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During the desorption process, as the relative pressure is reduced, systems in which capillary condensation occurs, generally show hysteresis. Sing et al. (1985) classified several shape groups of hysteresis loops, which are shown in Figure 3.4. In Type H1 hysteresis loops, both adsorption and desorption branches are steep at intermediate relative pressures. Examples of pores that give such shapes include tubular capillaries open at both ends, tubular capillaries with slightly wider parts, wide ink bottle ores and wedged shaped capillaries (Allen, 1998). Also, Type H1 can be associated with agglomerates or compacts of uniform spheres in somewhat regular array that consequently have a narrow pore size distribution (Sing et al., 1985). Furthermore, the Type H1 loop could be a result of either condensation in independent capillaries within a narrow size distribution or by globular solids made of spherical particles of about the same diameter (Broekhoff and de Boer, 1968). In Type H2 hysteresis loops, the adsorption branch has a slopping character and the desorption branch is steep at intermediate relative pressures (Allen, 1998). The Type H2 hysteresis loop has been traditionally ascribed to a collection of ink-bottle type pores (Allen, 1998). However, this may well be an oversimplification since network effects that usually occur in a wide variety of adsorbents, can produce the same result (Sing et al., 1985). The dimensions responsible for the adsorption branch of the isotherm is heterogeneously distributed and the dimensions responsible for desorption are of equal size (Allen, 1998). This is a classic percolation shape.

In Type H3 hysteresis loops, the adsorption branch is steep at intermediate pressures whilst desorption branch is slopping. The ascending and descending boundary curves are sloping and usually the desorption branch also includes a steep region at which the remaining condensate comes out almost suddenly out of the pores as a consequence of the so-called tensile strength effect (Everett, 1967). Type H3 is normally associated with open-slit shaped capillaries with parallel walls (Allen, 1998). Aggregates of plate like particles give rise to slit shaped pores (Sing et al., 1985). Type H4 does not exhibit any limiting adsorption at high x which indicates that the adsorbent does not possess a well-defined

Figure 3.4

Types of hysteresis loops (Sing et al., 1985)

The hysteresis of capillary condensable vapour has several possible origins. Most of the early theories of adsorption hysteresis made explicit use of the Kelvin Equation (3.18). Zsigmondy (1911) was the first to suggest the hysteresis phenomenon was due to a difference in the contact angles of the condensing and evaporating liquid. This explanation may account for some of the effects produced by the presence of surface impurities, but in its original form cannot explain the reproducibility of the majority of recorded loops. Other earlier explanation of hysteresis phenomena were based on ink-bottle theory that was developed by Mcbain (1935). It was based on the principle that the rate of evaporation of a liquid in a relatively large pore is likely to be retarded if there is only one narrow exit. This led Brunauer et al. (1938) to conclude that the liquid in the pore cannot be in true equilibrium with its vapour during the desorption process, and therefore, it is the adsorption branch of the loop which represents thermodynamic reversibility.

In contrast, network theory describes isotherms in terms of pore connectivity and pore size distribution. Therefore, at the end of adsorption, the adsorption isotherm forms a plateau

desorption process reflects the distribution of channels (pore necks) rather than the distribution of pores (Allen, 1998). The resulting hysteresis between the adsorption and desorption isotherm is therefore as a result of pore interconnectivity. As the pressure is reduced, a liquid-filled cavity cannot evaporate until at least one of the channels to the outside has evaporated. Therefore, the geometry of the network determines the shape of the adsorption branch of the isotherm.

A classic hysteresis theory is the ‘independent pore model’. It regards the difference between the relative vapour pressures at which ink-bottle pores fill and empty as the primary cause of hysteresis (Sing et al., 1985). A single relative vapour pressure corresponding to the filling of the pore site with radius. However, there is a different single relative vapour pressure corresponding to the emptying of the pore connection with radius. The concept of pore blocking in ink-bottle type pores was successfully employed to describe the H2 hysteresis loops (Everett, 1967; and Ravikovitch et al., 2002), indicating exclusive dependence on the pore neck diameter. Pore blocking was reported to only take effect when the neck diameter is greater than a certain characteristic size of ~ 4 nm for nitrogen. These authors proposed a cavitation theory which states that for this critical 4 nm pore neck size and above pore blocking will take effect and for anything below there will be no pore blocking effects. Furthermore, the analysis of the hysteresis loops and scanning desorption isotherm on spherical pores of ~ 15 nm revealed three mechanisms of evaporation: evaporation from blocked cavities controlled by the size of connecting pores (classical ink-bottle or pore blocking effect); spontaneous evaporation caused by cavitation of the stretched metastable liquid; and finally that of near equilibrium evaporation in the region of hysteresis from unblocked cavities that have access to the vapour phase (Ravikovitch et al., 2002).

Sing et al., 1985 work on hysteresis loops was reviewed by Seaton (1991). This researcher proposed that for the Type H1 isotherm the formation of the hysteresis loop is governed by delayed condensation of nitrogen in the pores, whereas the H2 isotherm is dependent on network percolation effects. Similar observations were later made by Rajniak and Yang (1993), Soos and Rajniak (2001). The Type H3 and Type H4 are less common, and the reversibility depends on the adsorptive and operational temperature and parameters. Furthermore, Seaton (1991) determined pore connectivity based on the use of percolation

theory to analyse adsorption isotherms. The researcher illustrated the role of connectivity for three pores namely A, B, and C where only B has communication with the outside of the material as shown in Figure 3.5. As the pressure is increased during adsorption, nitrogen condenses into the pores in order of increasing pore size in the sequence of A, B and C. During desorption, the order in which the liquid in pores become thermodynamically unstable with respect to vapour phase is C, B, and A. However, the nitrogen condensed in contact with pore C is not in contact with the vapour phase which makes it difficult to vaporise at its condensation pressure. Consequently, metastable liquid nitrogen persists in pore C below its condensation pressure, until the liquid in B that is in contact with vapour phase, vaporises. Therefore, the nitrogen in Pore A is then in contact with its vapour and consequently able to vaporise at its condensation pressure. As a result, order of vaporisation during desorption in pore size is in the sequence of B and C together followed by A with the delay in C giving rise to the hysteresis.

Figure 3.5

Network of three pores in porous solid (Seaton, 1991)

In recent years, Androutsopoulos and Salmas (2000) used a random corrugated pore concept to develop a new statistical model called the corrugated pore structure model (CPSM) to simulate capillary condensation-evaporation hysteresis. The pore structure was envisaged to be composed of a statistically large number of independent (non-intersected) corrugated pores. This corrugated pore was assumed to be made of a series of

condensate in a corrugated pore, which is fully saturated upon the termination of the condensation process        1 0 P P

, a hemispherical interface is anticipated to be present at

either pore necks. When the pressure is decreased gas desorption occurs, resulting in the retreat of the two interfaces in opposite directions and their convergence towards the pore centre. A continual film of liquid interface to the wide pore segments positioned in the interior parts of the corrugated pore was restricted by the intervention of smaller segments or throats where evaporation should occur prior to the interface motion toward the wider sections of the corrugated pore interior. Therefore, according to the work of Androutsopoulos and Salmas (2000), it is the restricted access of the vapour towards the wider segments that induces hysteresis during capillary desorption-evaporation.

In subsequent years, Monte Carlo (MC), and molecular dynamics (MD) simulations have been used to investigate pore blocking effects in capillary condensation hysteresis. The emphasis of these simulations is on disordered materials. Porous glasses and silica gels have been considered as case study systems for researching capillary hysteresis and networking effects. Sarkisov and Monson (2001) performed MC and MD simulations of capillary condensation in a single ink-bottle pore composed by a central rectangular cavity connected with the bulk phase by slit micropores half its size in width. The hysteresis observed in this model was not related to the classical pore blocking effect, as desorption from the central cavity occurred with the connecting pores remaining filled. During the desorption process, evaporation occurs from the pore throat into the bulk vapour phase and the molecules removed were then replaced by those coming from the pore body. Therefore, hysteresis arises when the pressure is decreased on desorption, and there is initially nothing to nucleate a large scale evaporation process in the large cavity until the pore liquid reaches a sufficiently expanded state such that spontaneous local density fluctuations lead to cavitation (spontaneous nucleation of a bubble). Furthermore, independent MC Simulations work on adsorption-desorption cycles in model porous glasses by Gelb and Gubbins (2002) and Pellenq et al. (2000) did not display any appreciable pore blocking effects.

In a subsequent report, Woo et al. (2001) developed density functional theory DFT models of sorption in disordered media and constructed hysteresis loops resembling the shape of experimental adsorption isotherms on porous glasses, without invoking pore-blocking

effects. Classical pore-blocking effects take place when the neck size is greater than a certain characteristic value (50 Å for nitrogen at 77 K, assuming that the neck can be considered as a cylindrical pore). Thommes et al. (2006) studied nitrogen and argon adsorption experiments performed at 77.4 and 87.3K on novel micro/mesoporous silica materials with morphologically different networks of mesopores embedded into microporous matrixes. These researchers showed that the type of hysteresis loop formed by adsorption/desorption isotherms is determined by different mechanisms of condensation and evaporation, and depends on the shape and sizes of featured pores (see Figure 3.6). This finding confirmed that cavitation-controlled evaporation occurs in ink-bottle pores with the neck size smaller than a certain value, and in this case, the pressure of evaporation does not depend upon the neck size. Therefore, desorption in a structure consisting of large mesopores occurs first by cavitation of the liquid in the large mesopores followed by desorption from smaller mesopores. In contrast, for pores with larger necks, they confirmed that percolation-controlled evaporation occurs, as observed for nitrogen and argon adsorption on porous Vycor glass. Hence, percolation effects contribute to hysteresis and hence information about the size of the pore necks can be obtained from the desorption branch.