Improvement of the Derjaguin-Broekhoff-de Boer theory for capillary condensation/evaporation of nitrogen in mesoporous systems and its implications for pore size analysis of MCM-41 silicas and related materials

In this work, we propose an improvement of the classical Derjaguin-Broekhoff-de Boer (DBdB) theory for capillary condensation/evaporation in mesoporous systems. The primary idea of this improvement is to employ the Gibbs-Tolman-Koenig-Buff equation to predict the surface tension changes in mesopores. In addition, the statistical film thickness (so-called t-curve) evaluated accurately on the basis of the adsorption isotherms measured for the MCM-41 materials is used instead of the originally proposed t-curve (to take into account the excess of the chemical potential due to the surface forces). It is shown that the aforementioned modifications of the original DBdB theory have significant implications for the pore size analysis of mesoporous solids. To verify our improvement of the DBdB pore size analysis method (IDBdB), a series of the calcined MCM-41 samples, which are well-defined materials with hexagonally ordered cylindrical mesopores, were used for the evaluation of the pore size distributions. The correlation of the IDBdB method with the empirically calibrated Kruk-Jaroniec-Sayari (KJS) relationship is very good in the range of small mesopores. So, a major advantage of the IDBdB method is its applicability for small mesopores as well as for the mesopore range beyond that established by the KJS calibration, i.e., for mesopore radii greater than approximately 4.5 nm. The comparison of the IDBdB results with experimental data reported by Kruk and Jaroniec for capillary condensation/evaporation as well as with the results from nonlocal density functional theory developed by Neimark et al. clearly justifies our approach. Note that the proposed improvement of the classical DBdB method preserves its original simplicity and simultaneously ensures a significant improvement of the pore size analysis, which is confirmed by the independent estimation of the mean pore size by the powder X-ray diffraction method.     

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.     

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.     

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).     

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.     

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.     

Accurate relations between pore size and the pressure of capillary condensation and the evaporation of nitrogen in cylindrical pores

To examine the theoretical and semiempirical relations between pore size and the pressure of capillary condensation or evaporation proposed so far, we constructed an accurate relation between the pore radius and the capillary condensation and evaporation pressure of nitrogen at 77 K for the cylindrical pores of the ordered mesoporous MCM-41 and SBA-15 silicas. Here, the pore size was determined from a comparison between the experimental and calculated X-ray diffraction patterns due to X-ray structural modeling recently developed. Among the many theoretical relations that differ from each other in the degree of theoretical improvements, a macroscopic thermodynamic approach based on Broekhoff-de Boer equations was found to be in fair agreement with the experimental relation obtained in the present study.     

A Stand-Alone Mesoporous Crystal Structure Model from in situ X-ray Diffraction: Nitrogen Adsorption on 3 D Cagelike Mesoporous Silica SBA-16

We present a modeling scheme to analyze cagelike silica mesoporous crystals based on in situ X-ray diffraction (XRD) data collected during gas adsorption-desorption (physisorption) processes. Nitrogen physisorption on a silica mesoporous crystal of SBA-16 was directly monitored by using synchrotron in situ powder XRD measurements conducted at SPring-8. SBA-16 is a well-ordered mesoporous silica in which three-dimensional interconnected cagelike primary mesopores are located at the body-centered cubic lattice points. In addition, the surrounding silica matrix contains random microporous and mesoporous intrawall porosities that are significantly influential to the diffusion properties, and thus important to be quantified for this media. The in situ XRD data exhibits seven Bragg reflections throughout the measurements, and the present method allows one to obtain the maximal and stand-alone information about the pore structure (for example, the mesopore size, the matrix density, the intrawall porosity, and pore surface roughness) together with the nitrogen film evolution in the primary mesopores and the intrawall pore-filling in the silica matrix. We furthermore observe a macroscopic amount of nitrogen adsorbed assuming the density of the fluid, and confirm that the XRD "isotherm" recalculated from the analysis result is consistent with the conventional nitrogen isotherm on a semi-quantitative level; however, these results suggest that the intrawall pores would have a greater contribution to the adsorption than considered based on the conventional isotherm analyses. The present method is readily extendable to any ordered mesopores wrapped by the wall matrix containing a certain intrawall porosity.     

Evaluation of pore size distribution in boundary region of micropore and mesopore using gas adsorption method

This paper discusses an accurate method of pore size distribution evaluation in boundary regions of micropores and mesopores using the gas adsorption process on the basis of the capillary condensation theory, which is liable to be underestimated with the existing BJH and DH methods. A typical nitrogen adsorption isotherm for highly ordered mesoporous silica, which has cylindrical pores with diameter smaller than 4 nm, is considered to be type IV and it is well known for the steep increase of the amount adsorbed through capillary condensation in the region of the relative pressure P/P0 smaller than 0.4. In calculating the distribution of the pore size from the change of the amount adsorbed due to capillary condensation, it is important to accurately predict both the multilayer thickness t of the adsorbed nitrogen molecules and the critical radius rc where capillary condensation occurs. It is necessary to consider the curvature of the adsorption layer-gas phase interface when predicting the multilayer thickness t of nitrogen adsorbed within the pore of highly ordered mesoporous silica. Revision of the Kelvin equation is also required when rc is to be predicted. While the predicted value of t based on the Broekhoff and de Boer theory is matched well with the value of t which is actually measured using highly ordered mesoporous silica, and the predicted value of rc based on the GTKB-Kelvin-cylindrical equation that has been revised considering the effect of the interfacial curvature on the interfacial tension of the adsorption layer-gas phase interface is matched with the value of rc which is actually measured using highly ordered mesoporous silica. A combination method of the Broekhoff and de Boer equation and the GTKB-Kelvin-cylindrical equation is proposed as a means of accurately evaluating, from the nitrogen adsorption isotherm, the pore size distribution in the highly ordered mesoporous silica in boundary region of micropore and mesopore. The proposed new method of pore size evaluation features high accuracy and offers the convenience of obtaining the pore size distribution without repeated calculations by employing the same algorithm as DH method. The pore size predicted by the Halsey equation and the Kelvin equation of the conventional DH method is about 20% smaller than the pore size predicted by the newly proposed evaluation method.     

Mechanism of freezing of water in contact with mesoporous silicas MCM-41, SBA-15 and SBA-16: role of boundary water of pore outlets in freezing

The freezing mechanism of water contacted with mesoporous silicas with uniform pore shapes, both cylindrical and cagelike, was studied by thermodynamic and structural analyses with differential scanning calorimetry (DSC) and X-ray diffraction (XRD) together with adsorption measurements. In the DSC data extra exothermic peaks were found at around 230 K for water confined in SBA-15, in addition to that due to the freezing of pore water. These peaks are most likely to be ascribed to the freezing of water present over the micropore and/or mesopore outlets of coronas in SBA-15. Freezing of water confined in SBA-16 was systematically analysed by DSC with changing the pore size. The freezing temperature was found to be around 232 K, close to the homogeneous nucleation temperature of bulk water, independent of the pore size when the pore diameter (d) < 7.0 nm. Water confined in the cagelike pores of SBA-16 is probably surrounded by a water layer (boundary water) at the outlets of channels to interconnect the pores and of fine corona-like pores, which is similar to that present at the outlet of cylindrical pores in MCM-41 and of cylindrical channels in SBA-15. The presence of the boundary water would be a key for water in SBA-16 to freeze at the homogeneous nucleation temperature. This phenomenon is similar to those well known for water droplets in oil and water droplets of clouds in the sky. The XRD data showed that the cubic ice I(c) was formed in SBA-16 as previously found in SBA-15 when d < 8.0 nm.     

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.     

Sol–gel synthesis,characterization and catalytic activity of mesoporous γ-alumina prepared from boehmite sol by different methods

In this study, boehmite sols were used as alumina precursors for preparing mesoporous γ-aluminas by two different methods. In one case polyethylenimine was used as a structure-directing agent, and in another case ultrasound treatment was applied. Nitrogen physisorption showed that aluminas that had been prepared by these methods demonstrated different porous structures. The sample obtained without additional treatment had closely packed spherical particles and pores had ink-bottle neck morphology. Ultrasound treatment led to the transformation of ink-bottle pores into cylindrical form and to the increase in surface area and pore volume. Aluminas prepared using polyethylenimine as a template showed larger cylindrical wormhole-like mesopores with a broader pore size distribution, high surface area and pore volume. Catalytic tests showed that textural properties as well as crystallite size were very important parameters of synthesized samples which affected the catalytic activity in the methanol dehydration reaction. It was found that γ-Al₂O₃ prepared by ultrasound treatment had large crystallite size and demonstrated high catalytic activity.     

Spherical particles in small porous sphero-cylindrical systems: phase diagrams and transfer coefficients

The equilibrium and transport characteristics of spherical particles in sphero-cylindrical porous systems were studied in terms of the lattice-gas model. The supramolecular structure of these systems is modeled by segments with a simple regular geometry (cylindrical and spherical) with additional inclusion of the interconnecting areas between different pore segments. Thus, one can model various types of porous systems ranging from zeolite cavities to stackings of long cylindrical sections in new mesoporous materials such as MCM-41 and MCM-49. The distribution of molecules is described in the quasichemical approximation with allowance for intermolecular interactions. The concentration dependences for local self-diffusion and shear viscosity coefficients were calculated. The contributions of the near-wall regions caused by the molecule—wall potential to the general pattern of phase diagrams, the effect of the pore size on the capillary condensation conditions, and the role of the molecular mobility on pore walls were discussed.     

Adsorption hysteresis in ordered mesoporous silicas

We review some recent progress in experimental studies of the adsorption hysteresis of simple molecules in ordered mesoporous silicas. We show that the nature of the adsorption hysteresis due to capillary condensation can be examined with less ambiguity by measuring the hysteresis loop for the ordered mesoporous silicas with three types of pore geometries (cylindrical, interconnected cylindrical, and interconnected spherical) over a wide temperature range. The adsorption hysteresis arises from the metastability of a confined phase and the temperature at which the hysteresis disappears is lower than the critical temperature of vapor-liquid equilibrium in pores. The hysteresis occurs mainly on the desorption rather than adsorption branch, irrespective of the pore geometries.     

Gas adsorption in mesoporous micelle-templated silicas: MCM-41, MCM-48, and SBA-15

This paper reports a molecular simulation and experimental study on the adsorption and condensation of simple fluids in mesoporous micelle-templated silicas MCM-41, MCM-48, and SBA-15. MCM-41 is described as a regular cylindrical silica nanopore, while SBA-15 is assumed to be made up of cylindrical nanopores that are connected through lateral channels. The 3D-connected topology of MCM-48 is described using a gyroid periodic minimal surface. Argon adsorption at 77 K is calculated for the three materials using Grand Canonical Monte Carlo simulations. Qualitative comparison with experiments for nitrogen adsorption in mesoporous micelle-templated silicas is made. The adsorption isotherm for SBA-15 resembles that for MCM-41. In particular, capillary condensation and evaporation are not affected by the presence of the connecting lateral channels. In contrast, the argon adsorption isotherm for MCM-48 departs from that for MCM-41 having the same pore size. While condensation in MCM-41 is a one-step process, filling of MCM-48 involves two successive jumps in the adsorbed amounts which correspond to condensation in different domains of the porosity. The condensation pressure for MCM-48 is larger than that for MCM-41. We attribute this result to the morphology of the MCM-48 surface (made up of both concave and convex regions) that differs from that for MCM-41 (concave only). Our results suggest that the pore connectivity affects pore filling when the size of the connections is comparable to that of the nanopores.     

Distribution of carbon nanotube sizes from adsorption measurements and computer simulation

The method for the evaluation of the distribution of carbon nanotube sizes from the static adsorption measurements and computer simulation of nitrogen at 77 K is developed. We obtain the condensation/evaporation pressure as a function of pore size of a cylindrical carbon tube using Gauge Cell Monte Carlo Simulation (Gauge Cell MC). To obtain the analytical form of the relationships mentioned above we use Derjaguin-Broekhoff-deBoer theory. Finally, the pore size distribution (PSD) of the single-walled carbon nanohorns (SWNHs) is determined from a single nitrogen adsorption isotherm measured at 77 K. We neglect the conical part of an isolated SWNH tube and assume a structureless wall of a carbon nanotube. We find that the distribution of SWNH sizes is broad (internal pore radii varied in the range 1.0-3.6 nm with the maximum at 1.3 nm). Our method can be used for the determination of the pore size distribution of the other tubular carbon materials, like, for example, multiwalled or double-walled carbon nanotubes. Besides the applicable aspect of the current work the deep insight into the problem of capillary condensation/evaporation in confined carbon cylindrical geometry is presented. As a result, the critical pore radius in structureless single-walled carbon tubes is determined as being equal to three nitrogen collision diameters. Below that size the adsorption-desorption isotherm is reversible (i.e., supercritical in nature). We show that the classical static adsorption measurements combined with the proper modeling of the capillary condensation/evaporation phenomena is a powerful method that can be applied for the determination of the distribution of nanotube sizes.     

Poly(ethylene oxide)-based surfactants as templates for the synthesis of mesoporous silica materials

Various mesoporous silica solids were prepared by using poly(ethylene oxide)-based surfactants as templates in a neutral, fluoride, or moderately acidic medium, and their properties examined by different physical techniques. Precipitation in an acid or neutral medium provided materials of pore size in between those of micropores and mesopores irrespective of the molecular size of the surfactant. On the other hand, syntheses in a fluoride-containing medium yielded mesoporous materials with pore diameters over the range 36-84 A that increased with increasing surfactant size. All materials possessed specific surface areas above 650 m(2)g(-1) and high pore volumes-particularly those obtained in a fluorinated medium. The conditions used in the syntheses and the fact that all produced highly disordered porous materials suggest that their mechanism of formation is essentially of the N(0)I(0) neutral type. The materials obtained in the presence of fluoride ion, which promote the condensation of siliceous species, retain greater amounts of surfactant and exhibit increased cross-linking and decreased particle sizes, which results in textural mesoporosity.     

Nucleation in hydrophobic cylindrical pores: a lattice model

We consider the nucleation process associated with capillary condensation of a vapor in a hydrophobic cylindrical pore (capillary evaporation). The liquid-vapor transition is described within the framework of a simple lattice model. The phase properties are characterized both at the mean-field level and with Monte Carlo simulations. The nucleation process for the liquid to vapor transition is then specifically considered. Using umbrella sampling techniques, we show that nucleation occurs through the condensation of an asymmetric vapor bubble at the pore surface. Even for highly confined systems, good agreement is found with macroscopic considerations based on classical nucleation theory. The results are discussed in the context of recent experimental work on the extrusion of water in hydrophobic pores.     

Using Polymer-Colloid Complexes for Obtaining Mesoporous Aluminium Oxide by the Template Sol-Gel Method

The paper reports the results of using polymer-colloid complexes in solutions in order to control textural properties of mesoporous aluminium oxide in the sol–gel synthesis process. Polyethyleneimine, cetyltrimethylammonium chloride, as well as a polymer-colloid complex formed by their interaction in the solution were used as pore-forming templates. The mesoporous aluminium oxides synthesized in this work had a narrow pore size distribution and a large surface area. The application of different templates made it possible to affect the mechanism of supramolecular self–assembly of materials, namely by controlling the pore sizes. When the polymer-colloid complex was used as the template for the formation of aluminium oxide nanostructures, 6 nm cylindrical pores were formed, while using individual templates led to the formation of 8–13 nm mesopores. Identifying the formation mechanism of a certain pore type will make it possible to use these materials in specific reactions.     

Grand canonical Monte Carlo simulation of argon adsorption at the surface of silica nanopores: effect of pore size, pore morphology, and surface roughness

Argon adsorption (77 K) in atomistic silica nanopores of various sizes and shapes has been studied by means of grand canonical Monte Carlo simulations (GCMC). We discuss the effects of confinement (pore size), pore morphology (ellipsoidal, hexagonal, constricted pore), and surface texture (rough/smooth) on the thickness variation of the adsorbed film with pressure onto the disordered inner surface of porous materials (usually called t-plot or t-curve). We show that no confinement effect occurs when the diameter of the regular cylindrical pore is larger than 10 nm. For pores smaller than 6 nm, we find that the film thickness increases as the pore size decreases. We show that the adsorption isotherm in the rough pore can be described as the sum of an adsorbed amount similar to that found for a smooth pore (of the same radius) and a constant contribution due to atoms "trapped" in the infractuosities of the rough surface which act as a microporous texture. Simulation snapshots for Ar adsorption in hexagonal and ellipsoidal smooth pores indicate that at low pressures the gas/adsorbate interface retains memory of the pore shape and becomes cylindrical prior to the capillary condensation of the fluid in the pore. The film thickness in the hexagonal pore is close to that obtained for a cylindrical pore having a similar dimension. By contrast, we find that the film thickness for an ellipsoidal pore is always larger than that for an equivalent cylindrical pore (having the same length and volume but a circular section). We show that this effect strengthens as the pore size decreases and/or the pore asymmetry increases. Ar adsorption in a cylindrical constricted pore shows that the presence of the narrower part considerably modifies the adsorption mechanism. Finally, we report GCMC simulations of Ar adsorption (77 K) on a plane silica reference substrate for different intermolecular potentials. We discuss the effect of the interaction on the shape of the adsorption isotherm and compare our results with experiments.