Pore-size distributions

When the radius of a cylindrical pore is changed from r to r —dr the corresponding decremental change in the pore volume V is 

Differentiation of the Washburn equation (10.24), assuming constancy of y and 6, yields 

Equation (11.16) represents a convenient means of reducing the cumulative curve to the distribution curve which gives the pore volume per unit radius interval. 

Unlike surface area calculations, the volume distribution function and all subsequently discussed functions are based on the model of cylindrical pore geometry. 

A narrow pore size distribution causes a good separation for a small MW range of peptides and proteins. Broad pore size distribution allows separations over a 

As for the pore volume, the pore sizes of meso- and microporous adsorbents are characterised using gas sorption measurements, whereas the pore sizes of macroporous adsorbents are best estimated using mercury penetration measurements or by electron microscopy. The function dvpjddp = f dp) can be calculated using various models. In gas sorption on sorbents with mesopores the function is obtained using the Kelvin equation describing capillary condensation. 

Historically, the calculation of pore size distributions in porous materials has been primarily based on various forms and modifications of capillary theory 

Equation (3.27) can be applied to both adsorption and desorption branches of the isotherm. For the model of a bundle of capillary tubes, it is more appropriate to use the desorption branch of the isotherm for the determination of the pore size distribution. The basic idea is that the effective meniscus radius is the difference between the capillary radius and the thickness of the multilayer adsorption at p/p°, which can be obtained from de Boer s t-plot. In practice, at each desorption pressure, P, the capillary radius can be calculated from Eq. (3.27). The actual pore radius is then the sum of the calculated capillary radius and the estimated thickness of the multilayer. The exposed pore volume and surface area can be obtained from the volume desorbed at that specific desorption pressure. This step can be repeated at different desorption pressures. Except for the first desorption step, the desorbed volume should be corrected for the multilayer thinning on the sum of the area of the previously exposed pores. The pore size distribution can then be determined from the slope of the cumulative volume versus r curve. 

A severe limitation of the bundle of capillaries model is that it can give erroneous readings for materials with ink-bottle pores. In this case, the pores are emptied at the capillary pressure of the neck followed by the discharge of the large cavity, resulting in a large reading of the desorbed volume at the capillary pressure of the ink-bottle neck. 

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) 

Eor meso- and macro-pore materials, the Laplace [24] equation has also been applied for the determination of pore size distribution with the assumption that the pores are cylindrical, resulting in the equality of the two radii of curvature in the Laplace equation. In practice, the penetration of a non-wetting liquid such as mercury into the pores at a specific pressure is related to the pore radius through the following equation, with the assumption that all pores are equally accessible 

A critical feature affecting selectivity in SEC is the minimisation of pore size distribution. Control of this parameter facilitates separation of molecules with a particular size distribution. Stationary phases are available over a wide range of pore sizes (Table 5.2) and often these columns can be used in series (Mori, 1979). Thus, effective selection within a broad range is accomplished by the first column and fractionation within a more defined range is achieved on the second column. 

The molecular weight range (in Daltons) for the separation of proteins and random coil molecules using currently available supports 

According to the lUPAC classification of pores, the size ranges are micropoies ( 2 nm), tnesopores (2-50 nm), and macropores ( 50 mn) (lUPAC, 1972). All useful sorbents have micropores. The quantitative estimation of pore size distribution (PSD), particularly for the micropores, is a crucial problem in the characterization of sorbents. Numerous methods exist, of which three main methods will be described Kelvin equation (and the BJH method), Horvath-Kawazoe approach, and the integral equation approach. 

A curved liquid-vapor interface gives rise to a change in the saturation vapor pressure. This phenomenon is governed by the Kelvin equation 

In a circular capillary of radius r partly filled with a liquid rl = r2 = r/cos(0), where 6 is the contact angle between adsorbate and solid, the Kelvin equation gives 

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 

The capillary condensation method is widely used for the investigation of mesopores. The size of the largest pores, that can be measured, is limited by the rapid change of the meniscus radius with pressure as the relative pressure P/Pt nears unity (because of massive condensation of the adsorbent around the boiling point). This holds for radii of about 100 nm. The smallest pore sizes that can be determined by this method are about 1.5 nm. 

A somewhat different relationship between the amount of vapor sorbed and the pressure is often experimentally obtained upon decreasing rather than increasing the pressure. This is explained by the different curvature of the meniscus during adsorption and desorption. For example, consider a cylindrical pore. The adsorption is determined by the curvature of the liquid film on the pore surface. If the film thickness is much lower than the pore radius r, the curvature of the surface is l/(2r) and the process is described by Equation 3.10. During desorption the liquid covers the entire cross-section of the pore and has a spherical meniscus of curvature 1/r, and Equation 3.9 holds. Comparison of Equations 3.9 and 3.10 shows that a higher pressure is required to fill pores. 

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For a single fluid flowing through a section of reservoir rock, Darcy showed that the superficial velocity of the fluid (u) is proportional to the pressure drop applied (the hydrodynamic pressure gradient), and inversely proportional to the viscosity of the fluid. The constant of proportionality is called the absolute permeability which is a rock property, and is dependent upon the pore size distribution. The superficial velocity is the average flowrate... 

SANS Small-angle neutron scattering [175, 176] Thermal or cold neutrons are scattered elastically or inelastically Incident-Beam Spectroscopy Surface vibrational states, pore size distribution suspension structure... 

The method to be described determines the pore size distribution in a porous material or compacted powder surface areas may be inferred from the results. 

We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

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]... 

Horvath G and Kawazoe K 1983 Method for oaloulation of effeotive pore size distribution in moleoular sieve oarbon J. Chem. Eng. Japan 16 470-5... 

Physical properties affecting catalyst perfoniiance include tlie surface area, pore volume and pore size distribution (section B1.26). These properties regulate tlie tradeoff between tlie rate of tlie catalytic reaction on tlie internal surface and tlie rate of transport (e.g., by diffusion) of tlie reactant molecules into tlie pores and tlie product molecules out of tlie pores tlie higher tlie internal area of tlie catalytic material per unit volume, tlie higher the rate of tlie reaction... 

The relation between the dusty gas model and the physical structure of a real porous medium is rather obscure. Since the dusty gas model does not even contain any explicit representation of the void fraction, it certainly cannot be adjusted to reflect features of the pore size distributions of different porous media. For example, porous catalysts often show a strongly bimodal pore size distribution, and their flux relations might be expected to reflect this, but the dusty gas model can respond only to changes in the... 

Che pore size distribution and Che pore geometry. Condition (iil). For isobaric diffusion in a binary mixture Che flux vectors of Che two species must satisfy Graham s relation... 

The simplest way of introducing Che pore size distribution into the model is to permit just two possible sizes--Tnlcropores and macropotes--and this simple pore size distribution is not wholly unrealistic, since pelleted materials are prepared by compressing powder particles which are themselves porous on a much smaller scale. The small pores within the powder grains are then the micropores, while the interstices between adjacent grains form the macropores. An early and well known model due to Wakao and Smith [32] represents such a material by the Idealized structure shown in Figure 8,2,... 

Of course, these shortcomings of the Wakao-Smith flux relations induced by the use of equations (8.7) and (8.8) can be removed by replacing these with the corresponding dusty gas model equations, whose validity is not restricted to isobaric systems. However, since the influence of a strongly bidisperse pore size distribution can now be accounted for more simply within the class of smooth field models proposed by Feng and Stewart [49], it is hardly worthwhile pursuing this."... 

Case (c). The pore size distribution is strictly bimodal, with macropores... 

The pore size distribution function (a) appears parametrically in the flux relations of Feng and Stewart, so their models certainly cannot be completely predictive in nature unless this distribution is known. It is... 

An interesting example of a large specific surface which is wholly external in nature is provided by a dispersed aerosol composed of fine particles free of cracks and fissures. As soon as the aerosol settles out, of course, its particles come into contact with one another and form aggregates but if the particles are spherical, more particularly if the material is hard, the particle-to-particle contacts will be very small in area the interparticulate junctions will then be so weak that many of them will become broken apart during mechanical handling, or be prized open by the film of adsorbate during an adsorption experiment. In favourable cases the flocculated specimen may have so open a structure that it behaves, as far as its adsorptive properties are concerned, as a completely non-porous material. Solids of this kind are of importance because of their relevance to standard adsorption isotherms (cf. Section 2.12) which play a fundamental role in procedures for the evaluation of specific surface area and pore size distribution by adsorption methods. 

A Type II isotherm indicates that the solid is non-porous, whilst the Type IV isotherm is characteristic of a mesoporous solid. From both types of isotherm it is possible, provided certain complications are absent, to calculate the specific surface of the solid, as is explained in Chapter 2. Indeed, the method most widely used at the present time for the determination of the surface area of finely divided solids is based on the adsorption of nitrogen at its boiling point. From the Type IV isotherm the pore size distribution may also be evaluated, using procedures outlined in Chapter 3. 

Type 1 isotherms, as will be demonstrated in Chapter 4, are characteristic of microporous adsorbents. The detailed interpretation of such isotherms is controversial, but the majority of workers would probably agree that the very concept of the surface area of a microporous solid is of doubtful validity, and that whilst it is possible to obtain an estimate of the total micropore volume from a Type I isotherm, only the crudest guesses can be made as to the pore size distribution. 

Isotherms of Type 111 and Type V, which are the subject of Chapter 5, seem to be characteristic of systems where the adsorbent-adsorbate interaction is unusually weak, and are much less common than those of the other three types. Type III isotherms are indicative of a non-porous solid, and some halting steps have been taken towards their use for the estimation of specific surface but Type V isotherms, which betoken the presence of porosity, offer little if any scope at present for the evaluation of either surface area or pore size distribution. 

The f-curve and its associated t-plot were originally devised as a means of allowing for the thickness of the adsorbed layer on the walls of the pores when calculating pore size distribution from the (Type IV) isotherm (Chapter 3). For the purpose of testing for conformity to the standard isotherm, however, a knowledge of the numerical thickness is irrelevant since the object is merely to compare the shape of the isotherm under test with that of the standard isotherm, it is not necessary to involve the number of molecular layers n/fi or even the monolayer capacity itself. 

Type IV isotherms are often found with inorganic oxide xerogels and other porous solids. With certain qualifications, which will be discussed in this chapter, it is possible to analyse Type IV isotherms (notably those of nitrogen at 77 K) so as to obtain a reasonable estimate of the specific surface and an approximate assessment of the pore size distribution. 

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

Foster s neglect of the role of the adsorbed film was unavoidable in the then absence of any reliable information as to the thickness of the film. It is now known that in fact the effect of the film on the calculated result is far from negligible, as will be demonstrated shortly. Since, however, all the methods of calculating pore size distributions involve a decision as to the upper limit of the range to be studied, this question needs to be discussed first. In effect one has to choose a point corresponding to point G in Fig. 3.1, where the mesopores are deemed to be full up. If the isotherm takes the course GH there are no further cores to be considered in any case but if it swings upwards as at GH, the isotherm is usually so steep that the Kelvin-type approach becomes too inaccurate (cf. p. 114) to be useful. 

When the relative pressure falls to pj/p", the second group of pores loses its capillary condensate, but in addition the film on the walls of the first group of pores yields up some adsorbate, owing to the decrease in its thickness from t, to t. Similarly, when the relative pressure is further reduced to pj/p°, the decrement (nj-Wj) in the uptake will include contributions from the walls of both groups 1 and 2 (as the film thins down from tj to fj), in addition to the amount of capillary condensate lost from the cores of group 3. It is this composite nature of the amount given up at each step which complicates the calculation of the pore size distribution. 

In the pioneer work of Foster the correction due to film thinning had to be neglected, but with the coming of the BET and related methods for the evaluation of specific surface, it became possible to estimate the thickness of the adsorbed film on the walls. A number of procedures have been devised for the calculation of pore size distribution, in which the adsorption contribution is allowed for. All of them are necessarily somewhat tedious and require close attention to detail, and at some stage or another involve the assumption of a pore model. The model-less method of Brunauer and his colleagues represents an attempt to postpone the introduction of a model to a late stage in the calculations. 

Various methods have been devised for incorporating the bv correction into calculations of pore size distribution. Some of them involve the length of the pores and the area of their walls others the area of the walls only and yet others avoid the direct involvement of either the length or the area. Up to the present, virtually all the procedures have been restricted to nitrogen as the adsorptive. 

The significance of the various columns is explained in the notes below the table, which enable the calculations of 6v l6r to be followed through. Only the first few lines are reproduced, by way of illustration the pore size distribution curve resulting from the complete table is given in Fig. 3.18 (Curve A), as a plot of 6i j6r against f. 

Calculation of pore size distribution (Method of Pierce, also of Orr and DallaValle )... 

Fig. 3.18 Pore size distributions of a silica geP GSSO, calculated from the desorption branch of the isotherm at 77 K by dilTerent methods. (A) x,...

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