Capillary condensation, 5.21

Capillary condensation is the condensation of vapor into capillaries or fine pores even at vapor pressures below Pq. Lord Kelvin was the one who realized that the vapor pressure ofaliquid depends on the curvature of its surface. In his words, this explains why moisture is retained by vegetable substances, such as cotton cloth or oatmeal, or wheat-flour biscuits, at temperatures far above the dew point of the surrounding atmosphere [505]. 

r corresponds to the capillary radius at the point where the meniscus is in equilibrium. The curvature of the liquid surface is —2/r. 

We have a porous solid with cylindrical pores of all dimensions. It is in water vapor at 20 °C. The humidity is 90%. What is the size of the pores, which fill up with waterThe solid is supposed to be hydrophilic with 0 = 0. 

Capillary condensation has been studied by various methods, and the validity of the previous description has been confirmed for several liquids and radii of curvature down to a few nanometers [506, 507]. 

On the other hand, the Kelvin equation has been extensively used in research on gas adsorption onto porous solids (see Sections 8.4 and 8.5) and capillary condensation. 

If a fissure or crack is present in any pore, the pore is no longer assumed to be spherical, and if rc is chosen as the radius of curvature perpendicular to the fissure direction, then Equation (349) is used, where Pi = rc and P2 = 00 for these conditions. 

Equation (352) shows that the capillary force depends mostly only on the size of the particle and the surface tension of the liquid. However, in reality, the surfaces of most of the particles are rough and they touch only at some points, having much smaller liquid menisci between them (and giving much smaller z values), so that the resultant capillary force, Fr, is much smaller than the value calculated by Equation (352). 

r is the radius of the bubble. The vapor pressure inside a bubble is therefore reduced. This explains why it is possible to overheat liquids When the temperature is increased above the boiling point (at a given external pressure) occasionally, tiny bubbles are formed. Inside the bubble the vapor pressure is reduced, the vapor condenses, and the bubble collapses. Only if a bubble larger than a certain critical size is formed, is it more likely to increases in size rather than to collapse. As an example, vapor pressures for water drops and bubbles in water are given in Table 2.2. 

Capillary condensation can be illustrated by the model of a conical pore with a totally wetting surface (Fig. 2.12). Liquid will immediately condense in the tip of the pore. Condensation continues until the bending radius of the liquid has reached the value given by the Kelvin equation. The situation is analogous to that of a bubble and we can write 

Many surfaces are not totally wetted, but they form a certain contact angle 0 with the liquid. In this case the radius of curvature increases. It is not longer equal to the capillary radius, but tor = rc/cos 0. 

The hemispherical-shaped-interface-in-a-cylindrical-tube is the simplest mathematically of the possible geometries. For it we can use Equation 14.16 to find that the pressure inside the fluid is less than the pressure of the gas or vapor at the mouth of the pore by AcrlD. From Example 14.6 we know that for ordinary sized holes this difference is negligible, but for microscopic ones it becomes quite large. If we repeat Example 14.7 for this geometry (D = 0.01 p case only), we find almost the same numbers, but with different meanings. As before the pressure difference across the hemispherical 0.01 p gas-liquid interface must be 3420 psi, but in this case that makes the pressure in the column of fluid inside the pore = —3240 psig  

This is certainly a surprise, but in this situation the negative pressure simply indicates that the trapped liquid is in tension. An ordinary liquid under tension will boil and convert to a gas-liquid mixture, but at this extremely small size the needed bubble cannot form, so the liquid under tension is quite stable. 

867 atm if it is constrained by the surface forces in an 0.01 p, pore. This behavior, called capillary condensation plays a significant role in adsorption. 

Here we have used the Poynting Factor (Section 7.4.2) to show the change in vapor pressure for decrease in system pressure. That is uncommon, but correct. The result is the Kelvin equation (14.20) with a minus sign inserted in the argument of the exponent (also called the Kelvin equation.) 

If we take into account the effects of gravity, magnetic, tensile, or electrostatic energies we find that the criterion of equilibrium is expanded, as shown in Eq. 14.3. 

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There are two approaches to explain physical mechanism of the phenomenon. The first model is based on the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

Thus D(r) is given by the slope of the V versus P plot. The same distribution function can be calculated from an analysis of vapor adsorption data showing hysteresis due to capillary condensation (see Section XVII-16). Joyner and co-woikers [38] found that the two methods gave very similar results in the case of charcoal, as illustrated in Fig. XVI-2. See Refs. 36 and 39 for more recent such comparisons. There can be some question as to what the local contact angle is [31,40] an error here would shift the distribution curve. 

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]. 

Below the critical temperature of the adsorbate, adsorption is generally multilayer in type, and the presence of pores may have the effect not only of limiting the possible number of layers of adsorbate (see Eq. XVII-65) but also of introducing capillary condensation phenomena. A wide range of porous adsorbents is now involved and usually having a broad distribution of pore sizes and shapes, unlike the zeolites. The most general characteristic of such adsorption systems is that of hysteresis as illustrated in Fig. XVII-27 and, more gener-... 

Adsorbents such as some silica gels and types of carbons and zeolites have pores of the order of molecular dimensions, that is, from several up to 10-15 A in diameter. Adsorption in such pores is not readily treated as a capillary condensation phenomenon—in fact, there is typically no hysteresis loop. What happens physically is that as multilayer adsorption develops, the pore becomes filled by a meeting of the adsorbed films from opposing walls. Pores showing this type of adsorption behavior have come to be called micropores—a conventional definition is that micropore diameters are of width not exceeding 20 A (larger pores are called mesopores), see Ref. 221a. 

Wanless E J and Christenson H K 1994 Interaction between surfaces in ethanol adsorption, capillary condensation, and solvation forces J. Chem. Rhys. 101 4260-7... 

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. 

The rather low value obtained with the copper phthalocyanine, a low-energy solid (line (v)), is probably explicable by some reversible capillary condensation in the crevices of the aggregate, the effect of which would be to increase the uptake at a given relative pressure the plausibility of this explanation is supported by the fact that very low values of s, 1-47-1-77, were obtained with certain other phthalocyanines known to be meso-porous (cf. Chapter 3). 

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... 

As already indicated in Section 3.1, the study of mesoporous solids is closely bound up with the concept of capillary condensation and its quantitative expression in the Kelvin equation. This equation is, indeed, the basis of virtually all the various procedures for the calculation of pore size... 

As with all thermodynamic relations, the Kelvin equation may be arrived at along several paths. Since the occurrence of capillary condensation is intimately, bound up with the curvature of a liquid meniscus, it is helpful to start out from the Young-Laplace equation, the relationship between the pressures on opposite sides of a liquid-vapour interface. 

Let us now consider the process of capillary condensation. For the pure liquid (a) in equilibrium with its vapour fi), the condition for mechanical equilibrium is given by Equation (3.6) and that for physicochemical equilibrium by... 

From the Kelvin equation it follows that the vapour pressure p over a concave meniscus must be less than the saturation vapour pressure p°. Consequently capillary condensation of a vapour to a liquid should occur within a pore at some pressure p determined by the value of r for the pore, and less than the saturation vapour pressure—always provided that the meniscus is concave (i.e. angle of contact <90°). 

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 general there are two factors capable of bringing about the reduction in chemical potential of the adsorbate, which is responsible for capillary condensation the proximity of the solid surface on the one hand (adsorption effect) and the curvature of the liquid meniscus on the other (Kelvin effect). From considerations advanced in Chapter 1 the adsorption effect should be limited to a distance of a few molecular diameters from the surface of the solid. Only at distances in excess of this would the film acquire the completely liquid-like properties which would enable its angle of contact with the bulk liquid to become zero thinner films would differ in structure from the bulk liquid and should therefore display a finite angle of contact with it. 

Figure 3.10 is a plot of potential against distance from the wall for a liquid in a capillary of sufficient width for its middle A to be outside the range of forces from the wall. Since the capillary condensate is in equilibrium with the vapour, its chemical potential (=p represented by the horizontal line GF, will be lower than that of the free liquid the difference in chemical potential of the condensate at A, represented by the vertical distance AF, is brought about entirely by the pressure drop, Ap = 2y/r , across the meniscus (cf. Equation (3.6)) but at some point B. say, nearer the wall, the chemical potential receives a contribution represented by the line BC, from the adsorption potential. Consequently, the reduction Ap in pressure across the meniscus must be less at B than at A, so that again... 

Fig. 3.10 Contributions to the lowering of chemical potential of the condensed liquid in a capillary, arising from adsorption forces (c) and meniscus curvature (Ap). The chemical potential of the free liquid is , and that of the capillary condensed liquid is (= ) z is the distance from the capillary wall. (After Everett. )...

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. 

We consider first a cylinder closed at one end, B (Fig. 3.11(a)). Capillary condensation commences at that end to form a hemispherical meniscus r, and are equal to one another and therefore to r , which in turn is equal to r, the radius of the core (cf. Equation (3.7) and Fig. 3.7). Thus capillary condensation, to fill the whole pore, takes place at the relative pressure... 

Fig. 3.11 Capillary condensation in cylindrical pores, (a) Cylinder closed at one end, B. The meniscus is hemispherical during both capillary condensation and capillary evaporation, (h) and (c) Cylinder open at both ends. The meniscus is cylindrical during capillary condensation and hemispherical during capillary evaporation. Dotted lines denote the...

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