Monte-Carlo multiscale simulation study of argon adsorption/desorption hysteresis in mesoporous heterogeneous tubular pores like MCM-41 or oxidized porous silicon

Development and Application of an Advanced Percolation Model for Pore Network Characterization by Physical Adsorption

Physical adsorption is one of the most widely used techniques to characterize porous materials because it is reliable and able to assess micro- and mesopores within one approach. Challenges and open questions persist in characterizing disordered and hierarchically structured porous materials. This study introduces a pore network model aimed at enhancing the textural characterization of nanoporous materials. The model, based on percolation theory on a finite-sized Bethe lattice, includes all mechanisms known to contribute to adsorption hysteresis in mesoporous pore networks. The model accounts for delayed and initiated condensation during adsorption as well as equilibrium evaporation, pore blocking, and cavitation during desorption. Coupled with dedicated nonlocal-density functional theory kernels, the proposed method provides a unified framework for modeling the entire experimental adsorption-desorption isotherm, including desorption hysteresis scans. The applicability of the method is demonstrated on a selected set of nanoporous silica materials exhibiting distinct types of hysteresis loops (types H1, H2a, H1/H2a, and H5), including ordered mesoporous silica networks (KIT-6 and SBA-15/MCM-41 hybrid silica with plugged pores) and disordered mesoporous silica networks (hierarchical meso-macroporous monolith and porous Vycor glass). For all materials, a good correlation is found between calculated and experimental primary isotherms as well as desorption scans. The model allows us to determine key pore network characteristics such as pore connectivity and pore size distributions as well as a parameter correlated with the impact of pore network disorder on the adsorption behavior. The versatility and enriched textural insights provided by the proposed novel network model allow for a comprehensive characterization previously inaccessible and hence will contribute to further advancement in the textural characterization of novel nanoporous materials. It has the potential to provide important guidance for the design and selection of porous materials for optimizing various applications, including separation processes such as chromatography, heterogeneous catalysis and gas and energy storage.

Artifacts and misinterpretations in gas physisorption measurements and characterization of porous solids

This contribution provides a critical review of gas physisorption in the textural characterization of porous solids, with the focus on the artifacts in experimental data that lead to serious misinterpretation of the results derived from the analysis of adsorption isotherms. Apart from the problems related to the determination and interpretation of the BET area, we paid particular attention to the issues associated with the determination of pore size distribution; for example, the choice of the correct branch of the hysteresis loop and the network effects. Pitfalls in the analyses using either the classical macroscopic or the advanced microscopic (DFT, GCMC) methodology are addressed. The ultimate aim is to provide guidance for proper calculations and correct interpretation of physisorption data.

Pore Size Distribution Analysis Using Developing Hysteresis of Nitrogen in the Cylindrical Pores of Silica

A new method of mesopore size analysis was developed for the cylindrical pores of silica using a developing hysteresis of nitrogen measured at liquid nitrogen temperature. The method is based on the semimacroscopic approach of a modified Bonnet-Wolf model that deals with the grand potential of a vapor bubble in the cylindrical pore. It is capable of assessing correctly the pore structures of mesoporous materials with cylindrical pores using the narrow hysteresis loop of the developing hysteresis. When the mesoporous materials possess cylindrical pores of minor imperfections, two pore size distributions (PSDs) from the adsorption and desorption branches overlap. On the other hand, for cylindrical pores with an enhanced amplitude of pore corrugations and/or constrictions, PSD from the desorption branch is shifted to smaller pore sizes compared to the one from the adsorption branch, and at the same time, both the PSDs evaluated are shifted to lower pore sizes compared to the actual ones. The actual PSD can be assessed from the reversible isotherm measured at a hysteresis critical temperature. In principle, the present method may enable the determination of the PSDs from the adsorption hysteresis measured at any given temperature for mesoporous materials with cylindrical pores.

Machine-Learned Free Energy Surfaces for Capillary Condensation and Evaporation in Mesopores

Using molecular simulations, we study the processes of capillary condensation and capillary evaporation in model mesopores. To determine the phase transition pathway, as well as the corresponding free energy profile, we carry out enhanced sampling molecular simulations using entropy as a reaction coordinate to map the onset of order during the condensation process and of disorder during the evaporation process. The structural analysis shows the role played by intermediate states, characterized by the onset of capillary liquid bridges and bubbles. We also analyze the dependence of the free energy barrier on the pore width. Furthermore, we propose a method to build a machine learning model for the prediction of the free energy surfaces underlying capillary phase transition processes in mesopores.

Revisiting the Nature of Adsorption and Desorption Branches: Temperature Dependence of Adsorption Hysteresis in Ordered Mesoporous Silica

To gain a deeper understanding as to the nature of the adsorption hysteresis due to capillary condensation of nitrogen in ordered mesoporous silicas, we calculated the temperature dependences of the activated condensation, equilibrium transition, and activated desorption pressures for nitrogen in spherical and cylindrical silica pores with several different pore sizes on the basis of semimacroscopic continuum models. The results clearly indicate that the models capture the exact nature of capillary condensation and evaporation phenomena of a fluid in cagelike and cylindrical mesopores. The temperature dependences of the adsorption hysteresis of nitrogen measured confirm previous theoretical predictions for cylindrical pores: for the ordered mesoporous silicas with cylindrical mesopores at least greater than ∼7 nm in diameter, the capillary condensation takes place via a nucleation process followed by a growth process of a bridging meniscus at pressures higher than the equilibrium transition, while the capillary evaporation takes place via a receding meniscus from pore ends at the equilibrium. For SBA-15 and MCM-41 with smaller mesopore sizes, on the other hand, the capillary condensation takes place close to the equilibrium transition pressures, while the capillary evaporation takes place at pressures lower than the equilibrium, owing to single pore blocking due to corrugation of the cylindrical pores. We discuss the effect of curvature on surface tension in capillary condensation, as well as the relation between a change in the mechanisms of adsorption and desorption and the pore corrugation in the cylindrical pores.

Impact of Geometrical Disorder on Phase Equilibria of Fluids and Solids Confined in Mesoporous Materials

Porous solids used in practical applications often possess structural disorder over broad length scales. This disorder strongly affects different properties of the substances confined in their pore spaces. Quantifying structural disorder and correlating it with the physical properties of confined matter is thus a necessary step toward the rational use of porous solids in practical applications and process optimization. The present work focuses on recent advances made in the understanding of correlations between the phase state and geometric disorder in nanoporous solids. We overview the recently developed statistical theory for phase transitions in a minimalistic model of disordered pore networks: linear chains of pores with statistical disorder. By correlating its predictions with various experimental observations, we show that this model gives notable insight into collective phenomena in phase-transition processes in disordered materials and is capable of explaining self-consistently the majority of the experimental results obtained for gas-liquid and solid-liquid equilibria in mesoporous solids. The potentials of the theory for improving the gas sorption and thermoporometry characterization of porous materials are discussed.

Nucleation of Capillary Bridges and Bubbles in Nanoconfined CO2

Using molecular simulation, we examine the capillary condensation and the capillary evaporation of CO₂ in cylindrical nanopores. More specifically, we employ the recently developed μV T-S method to determine the microscopic mechanism associated with these processes and the corresponding free energy profiles. We calculate the free energy barrier for capillary condensation and identify that the key step consists in the nucleation of a liquid bridge of a critical size. Similarly, the free energy maximum found for the capillary evaporation process is found to correspond to the nucleation of a vapor bubble of a critical size. In addition, we assess the impact of the strength of the wall-fluid on the height of the free energy barrier and on the critical size of liquid bridges (condensation process) and vapor bubbles (evaporation process). We observe that the height of the free energy barrier increases with the strength of the wall-fluid interactions. Finally, we build a theoretical model, based on capillary theory, to rationalize our findings. In particular, the simulation results reveal a linear scaling of the free energy barrier with the critical size, in excellent agreement with the theoretical predictions.

On Condensation and Evaporation Mechanisms in Disordered Porous Materials

Sorption isotherm measurement is a standard method for characterizing porous materials. However, such isotherms are generally hysteretic, differing between condensation and evaporation. Quantitative measurement of pore diameter distributions requires proper identification of the mechanisms at play, a topic which has been and remains the subject of intensive studies. In this paper, we compare high-precision measurements of condensation and evaporation of helium in Vycor, a prototypical disordered porous glass, to a model incorporating mechanisms on the single pore level through a semimacroscopic description and collective effects through lattice simulations. Our experiment determines both the average of the fluid density through volumetric measurements and its spatial fluctuations through light scattering. We show that the model consistently accounts for the temperature dependence of the isotherm shape and of the optical signal over a wide temperature range as well as for the existence of thermally activated relaxation effects. This demonstrates that the evaporation mechanism evolves from pure invasion percolation from the sample's surfaces at the lowest temperature to percolation from bulk cavitated sites at larger temperatures. The model also shows that the experimental lack of optical signals during condensation does not imply that condensation is unaffected by network effects. In fact, these effects are strong enough to make most pores to fill at their equilibrium pressure, a situation deeply contrasting the behavior for isolated pores. This implies that, for disordered porous materials, the classical Barrett-Joyner-Halenda approach, when applied to the condensation branch using an extended version of the Kelvin equation, should properly measure the true pore diameter distribution. Our experimental results support this conclusion.

Effects of carbon coating and pore corrugation on capillary condensation of nitrogen in SBA-15 mesoporous silica

To examine the origin of an ink-bottle-like structure in SBA-15 formed by carbon coating and the effects of pore corrugation on capillary condensation and evaporation of a vapor in the cylindrical pores, we measured the adsorption isotherms of nitrogen at 77 K on 10 kinds of SBA-15 samples before and after a carbon coating process by the exposure to acetylene at 1073 K, as well as desorption scanning curves and subloops on the untreated samples. These SBA-15 samples were synthesized under the different conditions of initial SiO2/P123 ratio and hydrothermal treatment. SBA-15 with relatively large microporosity tends to form easily constrictions inside the main channels by the carbon coating. This strongly suggests that the rough pore walls of SBA-15 may induce the incomplete wetting of carbon layers on the pore walls to form the constrictions inside the cylindrical pores. A comparison of two subloops implies that the pores of SBA-15 synthesized with a SiO2/P123 ratio of 75 consist of an assembly of connecting domains of different diameters; that is, the pores are highly corrugated. For SBA-15 synthesized with a SiO2/P123 ratio of 60, the amplitude of the pore corrugation is significantly decreased by the prolonged hydrothermal treatment at 373 K. On the other hand, for SBA-15 synthesized with a SiO2/P123 ratio of 45, the amplitude of the corrugation is negligibly small, although the cylindrical pores are interconnected through narrow necks with each other. It is found that the smaller the amplitude of the pore corrugation, the smaller the width of the hysteresis loop.

Capillary condensation in one-dimensional irregular confinement

A lattice-gas model with heterogeneity is developed for the description of fluid condensation in finite sized one-dimensional pores of arbitrary shape. Mapping to the random-field Ising model allows an exact solution of the model to be obtained at zero-temperature, reproducing the experimentally observed dependence of the amount of fluid adsorbed in the pore on external pressure. It is demonstrated that the disorder controls the sorption for long pores and can result in H2-type hysteresis. Finite-temperature Metropolis dynamics simulations support analytical findings in the limit of low temperatures. The proposed framework is viewed as a fundamental building block of the theory of capillary condensation necessary for reliable structural analysis of complex porous media from adsorption-desorption data.

On the irreversibility of the adsorption isotherm in a closed-end pore

We present a simulation study of argon adsorption in a closed-end mesopore of uniform diameter in order to investigate the occurrence of hysteresis and propose two principal reasons for its existence: the variation in the shape and radius of curvature of the meniscus and the change in the packing of adsorbate during adsorption and desorption. This interpretation differs from classical theories that neglect both of these factors, and therefore find that adsorption-desorption in a closed-end pore is reversible. A detailed simulation study of the effects of temperature on the microscopic behavior of the adsorbate supports the interpretation proposed here.

Controlling the role of nanopore morphology in capillary condensation

The effect of pore morphology on capillary condensation and evaporation in nanoporous silicon is studied experimentally. A variety of cooperative and local effects are observed in tailored nanopores with well-defined regions by directly probing gas adsorption in each region using optical interferometry. All observations are ascribed to the ability of the nanopore region to access the gas reservoir directly and the nucleation of liquid bridges at local heterogeneities within the nanopore region. These assumptions, consistent with recent simulations, can be extended to any real nanoporous system.

Stability intervals of metastable states in hysteretic systems

Hysteresis in disordered systems originates in a plethora of metastable states. Previous works focused on their distribution inside the hysteresis. In contrast, an analysis of their range of metastability is proposed. This model, designed to catch the main features of fluid adsorption in porous materials, shows strong evidence that, in the thermodynamic limit, despite that metastable states of finite range can be found, they are exponentially dominated by those infinitely localized states.

Capillary condensation of adsorbates in porous materials

Hysteresis in capillary condensation is important for the fundamental study and application of porous materials, and yet experiments on porous materials are sometimes difficult to interpret because of the many interactions and complex solid structures involved in the condensation and evaporation processes. Here we make an overview of the significant progress in understanding capillary condensation and hysteresis phenomena in mesopores that have followed from experiment and simulation applied to highly ordered mesoporous materials such as MCM-41 and SBA-15 over the last few decades.

On the cavitation and pore blocking in slit-shaped ink-bottle pores

We present GCMC simulations of argon adsorption in slit pores of different channel geometry. We show that the isotherm for an ink-bottle pore can be reconstructed as a linear combination of the local isotherms of appropriately chosen independent unit cells. Second, depending on the system parameters and operating conditions, the phenomena of cavitation and pore blocking can occur for a given configuration of the ink-bottle pore by varying the geometrical aspect ratio. Although it has been argued in the literature that the geometrical aspects of the system govern the evaporation mechanism (either cavitation or pore blocking), we here put forward an argument that the local compressibility in different parts of the ink-bottle pore is the deciding factor for evaporation. When the fluid in the small neck is strongly bound, cavitation is the governing process, and molecules in the cavity evaporate to the surrounding bulk gas via a mass transfer mechanism through the pore neck. When the pore neck is sufficiently large, the system of neck and cavity evaporates at the same pressure, which is a consequence of the comparable compressibility between the fluid in the neck and that in the cavity. This suggests that local compressibility is the measure of cohesiveness of the fluid prior to evaporation. One consequence that we derive from the analysis of isotherms of a number of connected pores is that by analyzing the adsorption branch or the desorption branch of an experimental isotherm may not lead to the correct pore sizes and the correct pore volume distribution.

Cavitation in metastable fluids confined to linear mesopores

We study the adsorption process of nitrogen (at 77.4 and 51.3 K) and argon (at 60 K) in porous silicon duplex layers, Si/A/B and Si/B/A, where the pores of A are on average narrower than the pores of B. We compare the experimental isotherms to that calculated from elemental isotherms measured in layers A and B supported by or detached from the silicon substrate. This allows us to confirm our previous studies which show that the relaxation of the substrate constraint modifies the adsorption strains and leads to a decrease of the adsorbed amount before condensation and consequently increases the condensation pressure. In the so-called ink-bottle Si/B/A configuration, layer B empties while layer A remains filled which proves that layer B empties via cavitation. The vapor pressure at which cavitation occurs in layer B in Si/B/A configuration is close to the pressure at which the same layer empties when it is in direct contact with the gas reservoir (Si/A/B configuration) which indicates that layer B contains all the ingredients necessary for cavitation to occur. The absolute value of the liquid pressure at which cavitation occurs is much lower than the value predicted by the theory of homogeneous nucleation. Nucleation of gas bubbles thus takes place on the surface of the pore walls. This is the crucial point of the paper. A receding meniscus with a contact angle lower than π/2 inside a pore and a gas bubble with a contact angle higher than π/2 are thus mutually exclusive. A receding meniscus cannot enter a pore. This has nothing to do with a pore-blocking effect; this is related to the physical parameters which define the contact angle inside the pores, that is, the surface energies at the solid-liquid, solid-vapor, and liquid-vapor interfaces. For argon at 60 K in the Si/B/A duplex layer, cavitation in layer B activates the emptying of a fraction of pores of layer A which constitutes a direct observation of metastable states.

Numerical characterization of the density of metastable states within the hysteresis loop in disordered systems

An improved approach is proposed to analyze the density of metastable states within any hysteresis loop, such as those observed in magnetic materials or for adsorption in porous materials. Except for a few analytically tractable models, most calculations have to be performed numerically on finite systems. The main points to be addressed thus concern the average over various material samples (the so-called realizations of the disorder), and the finite size analysis to estimate the thermodynamic limit. As an improvement of previously existing methods, it is proposed to introduce the Fourier transform of the density of metastable states (characteristic function). Its logarithm is shown to be additive and can straightforwardly be averaged over disorder. This procedure leads to a new definition of the complexity in finite size, giving the usual quenched complexity in the thermodynamic limit, while being better suited to performing finite size analysis. The calculations are illustrated on a molecular simulation based model for a simple fluid adsorbed in heterogeneous siliceous tubular pores mimicking mesoporous materials like MCM-41 or porous silicon. This approach is expected to be of general interest for hysteresis phenomena, including magnetic materials.

Influence of elastic strains on the adsorption process in porous materials: an experimental approach

The experimental results presented in this paper show the influence of the elastic deformation of porous solids on the adsorption process. With p(+)-type porous silicon formed on highly boron doped (100) Si single crystal, we can make identical porous layers, either supported by or detached from the substrate. The pores are perpendicular to the substrate. The adsorption isotherms corresponding to these two layers are distinct. In the region preceding capillary condensation, the adsorbed amount is lower for the membrane than for the supported layer and the hysteresis loop is observed at higher pressure. We attribute this phenomenon to different elastic strains undergone by the two layers during the adsorption process. For the supported layer, the planes perpendicular to the substrate are constrained to have the same interatomic spacing as that of the substrate so that the elastic deformation is unilateral, at an atomic scale, and along the pore axis. When the substrate is removed, tridimensional deformations occur and the porous system can find a new configuration for the solid atoms which decreases the free energy of the system adsorbate-solid. This results in a decrease of the adsorbed amount and in an increase of the condensation pressure. The isotherms for the supported porous layers shift toward that of the membrane when the layer thickness is increased from 30 to 100 mum. This is due to the relaxation of the stress exerted by the substrate as a result of the breaking of Si-Si bonds at the interface between the substrate and the porous layer. The membrane is the relaxed state of the supported layer.

Hysteresis critical point of nitrogen in porous glass: occurrence of sample spanning transition in capillary condensation

To examine the mechanisms for capillary condensation and for capillary evaporation in porous glass, we measured the hysteresis critical points and desorption scanning curves of nitrogen in four kinds of porous glasses with different pore sizes (Vycor, CPG75A, CPG120A, and CPG170A). The shapes of the hysteresis loop in the adsorption isotherm of nitrogen for the Vycor and the CPG75A changed with temperature, whereas those for the CPG120A and the CPG170A remained almost unchanged with temperature. The hysteresis critical points for the Vycor and the CPG75A fell on the common line observed previously for ordered mesoporous silicas. On the other hand, the hysteresis critical points for the CPG120A and the CPG170A deviated appreciably from the common line. This strongly suggests that capillary evaporation of nitrogen in the interconnected and disordered pores of both the Vycor and the CPG75A follows a cavitation process at least in the vicinity of their hysteresis critical temperatures in the same way as that in the cagelike pores of the ordered silicas, whereas the hysteresis critical points in the CPG120A and the CPG170A have origin different from that in the cagelike pores. The desorption scanning curves for the CPG75A indicated the nonindependence of the porous domains. On the other hand, for both the CPG120A and the CPG170A, we obtained the scanning curves that are expected from the independent domain theory. All these results suggest that sample spanning transitions in capillary condensation and evaporation take place inside the interconnected pores of both the CPG120A and the CPG170A.