Understanding capillary condensation and hysteresis in porous silicon: network effects within independent pores

Small-Angle Scattering Indicates Equilibrium Instead of Metastable Capillary Condensation in SBA-15 Mesoporous Silica

Questions about the origin of the adsorption/desorption hysteresis in mesoporous materials are as old as sorption experiments themselves. The historical conception that underlines most existing methods to extract pore size distributions from sorption data assumes that adsorption is a metastable process and that desorption takes place at thermodynamic equilibrium. In this work, we measure nitrogen and argon sorption on a series of 14 SBA-15 ordered mesoporous silicas and use small-angle X-ray scattering to independently determine their pore sizes. We find that capillary condensation systematically occurs close to thermodynamic equilibrium according to a Derjaguin-Broekhoff-de Boer calculation. Our analysis suggests that many earlier works have significantly underestimated the actual pore size in SBA-15 materials. It also highlights the critical role of the reference isotherm used to calibrate the fluid-solid interaction in the models.

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.

Signatures of nanoemulsion jamming and unjamming in stimulated-echo NMR

The unjamming of elastic concentrated nanoemulsions into viscous dilute nanoemulsions, through dilution with the continuous phase, offers interesting opportunities for a pulsed-field gradient (PFG) NMR, particularly if the nanoemulsion is designed to take advantage of the nuclear specificity offered by NMR. Here, we make and study size-fractionated oil-in-water nanoemulsions using a perfluorinated copolymer silicone oil that is highly insoluble in the aqueous continuous phase. By studying these nanoemulsions using ^{19}F stimulated-echo PFG-NMR, we avoid any contribution from the aqueous continuous phase, which contains a nonfluorinated ionic surfactant. We find a dramatic change in the ^{19}F PFG-NMR decays at high field-gradient strengths as the droplet volume fraction, ϕ, is lowered through dilution. At high ϕ, observed decays as a function of field-gradient strength exhibit decay-to-plateau behavior indicating the jamming of nanodroplets, which contain ^{19}F probe molecules, in an elastic material reminiscent of a nanoporous solid. In contrast, at lower ϕ, only a simple decay is observed, indicating that the nanodroplets have unjammed and can diffuse over much larger distances. Through a comparison with bulk mechanical rheometry, we show that this dramatic change coincides with the loss of low-frequency shear elasticity of the nanoemulsion.

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.

Direct Observation of the Xenon Physisorption Process in Mesopores by Combining In Situ Anomalous Small-Angle X-ray Scattering and X-ray Absorption Spectroscopy

The morphology and structural changes of confined matter are still far from being understood. This report deals with the development of a novel in situ method based on the combination of anomalous small-angle X-ray scattering (ASAXS) and X-ray absorption near edge structure (XANES) spectroscopy to directly probe the evolution of the xenon adsorbate phase in mesoporous silicon during gas adsorption at 165 K. The interface area and size evolution of the confined xenon phase were determined via ASAXS demonstrating that filling and emptying the pores follow two distinct mechanisms. The mass density of the confined xenon was found to decrease prior to pore emptying. XANES analyses showed that Xe exists in two different states when confined in mesopores. This combination of methods provides a smart new tool for the study of nanoconfined matter for catalysis, gas, and energy storage applications.

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.

Effect of Pore Size Distribution on Capillary Condensation in Nanoporous Media

The occurrence of capillary condensation is often ignored in many naturally occurring nanoporous media, such as shale rock, simply because their isotherms do not adhere to the prescribed shapes presented in the literature. In particular, it is apparent from the literature that most shale isotherms do not display a clear capillary condensation step, which is commonly observed for much simpler adsorbents, such as MCM-41. We contend that the absence of this step from the isotherms for natural adsorbents is not due to the absence of nanoconfinement-induced phase behavior. Rather, it is due to the broad pore size distribution characteristic of such materials. By mechanically mixing different sizes of MCM-41 together and measuring isotherms for propane and n-butane in them at a variety of temperatures, we show that phase behavior in different pore sizes is additive and suppresses the commonly observed appearance of capillary condensation. By comparing the isotherms in the mixtures of MCM-41 to those measured in single pore sizes of MCM-41, we develop correlations, using the Lorentzian function, that make the determinations of porosity and fluid density from the mixture isotherms straightforward.

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.

Position-Dependent Dynamics Explain Pore-Averaged Diffusion in Strongly Attractive Adsorptive Systems

Using molecular simulations, we investigate the relationship between the pore-averaged and position-dependent self-diffusivity of a fluid adsorbed in a strongly attractive pore as a function of loading. Previous work (Krekelberg, W. P.; Siderius, D. W.; Shen, V. K.; Truskett, T. M.; Errington, J. R. Connection between thermodynamics and dynamics of simple fluids in highly attractive pores. Langmuir 2013, 29, 14527-14535, doi: 10.1021/la4037327) established that pore-averaged self-diffusivity in the multilayer adsorption regime, where the fluid exhibits a dense film at the pore surface and a lower density interior pore region, is nearly constant as a function of loading. Here we show that this puzzling behavior can be understood in terms of how loading affects the fraction of particles that reside in the film and interior pore regions as well as their distinct dynamics. Specifically, the insensitivity of pore-averaged diffusivity to loading arises from the approximate cancellation of two factors: an increase in the fraction of particles in the higher diffusivity interior pore region with loading and a corresponding decrease in the particle diffusivity in that region. We also find that the position-dependent self-diffusivities scale with the position-dependent density. We present a model for predicting the pore-average self-diffusivity based on the position-dependent self-diffusivity, which captures the unusual characteristics of pore-averaged self-diffusivity in strongly attractive pores over several orders of magnitude.

Phase transitions in disordered mesoporous solids

Fluids confined in mesoporous solids exhibit a wide range of physical behavior including rich phase equilibria. While a notable progress in their understanding has been achieved for fluids in materials with geometrically ordered pore systems, mesoporous solids with complex pore geometries still remain a topic of active research. In this work we study phase transitions occurring in statistically disordered linear chains of pores with different pore sizes. By considering, quite generally, two phase change mechanisms, nucleation and phase growth, occurring simultaneously we obtain the boundary transitions and the scanning curves resulting upon reversing the sign of the evolution of the chemical potential at different points along the main transition branches. The results obtained are found to reproduces the key experimental observations, including the emergence of hysteresis and the scanning behavior. By deriving the serial pore model isotherm we suggest a robust framework for reliable structural analysis of disordered mesoporous solids.

Connection Between Thermodynamics and Dynamics of Simple Fluids in Pores: Impact of Fluid-Fluid Interaction Range and Fluid-Solid Interaction Strength

Using molecular simulations, we investigate how the range of fluid-fluid (adsorbate-adsorbate) interactions and the strength of fluid-solid (adsorbate-adsorbent) interactions impact the strong connection between distinct adsorptive regimes and distinct self-diffusivity regimes reported in [Krekelberg, W. P.; Siderius, D. W.; Shen, V. K.; Truskett, T. M.; Errington, J. R. Langmuir2013, 29, 14527-14535]. Although increasing the fluid-fluid interaction range changes both the thermodynamics and the dynamic properties of adsorbed fluids, the previously reported connection between adsorptive filling regimes and self-diffusivity regimes remains. Increasing the fluid-fluid interaction range leads to enhanced layering and decreased self-diffusivity in the multilayer-formation regime but has little effect on the properties within film-formation and pore-filling regimes. We also find that weakly attractive adsorbents, which do not display distinct multilayer formation, are hard-sphere-like at super- and subcritical temperatures. In this case, the self-diffusivity of the confined and bulk fluid has a nearly identical scaling-relationship with effective density.

Scale-dependent diffusion anisotropy in nanoporous silicon

Nanoporous silicon produced by electrochemical etching of highly B-doped p-type silicon wafers can be prepared with tubular pores imbedded in a silicon matrix. Such materials have found many technological applications and provide a useful model system for studying phase transitions under confinement. This paper reports a joint experimental and simulation study of diffusion in such materials, covering displacements from molecular dimensions up to tens of micrometers with carefully selected probe molecules. In addition to mass transfer through the channels, diffusion (at much smaller rates) is also found to occur in directions perpendicular to the channels, thus providing clear evidence of connectivity. With increasing displacements, propagation in both axial and transversal directions is progressively retarded, suggesting a scale-dependent, hierarchical distribution of transport resistances (“constrictions” in the channels) and of shortcuts (connecting “bridges”) between adjacent channels. The experimental evidence from these studies is confirmed by molecular dynamics (MD) simulation in the range of atomistic displacements and rationalized with a simple model of statistically distributed “constrictions” and “bridges” for displacements in the micrometer range via dynamic Monte Carlo (DMC) simulation. Both ranges are demonstrated to be mutually transferrable by DMC simulations based on the pore space topology determined by electron tomography.

Control of the Pore Texture in Nanoporous Silicon via Chemical Dissolution

The surface and textural properties of porous silicon (pSi) control many of its physical properties essential to its performance in key applications such as optoelectronics, energy storage, luminescence, sensing, and drug delivery. Here, we combine experimental and theoretical tools to demonstrate that the surface roughness at the nanometer scale of pSi can be tuned in a controlled fashion using partial thermal oxidation followed by removal of the resulting silicon oxide layer with hydrofluoric acid (HF) solution. Such a process is shown to smooth the pSi surface by means of nitrogen adsorption, electron microscopy, and small-angle X-ray and neutron scattering. Statistical mechanics Monte Carlo simulations, which are consistent with the experimental data, support the interpretation that the pore surface is initially rough and that the oxidation/oxide removal procedure diminishes the surface roughness while increasing the pore diameter. As a specific example considered in this work, the initial roughness ξ ∼ 3.2 nm of pSi pores having a diameter of 7.6 nm can be decreased to 1.0 nm following the simple procedure above. This study allows envisioning the design of pSi samples with optimal surface properties toward a specific process.

Inhomogeneous relaxation dynamics and phase behaviour of a liquid crystal confined in a nanoporous solid

We report filling-fraction dependent dielectric spectroscopy measurements on the relaxation dynamics of the rod-like nematogen 7CB condensed in 13 nm silica nanochannels. In the film-condensed regime, a slow interface relaxation dominates the dielectric spectra, whereas from the capillary-condensed state up to complete filling an additional, fast relaxation in the core of the channels is found. The temperature-dependence of the static capacitance, representative of the averaged, collective molecular orientational ordering, indicates a continuous, paranematic-to-nematic (P-N) transition, in contrast to the discontinuous bulk behaviour. It is well described by a Landau-de-Gennes free energy model for a phase transition in cylindrical confinement. The large tensile pressure of 10 MPa in the capillary-condensed state, resulting from the Young-Laplace pressure at highly curved liquid menisci, quantitatively accounts for a downward-shift of the P-N transition and an increased molecular mobility in comparison to the unstretched liquid state of the complete filling. The strengths of the slow and fast relaxations provide local information on the orientational order: the thermotropic behaviour in the core region is bulk-like, i.e. it is characterized by an abrupt onset of the nematic order at the P-N transition. By contrast, the interface ordering exhibits a continuous evolution at the P-N transition. Thus, the phase behaviour of the entirely filled liquid crystal-silica nanocomposite can be quantitatively described by a linear superposition of these distinct nematic order contributions.

Soft matter in hard confinement: phase transition thermodynamics, structure, texture, diffusion and flow in nanoporous media

Spatial confinement in nanoporous media affects the structure, thermodynamics and mobility of molecular soft matter often markedly. This article reviews thermodynamic equilibrium phenomena, such as physisorption, capillary condensation, crystallisation, self-diffusion, and structural phase transitions as well as selected aspects of the emerging field of spatially confined, non-equilibrium physics, i.e. the rheology of liquids, capillarity-driven flow phenomena, and imbibition front broadening in nanoporous materials. The observations in the nanoscale systems are related to the corresponding bulk phenomenologies. The complexity of the confined molecular species is varied from simple building blocks, like noble gas atoms, normal alkanes and alcohols to liquid crystals, polymers, ionic liquids, proteins and water. Mostly, experiments with mesoporous solids of alumina, gold, carbon, silica, and silicon with pore diameters ranging from a few up to 50 nm are presented. The observed peculiarities of nanopore-confined condensed matter are also discussed with regard to applications. A particular emphasis is put on texture formation upon crystallisation in nanoporous media, a topic both of high fundamental interest and of increasing nanotechnological importance, e.g. for the synthesis of organic/inorganic hybrid materials by melt infiltration, the usage of nanoporous solids in crystal nucleation or in template-assisted electrochemical deposition of nano structures.

Adsorption in alumina pores open at one and at both ends

We have studied adsorption in regular, self-ordered alumina pores open at both ends or only at one end. The straight, non-connected pores have diameters ranging from 22 to 83 nm, with a relative dispersion below 1% in the pore size. Adsorption isotherms measured in open pores with a torsional microbalance show pronounced hysteresis loops characterized by nearly vertical and parallel adsorption and desorption branches. Blocking one end of the pores with glue has a strong influence on adsorption, as expected from classical macroscopic arguments. However, the experimental measurements show an unexpectedly rich phenomenology dependent on the pore size. For large pores (Dp ≥ 67 nm), the isotherms for closed end pores present much narrower hysteresis loops whose adsorption and desorption boundaries envelop the desorption branches of the isotherms for the corresponding open pores of the same size. The loop for small closed end pores (Dp = 22 nm) is slightly wider than that for open pores while the adsorption branches coincide. For large pores, in contrast, the desorption branches of pores with the same Dp overlap regardless of the pore opening. These observations are in agreement with our grand canonical Monte Carlo (GCMC) simulations for a cylindrical pore model with constrictions, suggesting that the alumina pores could be modeled using a constricted pore model whose adsorption isotherm depends on the ratio of the constriction size to the pore size (Dc/Dp).

Modeling the influence of side stream and ink bottle structures on adsorption/desorption dynamics of fluids in long pores

We apply dynamic mean field theory to study relaxation dynamics for lattice models of fluids confined in linear pores with side streams and with ink bottle structures. Our results show several mechanisms for how the pore structure affects the dynamics, and these are amplified in longer pores. An important conclusion of this work is that features such as side streams and ink bottle segments can substantially slow the equilibration of fluids confined in long pore systems where the pore lengths can be more than 100 micrometers, such as in porous silicon. This may make it difficult to properly equilibrate these systems for states close to those where the pores should be completely filled with liquids. The presence of trapped bubbles in the system may change the desorption characteristics of the system and the shape of the hysteresis loops.

Filling dynamics of closed end nanocapillaries

We have studied the filling dynamics of model capillaries using dynamic mean field theory for a confined lattice gas and Kawasaki dynamics simulations. We have found two different scenarios for filling of capped nanocapillaries from the vapor phase. As compared to channels with macroscopic width, in which the filling process occurs by the detachment of the meniscus from the cap, in mesoscopic channels there is an alternative mechanism associated with the spontaneous condensation of the liquid close to the pore opening and its subsequent growth toward the closed pore end. We show that these two scenarios have totally different filling dynamics, providing an additional mechanism for slow capillary condensation kinetics in nanoscopic objects.

Connection between thermodynamics and dynamics of simple fluids in highly attractive pores

Using molecular simulations, we investigate the structural and diffusive dynamics properties of a model fluid in highly absorptive cylindrical pores. At subcritical temperatures, self-diffusivity displays three distinct regimes as a function of average pore density ρ: (1) a decrease in self-diffusivity with increasing ρ at low ρ, (2) constant self-diffusivity with respect to varying ρ at moderate density, and (3) a decrease in self-diffusivity with increasing ρ at high density. These regimes are closely linked to the thermodynamic properties of the fluid in the pore, specifically, the adsorption isotherm, isosteric heat of adsorption, and the density profile. We show that these three diffusivity regimes qualitatively correspond to three distinct adsorption regimes: monolayer formation, multilayer adsorption, and pore filling, respectively. In addition, we find that the self-diffusivity is a universal function of the local film density in the monolayer formation regime at subcritical temperatures. The results of this work suggest a potential means to estimate the self-diffusivity over a broad pressure range using a limited number of experiments.

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.

Thermotropic nematic order upon nanocapillary filling

Optical birefringence and light absorption measurements reveal four regimes for the thermotropic behavior of a nematogen liquid (7CB) upon sequential filling of parallel-aligned capillaries of 12 nm diameter in a monolithic, mesoporous silica membrane. No molecular reorientation is observed for the first adsorbed monolayer. In the film-condensed state (up to 1 nm thickness), a weak, continuous paranematic-to-nematic (P-N) transition is found, which is shifted by 10 K below the discontinuous bulk transition at T(IN)=305 K. The capillary-condensed state exhibits a more pronounced, albeit still continuous P-N reordering, located 4 K below T(IN). This shift vanishes abruptly upon complete filling of the capillaries. It could originate in competing anchoring conditions at the free inner surfaces and at the pore walls or result from the 10-MPa tensile pressure release associated with the disappearance of concave menisci in the confined liquid upon complete filling. The study documents that the thermo-optical properties of nanoporous systems (or single nanocapillaries) can be tailored over a surprisingly wide range simply by variation of the filling fraction with liquid crystals.

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.

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

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

Monte Carlo simulation of cavitation in pores with nonwetting defects

We investigate the onset of cavitation in a metastable fluid confined to nanoscale pores with nonwetting defects present. Using grand canonical and gauge cell mesocanonical Monte Carlo simulations, we study the degree of metastability (relative vapor pressure), at which the critical bubble forms in a spherical pore with a circular nonwetting defect. It is shown that an increase of the defect size leads to a transition from homogeneous to heterogeneous nucleation of critical bubbles formed at the defect site. In this case, the desorption process may be initiated at larger relative vapor pressures than those predicted by the theories of homogeneous cavitation.

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.

Capillary condensation and evaporation in alumina nanopores with controlled modulations

Capillary condensation in nanoporous anodic aluminum oxide presenting not interconnected pores with controlled modulations is studied using adsorption experiments and molecular simulations. Both the experimental and simulation data show that capillary condensation and evaporation are driven by the smallest size of the nanopore (constriction). The adsorption isotherms for the open and closed pores are almost identical if constrictions are added to the system. The latter result implies that the type of pore ending does not matter in modulated pores. Thus, the presence of hysteresis loops observed in adsorption isotherms measured in straight nanopores with closed bottom ends can be explained in terms of geometrical inhomogeneities along the pore axis. More generally, these results provide a general picture of capillary condensation and evaporation in constricted or modulated pores that can be used for the interpretation of adsorption in disordered porous materials.

Numerical solutions of a generalized theory for macroscopic capillarity

A recent macroscopic theory of biphasic flow in porous media [R. Hilfer, Phys. Rev. E 73, 016307 (2006)] has proposed to treat microscopically percolating fluid regions differently from microscopically nonpercolating regions. Even in one dimension the theory reduces to an analytically intractable set of ten coupled nonlinear partial differential equations. This paper reports numerical solutions for three different initial and boundary value problems that simulate realistic laboratory experiments. All three simulations concern a closed column containing a homogeneous porous medium filled with two immiscible fluids of different densities. In the first simulation the column is raised from a horizontal to a vertical orientation inducing a buoyancy-driven fluid flow that separates the two fluids. In the second simulation the column is first raised from a horizontal to a vertical orientation and subsequently rotated twice by 180 degrees to compare the resulting stationary saturation profiles. In the third simulation the column is first raised from horizontal to vertical orientation and then returned to its original horizontal orientation. In all three simulations imbibition and drainage processes occur simultaneously inside the column. This distinguishes the results reported here from conventional simulations based on existing theories of biphasic flows. Existing theories are unable to predict flow processes where imbibition and drainage occur simultaneously. The approximate numerical results presented here show the same process dependence and hysteresis as one would expect from an experiment.

Criticality of an isotropic-to-smectic transition induced by anisotropic quenched disorder

We report combined optical birefringence and neutron scattering measurements on the liquid crystal 12CB nanoconfined in mesoporous silicon layers. This liquid crystal exhibits strong nematic-smectic coupling responsible for a discontinuous isotropic-to-smectic phase transition in the bulk state. Confined in porous silicon, 12CB is subjected to strong anisotropic quenched disorder: a short-ranged smectic state evolves out of a paranematic phase. This transformation appears continuous, losing its bulk first-order character. This contrasts with previously reported observations on liquid crystals under isotropic quenched disorder. In the low temperature phase, both orientational and translational order parameters obey the same power law.

Understanding adsorption and desorption processes in mesoporous materials with independent disordered channels

Using a lattice-gas model in mean-field theory, we discuss the problem of how adsorption and desorption of fluids in independent cylinderlike pores is influenced by variations in the pore diameter along the length of the pore, surface roughness of the pore walls, and chemical heterogeneity. We also consider the impact of contact with the bulk phase via the pore opening and the possibility of interactions between neighboring pores via a liquid film on the external surface of the material. We find that a combination of pore size variation along the length of the pore and surface roughness yields sorption hysteresis similar to that found in systems with three-dimensional disordered pore networks such as porous glasses. Our results are especially relevant to adsorption and desorption in porous silicon materials with independent linear pores and apparently anomalous features of the behavior in these systems can be accounted for within the context of the present model.

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.