Frenkel-Halsey-Hill theory

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 exception is represented by the Frenkel-Halsey-Hill theory, which describes the multilayer adsorbed film as a liquid perturbed by the presence of a surface. 

The Frenkel-Halsey-Hill theory considers the adsorbed film to have the bulk liquid density out from the surface to a point at which there is a... 

The surface waves of MacMillan and Teller (71) would give a mean density transition curve p(z) that is continuous, although each individual wave has the same type of discontinuity as in the Frenkel-Halsey-Hill theory. The MacMillan and Teller approach is essentially one way of classifying a oertain restricted set of configurations consistent with a continuous mean density transition. 

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. ... 

In Eq. (31), Pq is the saturation pressure and Pc is the pore condensation pressure. The assumed exponential dependence of the condensation pressure on the adsorption free energy change is similar in basis to the Polanyi potential theory [101] and the Frenkel-Halsey-Hill (FHH) theory [102-104]. In the HK method, the mean free energy change due to adsorption is calculated... 

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 potential concept just formulated is used in the slab theory of multilayer adsorption on homogeneous and heterogeneous surfaces. Several dependencies for the adsorption potential e(h) have been suggested (see the analysis in Refs. 3, 8, 94 and 95). It is often described by the Frenkel-Halsey-Hill (FHH) power dependence [96,97]. A general form of this dependence, avoiding singularity at the surface, has the form [3]... 

The results of the said simulations have been compared with theoretical and/or experimental results. Good agreement was found, for instance, between the simulation results of Lane and Spurling [318] and the experimental results of Thorny and Duval [37] for Kr on graphite. Another example is the possibility of introducing improvements in the classical adsorption equations. Thus, Steele [219] has developed a modified Frenkel-Halsey-Hill (FHH) theory based on computer simulation results for the multilayer adsorption of argon on graphite performed by Rowley et al. [179,180]. 

The form of an adsorption isotherm on a geometrically flat substrate in the thick film regime can be described by the so called Frenkel-Halsey-Hill (FHH) theory, under the eussumption of complete wetting of the surface by the film (ref. 4). The FHH vapor pressure is of the form ... 

The calculation of firactal dimensions from sorptometry data is based on the theory by Frenkel, Halsey and Hill and also of Kisielev [55], The fi nctal dimension Fj can be calculated from the following equations ... 

Frenkel, Halsey and Hill theory of adsorption and isotherm equation... 

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. 

After Ya. I. Frenkel, Kineticheskaya Teoriya Zhidkostei (1953), chapter VI. (English transl. J, Frenkel, Kinetic Theory of Liquids, Oxford Clarendon Press (1946), also available as a Dover reprint (Dover, 1955)), G.D. Halsey Jr.. J. Chem. Phys. 16 (1948) 931 and T.L. Hill, Adv. Catal 4 (1952) 211. 

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