Crystallization in charged monodisperse samples

In order to keep theoretical calculations and comparisons as simple as possible first attempts aimed to find glassy suspensions that consisted of one particle species. Already from experiments with hard spheres it is known that “too perfect” systems, where all the particles have exactly the same size, do not show a disordered solid state. They crystallize at a volume fraction of Φ = 49 % [14, 15]. According to simulations [15] a size polydispersity of more than7 %(standard deviation divided by mean) is enough to avoid crystallization. Experiments confirmed this statement [91,92].

5.4 Searching for the glass

Figure 5.11: Micrographs of crystallizing samples. Left and middle panel: NI4 (2.0µm, NIR) particles at20 %volume fraction showing bcc and fcc crystals. Right panel: FL4 (2.4µm, fluorescine) particles atΦ = 30 %crystallize despite of the large size polydispersity in this region.

For charged systems the more important criterion is the charge polydispersity, since the particles usu-ally do not touch each other due to their electrostatic repulsion. Tata and Arora [93] showed in their simulations that a charge number polydispersity of26%is required to prevent crystallization in systems of charged particles. Since the surface area of a sphere goes quadratically with its radius one might expect that the effective charge number should also increase quadratically. However, experiments with polystyrene beads [62] revealed that the charge number rather goes linearly with the particle radius. In principle this implies that all the monodisperse samples investigated in this work (see Table5.1) should tend to crystallize. For all of them the size polydispersity was clearly less than26 %. Nevertheless, Beck et al. [25] claimed to observe glassy behaviour in suspensions of charged silica colloids at10.5 %size polydispersity. With those results in mind, attempts to find glassy samples of one particle species were also made in this work. But as one might expect, most of those monodisperse systems showed immediate crystallization.

Criteria for crystallinity

While many samples simply showed large crystalline structures easily visible to the eye (see Figure5.11) the crystallinity of others was more difficult to recognize and required the determination of the particle positions. With those it is possible to compute local bond-order parameters and the three-dimensional representation of the pair distribution function as a scalar field 𝑔(⃗𝑟). Both quantities can be used to determine the crystallinity and the latter is further useful to verify the isotropy of the sample.

The most common method today [68,15,94,76] is to compute local bond-orientational order parameters according to Steinhardt et al. [95]. For a particle𝑎with𝑁(𝑎)nearest neighbours, one defines a complex-valued vector with2𝑙+ 1components as:

𝑞_𝑙𝑚(𝑎) = 1 𝑁(𝑎)

𝑁∑(𝑎) 𝑏=1

𝑌_𝑙𝑚(𝜃_𝑎𝑏, 𝜑_𝑎𝑏), (5.6)

with the spherical harmonics𝑌_𝑙𝑚(𝜃_𝑎𝑏, 𝜑_𝑎𝑏)of order𝑙. The latter depend on the angular directions𝜃_𝑎𝑏and 𝜑_𝑎𝑏of the vector⃗𝑟_𝑎𝑏that connects particle𝑎to its neighbour𝑏. It turned out that the most useful choice for the order is𝑙= 6because it is sensitive to all 3D crystal types. Next step is to define the connection between two neighbouring particles ascrystallineif the scalar product of the normalized vectors

𝑠_𝑞(𝑎, 𝑏) =

∑6 𝑚=−6

𝑞_6𝑚(𝑎)[

𝑞_6𝑚(𝑏)]∗ ( ₆

∑

𝑚=−6|𝑞_6𝑚(𝑎)|²⁾

1∕2( ₆

∑

𝑚=−6|𝑞_6𝑚(𝑏)|²⁾

1∕2 (5.7)

0 2000 4000 6000 8000

Figure 5.12:Illustrating the crystallization of a NI4 (2µm) monodisperse sample at 20% volume fraction.

Left panel: The fraction𝑓_𝑥of crystalline particles (blue) and the mean𝑞₆bond order parameter of all particles (green) grow in parallel. Mid panel: A histogram of the number of crystalline bonds𝑛_𝑘for each particle reveals the ongoing crystallization. Right panel: The distribution of𝑞₆values is shifted to greater values at later time. Lines in the two right panels correspond to the coloured areas in the left plot.

exceeds a certain threshold. Following Ziese [76] and Gasser et al. [68] a particle is defined to be crys-talline if it has𝑛_𝑘 ≥ 8connections to next nearest neighbours fulfilling𝑠_𝑞 ≥0.5. With the definition of the order parameter𝑞_𝑙as

the vector of Equation5.6becomes a positive scalar that does not depend on the orientation of the coordi-nate system. With𝑙= 6one obtains𝑞₆, which is the most important bond-orientational order parameter due to its sensitivity to bcc, fcc and hcp crystal structures. However, one should keep in mind that the𝑞₆ value in an icosahedral configuration is large as well. While𝑞₆ = 0.484in a perfect hcp crystal, for the other three structures (bcc,fcc,icosahedral) one obtains𝑞₆>0.6in the perfect case [96].

The Voronoi decomposition was used for the determination of the next nearest neighbours. For each particle one obtains a polyhedron (Voronoi cell) consisting of all the points that are closer to this particle than to any other particle. Particle𝑏is considered as next nearest neighbour of𝑎if the Voronoi cells of the two share a common face. The bonds between particles are thus the same as those of a Delaunay triangulation. This definition has the advantage that there is no need for an arbitrary parameter e.g. a cutoff-distance. An discussion about the influence of the neighbour definition on bond order parameters can be found in [96].

Left panel of Figure5.12shows how the fraction𝑓_𝑥of crystalline particles grows during crystallization, from about 5% in the supercooled liquid up to more than 80% in the crystal. At the same time, the mean𝑞₆bond order parameter runs parallel to𝑓_𝑥but only shows a quite small change from0.35to0.39. A histogram of the number of crystalline bonds proves to be very sensitive. An ongoing crystallization (green curve in the mid panel of Fig.5.12) is clearly visible as an increased number of particles with more than 10 crystalline connections to its next neighbours. On the other hand, the ongoing crystallization only has a quite small effect on the histogram of𝑞₆values (right panel).

Another possibility to recognize crystallinity is the pair distribution function defined as a scalar field in the 3D space:

In order to compute it numerically, the distance vectors from each particle𝑖to any of the other particles 𝑗 ≠𝑖are calculated and then counted in a three-dimensional histogram of the distance coordinates𝑥_𝑖𝑗, 𝑦_𝑖𝑗 and𝑧_𝑖𝑗. Usually a time-average over 10 up to 1000 frames is done to reduce noise. 10 frames with 5000

5.4 Searching for the glass

Figure 5.13: Slices at𝑧 = 0through the 3D pair distribution function𝑔(⃗𝑟)of the crystallizing sample already presented in Figure5.12at the same four stages. Already in the beginning of the crystallization process (second panel) the six-fold symmetry is very prominent.

particles are enough to obtain reliable results. In order to make the histogram values comparable to the usual radial distribution function𝑔(𝑟) one has to divide the counts by the number of particles𝑁, the volume of the binsΔ𝑥Δ𝑦Δ𝑧and the particle density𝑛 = 𝑁∕𝑉. In this way one obtains𝑔(⃗𝑟)with the property that the value𝑛𝑔(⃗𝑟) d⃗𝑟 contains the number of other particles found in a volume of size d⃗𝑟= Δ𝑥Δ𝑦Δ𝑧at the vector𝑟⃗, if there is a particle at the origin.

In Figure5.13we can see slices through𝑔(⃗𝑟)for the sample already discussed in Figure5.12. One can see that the six-fold symmetry of a crystallizing sample is very prominent already for a fraction of crystalline particles of about 15% (second panel). In the liquid phase (first panel) the particles are really isotropically distributed. There are no peaks in the pair distribution function and the rings are perfectly circular.