Pore Size distribution: relation adsorption

The objective of this chapter is to present the fundamental theories of adsorption followed by the description and discussion of experimental techniques for the measurements of adsorption isotherms and for the determination of surface area and pore size distribution. The adsorption of gases on microporous membranes and the inter-relation between adsorption and permeation are then discussed. The adsorption in liquid phase is briefly presented. The chapter concludes with a brief summary. 

An important issue in the classical methods of pore size analysis is the selection of the t-ctirve, i.e., the statistical film thickness on the carbon surface. In the original method [136], it was assumed that the film thickness on the pore walls does not depend on the pore radius but only on the relative pressure. Thus, this BJH method required two relations for the evaluation of the pore-size distribution from adsorption isotherms. The first, represented by the Kelvin Eq. (70), is the relation between the pore radius and the relative pressure at which capillary condensation occurs in the pores. The second represents the functional dependence of the statistical film thickness on the relative pressure. It should be noted that existing relationships for t-curves reported some time ago do not represent low-pressure adsorption behavior because the relevant low-pressure data were not available at that time [13], The corresponding low-pressure adsorption isotherms on carbon surfaces are now available and can be used to evaluate the t-curve for the entire pressure range. 

As the size of a particle decreases, the contribution of the surface to the total flee energy becomes increasingly significant. This trend is evident in the Kelvin equation, a classic thermodynamic equation relating the vapor pressure of a droplet to its physical size. This equation is often applied in the determinalion of the pore size distribution through adsorption porosimetry. For this appHcalion, the equation can be written ... 

Since NO adsorption appears to be the two step process of conversion to NO2 and subsequent storage within the pores, the correlation between pore size distribution and adsorption capacity and rate may well be related to the storage capacity and not the active sites. 

Recent progress in the theory of adsorption on porous solids, in general, and in the adsorption methods of pore structure characterization, in particular, has been related, to a large extent, to the application of the density functional theory (DFT) of Inhomogeneous fluids [1]. DFT has helped qualitatively describe and classify the specifics of adsorption and capillary condensation in pores of different geometries [2-4]. Moreover, it has been shown that the non-local density functional theory (NLDFT) with suitably chosen parameters of fluid-fluid and fluid-solid interactions quantitatively predicts the positions of capillary condensation and desorption transitions of argon and nitrogen in cylindrical pores of ordered mesoporous molecular sieves of MCM-41 and SBA-15 types [5,6]. NLDFT methods have been already commercialized by the producers of adsorption equipment for the interpretation of experimental data and the calculation of pore size distributions from adsorption isotherms [7-9]. 

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. 

Another property of importance is the pore volume. It can be measured indirectly from the adsorption and/or desorption isotherms of equilibrium quantities of gas absorbed or desorbed over a range of relative pressures. Pore volume can also be measured by mercury intrusion techniques, whereby a hydrostatic pressure is used to force mercury into the pores to generate a plot of penetration volume versus pres- sure. Since the size of the pore openings is related to the pressure, mercury intrusion techniques provide information on the pore size distribution and the total pore volume. 

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 ... 

N2 adsorption-desorption isotherms and pore size distribution of sample II-IV are shown in Fig. 4. Its isotherm in Fig. 4a corresponds to a reversible type IV isotherm which is typical for mesoporous solids. Two definite steps occur at p/po = 0.18, and 0.3, which indicates the filling of the bimodal mesopores. Using the BJH procedure with the desorption isotherm, the pore diameter in Fig. 4a is approximately 1.74, and 2.5 nm. Furthermore, with the increasing of synthesis time, the isotherm in Fig. 4c presents the silicalite-1 material related to a reversible type I isotherm and mesoporous solids related to type IV isotherm, simultaneously. These isotherms reveals the gradual transition from type IV to type I. In addition, with the increase of microwave irradiation time, Fig. 4c shows a hysteresis loop indicating a partial disintegration of the mesopore structure. These results seem to show a gradual transformation... 

The particle size and surface area distributions of pharmaceutical powders can be obtained by microcomputerized mercury porosimetry. Mercury porosimetry gives the volume of the pores of a powder, which is penetrated by mercury at each successive pressure the pore volume is converted into a pore size distribution. Two other methods, adsorption and air permeability, are also available that permit direct calculation of surface area. In the adsorption method, the amount of a gas or liquid solute that is adsorbed onto the sample of powder to form a monolayer is a direct function of the surface area of the sample. The air permeability method depends on the fact that the rate at which a gas or liquid permeates a bed of powder is related, among other factors, to the surface area exposed to the permeant. The determination of surface area is well described by the BET (Brunauer, Emmett, and Teller) equation. 

Nitrogen adsorption/condensation measurements were performed using an Autosorb-1 analyzer to calculate sample surface area and pore size distribution. BET analysis at 77 K was applied for extracting the monolayer capacity from the adsorption isotherm and a N molecular cross-sectional area of 0.162 nm2 was used to relate tne monolayer capacity to surface area. PSD s were calculated from the desorption branches of the isotherms using a modified form of the BJH method [18]. Mercury intrusion measurements were performed using an Autoscan-33 continuous scanning mercury porosimeter (12-33000 psia) and a contact angle of 140°. 

In the last decade, a variety of microporous and mesoporous materials have been developed for applications in catalysis, chromatography and adsorption. Great attention has been paid to the control of (i) pore surface chemistry and (ii) textural properties such as pore size distribution, pore size and shape. Recently, a new field of applications for these materials has been highlighted [1-3] by forcing a non-wetting liquid to invade a porous solid by means of an external pressure, mechanical energy can be converted to interfacial energy. The capillary pressure, Pc p, required for pore intrusion can be written from the Laplace-Washbum relation,... 

From these results it can be said that the shape of water isotherm up to relative pressures around 0.6 is mainly related to the micropore size distribution. On the other hand, the adsorption at higher P/Po is due to the presence of mesoporosity. Depending on the pore size distribution and the contribution of the different pore volumes, the shape of water isotherms can be more similar to the sample 1A or 3B. 

Now, an important question is How, those parameters Sp, Vp, (Dmax/2c), c, and x are affected by the gradual narrowing of pores due to some kind of surface functionalization A second question is which of, and how, the parameters t, c and (Dmax/2o) are interrelated. The question becomes more interesting, and perhaps intriguing, since all the above quantities are calculated just from one kind of measurement, namely the N2 adsorption/desorption data. A partly answer to the above question was attempted in a previous work [10] in which sixteen mesoporous vanado-phoshoro-aluminates solids were tested and some relationships between c and (Dmax/2o) were established. A first target of this paper is to extend the search for such possible inter-relations to a class of mesoporous silicas, with a random pore size distribution whose porosity has been systematically and gradually modified by surface fiinctionalization... 

In the context of microporous and mesoporous materials, lUPAC has provided a variety of recommendations for nomenclature and characterization of porous materals, that can be found in the literature. Microporosity should not be based on structural data but on adsorption data. Sorption by materials that show Type 1 isotherms is an indication of a microporous material. Pore size distributions less than 20 A are related to microporous materials like zeolites. Materials having pores between 20 A and 500 A are refered to as mesoporous materials. Materials that have pores larger than 500 A are refered to as macroporous. 

The novel approach for calculation of pore size distributions, which is reported in the current study is based on recent developments in the materials science and in the theory of inhomogeneous fluids. First, an application of experimental adsorption data for well-characterized MCM-41 silicas enabled proper calibration of the pore size analysis. Second, an application of a modem theory to describe the behavior of inhomogeneous fluids in confined spaces, that is the non-local density functional theory [6], allowed the numerical calculation of model isotherms for various pore sizes. In addition, a practical numerical deconvolution method that provides a "best fit" solution representing the pore distribution of the sample was implemented [7, 8]. In this paper we describe a deconvolution method for estimating mesopore size distribution that explicitly allows for unfilled large pores, and a method for creating composite, or hybrid, models that incorporate both theoretical calculations and experimental observations. Moreover, we showed the applicability of the new approach in characterization of MCM-41 and related materials. 

Two series of carbon molecular sieves have been prepared from coconut shells, with different pore size distribution. They have been characterised by carbon dioxide adsorption at 273 K and immersion calorimetry into liquids of different molecular sizes. The results have been related with the abihty of the CMS to separate the components of O2/N2, CO2/CH4 and n-C4H4/i-C4H4 gas mixtures. 

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