In practice objects exhibiting random properties are encountered. It is often assumed that these objects exhibit the self-affine properties in a certain range of scales. Self-affinity is a generalization of self-similarity which is the basic property of most of the deterministic fractals. A part of self-affine object is similar to whole object after anisotropic scaling. Many randomly rough surfaces are assumed to belong to the random objects that exhibit the self-affine properties and they are treated self-affine statistical fractals. Of course, these surfaces can be studied using atomic force microscopy (AFM). The results of the fractal analysis of the self-affine random surfaces using AFM are often used to classify these surfaces prepared by various technological procedures [1,2,3,4].
Within Gwyddion, there are different methods of fractal analysis implemented within → → .
The axes in Fractal Dimension graphs always show already logarithmed quantities, therefore the linear dependencies mentioned above correspond to straight lines there. Frequently only a part of the plotted curve is a straight line as the end tend to be influenced by various systematic errors and artefacts. The measure of the axes should be treated as arbitrary.
Note that that results of different methods differ. This fact is caused by systematic error of different fractal analysis approaches.
Moreover, the results of the fractal analysis can be influenced strongly by the tip convolution. We recommend therefore to check the certainty map before fractal analysis. In cases when the surface is influenced a lot by tip imaging, the results of the fractal analysis can be misrepresented strongly.
Note, that algorithms that can be used within the fractal analysis module are also used in Fractal Correction module and Fractal Correction option of Remove Spots tool.
[1] C. Douketis, Z. Wang, T. L. Haslett, M. Moskovits: Fractal character of cold-deposited silver films determined by low-temperature scanning tunneling microscopy. Physical Review B 51 (1995) 11022, doi:10.1103/PhysRevB.51.11022
[2] W. Zahn, A. Zösch: The dependence of fractal dimension on measuring conditions of scanning probe microscopy. Fresenius J Analen Chem 365 (1999) 168-172, doi:10.1007/s002160051466
[3] A. Van Put, A. Vertes, D. Wegrzynek, B. Treiger, R. Van Grieken: Quantitative characterization of individual particle surfaces by fractal analysis of scanning electron microscope images. Fresenius J Analen Chem 350 (1994) 440-447, doi:10.1007/BF00321787
[4] A. Mannelquist, N. Almquist, S. Fredriksson: Influence of tip geometry on fractal analysis of atomic force microscopy images. Appl. Phys. A 66 (1998) 891-895, doi:10.1007/s003390051262
[5] W. Zahn, A. Zösch: Characterization of thin film surfaces by fractal geometry. Fresenius J Anal Chem 358 (1997) 119-121, doi:10.1007/s002160050360