The MP method

Mikhail, Brunauer, and Bodor proposed an extension of deBoer s r-method for the analysis of micropores which offers several advantages. These include the ability to obtain the micropore volume, surface, and their distributions from one experimental isotherm. Data for the MP (micropore analysis) method need not be measured at the very low pressures needed for the Dubinin and Kaganer theories. The method 

When measuring mesopores the statistical thickness t is used as a correction to allow for desorption from the adsorbed film. However, the value of t is much more critical when measuring micropores because in the MP method, t is the actual measure of the pore size. 

The usual method of measuring the statistical thickness is to divide the liquid volume of nitrogen adsorbed, by the BET surface area 

The assumption usually made is that the ratio Fu /Sbet has the same value at a given relative pressure independent of the solid. A plot therefore of t versus P/Pq should give the same curve for any non-porous solid (see Fig. 8.6). In fact, plots of the number of adsorbed layers versus P/Pq show some discrepancies which for the analysis of large pores is not significant. Therefore, the Halsey equation can be used for the statistical thickness in that application. However, for micropore analysis, a statistical thickness must be taken from a t versus P/Pq curve that has approximately the same BET C value as the test sample. The unavailability of t versus P/Pq plots on numerous surfaces with various C values would make the MP method of passing interest were it not for the fact that t can be calculated from equation (8.36). This implies that surface area can be accurately measured on microporous samples. Brunauer points out that in most instances the BET equation does correctly measure the micropore surface area. 

As stated earlier the C value in micropores will be large due to the overlapping wall potentials. Under these circumstances, the surface Will be covered well over 90% by stacks of adsorbate not in excess of two molecules in depth as shown by equation (4.45) and Table 4.1. Therefore, the close proximity of the walls offer no special condition which is not already allowed for by the BET theory. 

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Fig. XVII-31. (a) Nitrogen adsorption isotherms expressed as /-plots for various samples of a-FeOOH dispersed on carbon fibers, (h) Micropore size distributions as obtained by the MP method. [Reprinted with permission from K. Kaneko, Langmuir, 3, 357 (1987) (Ref. 231.) Copyright 1987, American Chemical Society.]...

There has been fierce debate (see Refs. 232, 235-237) over the usefulness of the preceding methods and the matter is far from resolved. On the one hand, the use of algebraic models such as modified DR equations imposes artificial constraints, while on the other hand, the assumption of the validity of the /-plot in the MP method is least tenable just in the relatively low region where micropore filling should occur. 

Pople, Binkley and Seeger (amongst others) have made a systematic study of the MP method. Here is their summary. 

Figure 4.12 Typical oscillating behaviour of results obtained with the MP method...

Here, the surface fractal dimension determined by using the MP method is denoted by dF MP. The probe molecules need not be... 

The MP method is widely applicable to various solid surfaces in whose system Nmoao can be accurately determined. However, the MP method can not be routinely applied to the microporous system due to a serious problem in the concept of monolayer adsorption for the microporous system.101 This problem is whether the monolayer formation on the surface of the micropore is eligible or not. 

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. 

The MP method has also been adapted for classification of bioactive compounds, a task that substantially differs from diversity analysis. Here the key is to find descriptor combinations that place compounds with similar activity into the same partition, separate them from others, and avoid the creation of partitions containing molecules having different activity. Therefore, the following scoring function was optimized (14) ... 

To illustrate the MP method, consider the isotherm shown in Fig. 9.4. The volumes of adsorbed gas are converted to liquid volumes from which t is calculated using equation (8.36). The V t plot shown in Fig. 9.5 is then constructed using relative pressure intervals of 0.05. 

The MP method treats the correlation part of the Hamiltonian as a perturbation on the Hartree-Fock part, and truncates the perturbation expansion at some order, typically n = 4. MP4 theory incorporates the effect of single, double, triple and quadruple substitutions. The method is size-consistent but not variational. It is commonly believed that the series MP1, MP2, MP3,. .. converges very slowly. 

Here, Nmono is the number of adsorbed molecules to form a monolayer for each probe molecule and surface fractal dimension determined by using the MP method. The probe molecules need not to be spherical, provided they belong to a homologous series for which the ratio [linear extent rm]2 to molecular cross-sectional area Ac is the same for all members, i.e., an isotropic series. In this case, Eq. (13) turns into... 

The MP method is widely applicable to various solid surfaces in whose system Nmono can be accurately determined. It was recently reported64,134 136 that even for the micropore whose pore width is greater than the bilayer thickness of N2 (about 0.7 nm), 4urf,MP can be determined using Eq. (14) if the monolayer evaluation fit for the micropore is introduced. 

Fig. 2.9 (Continued) identical trees were obtained using the ProML- [16] and the MP-methods [13]. Numbers in the nodes indicate bootstrap values forNJ/MP/ML methods, respectively. Arrow indicates the occurrence of LGT from Bacteroidetes towards sac fungi. Abbreviations of protein names correspond to PeroxiBase...

However a recent study has shown that the Stoeckli method (based on the Dubinin-Astakhov theory) [6] gave results similar to those obtained from the molecular simulation methods [9]. On the other hand, the H-K and the MP methods are known to be rather inconsistent. 

From the results presented in Table 1 the wide variation in the Sbet with the pH of formation was most obvious. The Sbet was lowest for the sample prepared at pH 1 but rose to a maximum at pH 4.5 and then fell. All of the materials dried at 60°C were microporous, giving rise to type I isotherms with very little external surface area, or mesopore volume. However, the width of the micropores determined by the MP method was found to progressively increase as the pH at which the sol was formed was raised. This gradual widening of the micropores was apparent from the change in the shape of the nitrogen isotherms shown in Fig. 1. 

The average pore size for sample SG 5.T in Table 2 was determined by the MP method. The average pore diameters for the heat-treated samples were taken from the maxima in the PSD curves drawn using the BJH method from the desorption branches of the corresponding isotherms presented in Fig. 3. It may be noted that the most significant change in pore diameter was caused by heat-treatment at 500°C. This shift along with the severe reduction in the Sbet to only c. 50 m g was mainly due to the transformation into rutile. 

We turn now to the analysis of pore structure. For this purpose, various optional computational procedures are incorporated in the software, which is now provided with most commercial adsorption equipment. For example, for micropore size analysis the isotherm can be converted into a t-plot and also displayed in either the Dubinin-Radushkevich (DR) or the Dubinin-Astakov (DA) coordinates. With some packages it is also possible to apply the MP method of Brunauer, the Horvath-Kawazoe (HK) method and/or density functional theory... 

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores ). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slitshaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution. 

Fig. 4.11. Comparison of the hydraulic radius distributions of pores (obtained by the MP method from N2 adsorption isotherms) of separate and mixed silica samples and the corresponding e...

The mean hydraulic pore radius RH (estimated by the MP method) varied between 3.2 and 7.0 A. This parameter was slightly affected by the surfactant/ precursor ratio (SAA/TEOS) but increased with the surfactant chain length X. 

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