Capillary isotherm

The experimental basis of sorption studies includes structural data (SANS, SAXS, USAXS), isopiestic vapor sorption isotherms,i and capillary isotherms, measured by the method of standard porosimetry. i 2-i44 Thermodynamic models for water uptake by vapor-equilibrated PEMs have been suggested by various groupThe models account for interfacial energies, elastic energies, and entropic contributions. They usually treat rate constants of interfacial water exchange and of bulk transport of water by diffusion and hydraulic permeation as empirical functions of temperature. 

Water uptake of Nafion 117. (a) Isopiestic water sorption data (extracted from T. E. Springer et al. Journal of the Electrochemical Society 138 (1991) 2334-2342) fitted by Equation (6.7) (b) capillary isotherms (extracted from J. Divisek et al.. Journal of the Electrochemical Society 145 (1998) 2677-2683) fitted by Equation (6.8). 

Gibbs free energy of water sorption by Nation 117. Comparison of energies obtained from sorption isotherms (solid line), corresponding to Figure 6.10(a), and from capillary isotherms (dashed line), corresponding to Figure 6.10(b). 

The experimental basis of sorption studies includes isopiestic vapor sorption isotherms (Morris and Sun, 1993 Pushpaet al., 1988 Rivin et al., 2001 Zawodzinski et al., 1993c) and capillary isotherms, measured by standard porosimetry (Divisek et al., 1998 Vol fkovich and Bagotsky, 1994 Vol fkovich et al., 1980). A number of thermodynamic models of water uptake by vapor-equilibrated PEMs have been... 

The very considerable success of the BET equation stimulated various investigators to consider modifications of it that would correct certain approximations and give a better fit to type II isotherms. Thus if it is assumed that multilayer formation is limited to n layers, perhaps because of the opposing walls of a capillary being involved, one... 

As a general rule, adsorbates above their critical temperatures do not give multilayer type isotherms. In such a situation, a porous absorbent behaves like any other, unless the pores are of molecular size, and at this point the distinction between adsorption and absorption dims. Below the critical temperature, multilayer formation is possible and capillary condensation can occur. These two aspects of the behavior of porous solids are discussed briefly in this section. Some lUPAC (International Union of Pure and Applied Chemistry) recommendations for the characterization of porous solids are given in Ref. 178. 

The adsorption isotherms are often Langmuirian in type (under conditions such that multilayer formation is not likely), and in the case of zeolites, both n and b vary with the cation present. At higher pressures, capillary condensation typically occurs, as discussed in the next section. Some N2 isotherms for M41S materials are shown in Fig. XVII-27 they are Langmuirian below P/P of about 0.2. In the case of a microporous carbon (prepared by carbonizing olive pits), the isotherms for He at 4.2 K and for N2 at 77 K were similar and Langmuirlike up to P/P near unity, but were fit to a modified Dubninin-Radushkevich (DR) equation (see Eq. XVII-75) to estimate micropore sizes around 40 A [186]. 

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

The basis of the classification is that each of the size ranges corresponds to characteristic adsorption effects as manifested in the isotherm. In micropores, the interaction potential is significantly higher than in wider pores owing to the proximity of the walls, and the amount adsorbed (at a given relative pressure) is correspondingly enhanced. In mesopores, capillary condensation, with its characteristic hysteresis loop, takes place. In the macropore range the pores are so wide that it is virtually impossible to map out the isotherm in detail because the relative pressures are so close to unity. 

Deviation from the standard isotherm in the high-pressure region offers a means of detecting the occurrence of capillary condensation in the crevices l>etween the particles of a solid and in any mesopores present within the particles themselves. A convenient device for detecting deviations from the standard is the t-plot . In the next section the nature and uses of t-plots will be discussed, together with a,-plots, a later development from them. As will l>e shown, both of these plots may l>e used not only for the detection of capillary condensation in mesopores, but also for showing up the presence of micropores and evaluating their volume. 

If the adsorbent contains mesopores, capillary condensation will occur in each pore when the relative pressure reaches a value which is related to the radius of the pore by the Kelvin equation, and a Type IV isotherm will... 

The model proposed by Zsigmondy—which in broad terms is still accepted to-day—assumed that along the initial part of the isotherm (ABC of Fig. 3.1), adsorption is restricted to a thin layer on the walls, until at D (the inception of the hysteresis loop) capillary condensation commences in the finest pores. As the pressure is progressively increased, wider and wider pores are filled until at the saturation pressure the entire system is full of condensate. 

This widespread conformity to the Gurvitsch rule constitutes powerful support for the capillary condensation hypothesis in relation to Type IV isotherms. It is perhaps hardly necessary to stress that in order to test data for conformity to the rule it is essential that the stage which corresponds to the complete filling of the pores shall be clearly identifiable—as in the... 

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... 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

Fig. 3.15 (a) A pore in the form of an interstice between close-packed and equal-sized spherical particles. The adsorbed him which precedes capillary condensation is indicated, (b) Adsorption isotherm (idealized). 

It was noted earlier (p. 115) that the upward swing in the Type IV isotherm characteristic of capillary condensation not infrequently commences in the region prior to the lower closure point of the hysteresis loop. This feature can be detected by means of an a,-plot or a comparison plot (p. 100). Thus Fig. 3.25(a) shows the nitrogen isotherm and Fig. 3.25(h) the a,-plot for a particular silica gel the isotherm is clearly of Type IV and the closure point is situated around 0 4p° the a,-plot shows an upward swing commencing at a = 0-73, corresponding to relative pressures of 013 and therefore well below the closure point. 

At the point where capillary condensation commences in the finest mesopores, the walls of the whole mesopore system are already coated with an adsorbed film of area A, say. The quantity A comprises the area of the core walls and is less than the specific surface A (unless the pores happen to be parallel-sided slits). When capillary condensation takes place within a pore, the film-gas interface in that pore is destroyed, and when the pore system is completely filled with capillary condensate (e.g. at F in Fig. 3.1) the whole of the film-gas interface will have disappeared. It should therefore be possible to determine the area by suitable treatment of the adsorption data for the region of the isotherm where capillary condensation is occurring. 

These various considerations led Pierce, Wiley and Smith in 1949, and independently, Dubinin, to postulate that in very fine pores the mechanism of adsorption is pore filling rather than surface coverage. Thus the plateau of the Type 1 isotherm represents the filling up of the pores with adsorbate by a process similar to but not identical with capillary condensation, rather than a layer-by-layer building up of a film on the pore walls. 

In general, therefore, there are three processes, prior to the kind of capillary condensation associated with the hysteresis loop of a Type IV isotherm, which may occur in a porous body containing micropores along with mesoporesia primary process taking place in very narrow micropores a secondary, cooperative process, taking place in wider micropores, succeeded by a tertiary process governed by a modified Kelvin equation. 

Type V isotherms of water on carbon display a considerable variety of detail, as may be gathered from the representative examples collected in Fig. 5.14. Hysteresis is invariably present, but in some cases there are well defined loops (Fig. 5.14(b). (t ), (capillary-condensed water. Extreme low-pressure hysteresis, as in Fig. 5.14(c) is very probably due to penetration effects of the kind discussed in Chapter 4. 

The first stage in the interpretation of a physisorption isotherm is to identify the isotherm type and hence the nature of the adsorption process(es) monolayer-multilayer adsorption, capillary condensation or micropore filling. If the isotherm exhibits low-pressure hysteresis (i.e. at p/p° < 0 4, with nitrogen at 77 K) the technique should be checked to establish the degree of accuracy and reproducibility of the measurements. In certain cases it is possible to relate the hysteresis loop to the morphology of the adsorbent (e.g. a Type B loop can be associated with slit-shaped pores or platey particles). 

In both of these pieces of apparatus, isothermal operation and optimum membrane area are obtained. Good temperature control is essential not only to provide a value for T in the equations, but also because the capillary attached to a larger reservoir behaves like a thermometer, with the column height varying with temperature fluctuations. The contact area must be maximized to speed up an otherwise slow equilibration process. Various practical strategies for presetting the osmometer to an approximate n value have been developed, and these also accelerate the equilibration process. 

Henry s law corresponds physically to the situation in which the adsorbed phase is so dilute that there is neither competition for surface sites nor any significant interaction between adsorbed molecules. At higher concentrations both of these effects become important and the form of the isotherm becomes more complex. The isotherms have been classified into five different types (9) (Eig. 4). Isotherms for a microporous adsorbent are generally of type I the more complex forms are associated with multilayer adsorption and capillary condensation. 

In Fig. 15 we show similar results, but for = 10. Part (a) displays some examples of the adsorption isotherms at three temperatures. The highest temperature, T = 1.27, is the critical temperature for this system. At any T > 0.7 the layering transition is not observed, always the condensation in the pore is via an instantaneous filling of the entire pore. Part (b) shows the density profiles at T = 1. The transition from gas to hquid occurs at p/, = 0.004 15. Before the capillary condensation point, only a thin film adjacent to a pore wall is formed. The capillary condensation is now competing with wetting. 

---