Surface roughness and fractal dimensions

The surface of a crystal often shows irregularities such as steps, Idnks and holes. As microscopic resolution increases, smaller and smaller irregularities become visible. If similar surface morphologies are observed on all scales, regardless of magnification, these surfaces are termed self similar or self affine. 

When measuring the area of such surfaces using the BET method and probe molecules of different cross sectional area, do. the specific surface area, SA, as a function of (7o will obey a power law (Avnir et ah, 1983) 

According to eq. (5.5), a linear relationship, the slope of which is characterized by Ds, is expected between (Tq and log A. To establish such a relationship, the size of the probe molecule should vary by at least one order of magnitude. If however, the variation is restricted to less than an order of magnitude, surface roughness instead of self-similarity might be a more appropriate term. 

To overcome the problems which arise from a limited size range and the different adsorption properties of different adsorbates, one may use the entire adsorption isotherm obtained with just one probe molecule instead. There are two approaches both leading to a Frenkel-Halsey-Hill type equation  

Other possible methods of obtaining fractal dimensions are light- and small-an-gle-X-ray-scattering (SAXS). The first method gives information about mass fractality of aggregates and aggregation kinetics (Fleischmann et al., 1990), whereas the latter allows mass and surface fractals to be differentiated (Schmidt, 1990). 

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