Frenkel, Halsey and Hill

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. 

In spite of this, there is no accurate description accounting for the formation of a liquid film at the surface of a solid starting from an adsorbed film, or conversely of the appearance of an adsorbed film in the final stages of evaporation of a liquid film. In fact, the two major models hitherto developed for multilayer adsorption, namely the model originally proposed by Brunauer, Emmett and Teller (BET) and the one proposed separately by Frenkel, Halsey and Hill (FHH), apply to the description of a thin adsorbed phase the former and of a liquid film the latter. 

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 general approach goes back to Frenkel [63] and has been elaborated on by Halsey [64], Hill [65], and McMillan and Teller [66]. A form of Eq. XVII-78, with a = 0,... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

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). 

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. 

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 ... 

Since z oc V, this may be written In (p0/p) = const. V where V is the volume of gas adsorbed. A more general form of this isotherm, called the Frenkel-Halsey-Hill (FHH) isotherm, treats the power dependence of V as an unknown n and writes... 

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... 

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 ... 

Important trends in N2 isotherm when the PS beads are used as a physical template are shown in Table 1 and Fig. 2. In Table 1, PI is the alumina prepared without any templates, P2 is prepared without ]4iysical template (PS bead), P3 is prepared without chemical template (stearic acid), and P4 is prepared with all templates. For above 10 nm of pore size and spherical pore system, the Barrett-Joyner-Halenda (BJH) method underestimates the characteristics for spherical pores, while the Broekhoff-de Boer-Frenkel-Halsey-Hill (BdB-FHH) model is more accurate than the BJH model at the range 10-100 nm [13]. Therefore, the pore size distribution between 1 and 10 nm and between 10 and 100 nm obtained from the BJH model and BdB-FHH model on the desorption branch of nitrogen isotherm, respectively. N2 isotherm of P2 has typical type IV and hysteresis loop, while that of P3 shows reduced hysteresis loop at P/Po ca. 0.5 and sharp lifting-up hysteresis loop at P/Po > 0.8. This sharp inflection implies a change in the texture, namely, textural macro-porosity [4,14]. It should be noted that P3 shows only macropore due to the PS bead-free from alumina framework. 

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. 

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... 

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]. 

An approach completely different from any discussed above has been used by Frenkel (69), Halsey (55), Hill (70), and MacMillan and Teller... 

Frenkel discussed this problem first, but, independently of Frenkel s work, it was also treated by Halsey and by Hill. 

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). 

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. 

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