It is widely accepted that there is a correlation between the shape of the hysteresis loop and the texture (e.g., pore size distribution, pore geometry, connectivity) of a mesoporous adsorbent. An empirical classification of hysteresis loops was given by the IUPAC, which is based on an earlier classification by de Boer. The IUP AC classification is shown in Fig 1. 

Figure 1: IUPAC classifications of hysteresis loops.

According to the IUPAC classification type HI is often associated with porous materials consisting of well-defined cylindrical-like pore channels or agglomerates of compacts of approximately uniform spheres. It was found that materials that give rise to H2 hysteresis are often disordered and the distribution of pore size and shape is not well defined. Isotherms revealing type H3 hysteresis do not exhibit any limiting adsorption at high P/Po, which is observed with nonrigid aggregates of plate-like particles giving rise to slit-shaped pores. The desorption branch for type H3 hysteresis contains also a steep region associated with a (forced) closure of the hysteresis loop, due to the so-called tensile strength effect. This phenomenon occurs for nitrogen at 77K in the relative pressure range from 0.4 - 0.45. Similarly, type H4 loops are also often associated with narrow slit pores, but now including pores in the micropore region (see for instance Fig. 2, which depicts the nitrogen sorption isotherm on activated carbon).

Figure 2: Semi-logarithmic isothenn plot of argon at 87K on a faujasite zeolite which clearly resolves the micropore filling in the low relative pressure range. The steep increase close to the saturation pressure represents the pore filling oflarge meso- and macro-pores.

The dashed curves in the hysteresis loops shown in Fig. 1 reflect low-pressure hysteresis, which may be observable down to very low relative pressure. Low-pressure hysteresis may be associated with the change in volume of the adsorbent, i.e. the swelling of non-rigid pores or with the irreversible uptake of molecules in pores of about the same width as that of the adsorptive molecule. In addition chemisorption will also lead to such "open" hysteresis loops. An interpretation of sorption isotherms showing low-pressure hysteresis is difficult and an accurate pore size analysis is not possible anymore. But also the hysteresis loops usually associated with pore condensation imposes, of course, a difficulty to the pore size analysis of the porous materials and the decision whether the adsorption -or desorption branch should be taken for calculation of the pore size distribution curve depends very much on the reason(s) which caused the hysteresis. Hence, we discuss the origin of pore condensation hysteresis in the following section.

As mentioned before the occurrence of pore condensation/evaporation in mesoporous adsorbents is often accompanied by hysteresis. However, the mechanism and origin of sorption hysteresis is still a matter of discussion. There are essentially three models that contribute to the understanding of sorption hysteresis: (a) independent (single) pore model (b) network model, and (c) disordered porous material model. In the following we will discuss some aspects of these models.

(a) Independent Pore Model. 

Sorption hysteresis is considered as an intrinsic property of a phase transition in a single, idealized pore, reflecting the existence of metastable gas states. The hysteresis loop expected for this case is of type HI, according to the IUPAC classification. Different approaches, which would explain the occurrence of hysteresis in a single pore, can be found in the literature since ca. 1900. Cohan assumed that pore condensation occurs by filling the pore from the wall inward (for a cylindrical pore model). It was suggested that pore condensation would be controlled by a cylindrical meniscus once the pore is filled, whereas evaporation of the liquid would occurs from a hemispherical meniscus, which would lead according to the Kelvin equation to different values of PIPo for condensation and evaporation.

Theories by Foster, Cassell, Everett, Cole and Saam (CS) and Ball and Evans suggested that hysteresis may be caused by the development of metastable states of the pore fluids associated with the capillary condensation transition in a manner analogous to superheating or supercooling of a bulk fluid. These ideas could be essentially confirmed by recent theoretical studies based on Non Local Density Functional Theory (NLDFT). These studies revealed that the HI hysteresis can indeed be attributed to the existence of metastable states of the pore fluid, associated with the nucleation of the liquid phase, i.e., pore condensation is delayed. In principle, both pore condensation and pore evaporation can be associated with metastable states of the pore fluid. This is consistent with the classical van der Waals picture, which predicts that the metastable adsorption branch terminates at a vapor-like spinodal, where the limit of stability for the metastable states is achieved and the fluid spontaneously condenses into a liquid-like state (so-called spinodal condensation). Accordingly, the desorption branch would tenninate at a liquid-like spinodal, which corresponds to spontaneous evaporation (spinodal evaporation) In practice however, metastabilities occur only on the adsorption branch. Assuming a pore of finite length evaporation can occur via a receding meniscus and therefore metastability is not expected to occur during desorption. The NLDFT prediction for pore condensation and hysteresis in comparison with the correspondent experimental sorption isothenn of argon at 87 K in MCM-41 silica is shown in Fig. 3. The experimental isothenn and the NLDFT isothenn agree quite well and the theoretical prediction for the position of the equilibrium liquid-gas transition (which corresponds to the condition at which the two states have equal grand potential) agrees quite well with the experimentally observed evaporation transition, i.e. the position of the desorption branch of the hysteresis loop. Hence, it was concluded that the desorption branch is associated with the equilibrium gas-liquid phase transition. In such a case the desorption branch should be chosen for pore size analysis if theories/methods are applied which describe the equilibrium transition (e.g., BJH, conventional NLDFT).

Figure 3：NLDFT adsorption isotherm of argon at 87K in a cylindrical pore of diameter 4.8 nm in comparison with the appropriate experimental sorption isotherm on MCM-41. It can be clearly seen that the experimental desorption branch is associated with the equilibrium gasliquid phase transition, whereas the condensation step corresponds to the spinodal spontaneous transition.

The small steps in the theoretical isotherm are a consequence of assuming a structureless (i.e., chemically and geometrically smooth) pore wall model, which neglects the heterogeneity of the MCM-4I pore walls. There is also some evidence that type HI hysteresis as observed in ordered three dimensional pore systems such as MCM-48 silica but also in highly ordered porous glasses (such as sol-gel glasses and controlled pore glasses) is predominantly caused by the existence of metastable states associated with pore condensation. The hysteresis loops could be described by applying models based on the independent pore model (e.g., Cole-Saam theory NLDFT etc.). Accordingly, classical networking and pore blocking effects are not necessarily present in an (ordered) interconnected pore system.

(b) Network Model. 

Sorption hysteresis is explained as a consequence of the interconnectivity of a real porous network with a wide distribution of pore sizes. If network and pore blocking effects are present typically a hysteresis loop of type H2 (IUPAC classification) is expected. Network models take into account that in many materials the pores are connected and form a three-dimensional network. An important feature of the network model is the possibility of pore blocking effects during evaporation, which occurs if a pore has access to the external gas phase only via narrow constrictions (e.g., an ink-bottle pore). The basis for the understanding of sorption hysteresis in inkbottle pores and networks can be found in the work of McBain. The wide inner portion of an inkbottle pore is filled at high relative pressures, but it cannot empty during desorption until the narrow neck of a pore first empties at lower relative pressure. Thus, in a network of inkbottle pores the capillary condensate in the pores is obstructed by liquid in the necks. The relative pressure at which a pore empties now depends on the size of the narrow neck, the connectivity of the network and the state of neighboring pores. Hence, the desorption branch of the hysteresis loops does not (in contrast to the single pore model) occur at thermodynamic equilibrium, but reflects a percolation transition instead. In such a case the desorption branch of the hysteresis loop is much steeper (compared to the adsorption branch) leading to H2 hysteresis according to the IUPAC classification.

Work by Everett and others have led to the development of several specific network models. Advanced network or percolation models were introduced for instance by Mason Wall and Brown, Neimark, Parlar and Yortsos, Ball and Evans, Seaton et al. and Rojas et al. Type H2 hysteresis is observed in many disordered porous materials such as, for instance, porous Vycor® glass, or disordered sol-gel glasses. By combining different experimental techniques such as adsorption measurements (volumetric, gravimetric), ultrasound and light scattering or gas adsorption and in situ neutron scattering some evidence for a percolation mechanism associated with pore evaporation could be obtained.

However, the existence of the conventional pore blocking mechanism as described above is under discussion. Sarkisov and Monson concluded that the H2 hysteresis loop (obtained from a molecular dynamics study of adsorption of a simple fluid) typically observed in inkbottle pores is not necessarily caused by the occurrence of conventional pore blocking. The large cavities could be emptied by a diffusional mass transport process from the fluid in the large cavity to the narrow neck and from there into the gas phase, hence the pore body can empty even while the pore neck remains filled. Further experimental and theoretical work by Ravikovitch et al. suggests that both conventional pore blocking and so-called cavitation can occur in inkbottle type pores depending on temperature and pore size.

Cavitation corresponds to the situation of spinodal evaporation, i.e., the condensed liquid evaporates when the limit of stability of metastable pore liquid is achieved and the pore fluid spontaneously evaporates into a vapour-like state as shown in Fig. 3. In such a case the desorption branch does not reflect the thermodynamic eqUilibrium liquid-gas transition. The cavitation effect is correlated with the occurrence of a lower limit of hysteresis in the sorption isotherm, which is within the classical picture correlated with the so-called tensile strength effect. This effect is believed to be the cause for the observation that for many disordered porous materials the hysteresis loop for nitrogen adsorption at 77.35 K is forced to close at relative pressure at or above 0.42, apparently independent of the porous material. The existence of a lower closure point affects primarily the position of the desorption branch with regard to its position and steepness. Despite the fact that the reasons for this phenomenon are still not sufficiently understood, it is clear that it leads to complications for pore size calculation.

(c) Disordered Porous Material Model. 

A more realistic picture takes into account that the thermodynamics of the pore fluid is determined by phenomena spanning the complete pore network. Even with the incorporation of network and percolation effects the adsorption thermodynamics is still modeled at a single pore level, i.e., the behavior of the fluid in the entire pore space is not assessed. In order to achieve this one needs to consider models which attempt to describe the microstructure of porous materials at length scales beyond that of a single pore. According to Gubbins there are two general approaches to construct a model of nanoporous materials by methods of molecular simulation. The first is the so-called mimetic simulation, and involves the development of a simulation strategy, that mimics the development of the pore structure in the materials preparation. In fact, Gelb and Gubbins have reproduced the complex network structure of porous glasses such as Vycor and controlled-pore glass by applying molecular simulation and have studied the sorption and hysteresis behavior of xenon in such systems. Grand Canonical Monte Carlo simulation results for xenon adsorption in these systems suggest strongly that the shape of the adsorption/desorption hysteresis does not depend on the connectivity of the material model, supporting the hypothesis that in materials of this type (e.g., a porous Vycor® glass with a porosity of 30%) the fluid in different pores behaves quasi-independently, and that no systemspanning phase transitions occur during adsorption or desorption.

The second approach is the reconstruction method. Here one seeks a molecular model, whose structure matches available experimental structure data. Monson and co-workers investigated by Monte Carlo simulation the condensation and hysteresis phenomena of a Lennard-Jones fluid in a reconstructed model of silica xerogel. Their adsorption isotherms exhibited hysteresis loops of type HI and H2 in agreement with experimental results obtained on the same type of material. The observed hysteresis was attributed with thermodynamic metastability of the low and high density phases of the adsorbed fluid - however these phases span the entire void space of the porous material and are therefore not associated with the individual pores.

However, it was also suggested that in disordered porous glass materials the origin of the hysteresis is associated with long time dynamics, which is so slow that on (experimentally) accessible time scales, the systems appear to be equilibrated, which leads to the observed reproducible results in the observation of the hysteresis loop. Theoretical and experimental work is necessary to (i) clarify what determines the shape of the hysteresis loop in such disordered systems and (ii) to obtain a clearer picture of the nature of phase behavior of fluids in disordered porous systems.