Pore size determination Kelvin equation

The classical Kelvin equation assumes that the surface tension can be defined and that the gas phase is ideal. This is accurate for mesopores, but fails if appUed to pores of narrow width. Stronger sohd-fluid attractive forces enhance adsorption in narrow pores. Simulation studies [86] suggest that the lower limit of pore sizes determined from classical thermodynamic analysis methods hes at about 15 nm. Correction of the Kelvin equation does lower this border to about 2 run, but finally also the texture of the fluid becomes so pronounced, that the concept of a smooth hquid-vapor interface cannot accurately be applied. Therefore, analysis based on the Kelvin equation is not applicable for micropores and different theories have to be applied for the different ranges of pore sizes. 

Before and after experiments the pore sizes of the membranes were measured by permpo-rometry [16], a technique based on blocking of smaller pores by capillary condensation of cyclohexane and the simultaneous measurement of the permeance of oxygen gas through the larger, open pores. The measurements are performed at 20°C on an area of 8.5 10 4 m2. The pore size distribution (Kelvin radii) is determined in the desorption stage using the Kelvin equation. More details on the permporometry technique can be found in [17] and all experimental details of the permporometry apparatus are presented in [16],... 

It must always be borne in mind that when capillary condensation takes place during the course of isotherm determination, the pore walls are already covered with an adsorbed him, having a thickness t determined by the value of the relative pressure (cf. Chapter 2). Thus capillary condensation occurs not directly in the pore itself but rather in the inner core (Fig. 3.7). Consequently the Kelvin equation leads in the first instance to values of the core size rather than the pore size. The conversion of an r value to a pore size involves recourse to a model of pore shape, and also a knowledge of the angle of contact 0 between the capillary condensate and the adsorbed film on the walls. The involvement of 0 may be appreciated by consideration... 

The evaluation of pore size distribution by application of the Kelvin equation to Type IV isotherms has hitherto been almost entirely restricted to nitrogen as adsorptive. This is largely a reflection of the widespread use of nitrogen for surface area determination, which has meant that both the pore size distribution and the specific surface can be derived from the same isotherm. 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

The principle underlying surface area measurements is simple physisorb an inert gas such as argon or nitrogen and determine how many molecules are needed to form a complete monolayer. As, for example, the N2 molecule occupies 0.162 nm at 77 K, the total surface area follows directly. Although this sounds straightforward, in practice molecules may adsorb beyond the monolayer to form multilayers. In addition, the molecules may condense in small pores. In fact, the narrower the pores, the easier N2 will condense in them. This phenomenon of capillary pore condensation, as described by the Kelvin equation, can be used to determine the types of pores and their size distribution inside a system. But first we need to know more about adsorption isotherms of physisorbed species. Thus, we will derive the isotherm of Brunauer Emmett and Teller, usually called BET isotherm. 

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-... 

As discussed in Section 1.4.2.1, the critical condensation pressure in mesopores as a function of pore radius is described by the Kelvin equation. Capillary condensation always follows after multilayer adsorption, and is therefore responsible for the second upwards trend in the S-shaped Type II or IV isotherms (Fig. 1.14). If it can be completed, i.e. all pores are filled below a relative pressure of 1, the isotherm reaches a plateau as in Type IV (mesoporous polymer support). Incomplete filling occurs with macroporous materials containing even larger pores, resulting in a Type II isotherm (macroporous polymer support), usually accompanied by a H3 hysteresis loop. Thus, the upper limit of pore size where capillary condensation can occur is determined by the vapor pressure of the adsorptive. Above this pressure, complete bulk condensation would occur. Pores greater than about 50-100 nm in diameter (macropores) cannot be measured by nitrogen adsorption. 

The most common method used for the determination of surface area and pore size distribution is physical gas adsorption (also see 1.4.1). Nitrogen, krypton, and argon are some of the typically used adsorptives. The amount of gas adsorbed is generally determined by a volumetric technique. A gravimetric technique may be used if changes in the mass of the adsorbent itself need to be measured at the same time. The nature of the adsorption process and the shape of the equilibrium adsorption isotherm depend on the nature of the solid and its internal structure. The Brunauer-Emmett-Teller (BET) method is generally used for the analysis of the surface area based on monolayer coverage, and the Kelvin equation is used for calculation of pore size distribution. 

The same equipment as that for measuring surface area can be used to determine the pore size distribution of porous materials with diameters of 20 to 500 A, except that high relative pressures are used to condense N2 in the catalyst pores. The procedure involves measuring the volume adsorbed in either the ascending or the descending branch of the BET plot at relative pressures close to 1. Capillary condensation occurs in the pores in accordance with the Kelvin equation,... 

The measurements were performed at 20°C and the oxygen concentration was determined with a gas chromatograph. From the oxygen permeance data as a function of cyclohexane partial pressure the pore-size distribution was calculated with the Kelvin equation [25],... 

Computer modelling of physisorption hysteresis is simplified if it is assumed that pore filling occurs reversibly (i.e. in accordance with the Kelvin equation) along the adsorption branch of the loop. Percolation theory has been applied by Mason (1988), Seaton (1991), Liu et al., (1993, 1994), Lopez-Ramon et al., (1997) and others (Zhdanov et al.,1987 Neimark 1991). One approach is to picture the pore space as a three-dimensional network (or lattice) of cavities and necks. If the total neck volume is relatively small, the location of the adsorption branch should be mainly determined by the cavity size distribution. On the other hand, if the evaporation process is controlled by percolation, the location of the desorption branch is determined by the network coordination number and neck size distribution. 

Hie pioneering work in this area was carried out by Seaton et al. (1989), who adapted a statistical mechanical approach originally known as mean field theory (Ball and Evans, 1989). At the time of their early work (Jessop et al., 1991) mean field theory was already known to become less accurate as the pore size was reduced, but even so it was claimed to offer a more realistic way of determining the pore size distribution than the classical methods based on the Kelvin equation. 

The phenomenon of capillary condensation provides a method for measuring pore-size distribution. Nitrogen vapor at the temperature of liquid nitrogen for which cos(0) = 1 is universally used. To determine the pore size distribution, the variation in the amount of nitrogen inside the porous particle is measured when the pressure is slightly increased or decreased. This variation is divided into two parts one part is due to true adsorption and the other to capillary condensation. The variation due to adsorption is known from adsorption experiments with nonporous substances of known surface area, so the variation due to condensation can be calculated. The volume of this amount of nitrogen is equal to the volume of pores with the size as determined by the Kelvin equation. Once a certain model has been selected for the complicated pore geometry, the size of the pores can be calculated. Usually it is assumed that an array of cylindrical capillaries of uniform but different radii, and randomly oriented represents the porous medium. So the Kelvin equation in the form of Equation 3.9 is used. Since condensation is combined with adsorption, the thickness of the adsorption layer... 

Pore size distributions are determined based on the fact that condensation in small pores occurs at a lower partial pressure than is required to obtain condensation on a flat surface. The relationship quantifying this effect is the Kelvin equation which, for a hemispherical meniscus of radius r, has the form [116] ... 

Table 1 presents the textural parameters of the different materials studied using adsorption/ desorption isotherms before and after modifications or catalytic testing, corresponding to BET surface area, the total pore volume and the proportion of the micropore volume. The adsorption isotherm was found to be in agreement with the ones reported for MCM-41 materials with similar pore sizes [5]. Pore condensation of N, signified by a steep increase in the adsorbed volume in the N2 adsorption isotherm, was observed at a relative pressure (P/Po) of 0.26. Using Kelvin s equation, compensating for the multilayer adsorption the pore size was determined to be 2.5 nm. 

As with experimental sorption data, it is possible to obtain the ratio Ff as a function of pressure (and thus pore size via the Kelvin equation) for simulations of the nitrogen sorption experiment on the model grids derived from NMR images. In order to determine the effect, if any, of the macroscopic heterogeneities (non-randomness) in the spatial distribution of voidage and pore size on the nitrogen desorption isotherm it is... 

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). 

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. 

Determination of the meximanr pore size in a porous solid can be performed using the Kelvin equation 1... 

BET adsorption and desorption is usually performed with liquid N2 and is a widespread technique for determining the specific surface area of porous materials [28]. In fact, only surfaces that are accessible to the N2 molecules are detected. The Kelvin equation correlates the curvature of the liquid surface with the applied partial pressure and pore size distributions can be derived [29]. However, this method is successful only for pore structures below about 20 nm. Thus, in aerogels with typical pore sizes in the 1—100 nm range, only a fraction of the total available pore space is detected. For an aerogel with a den-... 

For the actual catal5dic reaction, the distribution of meso- and micropores is of greater importance. The specific pore volume, pore size, and pore size distribution of microporous materials are determined by gas adsorption measurements at relatively low pressures (low values of p/po = pressure/saturation pressure). The method is based on the pressure dependence of capillary condensation on the diameter of the pores in which this condensation takes place. To calculate the pore size distribution, the desorption isotherm is also determined. Thus a distinction can be made between true adsorption and capillary condensation. The latter is described by the Kelvin equation (Eq. 5-95). 

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