Kelvin equation, pore size distributions mesopores adsorption

The basic description of a mesoporous sample requires two types of determinations X-ray diffraction and gas adsorption/dcsorption isotherm. The latter are usually represented as the amount of gas adsorbed by the sample as the function of relative pressure. This information characterizes pore size distribution. Nitrogen adsorption/desorplion isotherm at 77 K is most often used and relatively convenient to carry out. The adsorption of noble gases is used if accurate in-depth pore characterization is attempted, especially quantitative. The calculation of pore size distribution from the isotherms is carried out using appropriate formulas such as Kelvin and IIorwath-Kawazoe equations (e.g. as in Ref. 5 and [6]), which involve assumptions and approximations. A more detailed and rigorous treatments have been developed, as for example KJS (Kruk-Jaroniec-Sayari), which is relatively simple and accurate [42]. In practice, the diameter of mesopores can be quickly estimated directly from the position of the capillary condensation or, if not vertical, the p/p0 of the inflection point. The conversion table of p/po values to pore diameters can be found in Ref. [43] and is partially reproduced here in Table 2. 

Table 16-4 shows the IUPAC classification of pores by size. Micropores are small enough that a molecule is attracted to both of the opposing walls forming the pore. The potential energy functions for these walls superimpose to create a deep well, and strong adsorption results. Hysteresis is generally not observed. (However, water vapor adsorbed in the micropores of activated carbon shows a large hysteresis loop, and the desorption branch is sometimes used with the Kelvin equation to determine the pore size distribution.) Capillary condensation occurs in mesopores and a hysteresis loop is typically found. Macropores form important paths for molecules to diffuse into a par-... 

Nitrogen adsorption/desorption. One of the most common techniques used for analyzing size distributions of mesopores is the nitrogen adsorption/desorption method. The method is capable of describing pore diameters in the 1.5 nm to 100 nm (or 0.1 micron) range. Pore size distribution can be deteimined from either the adsorption or the desorption isotherm based on the Kelvin equation ... 

The type II isotherm is associated with solids with no apparent porosity or macropores (pore size > 50 nm). The adsorption phenomenon involved is interpreted in terms of single-layer adsorption up to an inversion point B, followed by a multi-layer type adsorption. The type IV isotherm is characteristic of solids with mesopores (2 nm < pore size < 50 nm). It has a hysteresis loop reflecting a capillary condensation type phenomenon. A phase transition occurs during which, under the eflcct of interactions with the surface of the solid, the gas phase abruptly condenses in the pore, accompanied by the formation of a meniscus at the liquid-gas interface. Modelling of this phenomenon, in the form of semi-empirical equations (BJH, Kelvin), can be used to ascertain the pore size distribution (cf. Paragr. 1.1.3.2). 

The method devised by Barrett, Joyner, and Halenda (BJH) [35] is one of the earhest methods developed to address the pore size distribution of mesoporous sohds. This method assumes that adsorption in mesoporous solid (cylindrical pore is assumed) follows two sequential processes — building up of adsorbed layer on the surface followed by a capillary condensation process. Karnaukhov and Kiselev [45] accounted for the curvature in the first process, but Bonnetain et al. [46] found that this improvement has httle influence on the determination of pore size distribution. The second process is described by either the Cohan equation (for adsorption branch) or the Kelvin equation (for desorption branch). 

In the discussion of the mesopore shape, the contact angle, is assumed to be zero (uniform adsorbed film formation). The lower hysteresis loop of file same adsorbate encloses at a common relative pressure depending to the stability of the adsorbed layer regardless of the different adsorbents due to the so called tensile strength effect. This tensile strength effect is not sufficiently considered for analysis of mesopore structures. The Kelvin equation provides the relationship between the pore radius and the amount of adsorption at a relative pressure. Many researchers developed a method for the calculation of the pore size distribution on the basis of the Kelvin equation with a correction term for the thickness of the multilayer adsorbed film. 

The pore size distribution from the Kelvin equation should be limited to mesopores due to the ambiguity of the meniscus in the microporous region. It is well known that the presence of micropores is essential for the adsorption of small gas molecules on activated carbons. However, when the adsorbate is polymer, dye or vitamin, only mesopores allow the adsorption of such giant molecules and can keep even bacteria. The importance of mesopores has been pointed out not only for the giant molecule adsorption, but also for the performance of new applications such as electric double layer capacitors. Thus, the design and control of mesoporosity is very desirable both for the improvement of performance of activated carbon and for the development of its new application fields [1-3],... 

A pore size distribution (PSD) of a sample is a measure of the cumulative or differential pore volume as a function of pore diameter. PSDs can be calculated from adsorption isotherms based on an analysis which accounts for capillary condensation into pores. This analysis (14.16) uses a model of the pore structure combined with the Kelvin equation (12) to relate the pore size to the value of p/Po at which pore "filling" occurs. Due to limitations in this technique, only pores with diameters from about 3 to 50 nm, called mesopores (14), can be characterized. This pore size range, however, is typical of many porous samples of interest. For samples with pores smaller or larger than this range, alternative techniques, such as mercury intrusion for large pores (14.16), are typically more suitable. 

The total surface area of a porous material is given by the sum of the internal and external surface areas. Pores are classified as micropores (pore width less than 2 nm), mesopores (pore width between 2 and 50 nm), and macropores (pore width greater than 50 nm) according to the definitions proposed by lUPAC (6). The specific pore volume, pore widths, and pore size distributions for micro-and mesopores are determined by gas adsorption. In the case of mesopores, the method is based on the relationship between the pressure of capillary condensation and the radius of a cylindrical pore in which condensation takes place, given by the Kelvin equation ... 

All of these methods assume the following [4] (i) Kelvin equation is applicable over the complete mesopore range (ii) the meniscus curvature is controlled by the pore size and shape, and 0 (contact angle) is 0 (iii) the pores are rigid and of well-defined shape (iv) the distribution is confined to the mesopore range (v) the filling (or emptying) of each pore does not depend on its location, and (vi) the adsorption on pore walls follows the same mechanism as on the open surface. 

It is well established that the pore space of a mesoporous solid fills with condensed adsorbate at pressures somewhat below the prevailing saturated vapor pressure of the adsorptive. When combined with a eorrelating function that relates pore size with a critical condensation pressure, this knowledge can be used to characterize the mesopore size distribution of an adsorbent from its adsorption isotherm. The correlating function most commonly used is the Kelvin equation [1], Refinements make allowance for the reduction of the physical pore size by the thickness of the adsorbed film existing at the critical condensation pressure [1-2]. Still further refinements adjust the film thickness for the curvature of the pore wall [3]. 

As seen fnxn Fig. 3, a region exists fdiere the desorption and adsorption isotherms diverge. This hysteresis is attributed to the differences in the mechanians of filling and enptying of mesopores (62). The locaticsi of the hysteresis loop will thus reflect the size distribution of mesopores. The pore size (or more exactly, the core size) may be calculated front the desorption branch with the aid of the Kelvin equation (62)... 

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