Roughness and Fractal Dimension

Average roughness (R ) can be directly obtained from AFM images for different explored areas using the definition expressed by the following equation  

The roughness versus scanned area pattern is characteristic of a given material and defines a fractal dimension, dp., which is evaluated as dp.= 3—a where a is the so-called roughness exponent that can be calculated as the slope of roughness versus scan size in a double log plot [42]. 

This fractal dimension experimentally gives a good represention of how roughness increases with scan size, with accurate fittings to experimental data [43]. Thus, a value of fractal dimension close to 2.0 means that the membrane has a reasonably flat surface while, if djj. is close to 3.0, the membrane surface has a 3D-like interface. 

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