Fractal dimension and complexity in the long-term dynamics of a monomolecular layer

The formation of three-dimensional domains in monomolecular layers (Nucleation Dynamics, ND) of four fatty acids, stearic, arachidic, behenic and lignoceric acids, containing the same carboxylic (–COOH) head and an alkyl chain with 18, 20, 22 and 24 carbon atoms, respectively, on water surface, has been studied through Specific Molecular Area (A) versus time (t) studies from Surface Pressure (π)-A isotherms and Brewster Angle Microscopy (BAM). To investigate the fractal nature, the gray-scale Brewster angle micrographs are converted to binary images containing only two pixel values – 0 for 2D phase and 255 for 3D phase before box-counting method is employed to compute the fractal dimension. The 3D phase in the background of 2D phase is found to be fractal in nature. In fact, 3D phase is an interpenetration of two fractal structures with two different fractal dimensions – one corresponding to smaller (intra-domain) structures and other corresponding to larger (inter-domain) structures. These fractal dimensions are seen to evolve with coverage as 3D phase grows. The two fractal dimensions and their evolution in ND dynamics are identical for the longest-tailed lignoceric acid which has a porous 3D phase.