Capillary Condensation Pressure Research Articles

Improving the estimation accuracy of confined vapor-liquid equilibria by fine-tuning the pure component parameter in the PC-SAFT equation of state.

We propose a thermodynamic model that combines the Young-Laplace equation and perturbed chain-statistical associating fluid theory (PC-SAFT) equation of state to estimate capillary condensation pressure in microporous and mesoporous sorbents. We adjust the PC-SAFT dispersion-energy parameter when the pore size becomes comparable to the molecular dimension. This modelling framework is applied to diverse systems containing associating and non-associating gases, various sorbents, and a wide range of temperatures. Our simulation results show that under extreme confinement, a higher value of the dispersion-energy parameter (ε) is required. Furthermore, using the experimental saturation pressure data for 18 different associating and non-associating confined fluids, we find that the shift in the PC-SAFT dispersion energy correlates with the ratio of the sorbent mean pore size to the PC-SAFT segment size (rp/σ). By fitting to the capillary condensation data, the relative deviation between the confined and bulk PC-SAFT dispersion energy parameter is only 0.1% at rp/σ = 15; however, this deviation starts to increase exponentially as rp/σ decreases. For a sorbent with large pores, when rp/σ > 15, the capillary condensation pressure results from our model are similar to the predictions from the Kelvin equation. Using a dataset containing 235 saturation pressure data points composed of 18 pure gases and 4 binary mixtures, the overall AARD% from our model is 12.26%, which verifies the good accuracy of our model. Because the mean sorbent pore radius (rp), the PC-SAFT energy parameter (ε), and segment size (σ) are known a priori, our model estimates the corrected energy parameter for small pores and, thus, extends its applicability.

Free-Energy-Averaged Potentials for Adsorption in Heterogeneous Slit Pores Using PC-SAFT Classical Density Functional Theory.

This study analyzes the adsorption behavior in two-dimensional heterogeneous slit pores using nonlocal density functional theory based on the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state. Both chemical heterogeneity and surface roughness on small atomistic scales are investigated. The solid structure is considered as individual solid interaction sites whereby chemical heterogeneity is introduced through the presence of different solid-fluid sites and molecular roughness by varying the position of the interaction sites in the first solid layers. The effect of both forms of heterogeneity on the adsorption behavior is assessed individually. Effective one-dimensional solid-fluid potentials provide a way to reduce the dimensionality and computational demand of the density functional theory (DFT) calculations. We determine one-dimensional free-energy-averaged (FEA) solid-fluid potentials of methane and n-butane in the low-density limit for solid systems with molecular roughness and chemical heterogeneity. Using this effective one-dimensional solid-fluid potential at any density, we find excellent agreement of adsorption isotherms for both solid descriptions in systems with homogeneous slit pores. Subcritical adsorption isotherms of n-butane in slit pores with surface roughness show deviations at higher pressures due to the formation of fluid layers in the one-dimensional FEA potential. Chemical heterogeneity introduces a shift of the capillary condensation pressure below the saturation pressure of the bulk liquid, which is well described by the free-energy-averaged system.

Capillary condensation and capillary pressure of methane in carbon nanopores: Molecular Dynamics simulations of nanoconfinement effects

Two main groups of thought have emerged over the past decade to improve thermodynamic modeling of capillary condensation of nanoconfined fluids. One approach has been developed on the premise of using shifted critical parameters for the condensed phase to account for the wall-fluid interactions, while the other one considers a pressure difference across the interface between the condensed and the bulk phases. This pressure difference is the capillary pressure that exists across a curved meniscus formed in a capillary pore. For nanoconfinement, a modified Laplace equation has been utilized to calculate the capillary pressure to take into account the confinement effects that become more prominent as the pore size reduces. For small pores of a few nanometers in size, however, the impact of structural forces known as nanoconfinement effects, on the pressure of the confined phase becomes more significant. In this work, we studied the capillary pressure of methane at the capillary condensation point in graphite pores smaller than 7 nm, to verify whether the confined phase can experience a negative pressure at capillary condensation point and whether we can accurately predict this pressure from thermodynamic equations. For this purpose, we used Molecular Dynamics (MD) simulations to investigate the pressure of methane molecules confined in graphite pores of various sizes. Furthermore, normal and tangential pressures of methane in selected pore sizes were obtained during capillary condensation at constant pressure. Our results indicated that for small pores there is a critical size below which capillary condensation did not occur. For larger pores, on the other hand, capillary condensation could be identified with an abrupt drop in the pressure of the confined phase. The capillary pressure attained from the MD simulations were similar to the thermodynamic calculations provided the adsorbed phase was thick enough to screen out wall-fluid interactions.

Comprehensive Modeling of Capillary Condensation in Open-Ended Nanopores: Equilibrium, Metastability, and Spinodal

Capillary condensation within open-ended cylindrical nanopores is modeled thermodynamically and kinetically, in order to provide reasonable estimates of both the equilibrium phase transition pressure and capillary condensation pressure from a metastable state. The thermodynamic model is established by adding to the conventional Derjaguin–Broekhoff–de Boer model the effects of curvature on the fluid–solid interaction potentials and surface tension at the vapor–liquid interface. The equilibrium vapor–liquid phase transition pressure estimated via the thermodynamic model is in close agreement with that from the nonlocal density functional theory. In contrast to the equilibrium phase transition, capillary condensation from a metastable state cannot be estimated by the conventional thermodynamic models, because it inherently includes an activated process. To overcome this challenge, we propose a kinetic model of metastable capillary condensation for the first time. The model is based on the finding in our previous molecular simulation study, that metastable capillary condensation occurs when the corresponding rate constant reaches a critical value. The proposed kinetic model allows us to quantitatively reproduce the experimental capillary condensation pressures over a wide range of temperatures and mesopore sizes without computationally expensive molecular simulations.

CARS detection of liquid-like phase appearance in small mesopores

Nonlinear-optical spectroscopic techniques that employ signals from the molecules located inside nanopores have promising potential for investigations of fluid behavior under nanoconfinement. Here, we apply coherent anti-Stokes spectroscopy to investigate the appearance of a liquid-like phase of carbon dioxide in mesoporous Vycor glass under isothermal compression. The spectra of the Q-branch (1388 cm−1) are registered at −11 °C in a wide pressure range, starting from submonolayer coverage of the pore wall up to the bulk saturation pressure. Results show that a spectral contribution, similar to that of the bulk liquid, appears at relatively low pressure that is several times lower than the capillary-condensation pressure. The Raman shift of the peak is equal to that of the bulk liquid, although the linewidth is somewhat increased. The peak is attributed to the layers adsorbed beyond the monolayer or to small liquid-like clusters appearing in specific areas of the porous network. The spectroscopic approach presented here demonstrates the ability to detect and estimate small amounts of the liquid-like phase and to distinguish it from the layers strongly interacting with the pore surface.

Mechanism of Kinetically Controlled Capillary Condensation in Nanopores: A Combined Experimental and Monte Carlo Approach.

We find the rule of capillary condensation from the metastable state in nanoscale pores based on the transition state theory. The conventional thermodynamic theories cannot achieve it because the metastable capillary condensation inherently includes an activated process. We thus compute argon adsorption isotherms on cylindrical pore models and atomistic silica pore models mimicking the MCM-41 materials by the grand canonical Monte Carlo and the gauge cell Monte Carlo methods and evaluate the rate constant for the capillary condensation by the transition state theory. The results reveal that the rate drastically increases with a small increase in the chemical potential of the system, and the metastable capillary condensation occurs for any mesopores when the rate constant reaches a universal critical value. Furthermore, a careful comparison between experimental adsorption isotherms and the simulated ones on the atomistic silica pore models reveals that the rate constant of the real system also has a universal value. With this finding, we can successfully estimate the experimental capillary condensation pressure over a wide range of temperatures and pore sizes by simply applying the critical rate constant.

Critical energy barrier for capillary condensation in mesopores: Hysteresis and reversibility.

Capillary condensation in the regime of developing hysteresis occurs at a vapor pressure, Pcond, that is less than that of the vapor-like spinodal. This is because the energy barrier for the vapor-liquid transition from a metastable state at Pcond becomes equal to the energy fluctuation of the system; however, a detailed mechanism of the spontaneous transition has not been acquired even through extensive experimental and simulation studies. We therefore construct accurate atomistic silica mesopore models for MCM-41 and perform molecular simulations (gauge cell Monte Carlo and grand canonical Monte Carlo) for argon adsorption on the models at subcritical temperatures. A careful comparison between the simulation and experiment reveals that the energy barrier for the capillary condensation has a critical dimensionless value, Wc (*) = 0.175, which corresponds to the thermal fluctuation of the system and depends neither on the mesopore size nor on the temperature. We show that the critical energy barrier Wc (*) controls the capillary condensation pressure Pcond and also determines a boundary between the reversible condensation/evaporation regime and the developing hysteresis regime.

Theoretically Based Model for Competitive Adsorption of Subcritical Mixtures

Numerous works have been achieved on the prediction of subcritical fluids, but most of them are about gas adsorption at low pressures or low concentrations, where adsorption consists in the formation of one adsorption layer. However, at higher pressures or concentrations, these mixtures can form an important number of adsorbed fluid layers and may exhibit a gas/liquid transition when the capillary condensation pressure is reached. In this work, we derive from the statistical mechanics a model predicting subcritical mixtures in such conditions. In the proposed model, the adsorbed molecules are supposed to be distributed in two distinct adsorption layers. Two parameters per adsorbed species characterize the interactions of the species in the adsorption layers. One additional parameter characterizes the adsorption capacity of the porous structures. To check the consistency of the model, Monte Carlo simulations on pure compound and mixtures adsorption are used as reference data. Our model shows improvements compared to the Ideal Adsorbed Solution Theory, whose adsorption isotherm are modeled with the BET model developed by Gritti et al. [J. Chromatogr. A 2002, 978, 81–107]. Especially, the predictions of the adsorbed phases compositions are in good agreement with molecular simulation results, in both gas and liquid states. The adsorbed amount in the gas phase are also correctly predicted. By introducing an additional empirical parameter which characterizes the liquid adsorption capacity, it is also possible to predict the adsorbed amounts for the liquid phase, providing a consistent approach to model the adsorption of liquid and gas mixtures.

Monte Carlo Simulation on Adsorption Properties of Benzene, Toluene, and p-Xylene in MCM-41

The adsorption properties of benzene, toluene, p-xylene in MCM-41 with heterogeneous and cylindrical pore were studied using grand canonical ensemble Monte Carlo simulation. The simulated isotherms were compared with experimental ones, and the different adsorption behaviors in MCM-41 with pore diameters of 2.2 and 3.2 nm were investigated. The simulated adsorption amounts above the capillary-condensation pressure agreed with the experimental ones. The simulation results showed that most molecular planes were nearly parallel to the pore axis. This orientation was not affected by the molecular position in the pore. The molecular planes were nearly parallel to the pore surface for the adsorbate molecules close to the pore wall, and the molecules in the MCM-41 with the pore diameter of 3.2 nm were ordered along the pore axis.

Adsorption Isoterms and Capillary Condensation in a Nanoslit with Rough Walls: A Density Functional Theory

Adsorption isoterms and capillary condensation in an open slit with walls decorated with arrays of pillars are examined using the density functional theory. Compared with the main substrate, the pillars can have the same or different parameters in the Lennard-Jones interaction potential between them and the fluid in the slit. The roughness of the solid surface, defined as the ratio between the area of the actual surface and the area of the surface free of pillars, is controlled by the height of the pillars. It is shown that the capillary condensation pressure first increases with increasing roughness, passes through a maximum, and then decreases. The amount of adsorbed fluid at constant volume of the slit has, in general, a nonmonotonic dependence on roughness. These features of adsorption and capillary condensation are results of increased surface area and changes in the fluid-solid potential energy due to changes in roughness.

Repeated sorption of water in SBA-15 investigated by means of in situ small-angle x-ray scattering

The effect of repeated cycles of water adsorption/desorption on the structural stability of ordered mesoporous silica SBA-15 is studied by small-angle x-ray scattering (SAXS). In situ sorption measurements are conducted using a custom-built sorption apparatus in connection with a laboratory SAXS setup. Two striking irreversible changes are observed in the sorption isotherms as derived from the integrated SAXS intensity. First, the capillary condensation pressure shifts progressively to lower relative pressure values with increasing number of sorption cycles. This effect is attributed to chemisorption of water at the silica walls, resulting in a change of the fluid–wall interaction. Second, the sorption cycles do not close completely at vanishing vapour pressure, suggesting that progressively more water remains trapped within the porous material after each cycle. This effect is interpreted to be the result of an irreversible collapse of parts of mesopores, originating from pore wall deformation due to the large Laplace pressure of water acting on the pore walls at capillary condensation and capillary evaporation.

10.2478/s11814-009-0183-x

The adsorption properties of carbon tetrachloride, neopentane, and cyclohexane in MCM-41 with heterogeneous and cylindrical pores have been studied by using grand canonical ensemble Monte Carlo simulation. The adsorption isotherm, average potential of adsorbate, isosteric heat of adsorption, and number density of molecules in MCM-41 were calculated. The simulated isotherms were compared with experimental ones. Also, different adsorption behaviors in MCM-41 with pore diameter of 2.2 and 3.2 nm were discussed. The capillary-condensation pressure increased for a given adsorbate with an increase in pore diameter. The average densities of carbon tetrachloride, neopentane, and cyclohexane in the two different pores above the capillary-condensation pressure were smaller than the corresponding liquid densities by about 12%. The adsorbate molecules did not form the multilayer in pore below the capillary-condensation pressure. The number of adsorption layers of molecules was constant in a given pore for the three adsorbates above the capillary-condensation pressure. Carbon tetrachloride molecules in pore were also ordered along the pore axis.

Large-Pore Ethylene-Bridged Periodic Mesoporous Organosilicas with Face-Centered Cubic Structure

Low-temperature (∼15 °C) Pluronic-F127-templated synthesis of periodic mesoporous organosilicas (PMOs) with ethylene (−CH2−CH2−) framework groups and a face-centered cubic structure of spherical mesopores (Fm3m symmetry) was greatly enhanced through the use of judiciously chosen swelling agents and the optimization of synthesis conditions. The resulting materials were characterized by using small-angle X-ray scattering (SAXS), nitrogen adsorption, transmission electron microscopy (TEM), and thermogravimetry. While it was confirmed that 1,3,5-trimethylbenzene is a facile swelling agent for F127-templated ethylene-bridged PMOs with cubic Fm3m structure and our optimization of the synthesis afforded hitherto unreported unit-cell size and pore size, it was also shown that swelling agents predicted to have a higher extent of solubilization in Pluronics provide vast new opportunities. In particular, xylene was found to afford highly ordered PMOs with large unit-cell size and pore diameter, and a wide range of moderately or weakly ordered organosilicas with very large unit-cell parameters (up to ∼50 nm) and pore diameters (up to ∼18 nm). In the case of xylene, the pore size and unit-cell size were tunable by adjusting the amount of inorganic salt (KCl) in the synthesis mixture. The use of toluene allowed us to obtain unprecedented well-ordered PMOs with large unit-cell size (∼42 nm), very large mesopores (nominal diameter 16−17 nm; capillary condensation pressure up to ∼0.87 p/p0), and enhanced primary mesopore volume, and also afforded large-pore (nominal diameter ∼15 nm) PMO in the absence of an inorganic salt. The use of benzene also afforded large-pore PMO under salt-free conditions. Ethylene-bridged PMO was successfully converted to closed-pore ordered mesoporous silica at a temperature as low as 400 °C. The identification of new swelling agents for large-pore ethylene-bridged PMO with spherical mesopores is likely to be extendable on PMOs of other framework compositions and on other related materials.

Effect of Pore Shape on Freezing and Melting Temperatures of Water

To examine the effect of pore shape on freezing and melting temperatures of water, we measured X-ray diffraction patterns from water confined in four kinds of ordered mesoporous silicas with thin carbon films on the pore wall and three kinds of the inverse carbon replicas during freezing and melting processes. The melting temperature of the pore ice revealed a good correlation with the capillary condensation pressure of nitrogen at 77 K, whereas the freezing temperature of the pore water did not. This indicates that the experimental melting point represents an equilibrium solid−liquid phase transition temperature, whereas the experimental freezing point is controlled by a kinetic factor. The curvature effect of the solid−liquid interface adjacent to the pore wall did not appreciably affect the melting behavior of the pore ice. The melting point depression of the pore ice revealed an almost linear relationship with ln(p0/p) of the capillary condensation pressure of nitrogen at 77 K for various pores of different sizes and shapes. This strongly suggests that the reciprocal of ln(p0/p) for capillary condensation of nitrogen at 77 K gives a good measure of pore size, irrespective of the pore shapes, and the melting-point depression of the pore ice is almost proportional to the surface-to-volume ratio (S/V) of the pores confining it.

Monte Carlo simulation on the adsorption properties of carbon tetrachloride, neopentane, and cyclohexane in MCM-41

The adsorption properties of carbon tetrachloride, neopentane, and cyclohexane in MCM-41 with heterogeneous and cylindrical pores have been studied by using grand canonical ensemble Monte Carlo simulation. The adsorption isotherm, average potential of adsorbate, isosteric heat of adsorption, and number density of molecules in MCM-41 were calculated. The simulated isotherms were compared with experimental ones. Also, different adsorption behaviors in MCM-41 with pore diameter of 2.2 and 3.2 nm were discussed. The capillary-condensation pressure increased for a given adsorbate with an increase in pore diameter. The average densities of carbon tetrachloride, neopentane, and cyclohexane in the two different pores above the capillary-condensation pressure were smaller than the corresponding liquid densities by about 12%. The adsorbate molecules did not form the multilayer in pore below the capillary-condensation pressure. The number of adsorption layers of molecules was constant in a given pore for the three adsorbates above the capillary-condensation pressure. Carbon tetrachloride molecules in pore were also ordered along the pore axis.

Nitrogen Adsorption on Silica Surfaces of Nonporous and Mesoporous Materials

We present an accurate comparative analysis of N 2 adsorption at 77 K on nonporous silica and the pore wall surface of MCM-41 materials. The analysis shows that in the low-pressure region of N 2 adsorption obeys a peculiar mechanism governed by short-ranged forces, which makes the surface curvature effect on the N 2 adsorption in mesopores nearly negligible. We used this observation to define more exactly compared to the BET technique the specific surface area of the reference adsorption isotherm on nonporous silica basing on XRD data and linear sections of t-plots. Calculation of the capillary evaporation and condensation pressures seems to confirm our previous finding that the capillary condensation pressure corresponds to the equilibrium transition rather than spinodal condensation at least for pore sizes less than 7 nm. It allowed us to provide more reliable pore size distribution (PSD) analysis of mesoporous silica materials. For example, the PSDs of MCM-41 samples do not show artificial peaks in the micropore range that we obtained in our earlier publications.

Fundamental studies of gas sorption within mesopores situated amidst an inter-connected, irregular network

There has been little, or no, direct testing of theories of gas sorption within particular pores situated amidst a highly inter-connected pore network. The concept of thermodynamically independent pores within networks has also been challenged. In this work, a novel integrated nitrogen sorption and mercury porosimetry technique has been used to deconvolve the condensation and evaporation processes within a specific subset of pores contained within a larger, irregular network. The sizes and geometry of these pores were obtained completely independently of gas sorption, using mercury porosimetry and NMR cryoporometry, respectively. Hence, various theories of capillary condensation, such as the Kelvin equation, the Broeckhoff-de Boer method, Saam-Cole theory, and NLDFT could be directly tested, and the potential influence of any collective network phenomena detected. It was found that, even for a shielded pore, the Cohan equation for a cylindrical meniscus gave rise to the best prediction for the relative pressure of capillary condensation, once the effects of surface chemical heterogeneity on multi-layer build-up had been taken into account. The results were also found to be incompatible with the presence of particular collective adsorption effects, such as advanced condensation.

Capillary Condensation of Nitrogen in Ordered Mesoporous Silica with Bicontinuous Gyroid Structure

To know the accurate relationship between pore size and the pressure of capillary condensation or evaporation and also to elucidate the pore-connectivity effects on adsorption hysteresis and pore criticality for the ordered mesoporous silicas (MCM-48 and KIT-6) with bicontinuous gyroid structure, we carried out X-ray structural study of MCM-48 and KIT-6 and measurements of the adsorption−desorption isotherms of nitrogen in comparison with MCM-41 and SBA-15 with unconnected cylindrical pores. The results clearly show that the hydraulic diameter describes incorrectly the mean pore size of these materials expressing the strength of confinement under the assumption of cylindrical pores with no networking effects. In addition, the average pore diameter at the middle point of pore intersections still underestimates the strength of confinement compared to the unconnected cylindrical pores of MCM-41 and SBA-15. Despite a large difference in porous structure, the shape and thermal behavior of the adsorption hysteresis for KIT-6, as well as the sorption scanning behavior, were indistinguishable from those for SBA-15, indicating that interconnections among pores of almost the same size do not have a significant effect on the adsorption hysteresis and pore criticality.

Improvement of the Kruk−Jaroniec−Sayari Method for Pore Size Analysis of Ordered Silicas with Cylindrical Mesopores

In this work, the X-ray diffraction structure modeling was employed for analysis of hexagonally ordered large-pore silicas, SBA-15, to determine their pore width independently of adsorption measurements. Nitrogen adsorption isotherms were used to evaluate the relative pressure of capillary condensation in cylindrical mesopores of these materials. This approach allowed us to extend the original Kruk-Jaroniec-Sayari (KJS) relation (Langmuir 1997, 13, 6267) between the pore width and capillary condensation pressure up to 10 nm instead of previously established range from 2 to 6.5 nm for a series of MCM-41 and to improve the KJS pore size analysis of large pore silicas.

Physisorbed films in periodic mesoporous silica studied byin situsynchrotron small-angle diffraction

Adsorption and capillary condensation of an organic fluid in a periodic mesoporous silica (SBA-15) are studied by in situ synchrotron diffraction. Powder diffraction patterns resulting from the two-dimensional hexagonal packing of the cylindrical pores of SBA-15 are collected as a function of vapor pressure during continuous adsorption and desorption of the fluid (perfluoropentane ${\mathrm{C}}_{5}{\mathrm{F}}_{12}$), using a specially designed sorption cell. Seven diffraction peaks with systematic changes of the intensity are resolved as the adsorbed film thickness increases along the adsorption isotherm. The integrated intensities of the diffraction peaks are analyzed with a structural model involving four levels of electron density (dense silica matrix, microporous corona around the pores, adsorbed film, and core space of the pores). The model provides quantitative information about the structure of the evacuated specimen, the filling of the corona, and the growing thickness of the liquid film with increasing pressure. A very good fit of the data by this model is found for relative pressures up to $p∕{p}_{0}\ensuremath{\approx}0.5$, but the fit of the high-indexed diffraction peaks becomes poor close to the capillary condensation pressure $(p∕{p}_{0}\ensuremath{\approx}0.68)$. Tentatively, this fact may be attributed to a deviation of the liquid film structure from the simple flat geometry close to the phase transformation, presumably caused by density fluctuations.