Frenkel-Halsey-Hill equation

This may be based on Eq. XVI-2 [232] or on related equations with film thickness given by some version of the Frenkel-Halsey-Hill equation (Eq. XVII-79) [233,234],... 

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). 

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 ... 

The usefulness of the Frenkel-Halsey-Hill (FHH) equation for multilayer analysis was discussed in Chapters 4 and 9. FHH plots for nitrogen on various pyrogenic silicas are given in Figure 10.2. As expected, each FHH plot is linear over a wide range of p/p°, but this is rather more extensive (i.e. pjp° 0.3-0.9) with the arc... 

This result is in a qualitative agreement with the experimental t-plot of Ar adsorption at 87 K on MCM 41 samples (see Figure 2(b)) using the data given in reference [13], As for simulation data, we assume that the density of the adsorbate equals that of the 3D-liquid and we have determined the thickness of the adsorbed film as the ratio of the adsorbed volume with the surface of the sample. Assuming pores of MCM 41 are cylinders, the specific surface S of each sample was determined via the relation between the porous volume V (given by the adsorbed amount after capillary condensation) and the diameter d of the pores S = 4V/d. Comparison of the different t-curves indicates that there is a pore size (5.1 nm) above which no confinement effect occurs on multilayer adsorption. Below this value, the thickness of the adsorbed film increases as the pore diameter decreases, t-curves are often analysed with the Frenkel-Halsey-Hill equation [14] /n... 

In order to complete the study of the structural differences in xerogels with various additive concentrations, fractal analysis has been performed on nitrogen adsorption-desorption data. The generalisation of the classical Frenkel-Halsey-Hill (FHH) equation [6,7] to fractal surfaces has been used. In this approach a film thickness of more than 1.5 layers, is used to probe the geometry of the surface. It describes the multilayer coverage of a fractal structure by the equation [7] ... 

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 ... 

Many different equations have been used to interpret monolayer—multilayer isotherms [7, 11, 18, 21, 22] (e.g., the equations associated with the names Langmuir, Vohner, HiU-de Boer, Fowler-Guggenheim, Brunauer-Emmett-Teller, and Frenkel-Halsey-Hill). Although these relations were originally based on adsorption models, they are generally applied to the experimental data in an empirical manner and they all have Hmitations of one sort or another [7, 10, 11]. 

The multilayer Frenkel-Halsey-Hill (FHH) equation is usually expressed in the following form ... 

It has been pointed out above that the Wheeler-Ono approach (see Sec. III.l) to the idealized mathematically plane surface problem is the rigorous approach, though actual numerical calculations based on the general equations are not practical. On the other hand, the Frenkel-Halsey-Hill method (see Sec. III.4) is essentially a very approximate solution of this same problem resulting in a simple and surprisingly successful isotherm equation, Eq. (38), for 0 not too small. This method can be applied to capillary condensation (see Sec. III.5) and is capable of accounting for isotherm types II to V (1,55,75). 

According to Frenkel-Halsey-Hill (FHH)model, the calculation equation of fractal dimension is (Pfeifer P Liu K 1997) ... 

Note DFT/PaSD with void model, self-consistent model of a mixture of voids, cylindrical and slit-shaped pores, self-consistent regularization with respect to both PoSD (fy(i p)) and PaSD ((l)( z)) with the model of voids, (5 ), Aw=Sgg j-/(5 ) - 1, f)pHH e fractal dimension with Frenkel-Halsey-Hill equation accounting for adsorbate surface tension effects (Quantachrome Instruments software), Ag j is the gelatin adsorption in mg per gram of silica. 

The Frenkel-Halsey-Hill (FHH) isotherm has found much utilization due to the range specified for its application. It seems especially handy for porosity determinations. It seems to work well between relative pressures in the range 0.4-0.9. The equation is... 

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 Frenkel-Halsey-Hill (FHH) equation has been used to analyse the behaviour of the isotherms at higher relative pressures (7). It has been applied in the form ... 

The equations of MacMillan and Teller are not designed to take into account the effect considered by Frenkel, Halsey, and Hill, but rather to provide the correction term necessary because of the unjustified assumption of a plane surface. Intuitively, it is not obvious whether this correction term is going to be small or large, perhaps larger than the term it is supposed to correct. 

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... 

This form of equation has been studied by Frenkel, Halsey and Hill in the study of multilayer adsorption (Frenkel, 1946 Halsey, 1948 Hill, 1949, 1952), and hence it is known in the literature as the FHH equation. The parameter r is regarded as a measure of the rate of decline in the adsorption potential with distance from the surface. For van der Waals forces, r is equal to 3. A value of about 2.7 is commonly observed in practice. 

Other equations are provided by Frenkel, Halsey, and Hill (FHH) as well as Broekhoff and de Boer. 

---