On the hysteresis loop and equilibrium transition in slit-shaped ink-bottle pores

Bin grand canonical Monte Carlo simulations have been carried out to study adsorption–desorption of argon at 87.3 K in a model ink-bottle mesoporous solid in order to investigate the interplay between the pore blocking process, controlled by the evaporation through the pore mouth via the meniscus separating the adsorbate and the bulk gas surroundings, and the cavitation process, governed by the instability of the stretched fluid (with a decrease in pressure) in the cavity. The evaporation mechanism switches from pore blocking to cavitation when the size of the pore neck is decreased, and is relatively insensitive to the neck length under conditions where cavitation is the controlling mechanism. We have applied the recently-developed Mid-Density scheme to determine the equilibrium branch of the hysteresis loop, and have found that, unlike ideal simple pores of constant size and infinite length, where the equilibrium transition is vertical, the equilibrium branch of an ink-bottle pore has three distinct sub-branches within the hysteresis loop. The first sub-branch is steep but continuous and is close to the desorption branch (which is typical for a pore with two open ends); this is associated with the equilibrium state in the neck. The third sub-branch is much steeper and is close to the adsorption branch (which is typical for either a pore with one end closed or a closed pore), and is associated with the equilibrium state in the cavity. The second sub-branch, connecting the other two sub-branches, has a more gradual slope. The domains of these three sub-branches depend on the relative sizes of the cavity and the neck, and their respective lengths. Our investigation of the effects of changing neck length clearly demonstrates that cavitation depends, not only on fluid properties, as frequently stated, but also on pore geometry.     

Adsorption hysteresis of nitrogen and argon in pore networks and characterization of novel micro- and mesoporous silicas

We report results of nitrogen and argon adsorption experiments performed at 77.4 and 87.3 K on novel micro/mesoporous silica materials with morphologically different networks of mesopores embedded into microporous matrixes: SE3030 silica with worm-like cylindrical channels of mode diameter of approximately 95 angstroms, KLE silica with cage-like spheroidal pores of ca. 140 angstroms, KLE/IL silica with spheroidal pores of approximately 140 angstroms connected by cylindrical channels of approximately 26 angstroms, and, also for a comparison, on Vycor glass with a disordered network of pores of mode diameter of approximately 70 angstroms. We show that the type of hysteresis loop formed by adsorption/desorption isotherms is determined by different mechanisms of condensation and evaporation and depends upon the shape and size of pores. We demonstrate that adsorption experiments performed with different adsorptives allow for detecting and separating the effects of pore blocking/percolation and cavitation in the course of evaporation. The results confirm that cavitation-controlled evaporation occurs in ink-bottle pores with the neck size smaller than a certain critical value. In this case, the pressure of evaporation does not depend upon the neck size. In pores with larger necks, percolation-controlled evaporation occurs, as observed for nitrogen (at 77.4 K) and argon (at 87.3 K) on porous Vycor glass. We elaborate a novel hybrid nonlocal density functional theory (NLDFT) method for calculations of pore size distributions from adsorption isotherms in the entire range of micro- and mesopores. The NLDFT method, applied to the adsorption branch of the isotherm, takes into account the effect of delayed capillary condensation in pores of different geometries. The pore size data obtained by the NLDFT method for SE3030, KLE, and KLE/IL silicas agree with the data of SANS/SAXS techniques.     

Change in desorption mechanism from pore blocking to cavitation with temperature for nitrogen in ordered silica with cagelike pores

To verify pore blocking controlled desorption in ink-bottle pores, we measured the temperature dependence of the adsorption-desorption isotherms of nitrogen on four kinds of KIT-5 samples with expanded cavities hydrothermally treated for different periods of time at 393 K. In the samples, almost spherical cavities are arranged in a face-centered cubic array and the cavities are connected through small channels. The pore size of the channels increased with an increase in the hydrothermal treatment time. At lower temperatures a steep desorption branch changed to a gradual one as the hydrothermal treatment was prolonged. For the sample hydrothermally treated only for 1 day, the rectangular hysteresis loop shrank gradually with increasing temperature while keeping its shape. The temperature dependence of the evaporation pressure observed was identical with that expected for cavitation-controlled desorption. On the other hand, for the samples hydrothermally treated for long times, the gradual desorption branch became a sharp one with increasing temperature. This strongly suggests that the desorption mechanism is altered from pore blocking to cavitation with temperature. Application of percolation theory to the pore blocking controlled desorption observed here is discussed.     

Equilibrium adsorption in cylindrical mesopores: a modified Broekhoff and de Boer theory versus density functional theory

This paper presents a thermodynamic analysis of capillary condensation phenomena in cylindrical pores. Here, we modified the Broekhoff and de Boer (BdB) model for cylindrical pores accounting for the effect of the pore radius on the potential exerted by the pore walls. The new approach incorporates the recently published standard nitrogen and argon adsorption isotherm on nonporous silica LiChrospher Si-1000. The developed model is tested against the nonlocal density functional theory (NLDFT), and the criterion for this comparison is the condensation/evaporation pressure versus the pore diameter. The quantitative agreement between the NLDFT and the refined version of the BdB theory is ascertained for pores larger than 2 nm. The modified BdB theory was applied to the experimental adsorption branch of adsorption isotherms of a number of MCM-41 samples to determine their pore size distributions (PSDs). It was found that the PSDs determined with the new BdB approach coincide with those determined with the NLDFT (also using the experimental adsorption branch). As opposed to the NLDFT, the modified BdB theory is very simple in its utilization and therefore can be used as a convenient tool to obtain PSDs of all mesoporous solids from the analysis of the adsorption branch of adsorption isotherms of any subcritical fluids.     

A grand canonical Monte Carlo study of capillary condensation in mesoporous media: effect of the pore morphology and topology

We study by means of Grand Canonical Monte Carlo simulations the condensation and evaporation of argon at 77 K in nanoporous silica media of different morphology or topology. For each porous material, our results are compared with data obtained for regular cylindrical pores. We show that both the filling and emptying mechanisms are significantly affected by the presence of a constriction. The simulation data for a constricted pore closed at one end reproduces the asymmetrical shape of the hysteresis loop that is observed for many real disordered porous materials. The adsorption process is a quasicontinuous mechanism that corresponds to the filling of the different parts of the porous material, cavity, and constriction. In contrast, the desorption branch for this pore closed at one end is brutal because the evaporation of Ar atoms confined in the largest cavity is triggered by the evaporation of the fluid confined in the constriction (which isolates the cavity from the gas reservoir). This evaporation process conforms to the classical picture of "pore blocking effect" proposed by Everett many years ago. We also simulate Ar adsorption in a disordered porous medium, which mimics a Vycor mesoporous silica glass. The adsorption isotherm for this disordered porous material having both topological and morphological defects presents the same features as that for the constricted pore (quasicontinuous adsorption and steep desorption process). However, the larger degree of disorder of the Vycor surface enhances these main characteristics. Finally, we show that the effect of the disorder, topological and/or morphological, leads to a significant lowering of the capillary condensation pressure compared to that for regular cylindrical nanopores. Also, our results suggest that confined fluids isolated from the bulk reservoir evaporate at a pressure driven by the smallest size of the pore.     

The effects of energy sites on adsorption of Lennard-Jones fluids and phase transition in carbon slit pore of finite length a computer simulation study

A Monte Carlo simulation method is used to study the effects of adsorption strength and topology of sites on adsorption of simple Lennard-Jones fluids in a carbon slit pore of finite length. Argon is used as a model adsorbate, while the adsorbent is modeled as a finite carbon slit pore whose two walls composed of three graphene layers with carbon atoms arranged in a hexagonal pattern. Impurities having well depth of interaction greater than that of carbon atom are assumed to be grafted onto the surface. Different topologies of the impurities; corner, centre, shell and random topologies are studied. Adsorption isotherms of argon at 87.3 K are obtained for pore having widths of 1, 1.5 and 3 nm using a Grand Canonical Monte Carlo simulation (GCMC). These results are compared with isotherms obtained for infinite pores. It is shown that the surface heterogeneity affects significantly the overall adsorption isotherm, particularly the phase transition. Basically it shifts the onset of adsorption to lower pressure and the adsorption isotherms for these four impurity models are generally greater than that for finite pore. The positions of impurities on solid surface also affect the shape of the adsorption isotherm and the phase transition. We have found that the impurities allocated at the centre of pore walls provide the greatest isotherm at low pressures. However when the pressure increases the impurities allocated along the edges of the graphene layers show the most significant effect on the adsorption isotherm. We have investigated the effect of surface heterogeneity on adsorption hysteresis loops of three models of impurity topology, it shows that the adsorption branches of these isotherms are different, while the desorption branches are quite close to each other. This suggests that the desorption branch is either the thermodynamic equilibrium branch or closer to it than the adsorption branch.     

Modeling of Adsorption in Finite Cylindrical Pores by Means of Density Functional Theory

Adsorption of argon at its boiling point in finite cylindrical pores is considered by means of the non-local density functional theory (NLDFT) with a reference to MCM-41 silica. The NLDFT was adjusted to amorphous solids, which allowed us to quantitatively describe argon adsorption isotherm on nonporous reference silica in the entire bulk pressure range. In contrast to the conventional NLDFT technique, application of the model to cylindrical pores does not show any layering before the phase transition in conformity with experimental data. The finite pore is modeled as a cylindrical cavity bounded from its mouth by an infinite flat surface perpendicular to the pore axis. The adsorption of argon in pores of 4 and 5 nm diameters is analyzed in canonical and grand canonical ensembles using a two-dimensional version of NLDFT, which accounts for the radial and longitudinal fluid density distributions. The simulation results did not show any unusual features associated with accounting for the outer surface and support the conclusions obtained from the classical analysis of capillary condensation and evaporation. That is, the spontaneous condensation occurs at the vapor-like spinodal point, which is the upper limit of mechanical stability of the liquid-like film wetting the pore wall, while the evaporation occurs via a mechanism of receding of the semispherical meniscus from the pore mouth and the complete evaporation of the core occurs at the equilibrium transition pressure. Visualization of the pore filling and empting in the form of contour lines is presented.     

The evaluation of the possibility of determining the function characterizing the connectedness of pores of various sizes from adsorption measurements

The possibility of determining the function characterizing the connectedness of pores of different types and sizes from the experimental adsorption isotherms is discussed. It is shown that the presence of joints of pores manifests itself by two indications in phase diagrams and adsorption isotherms that contain hysteresis loops. The first indication is related to the appearance of an additional phase transition in the phase diagram and/or a density jump in the isotherm. The second indication is related to different numbers of density jumps in the desorption and adsorption branches of adsorption isotherms. The character of the behavior of the adsorption branch of the isotherm in the case of the presence of regions of joints in a porous system is established.     

The Finite Pore Volume GAB Adsorption Isotherm Model as a Simple Tool to Estimate a Diameter of Cylindrical Nanopores

The finite pore volume Guggenheim–Anderson–de Boer (fpv-GAB) adsorption isotherm model has been considered as a simple tool which not only enables us to analyze the shape of isotherms theoretically, but also provides information about pore diameter. The proposed methodology is based on the geometrical considerations and the division of the adsorption space into two parts: the monolayer and the multilayer space. The ratio of the volumes of these two spaces is unambiguously related to the pore diameter. This ratio can be simply determined from the N₂ adsorption isotherm by its fitting with the use of fpv-GAB model. The volume ratio is equal to the ratio of the adsorption capacities in the monolayer and the multilayer—two of the best-fit parameters. The suggested approach has been verified using a series of isotherms simulated inside ideal carbon nanotubes. The adsorption data for some real adsorbents has also been used during tests. The studies performed have proven that diameters estimated with the use of the proposed method are comparable with the geometrical sizes or diameters published by others and based on the application of more sophisticated methods. For pores wider than 3 nm, the relative error does not exceed a few percent. The approach based on the fpv-GAB model reflects well the differences in pore sizes for the series of materials. Therefore, it can be treated as a convenient tool to compare various samples.     

Capillary condensation hysteresis in overlapping spherical pores: a monte carlo simulation study

The mechanisms of hysteretic phase transformations in fluids confined to porous bodies depend on the size and shape of pores, as well as their connectivity. We present a Monte Carlo simulation study of capillary condensation and evaporation cycles in the course of Lennard-Jones fluid adsorption in the system of overlapping spherical pores. This model system mimics pore shape and connectivity in some mesoporous materials obtained by templating cubic surfactant mesophases or colloidal crystals. We show different mechanisms of capillary hysteresis depending on the size of the window between the pores. For the system with a small window, the hysteresis cycle is similar to that in a single spherical pore: capillary condensation takes place upon achieving the limit of stability of adsorption film and evaporation is triggered by cavitation. When the window is large enough, the capillary condensation shifts to a pressure higher than that of the isolated pore, and the possibility for the equilibrium mechanism of desorption is revealed. These finding may have important implications for practical problems of assessment of the pore size distributions in mesoporous materials with cagelike pore networks.     

Contact angles, pore condensation, and hysteresis: insights from a simple molecular model

We discuss the thermodynamics of adsorption of fluids in pores when the solid-fluid interactions lead to partial wetting of the pore walls, a situation encountered, for example, in water adsorption in porous carbons. Our discussion is based on calculations for a lattice gas model of a fluid in a slit pore treated via mean field density functional theory (MFDFT). We calculate contact angles for pore walls as a function of solid-fluid interaction parameter, alpha, in the model, using Young's equation and the interfacial tensions calculated in MFDFT. We consider adsorption and desorption in both infinite pores and in finite length pores in contact with the bulk. In the latter case, contact with the bulk can promote evaporation or condensation, thereby dramatically reducing the width of hysteresis loops. We show how the observed behavior changes with alpha. By using a value of alpha that yields a contact angle of about 85 degrees and maintaining the bulk fluid in a supersaturated vapor state on adsorption, we find an adsorption/desorption isotherm qualitatively similar to those for graphitized carbon black where pore condensation occurs at supersaturated bulk vapor states in the spaces between the primary particles of the adsorbent.     

Lower closure point of adsorption hysteresis in ordered mesoporous silicas

To examine the nature of the lower closure point of adsorption hysteresis in ordered mesoporous silicas, we measured the temperature dependence of the adsorption-desorption isotherm of nitrogen for three kinds of ordered silicas with cagelike pores and three kinds of ordered silicas with cylindrical pores. The lower closure point pressure of nitrogen in the cagelike pores with sufficiently small necks, that is, the cavitation pressure of a confined liquid, did not depend appreciably on the cage size in the temperature region far away from a hysteresis critical temperature (Tch) but its cage-size dependence was noticeable in the vicinity of Tch. The lower closure point in the cylindrical pores depended on the pore size, and its thermal behavior was totally different from that in the cagelike pores. Nevertheless, the hysteresis critical points of nitrogen in the ordered mesoporous silicas, which are defined as a threshold of temperatures (Tch) and pressure above which reversible capillary condensation takes place in a given size and shape of pores, fell on a common line in a temperature-pressure diagram regardless of the pore geometries. We consider this finding as evidence that capillary evaporation in the cylindrical pores follows a cavitation process in the vicinity of Tch in the same way as that in the cagelike pores and also that the low limit of the hysteresis loop that has been long recognized since 1965 is due to the occurrence of a vapor bubble in a stretched metastable liquid confined to the pores with decreasing pressure (cavitation).     

Nature of adsorption and desorption branches in cylindrical pores

To examine the nature of the adsorption and desorption branches in hysteretic adsorption isotherms of gases on mesoporous materials, we measured the temperature dependence of the adsorption and desorption isotherms of argon, oxygen, and carbon dioxide onto MCM-41 with a pore diameter of 4.4 nm. The results clearly show that in the open-ended cylindrical pores of MCM-41, capillary condensation rather than evaporation takes place near a thermodynamical equilibrium transition, as opposed to the general statement that capillary evaporation can occur via a meniscus formed at the pore mouth, and, thus, takes place at equilibrium.     

What can positronium tell us about adsorption?

The condensation and evaporation of n-heptane at 298 K in mesopores of silica material obtained by the polymer templating method have been studied by PALS measurements. It is demonstrated that the ortho-positronium lifetimes and intensities provide valuable information on pore filling and emptying which are not accessible from a conventional adsorption experiment. The results confirm the specific adsorption mechanism of n-heptane in pores with narrow openings (ink-bottle shape) which is different from that known for other pore geometries. The results from PALS experiment are compared to those derived from the conventional n-heptane and nitrogen adsorption data.     

Consideration of the Multicomponent Nature of Adsorbents during Analysis of Their Structural and Energy Parameters

A method is proposed for the calculation of the distribution functions for the pore size, the adsorption energy, and the fractal dimensionality with due regard to the multicomponent nature of complex adsorbents based on a combined approach, including calculation of the constants of the equations in the approximation of the local adsorption isotherm for individual adsorbents with allowance for their specific characteristics (e.g., the type of pores). The integral equation, in which the kernel is the sum of the local adsorption isotherms with weighting factors determining the accessibility of the various phases in the multicomponent adsorbent, is solved using the method of constrained regularization.     

Computer simulation of Lennard-Jones fluid adsorption in wedge-shaped pores with smooth walls

The adsorption isotherms and local density distributions in wedge-shaped pores with smooth solid walls are obtained for a molecular fluid by the grand canonical Monte Carlo and multiple histogram reweighting methods. The effect of wedge opening angle on the phase behavior of the fluid is considered, and the peculiarities of the mechanism of wedge-shaped pore filling are established.     

Adsorption, capillary bridge formation, and cavitation in SBA-15 corrugated mesopores: a Derjaguin-Broekhoff-de Boer analysis

A Derjaguin-Broekhoff-de Boer analysis of adsorption and desorption in SBA-15 mesoporous silica is presented, using realistic geometrical models that account for the pore corrugation in these materials. The model parameters are derived from independent electron tomography and small-angle scattering characterization. A geometrical characteristic of the pore that is found to be important for adsorption is the corrugation length, l(C), which describes the longitudinal size of the geometrical defects along a given pore. Capillary bridges are possible only for large values of l(C). The results are explained in terms of two spinodal and two equilibrium pressures, characterizing the wide and the narrow sections of the pores. Simplified analytical expressions are obtained, which provide necessary conditions for bridge formation and for cavitation in terms of the radii of the narrow and wide sections of the pores, as well as of l(C). Quite generally, the results show that the deviation of the pore shape from that of ideal cylinders is key to understanding adsorption and desorption in corrugated mesopores, notably in SBA-15.     

Importance of molecular shape in the adsorption of nitrogen,carbon dioxide and methane on surfaces and in confined spaces

The importance of shape in the adsorption of nitrogen, carbon dioxide and methane (common molecular probes for solid characterization) on surfaces and in confined spaces is investigated for its effects on the adsorption capacity and isosteric heat. We study the possibility of using an equivalent pseudo-sphere model to describe the potential energy of interaction of these molecular probes. On a flat open surface, we find that the equivalent pseudo-sphere model describes adsorption of these species sufficiently well. However, in the confined space of pores, especially pores that accommodate three layers or less, the pseudo-sphere model describes the adsorption badly because of the geometrical constraint on the molecular packing. It is recommended that to study adsorption properly in small pores, potential models that correctly describe molecular shape should be used. In characterization, pseudo-sphere models are commonly used to generate the kernels (local isotherms) for the determination of pore size distribution which can lead to misleading results. We illustrate this with an example to show that the wrong pore size distribution results if pseudo-sphere kernels are used.