Modelless method

The pore shape of very few adsorbents is known and it is unlikely that any one solid will contain pores of only one shape. In the modelless method no pore shape is assumed. The analysis is based on the hysteresis region of the isotherm [71,72]. The method of analysis, strictly speaking, gives the distribution of core volumes and surfaces as functions of the core hydraulic radii which is defined as the ratio of the volume to the surface of the cores. 

The cumulative pore size distribution by volume is obtained by plotting Vj against r ,. This may be differentiated graphically to produce the relative pore size distribution by volume or the calculation may be carried out using the tabulated data. 

The cumulative pore size distribution by surface is obtained by plotting Sp against r , where Sp is obtained using equation (3.7) in the form  

Relative Volume Condensed Relative Residual Condensed Condensed Mean  

Core surface (uncorrected) ASuc = -4939log(xmean) Vc, 

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Brunauer and co-workers [211, 212] proposed a modelless method for obtaining pore size distributions no specific capillary shape is assumed. Use is made of the general thermodynamic relationship due to Kiselev [213]... 

Brunauer s modelless method uses pore volume and pore area not as functions of the Kelvin radius but rather as functions of hydraulic radii that he defines as ... 

Brunauer and co-workers [21,22] used the following general thermodynamic relation [23] to obtain the pore size distribution (referred to as the "modelless" method)... 

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. 

In the modelless method, the term hydraulic radius, is introduced where... 

This suffices to show the difference between this corrected modelless method and the Cranston-Inkley method. For detailed description and examples of calculations reference must be made to the original paper. 

Pore Size Standard. Use of the modelless method on Gasil (I), a mesoporous silica consisting of spheres with an average 4.38 nm radius packed with a coordination number of 4, has been described by Havard and Wilson (168). This is one of the SCI/IUPAC/NPL standards for surface area and can be used also as a standard for pore size determinations and for calibrating the BET apparatus over the full pressure range. 

The MP method was demonstrated by Mikhail, Brunauer, and Bodor (164) to be applicable to micropores and the corrected modelless method to large pores. When applied to gel with both micro- and mesopores, the method gave a cumulative area of pore surfaces agreeing with the BET value. This indicates that, in spite of arguments to the contrary, the BET method can be depended on to give reliable surface areas even on microporous silicas. 

Water can be used to measure the area of materials containing micropores into which the larger nitrogen molecules cannot penetrate. The MP. or corrected modelless method of analysis was used in this case for data obtained on hydrated calcium silicate by Bodor, Skalny, and Brunauer (179). 

Table 3.1 Evaluation of pore size distribution from nitrogen desorption (modelless method)... 

In the corrected modelless method [75] a correction is applied for the residual film thickness, i.e. the Kelvin radius is used as - r... 

As in the other methods, the modelless procedure visualizes the progressive emptying of the pore system, initially full up, by the stepwise reduction in relative pressure. At each step the value of additional core wall area which becomes exposed is calculated by means of the equation (see p. 170) ... 

The surface area calculated by the above method is modelless and assumes no specific pore geometry. 

Based on the above general principles, quite a number of models have been developed to estimate pore size distributions.29,30,31-32,33 They are based on different pore models (cylindrical, ink bottle, packed sphere,. ..). Even the so-called modelless calculation methods do need a pore model in the end to convert the results into an actual pore size distribution. Very often, the exact pore shape is not known, or the pores are very irregular, which makes the choice of the model rather arbitrary. The model of Barett, Joyner and Halenda34 (BJH model) is based on calculation methods for cylindrical pores. The method uses the desorption branch of the isotherm. The desorbed amount of gas is due either to the evaporation of the liquid core, or to the desorption of a multilayer. Both phenomena are related to the relative pressure, by means of the Kelvin and the Halsey equation. The exact computer algorithms35 are not discussed here. The calculations are rather tedious, but straightforward. 

The Modelless Pore (MP) Method. Brunauer, Mikhail, and Bodor (166) developed a method of characterizing pore size distribution even in part of the micropore (MP) range. 

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