Computing the pair distribution function g(r) of a cell geometry

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Preparation of the colloidal suspension

For this experimental study we found that a careful sample preparation proce-dure was extremely necessary before beginning our series of measurements. We noticed that the beads tend to aggregate in pure distilled water as shown in Fig. 1.3. To avoid this problem the addition of a surfactant sodium dodecyl sulfate (SDS) of concentration slightly less than the critical micelle concentration (CMC) was needed (about 2.1g/l-2.2g/l). However the suspension still contained some pair of aggregates (doublets). Sonication of the sample just before using it was the only way to break the doublets and obtain a suspension exclusively constituted of non aggregated particles. In general, short sonication is a good way to reduce aggrega-tion of the beads but with iron inclusions in the beads, more care is needed, as the binding might break. Sonication should not exceed 12 minutes at low amplitude, because the particles will be damaged and thus lose their magnetic properties. A picture of bad sonication trial is shown in Fig. 1.4.

General description of the Experimental Setup

Two typical types of cell were used(see Fig. 1.5). They were custom made by fixing a hollow cylinder on a glass slide, using a 2 components epoxy glue which shows no visible interaction with the particles. Bulk concentration is adjusted to keep surface concentration constant. Since the mass density of the colloidal particles is 1:7 10 3kg=m3 , the particles in suspension are subjected to sedimentation. As a consequence all the particles fall on the glass/water interface and remain confined there due to gravity, thus creating a monolayer on the glass slide. Placing the glass slide on the microscope support we ensure that our system is flat which is necessary to avoid gradients of particle concentration.
Before going on further with the description of the optical part, one should note that after several trials we notice that the cleanliness of the cell used is extremely important. It is essential to check that the cell is clean before starting our series of measurements. Without these precautions, if the glass slide or the inner part of the cylindrical cell is not completely clean, impurities (see Fig. 1.6) are generally suspended in water and contaminate the interface. The protocol used to clean the cell consists in rinsing the cell with SDS and distilled water for about 30 minutes, and if necessary repeating this process until no impurities can be detected under the microscope. After filling the cell with the suspension, we wait a sufficiently long time to ensure that all beads have sedimented. Failing to do so might lead to the formation of chains of beads upon application of the magnetic field. The external magnetic field is applied in the direction normal to the glass/water interface and leads to a mutual repulsion between the particles. The potential interaction energy (described in section 1.1.1) can be controlled by varying the magnitude of the magnetic field.

Image Processing

After certain equilibration time, series of images where recorded every 10 s. These images were then processed with the use of ImageJ software. The image of resolution 640 480 pixels2 is first converted to 8-bit black and white image and then adjusted to a certain threshold, see Fig. 1.8. We checked that changing the threshold does not affect at all the statistical properties we are interested in. The particle detection works by scanning the image until it finds the edge of an object. Particles in the image are then selected based on their size in (pixel2) and circularity (a circularity of 1 indicates a perfectly circular shape). This ensures that only beads are selected by the program. The center of mass of the beads, which is the brightness-weighted average of its x and y coordinates, is then output. It provides the precise coordinates of the center of the beads.
Finally a plug-in script is written to treat a batch of images and extract the co-ordinates of each bead. The text file containing the positions of the beads was used as an input in a C language program to calculate the structural parameters g(r) and g6(r).

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Computing the pair distribution function g(r) of a cell geometry

The implementation of our algorithm differs from some previously used meth- ods and it is worth paying them a careful attention. Often to avoid errors due to edge/size effects, the evaluation function around each particle i is limited to a dis- tance r less than a maximum cutoff radius distance Rmax which we take to be half the radius of the inner circle inscribed inside the hexagon Rmax = Rin=2 (Fig. 1.12). Thus each particle explores the same environment as neighboring particles and effects at the edges of the image are excluded. So here is assumed to be a scanned density or effective density and equal to NAk where Nk is the number of particles included within an area A = Rmax2. One might think that because of this procedure, the correlation function may depend on the value of Rmax. Indeed, the correlations in Fig. 1.13 do not show a significant dependence on the value of Rmax chosen. The function g(r) will always be normalized by multiplying the x-axis by p . For a square lattice 1 /p in fact corresponds to the nearest neighbour distance between the particles and thus g(r) takes its first maximum at a value of about 1. However, for a triangular lattice, the first nearest distance between particles is equal to b = (p23 )1=2. The first maximum of g(r) is at a value of about r = 1:075.

Equilibration and finite size effect check

As discussed before in section 1.2.2, we have carefully checked that equilibrium is attained upon starting from a perfect crystal or starting from a random particles arrangement. We performed a similar check for the pair distribution function g(r) and noticed that they are eventually identical for N = 547, see Fig. 1.15.
In addition we have checked finite size effects by varying the hexagonal box size from 547 to 2611 particles, so that according to the relation in Eq. 1.9, the number of layers p is varied from 14 to 30 see Fig. 1.14. The same structural properties given by the pair distribution function (Fig. 1.16) were obtained for large enough cells (hence from N = 1141).

Table of contents :

1 Methods 
1.1 System and Experimental Tools
1.1.1 Colloidal suspension of super-paramagnetic spheres Super-paramagnetic spheres and their magnetic moment Preparation of the colloidal suspension
1.1.2 General description of the Experimental Setup Sample holder The external magnetic field ~B Optical Device Image Processing
1.2 Simulation Methods
1.2.1 Cell Geometry
1.2.2 Monte Carlo Procedure
1.3 Order Parameters
1.3.1 Pair distribution function Basic formalism for liquid structure Computing the pair distribution function g(r) of a cell geometry Equilibration and finite size effect check
1.3.2 Bond orientational correlation function g6(r) General definitions Equilibrium and finite size effect check
2 Granular media vs Colloidal system 
2.1 Introduction
2.2 Methods
2.3 Results
2.4 Crystalline Defects Analysis
2.4.1 Comparison of Granular and Colloidal systems
2.4.2 Influence of equilibration time on ordering
2.5 Discussion
3 Crystallization of binary mixtures 
3.1 Introduction
3.2 Model
3.3 Results
3.4 Concluding remarks


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