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Application of Fractal Analysis and Percolation Theory in Determination of the Structure of Langmuir Monolayer

The structure/lattice and in plane molecular organization of Langmuir monolayer of amphiphilic material is usually determined from grazing incidence X-ray diffraction or neutron reflectivity. Here we present a different approach for determination of monolayer structure based on application of fractal analysis and percolation theory in combination with Brewster angle microscopy. A strong correlation shown to exist between the fractal dimension and the corresponding compressibility modulus of amphiphilic material in Langmuir monolayer confirms the link between the short ordered structure on molecular level and the macroscopic property - compressibility of the monolayer. This correlation enables identification of onset of percolation - a second order phase transition that is otherwise not easy and unambiguously detectable and consequently application of percolation theory for determination of monolayer structure. The results of such combined approach - use of Brewster angle microscopy, fractal analysis and percolation theory in determination of structure of amphiphilic monolayers are presented and discussed. For a monolayer of 1-hexadecanol at given experimental conditions correlation analysis of fractal dimension’s dynamics determined from BAM images and compressibility indicates occurrence of percolation at the surface pressure 1.4 mN m-1. According to percolation theory critical exponents related to fractal dimension are independent of local structure details but are characteristic for given dimensionality of the system. However, the percolation threshold does depend on structure. According to percolation theory the percolation threshold inferred from our analysis (pc - 0.65) corresponds to a honeycomb structure. This result is in perfect agreement with structure of hexadecanol monolayer determined from GIX measurements.

langmuir monolayers; fractal analysis; percolation; Brewster angle microscopy

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