Adsorption∕desorption hysteresis of simple fluids confined in realistic heterogeneous silica mesopores of micrometric length: A new analysis exploiting a multiscale Monte Carlo approach

Hysteresis Elimination for an Anisotropic Liquid-Crystal Model via Molecule Design and Replica-Exchange Optimization

The phenomenon of hysteresis in simulations, in which a system's current state is correlated to previous states and inhibits the transition to a more stable phase, may often lead to misleading results in physical chemistry. In this study, in addition to the replica exchange method (REM), a novel approach was taken by combining an evolution strategy based on the evolutionary principles of nature to predict phase transitions for the Hess-Su liquid-crystal model. In this model, an anisotropy term is added to the simple 6-12 Lennard-Jones model to intuitively reproduce the behavior of liquid crystals. We first applied the pressure-temperature REM to the Hess-Su model and optimized the replica spacing for the energy distribution to gain the maximum advantage from the REM. We then used the same approach as for the Hamiltonian REM, seeking to optimize the replica spacing in the same way. Based on both results, we attempted to predict this coarse-grained liquid-crystal model's exact phase transition point. In the Hamiltonian REM, replicas were prepared with different molecular aspect ratios corresponding to the values of the anisotropy terms in the potential function. The Hess-Su liquid-crystal model, which undergoes a direct transition from the nematic to the solid phase without going through a smectic phase, is a challenging research target for understanding phase transitions. Despite the tremendous computational difficulty in overcoming the strong hysteresis present in this system, our method could predict the phase transition point clearly and significantly reduce the extent of hysteresis. Our approach is beneficial when simulating more complex systems and, above all, shows great potential for more accurate and efficient phase transition predictions in the field of molecular simulation in the future.

Gas Sorption Characterization of Porous Materials Employing a Statistical Theory for Bethe Lattices

In the present work, a recently developed statistical theory for adsorption and desorption processes in mesoporous solids, modeled by random Bethe lattices, has been applied to obtain pore size distributions and interpore connectivity from sorption isotherms in real random porous materials, employing a robust and validated methodology. Using the experimental adsorption-desorption N2 isotherms at 77.4 K on Vycor glass, a porous material with random pore structure, we demonstrate the solution of the inverse problem resulting in extracted pore size distribution and interpore connectivity, notably different from the predictions of earlier theories. The results presented are corroborated by the analysis of 3D digital images of reconstructed Vycor porous glass, showing excellent agreement between the predictions of geometric analysis and the new statistical theory.

A New Statistical Theory for Constructing Sorption Isotherms in Mesoporous Structures Represented by Bethe Lattices

In the present work, a new statistical theory is developed to describe adsorption and desorption in mesoporous materials (pore sizes ranging from 2 to 50 nm) represented by pore networks in the form of Bethe lattices. The new theory is an extension of a previous theory applied for Statistically Disordered Chain Model (SDCM) structures and incorporates the cooperative effects emerging during phase transitions in pore networks. The theory is validated against simulations and algorithmic models that describe sorption of lattice and real fluids in Bethe lattices. It is seen that the pore network coordination number, or pore connectivity, z, has a significant impact on two important processes observed in pore networks: pore assisting condensation during adsorption and evaporation by percolation during desorption. The inclusion of pore connectivity in the earlier developed framework accounting for cooperativity effects is an important step, rendering the existing models to mimic fluid behavior in real materials more accurately. Hence, the new theory inherently contains all essential elements that may offer the extraction of more reliable pore size distributions utilizing both the adsorption and desorption branches of the isotherm.

Evaporation Process in Porous Silicon: Cavitation vs Pore Blocking

We measured sorption isotherms for helium and nitrogen in wide temperature ranges and for a series of porous silicon samples, both native samples and samples with reduced pore mouth, so that the pores have an ink-bottle shape. Combining volumetric measurements and sensitive optical techniques, we show that, at a high temperature, homogeneous cavitation is the relevant evaporation mechanism for all samples. At a low temperature, the evaporation is controlled by meniscus recession, the detailed mechanism being dependent on the pore length and mouth reduction. Native samples and samples with ink-bottle pores shorter than 1 μm behave as an array of independent pores. In contrast, samples with long ink-bottle pores exhibit long-range correlations between pores. In this latter case, evaporation takes place by a collective percolation process and not by heterogeneous cavitation as previously proposed. The variety of evaporation mechanisms points to porous silicon being an anisotropic three-dimensional pore network rather than an array of straight independent 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.

Direct Observation of Homogeneous Cavitation in Nanopores

We report on the evaporation of hexane from porous alumina and silicon membranes. These membranes contain billions of independent nanopores tailored to an ink-bottle shape, where a cavity several tens of nanometers in diameter is separated from the bulk vapor by a constriction. For alumina membranes with narrow enough constrictions, we demonstrate that cavity evaporation proceeds by cavitation. Measurements of the pressure dependence of the cavitation rate follow the predictions of the bulk, homogeneous, classical nucleation theory, definitively establishing the relevance of homogeneous cavitation as an evaporation mechanism in mesoporous materials. Our results imply that porous alumina membranes are a promising new system to study liquids in a deeply metastable state.

Stress or Strain Does Not Impact Sorption in Stiff Mesoporous Materials

The present paper investigates strain-induced sorption in mesoporous silicon. Contrarily to a previous report based on indirect evidence, we find that external mechanical strain or stress has no measurable impact on sorption isotherms, down to a relative accuracy of 10^-3. This conclusion is in agreement with the analysis of the sorption-induced strain of porous silicon and holds for other stiff mesoporous materials such as porous silicas.

Adsorption on ordered and disordered duplex layers of porous anodic alumina

We have carried out systematic experiments and numerical simulations of the adsorption on porous anodic aluminum oxide (AAO) duplex layers presenting either an ordered or a disordered interconnecting interface between the large (cavity) and small (constriction) sections of the structured pores. Selective blocking of the pore openings resulted in three different pore topologies: open structured pores, funnel pores, and ink-bottle pores. In the case of the structured pores having an ordered interface, the adsorption isotherms present a rich phenomenology characterized by the presence of two steps in the condensation branch and the opening of one (two) hysteresis loops during evaporation for the ink-bottle (open and funnel) pores. The isotherms can be obtained by summing the isotherms measured on uniform pores having the dimensions of the constrictions or of the cavities. The numerical analysis of the three different pore topologies indicates that the shape of the junction between the two pore sections is only important for the adsorption branch. In particular, a conic junction which resembles that of the AAO pores represents the experimental isotherms for the open and funnel pores better, but the shape of the junction in the ink bottle pores does not matter. The isotherms for the duplex layers with a disordered interface display the same general features found for the ordered duplex layers. In both cases, the adsorption branches coincide and have two steps which are shifted to lower relative pressures compared to those for the ordered duplex. Furthermore, the desorption branches comprise hysteresis loops much wider than those of the ordered duplex layers. Overall, this study highlights the important role played by morphologies where there are interconnections between large and small pores.

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.

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.

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.

Simple phenomenological model for phase transitions in confined geometry. 2. Capillary condensation/evaporation in cylindrical mesopores

A simple phenomenological model that describes capillary condensation and evaporation of pure fluids confined in cylindrical mesopores is presented. Following the work of Celestini (Celestini, F. Phys. Lett. A 1997, 228, 84), the free energy density of the system is derived using interfacial tensions and a corrective term that accounts for the interaction coupling between the vapor/adsorbed liquid and the adsorbed liquid/adsorbent interfaces. This corrective term is shown to be consistent with the Gibbs adsorption isotherm and assessed by standard adsorption tests. This model reveals that capillary condensation and evaporation are metastable and equilibrium processes, respectively, hence exhibiting the existence of a hysteresis loop inadsorption/desorption isotherm that is well-known in experiment. We extend the phenomenological model of Celestini to give a quantitative description of adsorption on the pore wall and hysteresis width evolution with temperature and confinement. Direct quantitative comparison is made with experimental data for confined argon. Used as a characterizing tool, this integrated model allows in a single fit of an experimental adsorption/desorption isotherm assessing essential characterization data such as the specific surface area, pore volume, and mean pore size.

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

In a recent paper [J. Chem. Phys. 2007, 127, 154701] a multiscale approach was introduced which allowed calculation of adsorption/desorption hysteresis for fluid confined in a single mesoporous, heterogeneous tubular pore. The main interest in using such an approach is that it allows one to reconcile a molecular simulation approach generally limited to the nanometer scale (atomistic description of the confined fluid and pore roughness) with the much larger scale (micrometer) relevant to understand the complexity of adsorption/desorption hysteresis (the numerous metastable states in the hysteresis loop are a consequence of the large-scale disorder in the porous material). In this paper, this multiscale approach is used to study adsorption phenomena in mesoporous models made of a collection of disordered, noninterconnected tubular pores, as MCM-41 or porous silicon. A double distribution is introduced: one to characterize the disorder in a given pore, and the other to characterize the disorder between the pores. We consider two distribution shapes: Gaussian and uniform truncated and two cases of pores open at one or both ends. These models are expected to cover a wide variety of real materials made of independent pores, as MCM-41 and oxidized porous silicon. A large variety of hysteresis shapes is obtained, ranging from almost parallel adsorption/desorption branches typical of MCM-41 adsorption to triangular hysteresis typical of porous silicon. The structure of the metastable states inside the hysteresis (scanning adsorption/desorption curves) is also examined. The results are expected to be useful to experimentalists who want to infer pore structure and level of disorder from experimental adsorption/desorption experiments.

A density functional theory for Lennard-Jones fluids in cylindrical pores and its applications to adsorption of nitrogen on MCM-41 materials

A density functional theory (DFT) constructed from the modified fundamental-measure theory and the modified Benedict-Webb-Rubin equation of state is presented. The Helmholtz free energy functional due to attractive interaction is expressed as a functional of attractive weighted-density in which the weight function is a mean-field-like type. An obvious advantage of the present theory is that it reproduces accurate bulk properties such as chemical potential, bulk pressure, vapor-liquid interfacial tension, and so forth when compared with molecular simulations and experiments with the same set of molecular parameters. Capabilities of the present DFT are demonstrated by its applicability to adsorption of argon and nitrogen on, respectively, a model cylindrical pore and mesoporous MCM-41 materials. Comparison of the theoretical results of argon in the model cylindrical pore with those from the newly published molecular simulations indicates that the present DFT predicts accurate average densities in the pore, slightly overestimates the pore pressure, and correctly describes the effect of the fluid-pore wall interaction on average densities and pressures in the pore. Application to adsorption of nitrogen on MCM-41 at 77.4 K shows that the present DFT predicts density profiles and adsorption isotherms in good agreement with those from molecular simulations and experiments. In contrast, the hysteresis loop of adsorption calculated from the mean-field theory shifts toward the low pressure region because a low bulk saturated pressure is produced from the mean-field equation of state. The present DFT offers a good way to describe the adsorption isotherms of porous materials as a function of temperature and pressure.

Microscopic mechanism of adsorption in cylindrical nanopores with heterogenous wall structure

We study the microscopic mechanism of adsorption in nanometric cylindrical pores with strongly heterogeneous walls using grand canonical Monte Carlo simulations. The pore surface structure is modeled by a new lattice-site approach. Each site is characterized by two amplitudes--structural and energetic--that locally modify the structural and energetic properties of the surface. The amplitudes are randomly distributed over the pore wall. We have shown that different structural and energetic distribution functions lead to different mechanism of adsorption. The energetic site distribution plays the most crucial role in the submonolayer region. The structural site distribution modifies the multilayer adsorption. A method to analyze the stability of the adsorbed system using static susceptibility is proposed. Potential applications in multiscale modeling are discussed.

Capillary condensation in porous materials. Hysteresis and interaction mechanism without pore blocking/percolation process

We have performed measurements of boundary hysteresis loops, reversal curves, and subloops in p+-type porous silicon, a porous material composed of straight non-interconnected pores. These data show that a strong interaction mechanism exists between the pores. The pores of porous silicon are non-independent, whereas they are not interconnected. This hysteretic behavior is very similar to that observed in porous glass, which consists of cavities connected to each other by constrictions. This questions the so-called pore blocking/percolation model developed to explain the behavior of fluid in porous glass. More generally, if we disregard the shape of the boundary hysteresis loops which depends on the porous material (H1 for MCM-41 and SBA-15, H2 for porous glass and p+-type porous silicon), the hysteretic features inside the main loop are qualitatively the same for all these porous systems. This shows that none of these systems are composed of independent pores. A coupling between the pores is always present whether they are interconnected or not and whatever the shape of the main loop is.