Statistical film thickness

Equation 1 can be used to determine the pore diameter of an MCM-41 sample which exhibits capillary condensation at a certain relative pressure, or to determine the capillary condensation pressure for an MCM-41 sample of a certain pore diameter. To construct model adsorption isotherms for MCM-41, one also needs a description of the monolayer-multilayer formation on the pore walls. This description can be based on the experimental finding that the statistical film thickness in MCM-41 pores of different sizes (especially above 3 nm) is relatively constant for pressures sufficiently lower from those of the capillaiy condensation and can be adequately approximated by the t-curve for a suitable reference silica [29-31], for instance that reported in Ref. 35. In these studies [29-31], the statistical film thickness in MCM-41 pores, tMcM-4i, was calculated according to the following equation [29] ... 

Figure 1. (a) Experimental relations between the capillary condensation pressure and the pore diameter (hollow circles) and between the capillary evaporation pressure and the pore diameter (filled circles) for nitrogen adsorption at 77 K. The dashed line corresponds to the Kelvin equation with the statistical film thickness correction. The solid line corresponds to Eq. 2 derived using the KJS approach, (b) Relation between pore diameters calculated on the basis of Eq. 1 and the KJS-calibrated BJH algorithm using nitrogen adsorption data at 77 K. 

Figure 1. (a) Comparison of the NLDFT isotherm in a 107 nm diameter cylindrical pore with the standard nitrogen isotherm on nonporous oxides [26], (b) corresponding statistical film thickness plot. 

Textural characterization was performed by N2 adsorption-desorption at 77 K using a Micromeritics ASAP 2010 analyzer. The samples were preheated under vacuum in three steps of Ih at 423 K, Ih at 513 K, and finally 4 h at 623 K. BET specific surface area, Sbet, was calculated using adsorption data in the relative pressure range, P/Po, from 0.05 to 0.2. Total pore volume, Vp , was estimated by Gurvitsch rule on the basis of the amount adsorbed at P/Po of about 0.95. The primary mesopore diameter, Dp, was evaluated using the BJH method from the desorption data of the isotherm. The primary mesopore volume, Vp, and the external surface area, Sext were determined using the t-plot method with the statistical film thickness curve of a macroporous silica gel [5]. 

Figure 2. Dependencies of the adsorption potential on the pore width calculated according to the equations (3) (SF curve), (4) (curve KJSe) and (5) (curve KJSc). The KJSc curve was calculated via equation (5) using the Harkins-Jura-type expression for the statistical film thickness t, which gives a good representation of the experimental t-curve only in the range of relative pressures from 0.1 to 0.95. Therefore, this curve deviates from points at high values of A, which correspond to low values of p/po. Data for the ZLZ curve are from Zhu et al. [29].

An important issue in the classical methods of pore size analysis is the selection of the t-ctirve, i.e., the statistical film thickness on the carbon surface. In the original method [136], it was assumed that the film thickness on the pore walls does not depend on the pore radius but only on the relative pressure. Thus, this BJH method required two relations for the evaluation of the pore-size distribution from adsorption isotherms. The first, represented by the Kelvin Eq. (70), is the relation between the pore radius and the relative pressure at which capillary condensation occurs in the pores. The second represents the functional dependence of the statistical film thickness on the relative pressure. It should be noted that existing relationships for t-curves reported some time ago do not represent low-pressure adsorption behavior because the relevant low-pressure data were not available at that time [13], The corresponding low-pressure adsorption isotherms on carbon surfaces are now available and can be used to evaluate the t-curve for the entire pressure range. 

The last section has shown the basic concepts of capillary condensation and how they can be utilized in the determination of pore size distribution (PSD). In this section, we address a number of practical approaches for PSD determination. One of the early approaches is that of Wheeler and Schull and this will be presented first. A more practical approach is that of Cranston and Inkley, and this will be discussed next. Finally, the de Boer method is presented, which accounts for the effect of pore shape on the calculation of the statistical film thickness and the critical pore radius. 

The Critical Pore Radius and the Statistical Film Thickness... 

To evaluate the volume adsorbed from eq. (3.10-1), we need to know the critical pore radius r and the statistical film thickness (t). Using eq.(3.9-15) for the Kelvin radius, the critical radius is calculated from ... 

The statistical film thickness may be calculated from eq. (3.9-25) or from the BET equation, that is ... 

Thus, the statistical film thickness for nitrogen calculated from the BET theory is ... 

It is known that the BET statistical film thickness of a practical porous solid is larger than the experimental thickness for flat surfaces in the high pressure region (Schull, 1948). Figure 3.10-1 shows a plot of the statistical thickness t calculated from eq. (3.10-9) with C= 100. 

Figure 3.10-1 Plots of the statistical film thickness versus the reduced pressure...

Also plotted in the same figure is the statistical thickness calculated from equation (3.9-25). The two curves deviate significantly in the high pressure region. The circle symbols on this figure are experimental data obtained on many non-porous solids (Cranston and Inkley, 1957). We see that eq. (3.9-25) agrees very well with the experimental data and it is then a better choice of equation for the calculation of the statistical film thickness. 

Eq. (3.10-1) is the equation allowing us to calculate the volume adsorbed V as a function of the reduced pressure. For a given reduced pressure P/Pq, the statistical film thickness is calculated from eq. (3.9-25) and the critical radius r is calculated from eq. (3.10-5), and hence the volume adsorbed can be calculated by integrating the integral in eq. (3.10-1). We illustrate this with a number of examples below. 

This method, like the other methods, requires the knowledge of the statistical film thickness of the adsorbed layer on a flat surface. The experimentally determined thickness of the adsorbed layer for nitrogen on a flat surface was obtained by Cranston and Inkley as a function of reduced pressure as shown in Figure 3.10-1 as symbols and are tabulated in Table 3.10-1. 

De Boer studied extensively the pore size distribution, and refined methods to determine it. Basically, he accounted for the pore shape in the calculation of the statistical film thickness as well as the critical pore radius. What we present below is the brief account of his series of papers published from early 60 to early 70. 

For pores of cylindrical or slit shape, the behaviour of the calculated statistical thickness does not follow that of a flat surface as the pore shape can influence the statistical film thickness. This is explained as follows. For cylindrical pores, the solid will take up more sorbates than a free surface, that is... 

A reference non-porous material with similar surface characteristics is chosen to obtain the information on the statistical film thickness as a function of the reduced pressure. The t values for different reference adsorbents as a function of the reduced pressure are available in literature, normally in the table form or a best fit equation. 

In the above table, V is the amount adsorbed at the relative pressure P/Po- For each value of the reduced pressure, the statistical film thickness t can be obtained from a reference material of the same characteristics or calculated using eq. (3.9-25). This has been included in the third column of the above table by using eq. (3.9-25). 

Figure 3.11-2 Plot of the amount adsorbed versus the statistical film thickness...

In many cases, information about structural properties of high surface area materials can be obtained by comparing the adsorption isotherm of a solid under study with the adsorption isotherm of a reference macroporous material of similar surface properties [74,82,83]. The comparison is made by plotting the amount adsorbed for the solid under study as a function of the amount adsorbed for the reference material (tVef) at the same pressure. The adsorption for the reference is usually expressed either as the statistical film thickness t of the adsorbate on the surface (in the t-plot method) or as the standard reduced adsorption (in the %-plot method) [74]. The quantity a, is defined as the ratio of the adsorbed amount at a given pressure to the adsorbed amount at a relative pressure of 0.4. 

Here is the condensation pressure for a slit-shaped pore of width L, Pq is the saturation pressure of the adsorbate fluid, p is the fluid density, y is the surface tension of the sorbate, and/ and T are the ideal gas constant and absolute temperature, respectively. However, the Kelvin equation seemed to fail as pore size approached very low values because it does not take the thickness of the monolayer formed within the pore into account. Several models for the estimation of a statistical film thickness have been proposed (cf. Refs. 7-9) with the view of improving the PSD prediction. A commonly used method based on the modified Kelvin equation is the technique by Barrett et al. (the BJH method) [10,11]. A detailed description of such methods is available in the reviews by Gregg and Sing [1] and Webb and Orr [9]. 

In addition to coordination number, film surface area measurements have been used to obtain qualitative information concerning smface roughness [6]. This process is illustrated in Figure 1 for which the surface available for surface area measurement of a rough surface is smoothed with increasing Him thickness. This idea could also be extended as a pore size distribution probe. In other words, as the thickness of a film approachs the pore size, the surface area of the pore would rapidly decrease to zero. Although Figure 1 iilustrates the him as a smooth layer of constant thickness, we do not mean to imply this is actually the case except for films of many monolayers. Instead, this is a conceptual model of the statistical film thickness. 

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