Gold Nanoparticles Under the Microscope. Patterned structures and surfaces are ubiquitous in nature: from the hide of a giraffe to the distribution of galaxies in the universe, similar repeating structural elements are seen on all length scales. These somewhat ordered structures can often be said to arise by a process of self-assembly or self-organisation. Self-assembly may prove to be an important tool for the future of nanotechnology; to design materials that build themselves rather than manually constructing devices from individual component parts is naturally compatible with mass-production. Atomic Force Microscopy (AFM) has shown that a wide range of quite complex structures can be produced simply by spin-casting a suspension of nanoparticles onto a substrate [1-3]. Morphological Image Analysis (MIA) techniques, in conjunction with Monte Carlo (MC) simulation, can lead us to a better understanding of the processes involved in this apparent self-organisation.

Far-from-Equilibrium Pattern Formation
Much research into self-assembly focuses on processes occurring close to equilibrium, such as the formation of self-assembled monolayers, or nanocrystal ‘superlattices’. In the latter case, solvent is evaporated very slowly, allowing each nanocrystal to find an equilibrium site. However, if we move to the extreme opposite end of the scale, driving the solvent off so rapidly that the system can be considered not only non-equilibrium, but far-from-equilibrium, then quite different results are obtained. Fig. 1a shows a schematic of the particles used in our experiments. Suspensions of these particles are generally stable for many months if not years, making them highly suitable for large-scale production and long-term study. When these suspensions are spincast onto native-oxide silicon substrates, a wide range of patterns is observed, with structures that vary depending on concentration, type of solvent, and surface chemistry of the substrate [1].
Fig. 1 shows two of these morphologies: panel b is an example of a bicontinuous structure, a morphology that is commonly associated with phase-separating systems that proceed by ‘spinodal’ mechanisms, and panel c, with a slightly higher particle concentration, shows what is commonly described as a cellular network.

This network looks much like the cracks that appear in drying mud, or the pattern on the hide of a giraffe. At higher concentration still, cellular structures begin to appear on two levels, with a larger network superimposed on a structure similar to Fig. 1c (not shown).

Morphological Image Analysis
The question of the origin of these structures can be investigated by the use of morphological image analysis (MIA) techniques. Perhaps the most commonly used of these is the two-dimensional fast Fourier transform (2DFFT), which can pick out a directional or size preference of features in an image. However, in cases where the structure can be broken down into a series of points, such as the case of a cellular network, the Voronoi tessellation can provide complementary quantitative information. The Voronoi tessellation reduces a cellular network to a collection of tessellating polygons [4]. Two useful numbers that can be obtained from this construction are the variance of the probability of finding a cell with a given number of sides, and the entropy of the distribution of probabilities. Both of these quantities can be compared to the values expected for a random tessellation, and hence the degree of order of the structure can be established.

Minkowski Functional Grain Growth
Another technique for analysing a distribution of points is Minkowski functional grain growth [5]. In two dimensions, there are three Minkowski functionals. These are the total covered area, A, the perimeter, U, and the Euler characteristic, χ, of an image. The latter equals the number of regions of connected black pixels minus the number of enclosed regions of white pixels. If the points used for the Voronoi tessellation are used as germs for the growth of a two dimensional grain (e.g. a disc), then the values of A, U, and χ as a function of grain size can give us information about the distribution of the point set. Plots of these values can be compared to the known plots for a random distribution of points, and as with the Voronoi tessellation, the degree of order can be established.
Results from both techniques point strongly to a relatively high degree of order in Au cellular networks. This doesn’t directly resolve the question of the origin of these structures, but it does raise another question: what is the mechanism that drives these structures to order? Simulations have provided some interesting answers.