Frenkel-Halsey-Hill theory, fractal

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

Tang, P. Chew, N.Y.K. Chan, H.-K. Raper, J. Limitation of determination of surface fractal dimension using N2 adsorption isotherms and modified Frenkel-Halsey-Hill theory. Langmuir 2003, 7, 2632-2638. 

An alternative approach to the characterization of surface morphology of carbon blacks is the consideration of film formation of adsorbed molecules in the multi-layer regime. In this case, the surface roughness is evaluated with respect to a fractal extension of the classical Frenkel-Halsey-Hill (FHH)-theory, where, beside the van der Waals surface potential, the vapor-liquid surface tension has to be taken into account [100, 101]. Then the... 

Fractal geometry has been used to describe the structure of porous solid and adsorption on heterogeneous solid surface [6-8]. The surface fractal dimension D was calculated from their nitrogen isotherms using both the fractal isotherm equations derived from the FHH theory. The Frenkel-Halsey-Hill (FHH) adsorption isotherm applies the Polanyi adsorption potential theory and is expressed as ... 

Unlike the first two methods, the third method requires only a single adsorption isotherm. The analysis of the single isotherm to obtain Ds is performed by using a modified Frenkel-Halsey-Hill (FHH) theory. The original FHH theory was developed by Frenkel, Halsey,and HilP° and was later extended to fractal surfaces by Pfeifer et al. ... 

The Frenkel-Halsey-Hill (FHH) isotherm was originally developed to describe the growth of thick films and wetting phenomena on a flat surface and was later extended to studying adsorption on fractal surfaces [3, 55]. In contrast to BET theory, FHH theory applies to long-range adsorbate-absorbent interactions and its approach is closely related to the so-called potential theory of adsorption of Eucken and Polanyi (see Ref. [35]). 

The other molecular probe method is the single-probe method (SP method), which is separately proposed by Avnir and Jaroniec,93 and Pfeifer et al.108-112 In the SP method, a single adsorption isotherm is analyzed using a modified FHH theory. The FHH model was developed independently by Frenkel,113 Halsey,114 and Hill,115 and describes the multilayer adsorption coverage. Since the SP method uses only one probe molecule, this method is more convenient than the MP method. However, there are many theoretical limitations in applying the SP method to determination of the surface fractal dimension. Therefore, it is really necessary to discuss whether the SP method is an adequate tool to investigate the surface fractal dimension or not before applying the SP method to certain system. 

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