Fractal patterns formed by thermal treatment in alkali halide crystals
Introduction
Many physical, technological and biological processes exhibit noninteger (or fractal) dimensionality as was recognized by Mandelbrot [1] in his studies on complex geometrical shapes. Fractal growth processes have attracted considerable attention [2] and have been the subject of extensive experimental, numerical and theoretical studies [3], [4], [5] with the realization that scaling concepts may be applied to describe many features of the phenomena [6]. In particular, it was recognized that the structure of the resulting clusters may be described as a fractal [6], leading to a new understanding about both the kinetic mechanisms themselves, as well as the physical properties of the aggregates. These concepts have been successfully applied to the experimental investigation of several aqueous colloid systems [7], [8], [9], [10].
The diffusion-limited aggregation (DLA) model, in which infinitely diluted concentration of particles mimic Brownian motion by moving about randomly until attaching upon first contact with a central cluster is a particularly attractive realization of such a growth process [11], [12], [13].
Studies on silver nanoclusters obtained in alkali halide crystals by thermal treatments showed that the nanoclusters are obtained in structures spatially arranged like “corals” [14]. Vasile et al. [15], [16] found the same “corals” arrangement, without any isolated particles, for samples of KCl:Ag with Ag clusters obtained by thermal conversion of Ag−-ions at 700°C, 645°C, 525°C and 400°C.
To date, the almost universally used parameter for characterizing the structure of aggregates as being fractals is their fractal dimension df which is a noninteger.
The aim of this paper is to present DLA type Ag nanoclusters obtained in three alkali halide crystals (KCl, NaCl and KBr). These are fractal structures obtained by the thermal conversion of Ag− ions in Ag nanoclusters, a fact proven by the calculated values of the fractal dimensions.
Section snippets
Samples preparation
Single crystals of KCl:Ag+, KBr:Ag+ and NaCl:Ag+ were grown in air and in nitrogen atmosphere by the Kyropoulos method in quartz or platinum crucibles. Ultrapure and p.a. salts were used as starting materials. AgCl was added to the melt in order to obtain various concentrations of Ag+ ions (1016–5×1017 ions/cm3).
Electrolytical colouring was performed in air at various temperatures (400–600°C) and for different electric fields (200–4000 V/cm), using a standard device. The reversal of the electric
Calculation method
One of the most important “practical” problems is to determine the fractal dimension df of either a computer generated fractal or a digitized fractal picture. The two most useful methods are: the sandbox and boxcounting method.
In the sandbox method, we first choose one site (or one pixel) of the fractal as the origin of n circles of radii R1<R2<⋯<Rn where Rn is smaller than the radius of the fractal and count the number of points (pixels) M(Ri) within each circle. We plot M(Ri) vs. Ri in a
Results and discussion
We measure M(Ri) for the sandbox method and S(Ni) for the boxcounting method by analyzing transmission electron microscope (TEM) images of the silver nanoclusters. We have used TEM images with magnifications ×21,000 and ×69,000. To limit the errors we first digitally selected only the nanoclusters in the TEM images in which the background contribution was eliminated. Thus, the accuracy of the calculated fractal dimensions was increased.
Typical TEM images showing Ag nanoclusters produced by
Conclusions
In conclusion, we demonstrate that we obtained fractal structures in an alkali halide matrix. Moreover we stress that the Ag nanoclusters are structured in a DLA manner, which is proven by the calculated fractal dimensions. They show that although we used bidimensional pictures to characterize tridimensional structures, typical df values for diffusion-limited cluster aggregation (DLCA) were found. These results can explain the short times in which the formation of Ag nanoclusters during a
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