Cylindrical pore 111, core model

The variant of the cylindrical model which has played a prominent part in the development of the subject is the ink-bottle , composed of a cylindrical pore closed one end and with a narrow neck at the other (Fig. 3.12(a)). The course of events is different according as the core radius r of the body is greater or less than twice the core radius r of the neck. Nucleation to give a hemispherical meniscus, can occur at the base B at the relative pressure p/p°)i = exp( —2K/r ) but a meniscus originating in the neck is necessarily cylindrical so that its formation would need the pressure (P/P°)n = exp(-K/r ). If now r /r, < 2, (p/p ), is lower than p/p°)n, so that condensation will commence at the base B and will All the whole pore, neck as well as body, at the relative pressure exp( —2K/r ). Evaporation from the full pore will commence from the hemispherical meniscus in the neck at the relative pressure p/p°) = cxp(-2K/r ) and will continue till the core of the body is also empty, since the pressure is already lower than the equilibrium value (p/p°)i) for evaporation from the body. Thus the adsorption branch of the loop leads to values of the core radius of the body, and the desorption branch to values of the core radius of the neck. 

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. 

Fig. 2. 2D hexagonal packing of cylindrical pores coated with an adsorbed film (left) box model of the radial density profile p(r) with core density Pc, film density pf and matrix density pm (right)... 

In order to calculate a pore size distribution, a model must be selected, e.g. cylindrical pores, wedge shaped pores, ink bottle pores and so on. It is also necessary to decide which branch of the isotherm to employ, adsorption or desorption. During desorption the cores of the pores are emptied leaving a residual layer, the thickness of which has to be known, in order to calculate a pore size distribution. All of these variables affect the derived distribution. 

In order to allow for the thinning of the multilayer, it is necessary to assume a pore model so as to be able to apply a correction to Uj, etc., in turn for re-insertion into Equation (3.52). Unfortunately, with the cylindrical model the correction becomes increasingly complicated as desorption proceeds, since the wall area of each group of cores changes progressively as the multilayer thins down. With the slit model, on the other hand, <5/l for a... 

To convert the core area into the pore area ( = specific surface, if the external area is negligible) necessitates the use of a conversion factor R which is a function not only of the pore model but also of both r and t (cf. p. 148). Thus, successive increments of the area under the curve have to be corrected, each with its appropriate value of R. For the commonly used cylindrical model,... 

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