When studying colloidal structures with particle diameters around (or bigger than) 1 μm, it is easy to get pictures with an optical microscope (for particles smaller, they could be taken by S.E.M. or similar, and all that follows still holds). This pictures, if taken at enough resolution characterize completely the morphology of the upper layer. A resolution like the one here it is enough. Anyway our starting point will be the following figure:

The first kind of analysis we report is called complex demodulation. In this technique, we conveniently Fourier transform the image (applying a suitable window) and represent the absolute value of the transform in a suitable scale:

If we filter in the Fourier space and make the inverse Fourier transform to the result we can obtain the following images:

• letting only the central peak. This helps in finding voids (vacancies) when the amplitude of the inverse transform is imaged.

• letting only one of the main peaks. This helps in finding domains of the considered orientation as well as dislocations inside them. You can do it with any of the relevant modes (peaks).

  • Fourier transform with the studied peak (mode) and Real part of the inverse transform (reconstruction of the corresponding mode).

• Module of the inverse transform (when clearly you can see a dislocation, in the point which the amplitude drops to zero) and the same with an overlay of the image and the real part (as in the previous row of images) but with an overlay of the image.

In the previous analysis it is of maximal importance to set correctly the window type and the filters (type, radii and position). So, when one has not expertise enough, the analysis should be done by manually adjusting the free parameters, once and more times (usually). With this kind of analysis it is possible to obtain quantitative information (size of domains, position of defects, relative amplitudes, orientations, and so on) as well as qualitative image information as above.

The second kind of analysis is called Voronoi analysis which makes use of the connectivity properties of the sites (in this case colloidal particles). To do that, first it is necessary to detect the colloidal particles (in the figure in red):

After that, it is made a triangulation based on Voronoi cells. In the figure each colloidal particle is joined to its first neighbors (green lines). The particles of the contour of the figure are not taken into account (we do not know how many particles of outside the micrography are connected to them):

There it is easy to see defects (dislocations, penta-heptas, …). Also based on the triangulation before, it is build the improved adjacency matrix. With it, it is easy to know that (in the figure) there are 126 hexagons, 13 pentagons, 11 heptagons and 1 octagon. Also, with the improvement in the adjacency matrix it is straightforward to evaluate local orientation order parameters.

For both kind of techniques, we are developing a GPL software in Octave (the GPL counterpart of Matlab^®) which will do this automatically. The software is available upon request, since it is not completely finished and operational (it is necessary to adjust parameters by hand).

This kind of analysis can be used in other systems concerning pattern formation and/or defects, among others.