Pore size distributions Kelvin equation

Use of the Kelvin equation for calculation of pore size distribution... 

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. 

The computation of mesopore size distribution is valid only if the isotherm is of Type IV. In view of the uncertainties inherent in the application of the Kelvin equation and the complexity of most pore systems, little is to be gained by recourse to an elaborate method of computation, and for most practical purposes the Roberts method (or an analogous procedure) is adequate—particularly in comparative studies. The decision as to which branch of the hysteresis loop to use in the calculation remains largely arbitrary. If the desorption branch is adopted (as appears to be favoured by most workers), it needs to be recognized that neither a Type B nor a Type E hysteresis loop is likely to yield a reliable estimate of pore size distribution, even for comparative purposes. 

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. 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

Fig. 3.45. Pore size distribution of a commercial y-alumina calculated using the Kelvin equation.

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-... 

Thus, either type I or type IV isotherms are obtained in sorption experiments on microporous or mesoporous materials. Of course, a material may contain both types of pores. In this case, a convolution of a type I and type IV isotherm is observed. From the amount of gas that is adsorbed in the micropores of a material, the micropore volume is directly accessible (e.g., from t plot of as plot [1]). The low-pressure part of the isotherm also contains information on the pore size distribution of a given material. Several methods have been proposed for this purpose (e.g., Horvath-Kawazoe method) but most of them give only rough estimates of the real pore sizes. Recently, nonlocal density functional theory (NLDFT) was employed to calculate model isotherms for specific materials with defined pore geometries. From such model isotherms, the calculation of more realistic pore size distributions seems to be feasible provided that appropriate model isotherms are available. The mesopore volume of a mesoporous material is also rather easy accessible. Barrett, Joyner, and Halenda (BJH) developed a method based on the Kelvin equation which allows the calculation of the mesopore size distribution and respective pore volume. Unfortunately, the BJH algorithm underestimates pore diameters, especially at... 

Though not a general adsorption equilibrium model the Kelvin equation does provide the relationship between the depression of the vapor pressure of a condensable sorbate and the radius (r) of the pores into which it is condensing. This equation is useful for characterization of pore size distribution by N2 adsorption at or near its dew point. The same equation can also describe the onset of capillary condensation the enhancement of sorption capacity in meso- and macro-pores of formed zeolite adsorbents. 

Adsorption studies leading to measurements of pore size and pore-size distributions generally make use of the Kelvin equation which relates the equilibrium vapor pressure of a curved surface, such as that of a liquid in a capillary or pore, to the equilibrium pressure of the same liquid on a plane surface. Equation (8.1) is a convenient form of the Kelvin equation ... 

Sbet - BET specific surface area V, - single-point total pore volume w - pore width at the maximum of the pore size distribution calculated using the BJH method with the corrected form of the Kelvin equation [34]. 

Specific surface areas of the materials under study were calculated using the BET method [22, 23]. Their pore size distributions were evaluated from adsorption branches of nitrogen isotherms using the BJH method [24] with the corrected form of the Kelvin equation for capillary condensation in cylindrical pores [25, 26]. In addition, adsorption energy distributions (AED) were evaluated from submonolayer parts of nitrogen adsorption isotherms using the algorithm reported in Ref. [27],... 

The pore volume and the pore size distribution can be estimated from gas adsorption [83], while the hysteresis of the adsorption isotherms can give an idea as to the pore shape. In the pores, because of the confined space, a gas will condense to a liquid at pressures below its saturated vapor pressure. The Kelvin equation (Eq. (4.5)) gives this pressure ratio for cylindrical pores of radius r, where y is the liquid surface tension, V is the molar volume of the liquid, R is the gas constant ( 2 cal mol-1 K-1), and T is the temperature. This equation forms the basis of several methods for obtaining pore-size distributions [84,85]. 

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. 

Most models to calculate the pore size distributions of mesoporous solids, are based on the Kelvin equation, based on Thomson s23 (later Lord Kelvin) thermodynamical statement that the equilibrium vapour pressure (p), over a concave meniscus of liquid, must be less than the saturation vapour pressure (p0) at the same temperature . This implies that a vapour will be able to condense to a liquid in the pore of a solid, even when the relative pressure is less than unity. This process is commonly called the capillary condensation. 

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 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,... 

Hysteresis in the adsorption-desorption isotherms (Fig. 4) is a common observation for supports with a large fraction of small pores. It results from desorption from the meniscus at the end of a filled pore. The vapor pressure above the liquid at the pore mouth defines the pore radius in the Kelvin equation. Therefore, it is the desorption branch of the isotherm that is preferred in calculations of pore size distributions. 

FIGURE 4 Nitrogen adsorption and desorption isotherms at 78 K. Pore size distributions in the micropore range are calculated from the isotherms using 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],... 

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],... 

The pore size distribution is the distribution of pore volume with respect to pore size. The computation of pore size distribution involves a number of assumptions (pore shape, mechanism of pore filling, validity of Kelvin equation etc.)... 

Over the period 1945-1970 many different mathematical procedures were proposed for the derivation of the pore size distribution from nitrogen adsorption isotherms. It is appropriate to refer to these computational methods as classical since they were all based on the application of the Kelvin equation for the estimation of pore size. Amongst the methods which remain in current use were those proposed by Barrett, Joyner and Halenda (1951), apparently still the most popular Cranston and Inkley... 

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] ... 

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 study of mesoporous solids is closely related to the concept of capillary condensation and its quantitative expression in the Kelvin equation (.54). This equation is the basis of virtually all the various procedures for calculating the pore-size distributions from the type IV isotherms that have appeared over the past 80 years, beginning with the first papers of Zsigmondy (55), Anderson (56), and Foster (57) (see also Refs. 6 and 49) A convenient form of the Kelvin equation is... 

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). 

Such more realistic models of porous materials can also be used to rigorously test existing characterization methods. The model material is precisely characterized (we know the location of every atom in the material, hence the pore sizes, surface area and so on). By simulating adsorption of simple molecules in the model material and then inverting the isotherm, we can obtain a pore size distribution for any particular theory or method. Such a test for porous glasses is shown in Figure 8, where the exactly known (geometric) PSD is compared to that predicted by the Barrett-Joyner-Halenda (BJH) method, which is based on the modified Kelvin equation. 

Pore-size distributions (PSD) are routinely obtained by an algorithm dating back to Barret, Joyner and Halenda [3-4]. Either cylindrical or slit-shaped pores are assumed in these calculations. The BJH method virtually represents numerical solution of an integral equation, which describes adsorption and capillary condensation of adsorbate in pores and utilizes the Kelvin equation. Because the validity of Kelvin equation in micropores can be questioned a new approach based on statistical physics is developing, viz. the density functional theory [5-7]. This approach can supply adsorption isotherms for cylindrical or slit-shaped pores of different sizes in carbonaceous or oxide matrix. The problem then is to sum up these isotherms so that the experimental isotherm is reproduced. Expensive commercial programs are available for this purpose. 

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