had a duration of10000s and mobility is defined via the individual localization length𝑟_𝑠. See Figure6.13for the selection of slow and fast particles. Again we see results from the supercooled sample RP44_D4 in the left panel and from the glassy sample RP44_B4 in the right panel.

6.1 Monodisperse systems origin. The statement is analogous for𝑔_slow. So the analysis of these partial RDFs allows us to say more about the structure in the neighbourhood of “slow” or “fast” particles.

One of the first important findings in Figure6.14is that the position of the structure peak is the same in both𝑔_slow and𝑔_fast and also in the complete RDF. Accordingly, the average distance to the nearest neighbours is the same throughout the sample. There is no distinction between “slow” and “fast” regions.

This underlines the result from above, that there is no correlation between particle density and mobility.

Furthermore, one can observe that the structure peaks are higher than average for the “slow” regions and lower than average for the “fast” regions. This finding applies to both samples, and it is more pronounced in the glassy sample RP44_B4 than in the supercooled sample RP44_D4. This alone is already a corre-lation between mobility and structure. Regions with “fast” particles have a lower degree of correcorre-lation with their neighbours than average, while regions with “slow” particles have a higher degree. The differ-ence in the height of the principal peak of𝑔_fast and𝑔_slow(∼ 0.4for RP44_D4 and∼ 0.7for RP44_B4) is comparable to the difference between the complete𝑔(𝑟) of a supercooled to a glassy sample (∼ 0.8 between RP44_D4 and RP44_B4). So the mobile regions in the glassy sample RP44_B4 do not only behave similar to a supercooled liquid, they also have a comparable structure.

Note that this finding is actually not that unexpected. The radial distribution function𝑔(𝑟)is a time aver-aged quantity connected to the correlation of particle positions. Therefore, already the higher mobility of the particles in “fast” regions must lead to lower structure peaks and the lower mobility in “slow” regions to higher ones. Nevertheless, interpreting𝑔(𝑟)as the signature of the average structure in a system, the finding above is indeed a correlation between mobility and structure.

However, there is another interesting observation in comparing the RDF in “fast” and “slow” regions that concerns the split in the second peak. In Figure6.14one can see that this split is much more pronounced in the neighbourhood of a “slow” particle than for other particles. Especially the left sub-peak grows considerably going from “fast” to “slow”. In contrast to the difference in the peak heights discussed above, this finding cannot simply be explained by the different mobility of the particles. It is a clear signature for real structural differences.

As mentioned before, the split in the second peak is a well-known feature seen in random close packings (rcp) or more precisely in polytetrahedral structures. With their common neighbour analysis (CNA) Clarke and Jónsson [98] managed to link the two sub-peaks to specific configurations, where a pair of two particles shares a certain number of common neighbours. The distance of such a pair corresponds to the position of a sub-peak. For the left sub-peak they found two specific configurations. In the first (named 211), 4 particles build two triangles with adjacent sides: The pair consists of the two particles not taking part in the adjacent side. In the second configuration (named 333), 5 particles form two tetrahedra that share one face. Here the pair consists of the two particles not taking part in the shared face. It was found that both configurations equally contribute to the left sub-peak [98]. For the right sub-peak a configuration named 100 is responsible, where 3 neighbour particles are aligned: The distance of the two ends of the trimer corresponds to the right sub-peak.

Figure 6.15: Configuration of common neighbours that contribute to the two sub-peaks of the second peak in𝑔(𝑟)of a glassy system. Considered particles are shown in red color. 211 and 333 are the main contributions to the left sub-peak, 100 is the main contribution to the right one.

Furthermore Clarke and Jónsson [98] observed that the two sub-peaks increase upon increasing the vol-ume fraction. In agreement with what is observed here going from “fast” to “slow”, they observed that the increase of the right sub-peak is smaller. One interpret the enhanced split of the second peak and es-pecially the increase of its first sub-peak to be connected to the formation of tetrahedra. The conclusion is that in “slow” regions there are more tetrahedra or they are more regularly shaped so that the observed sub-peaks are higher.

It should be mentioned that this observation agrees very well with the results of Cianci et al. [101] on hard spheres. They reported a (weak) correlation between the mobility and the shapes of tetrahedra formed by groups of 4 neighbouring particles. The more mobile a particle the more irregular is the shape of the tetrahedra where it takes part in. Here we basically have the same observation from the analysis of the partial radial distribution functions for “slow” and “fast” regions, seen as a decrease of the first sub-peak in the split second peak of 𝑔_fast going from “slow” to “fast”. Correlations between mobility and structure in monodisperse hard sphere systems have also been found in more recent works.

A very detailed analysis was made by Leocmach and Tanaka [102]. They found that “slow” regions are connected to local icosahedral and fcc-like order. They further investigated the length scale of ordered regions and found that on approaching the glass transition, a growing length scale is not seen for the icosahedral order but for the crystal-like order. This was somehow unexpected since it is the icosahedral configuration that cannot be used to fill space periodically.

Correlation between mobility and crystallinity?

The finding in the paragraph above suggests the idea that relaxed regions have a bigger crystallinity. This would also be an explanation for the higher structure peaks of𝑔_{𝑠𝑙𝑜𝑤}and its more pronounced split in the second peak. A more relaxed region might correspond to a crystalline region and the whole sample is simply crystallising and the correlation found above has nothing to do with the physics in glasses.

In the section5.4.1on crystallization of monodisperse charged systems the definition of a “crystalline bond” between neighbouring particles was already introduced. For each particle one computes the

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r

s

/ r

m

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 r

s

/ r

m

Figure 6.16:Correlation between individual localization length𝑟_𝑠and the average number of crystalline particles𝑛_𝑘. As in Figure6.13a time interval ofΔ𝑡= 10000s is used. Red and green vertical lines are the same as in Figure6.13and also the presented measurements are the same. (Note that actually𝑛_𝑘 can only take on integer values, only the time average produces numbers with a fractional part. Since some particles stay with the same integer number𝑛_𝑘for the whole measurement we see lines with higher counts for integer values of𝑛_𝑘.)

6.1 Monodisperse systems complex-valued vector𝑞₆, the so called bond order parameter (cf. Equ. 5.6). If the scalar product 𝑠_𝑞 (see Equ.5.7) of normalized𝑞₆vectors from two neighbouring particles exceeds the threshold𝑠_𝑞 >0.5 one speaks of a crystalline bond between these two particles. This definition is used here to determine the average number𝑛_𝑘 of crystalline bonds for each particle during a measurement. The result is presented in Figure6.16: For each particle the individual correlation length𝑟_𝑠and the average number𝑛_𝑘is used to fill a 2D histogram. The evaluation is done for the same samples as in Figures6.13and6.14.

Figure6.16only shows a weak correlation between crystallinity and mobility. One can see that those particles with the highest number of crystalline bonds𝑛_𝑘have a rather small localization length𝑟_𝑠which means they are indeed less mobile. Vice versa, the most mobile particles with a large𝑟_𝑠do not have a very large number of crystalline bonds. But still the distribution of𝑛_𝑘values among the20 %fastest particles (on the right side of the green line) is not very different from the distribution of the20 %slowest particles (on the left side of the red line). Particularly for the supercooled sample RP44_D4 one cannot claim that the “slow” particles have more crystalline bonds. For RP44_C4, the system deeper in the glassy phase, there are indeed slightly more “slow” particles with a number of crystalline bonds𝑛_𝑘 ≥7.

But these weak correlations are not enough to say that slow regions are crystalline or on the way to crystallize. One should note that the definition used here to define a crystalline bond is also sensitive to icosahedral structures [76]. Thus, one can only say that there is an enhanced tetrahedral ordering in “slow” regions, not necessarily an enhanced crystal ordering. It could therefore be interesting to investigate this topic further by trying to discriminate between icosahedral and crystalline ordering.

6.2 Binary systems

Basically the same analysis as presented above for monodisperse systems can also be done for binary systems. But there is one big advantage for the binary case. At least for the systems presented here crystallization is completely prohibited. This does not only enable the possibility to look at well-relaxed systems but also on the aging process towards them. Therefore, in addition to the comparisons to MCT and an investigation of the heterogeneous dynamics, there will also be a discussion on the aging phenom-ena at the end of this chapter.

Im Dokument Glass transition in charged colloidal suspensions (Seite 163-170)