Inside the hysteresis loop: multiplicity of internal states in confined fluids

Equation of state for confined fluids

Fluids confined in small volumes behave differently than fluids in bulk systems. For bulk systems, a compact summary of the system’s thermodynamic properties is provided by equations of state. However, there is currently a lack of successful methods to predict the thermodynamic properties of confined fluids by use of equations of state, since their thermodynamic state depends on additional parameters introduced by the enclosing surface. In this work, we present a consistent thermodynamic framework that represents an equation of state for pure, confined fluids. The total system is decomposed into a bulk phase in equilibrium with a surface phase. The equation of state is based on an existing, accurate description of the bulk fluid and uses Gibbs’ framework for surface excess properties to consistently incorporate contributions from the surface. We apply the equation of state to a Lennard-Jones spline fluid confined by a spherical surface with a Weeks–Chandler–Andersen wall-potential. The pressure and internal energy predicted from the equation of state are in good agreement with the properties obtained directly from molecular dynamics simulations. We find that when the location of the dividing surface is chosen appropriately, the properties of highly curved surfaces can be predicted from those of a planar surface. The choice of the dividing surface affects the magnitude of the surface excess properties and its curvature dependence, but the properties of the total system remain unchanged. The framework can predict the properties of confined systems with a wide range of geometries, sizes, interparticle interactions, and wall–particle interactions, and it is independent of ensemble. A targeted area of use is the prediction of thermodynamic properties in porous media, for which a possible application of the framework is elaborated.

A heuristic approach for nanodrops on a smooth solid surface

A heuristic approach is developed to obtain a simple equation for the contact angle of a nanodrop on a smooth planar solid surface. First, nanodrops of various fluids in contact with various solid surfaces are considered on the basis of nonlocal density functional theory (DFT). Along with the traditional (apparent) contact angle, θ_a, which the drop profile makes with the solid surface, another one, θ_d, formed by the smooth part of the drop profile and the horizontal plane separating that part from the oscillatory part of the profile was examined. For each of the contact angles, a separate simple equation resembling the Young equation for the macroscopic drops but containing, instead of surface tensions, the microscopic parameters of intermolecular interactions, temperature, and average density of the fluid was hypothesized and the parameters of this equation were determined using the results of DFT calculations. It was shown that predictions of these equations coincide with the results provided by DFT.

An analog to Bond number for pendant nanodrops

A new dimensionless number is introduced which characterizes the shape and stability of a pendant nanodrop.

Critical adsorption and critical Casimir forces in the canonical ensemble

Critical properties of a liquid film between two planar walls are investigated in the canonical ensemble, within which the total number of fluid particles, rather than their chemical potential, is kept constant. The effect of this constraint is analyzed within mean-field theory (MFT) based on a Ginzburg-Landau free-energy functional as well as via Monte Carlo simulations of the three-dimensional Ising model with fixed total magnetization. Within MFT and for finite adsorption strengths at the walls, the thermodynamic properties of the film in the canonical ensemble can be mapped exactly onto a grand canonical ensemble in which the corresponding chemical potential plays the role of the Lagrange multiplier associated with the constraint. However, due to a nonintegrable divergence of the mean-field order parameter profile near a wall, the limit of infinitely strong adsorption turns out to be not well-defined within MFT, because it would necessarily violate the constraint. The critical Casimir force (CCF) acting on the two planar walls of the film is generally found to behave differently in the canonical and grand canonical ensembles. For instance, the canonical CCF in the presence of equal preferential adsorption at the two walls is found to have the opposite sign and a slower decay behavior as a function of the film thickness compared to its grand canonical counterpart. We derive the stress tensor in the canonical ensemble and find that it has the same expression as in the grand canonical case, but with the chemical potential playing the role of the Lagrange multiplier associated with the constraint. The different behavior of the CCF in the two ensembles is rationalized within MFT by showing that, for a prescribed value of the thermodynamic control parameter of the film, i.e., density or chemical potential, the film pressures are identical in the two ensembles, while the corresponding bulk pressures are not.

Evaluation of finite-size effects in cavitation and droplet formation

Nucleation of bubbles and droplets is of fundamental interest in science and technology and has been widely investigated through experiments, theory, and simulations. Giving the rare event nature of these phenomena, nucleation simulations are computationally costly and require the use of a limited number of particles. Moreover, they are often performed in the canonical ensemble, i.e., by fixing the total volume and number of particles, to avoid the additional complexities of implementing a barostat. However, cavitation and droplet formation take place differently depending on the ensemble. Here, we analyze the importance of finite-size effects in cavitation and droplet formation. We present simple formulas which predict the finite-size corrections to the critical size, the nucleation barrier, and the nucleation rates in the canonical ensemble very accurately. These results can be used to select an appropriate system-size for simulations and to get a more precise evaluation of nucleation in complex substances, by using a small number of molecules and correcting for finite-size effects.

Communication: superstabilization of fluids in nanocontainers

One of the main challenges of thermodynamics is to predict and measure accurately the properties of metastable fluids. Investigation of these fluids is hindered by their spontaneous transformation by nucleation into a more stable phase. We show how small closed containers can be used to completely prevent nucleation, achieving infinitely long-lived metastable states. Using a general thermodynamic framework, we derive simple formulas to predict accurately the conditions (container sizes) at which this superstabilization takes place and it becomes impossible to form a new stable phase. This phenomenon opens the door to control nucleation of deeply metastable fluids at experimentally feasible conditions, having important implications in a wide variety of fields.

Influence of system size on the properties of a fluid adsorbed in a nanopore: Physical manifestations and methodological consequences

Hysteresis and discontinuities in the isotherms of a fluid adsorbed in a nanopore in general hamper the determination of equilibrium thermodynamic properties, even in computer simulations. A way around this has been to consider both a reservoir of small size and a pore of small extent in order to restrict the fluctuations of density and approach a classical van der Waals loop. We assess this suggestion by thoroughly studying through Monte Carlo simulations and density functional theory the influence of system size on the equilibrium configurations of the adsorbed fluid and on the resulting isotherms. We stress the importance of pore-symmetry-breaking states that even for modest pore sizes lead to discontinuous isotherms and we discuss the physical relevance of these states and the methodological consequences for computing thermodynamic quantities.

Diffusion Study by IR Micro-Imaging of Molecular Uptake and Release on Mesoporous Zeolites of Structure Type CHA and LTA

The presence of mesopores in the interior of microporous particles may significantly improve their transport properties. Complementing previous macroscopic transient sorption experiments and pulsed field gradient NMR self-diffusion studies with such materials, the present study is dedicated to an in-depth study of molecular uptake and release on the individual particles of mesoporous zeolitic specimens, notably with samples of the narrow-pore structure types, CHA and LTA. The investigations are focused on determining the time constants and functional dependences of uptake and release. They include a systematic variation of the architecture of the mesopores and of the guest molecules under study as well as a comparison of transient uptake with blocked and un-blocked mesopores. In addition to accelerating intracrystalline mass transfer, transport enhancement by mesopores is found to be, possibly, also caused by a reduction of transport resistances on the particle surfaces.

Asymmetric profiles and prewetting lines in the filling of planar slits with Ne

The filling of slits with identical planar walls is investigated in the frame of the density functional theory. For this kind of slit, the confining potential is symmetric with respect to its central plane. Closed and open systems are studied by applying, respectively, the canonical and grand canonical ensembles (CE and GCE). Results obtained for the confinement of fluid Ne by alkaline surfaces are reported. The behavior of these systems is analyzed by varying the strength of the Ne-substrate attraction, the temperature T, and the coverage Γ(l). It is assumed that the one-body density of the fluid, ρ(r), is uniform along the (x, y) planes parallel to the walls, becoming a function of the coordinate z perpendicular to those planes. Two sorts of solutions are found for the density profile: (i) symmetric ones that follow the left-right symmetry of the potential exerted by the walls and (ii) asymmetric ones that break the symmetry of the slit. The pores are wide enough for determining prewetting (PW) lines and wetting and critical PW temperatures, i.e., T(w) and T(cpw), from the analysis of symmetric solutions provided by both the CE and GCE schemes. Asymmetric species are examined in detail for T > T(w). It is shown that for a given Ne-substrate pair at a fixed T both the CE and GCE frames yield only one asymmetric 2-fold degenerate stable profile (formed by a "thin" wetting film at one wall and a "thick" wetting film at the other) coexisting with two symmetric profiles (formed by "thin" or "thick" wetting films at the two walls), while the remaining asymmetric states are at best metastable. This feature occurs along PW lines and disappears at T(cpw).

Translocation dynamics of freely jointed Lennard-Jones chains into adsorbing pores

Polymer translocation into adsorbing nanopores is studied by using the Fokker-Planck equation of chain diffusion along the energy landscape calculated with Monte Carlo simulations using the incremental gauge cell method. The free energy profile of a translocating chain was found by combining two independent sub-chains, one free but tethered to a hard wall, and the other tethered inside an adsorbing pore. Translocation dynamics were revealed by application of the Fokker-Planck equation for normal diffusion. Adsorption of polymer chains into nanopores involves a competition of attractive adsorption and repulsive steric hindrance contributions to the free energy. Translocation times fell into two regimes depending on the strength of the adsorbing pore. In addition, we found a non-monotonic dependence of translocation times with increasing adsorption strength, with sharp peak associated with local free energy minima along the translocation coordinate.

Capillary condensation hysteresis in overlapping spherical pores: a Monte Carlo simulation study

The mechanisms of hysteretic phase transformations in fluids confined to porous bodies depend on the size and shape of pores, as well as their connectivity. We present a Monte Carlo simulation study of capillary condensation and evaporation cycles in the course of Lennard-Jones fluid adsorption in the system of overlapping spherical pores. This model system mimics pore shape and connectivity in some mesoporous materials obtained by templating cubic surfactant mesophases or colloidal crystals. We show different mechanisms of capillary hysteresis depending on the size of the window between the pores. For the system with a small window, the hysteresis cycle is similar to that in a single spherical pore: capillary condensation takes place upon achieving the limit of stability of adsorption film and evaporation is triggered by cavitation. When the window is large enough, the capillary condensation shifts to a pressure higher than that of the isolated pore, and the possibility for the equilibrium mechanism of desorption is revealed. These finding may have important implications for practical problems of assessment of the pore size distributions in mesoporous materials with cagelike pore networks.

Adsorption-induced deformation of mesoporous solids: macroscopic approach and density functional theory

We present a theoretical study of the deformation of mesoporous solids during adsorption. The proposed thermodynamic model allows one to link the mechanical stress and strain to the solvation pressure exerted by the adsorbed molecules on the pore wall. Two approaches are employed for calculation of solvation pressure as a function of adsorbate pressure: the macroscopic Derjaguin-Broekhoff-de Boer theory of capillary condensation, and the microscopic density functional theory. We revealed that the macroscopic and microscopic theories are in quantitative agreement for the pores >8 nm diameter within the whole range of adsorbate pressures. For smaller pores, the macroscopic theory gradually deteriorates, and the density functional theory extends the thermodynamic model of adsorption-induced deformation to the nanometer scales.

Thermodynamic analysis of nucleation in confined space: generalized Gibbs approach

A general thermodynamic analysis of nucleation-growth processes in confined space in initially metastable states of the ambient phase is performed based on the generalized Gibbs approach to the description of heterogeneous systems. In particular, it is shown analytically how the parameters of critical clusters and clusters in stable equilibrium with the ambient phase depend on the volume of the system for initially fixed intensive state parameters of the ambient phase. Qualitatively, the results are shown to be similar independent on the boundary conditions employed. It is demonstrated further that the behavior of systems in confined space is directly related to the kinetics of phase transformation processes in spatially extended systems, when ensembles of clusters are formed. The results of the thermodynamic analysis of cluster formation and growth in a confined space are employed then, in particular, to the derivation of kinetic equations for the description of the process of coarsening or Ostwald ripening. In the analysis of both the nucleation in confined space and the description of Ostwald ripening, no specific assumptions concerning the equations of state of the system under consideration and the number of components both in the ambient and newly evolving phases are made. Consequently, the results are of very general nature and hold always as far as the necessary condition for the possibility of a phase transformation is fulfilled.

Size dependence of the contact angle of a nanodrop in a nanocavity: density functional theory considerations

The dependence of the contact angles of nanodrops of Lennard-Jones type fluids in nanocavities on their sizes are calculated using a nonlocal density functional theory in a canonical ensemble. Cavities of various radii and depths, various temperatures, as well as various values of the energy parameter of the fluid-solid interactions were considered. It is argued that this dependence might affect strongly, for instance, the rate of heterogeneous nucleation on rough surfaces, which is usually calculated under the assumption of constant contact angle.

Cavitation in metastable liquid nitrogen confined to nanoscale pores

We studied cavitation in metastable fluids drawing on the example of liquid nitrogen confined to spheroidal pores of specially prepared well-characterized mesoporous silica materials with mean pore diameters ranging from approximately 6 to approximately 35 nm. Cavitation was monitored in the process of evaporation/desorption from fully saturated samples with gradually decreasing vapor pressure at the isothermal conditions. The onset of cavitation was displayed by a sharp step on the desorption isotherm. We found that the vapor pressure at the onset of cavitation depended on the pore size for the samples with pores smaller than approximately 11 nm and remained practically unchanged for the samples with larger pores. We suggest that the observed independence of the cavitation pressure on the size of confinement indicates that the conditions of bubble nucleation in pores larger than approximately 11 nm approach the nucleation conditions in the bulk metastable liquid. To test this hypothesis and to evaluate the nucleation barriers, we performed grand canonical and gauge cell Monte Carlo simulations of nitrogen adsorption and desorption in spherical silica pores ranging from 5.5 to 10 nm in diameter. Simulated and experimental adsorption isotherms were in good agreement. Exploiting the correlation between the experimental cavitation pressure and the simulated nucleation barrier, we found that the nucleation barrier increased almost linearly from approximately 40 to approximately 70 k(B)T in the range of pores from approximately 6 to approximately 11 nm, and varied in diapason of 70-75 k(B)T in larger pores, up to 35 nm. We constructed the dependence of the nucleation barrier on the vapor pressure, which asymptotically approaches the predictions of the classical nucleation theory for the metastable bulk liquid at larger relative pressures (>0.6). Our findings suggest that there is a limit to the influence of the confinement on the onset of cavitation, and thus, cavitation of nanoconfined fluids may be employed to explore cavitation in macroscopic systems.

Microscopic description of a drop on a solid surface

Two approaches recently suggested for the treatment of macro- or nanodrops on smooth or rough, planar or curved, solid surfaces, based on fluid-fluid and fluid-solid interaction potentials are reviewed. The first one employs the minimization of the total potential energy of a drop by assuming that the drop has a well defined profile and a constant liquid density in its entire volume with the exception of the monolayer nearest to the surface where the density has a different value. As a result, a differential equation for the drop profile as well as the necessary boundary conditions are derived which involve the parameters of the interaction potentials and do not contain such macroscopic characteristics as the surface tensions. As a consequence, the macroscopic and microscopic contact angles which the drop profile makes with the surface can be calculated. The macroscopic angle is obtained via the extrapolation of the circular part of the drop profile valid at some distance from the surface up to the solid surface. The microscopic angle is formed at the intersection of the real profile (which is not circular near the surface) with the surface. The theory provides a relation between these two angles. The ranges of the microscopic parameters of the interaction potentials for which (i) the drop can have any height (volume), (ii) the drop can have a restricted height but unrestricted volume, and (iii) a drop cannot be formed on the surface were identified. The theory was also extended to the description of a drop on a rough surface. The second approach is based on a nonlocal density functional theory (DFT), which accounts for the inhomogeneity of the liquid density and temperature effects, features which are missing in the first approach. Although the computational difficulties restrict its application to drops of only several nanometers, the theory can be applied indirectly to macrodrops by calculating the surface tensions and using the Young equation to determine the contact angle. Employing the canonical ensemble version of the DFT, nanodrops on smooth and rough solid surfaces could be investigated and their characteristics, such as the drop profile, contact angle, as well as the fluid density distribution inside the drop can be determined as functions of the parameters of the interaction potentials and temperature. It was found that the contact angle of the drop has a simple (quasi)universal dependence on the energy parameter epsilon(fs) of the fluid-solid interaction potential and temperature. The main feature of this dependence is the existence of a fixed value theta(0) of the contact angle theta which separates the solid substrates (characterized by the energy parameter epsilon(fs) of the fluid-solid interaction potential) into two classes with respect to their temperature dependence. For theta>theta(0) the contact angle monotonously increases and for theta<theta(0) monotonously decreases with increasing temperature. For theta=theta(0) the contact angle is independent of temperature. The results obtained via DFT were also applied to check the validity of the macroscopic phenomenological equations (Cassie-Baxter and Wenzel equations) for drops on rough surfaces, and of the equation for the sticking force of a drop on an inclined surface.

Symmetry breaking in confined fluids

The recent progress in the theoretical investigation of the symmetry breaking (the existence of a stable state of a system, in which the symmetry is lower than the symmetry of the system itself) for classical and quantum fluids is reviewed. The emphasis is on the conditions which cause symmetry breaking in the density distribution for one component fluids and binary mixtures confined in a closed nanoslit between identical solid walls. The existing studies have revealed that two kinds of symmetry breaking can occur in such systems. First, a one-dimensional symmetry breaking occurs only in the direction normal to the walls as a fluid density profile asymmetric with respect of the middle of the slit and uniform in any direction parallel to the walls. Second, a two-dimensional symmetry breaking occurs in the fluid density distribution which is nonuniform in one of the directions parallel to the walls and asymmetrical in the direction normal to the walls. It manifests through liquid bumps and bridges in the fluid density distribution. For one component fluids, conditions of existence of symmetry breaking are provided in terms of the average fluid density, strength of fluid-solid interactions, distance at which the solid wall generates a hard core repulsion, and temperature. In the case of binary mixtures, the occurrence of symmetry breaking also depends on the composition of the confined mixtures.

Nucleation and hysteresis of vapor-liquid phase transitions in confined spaces: effects of fluid-wall interaction

In this work, we propose a method to stabilize a nucleus in the framework of lattice density-functional theory (LDFT) by imposing a suitable constraint. Using this method, the shape of critical nucleus and height of the nucleation barrier can be determined without using a predefined nucleus as input. As an application of this method, we study the nucleation behavior of vapor-liquid transition in nanosquare pores with infinite length and relate the observed hysteresis loop on an adsorption isotherm to the nucleation mechanism. According to the dependence of hysteresis and the nucleation mechanism on the fluid-wall interaction, w , in this work, we have classified w into three regions ( w>0.9 , 0.1< or =w< or =0.9 , and w<0.1 ), which are denoted as strongly, moderately, and weakly attractive fluid-wall interaction, respectively. The dependence of hysteresis on the fluid-wall interaction is interpreted by the different nucleation mechanisms. Our constrained LDFT calculations also show that the different transition paths may induce different nucleation behaviors. The transition path dependence should be considered if morphological transition of nuclei exists during a nucleation process.

Nanodrop on a nanorough solid surface: density functional theory considerations

The density distributions and contact angles of liquid nanodrops on nanorough solid surfaces are determined on the basis of a nonlocal density functional theory. Two kinds of roughness, chemical and physical, are examined. The former considers the substrate as a sequence of two kinds of semi-infinite vertical plates of equal thicknesses but of different natures with different strengths for the liquid-solid interactions. The physical roughness involves an ordered set of pillars on a flat homogeneous surface. Both hydrophobic and hydrophilic surfaces were considered. For the chemical roughness, the contact angle which the drop makes with the flat surface increases when the strength of the liquid-solid interaction for one kind of plates decreases with respect to the fixed value of the other kind of plates. Such a behavior is in agreement with the Cassie-Baxter expression derived from macroscopic considerations. For the physical roughness on a hydrophobic surface, the contact angle which a drop makes with the plane containing the tops of the pillars increases with increasing roughness. Such a behavior is consistent with the Wenzel formula developed for macroscopic drops. For hydrophilic surfaces, as the roughness increases the contact angle first increases, in contradiction with the Wenzel formula, which predicts for hydrophilic surfaces a decrease of the contact angle with increasing roughness. However, a further increase in roughness changes nonmonotonously the contact angle, and at some roughness, the drop disappears and only a liquid film is present on the surface. It was also found that the contact angle has a periodic dependence on the volume of the drop.

Adsorption-induced deformation of microporous carbons: pore size distribution effect

We present a thermodynamic model of adsorption-induced deformation of microporous carbons. The model represents the carbon structure as a macroscopically isotropic disordered three-dimensional medium composed of stacks of slit-shaped pores of different sizes embedded in an incompressible amorphous matrix. Adsorption stress in pores is calculated by means of Monte Carlo simulations. The proposed model reproduces qualitatively the experimental nonmonotonic dilatometric deformation curve for argon adsorption on carbide-derived activated carbon at 243 K and pressure up to 1.2 MPa. The elastic deformation (contraction at low pressures and swelling at higher pressures) results from the adsorption stress that depends strongly on the pore size. The pore size distribution determines the shape of the deformation curve, whereas the bulk modulus controls the extent of the sample deformation.

Macrophase and microphase separations for surfactants adsorbed on solid surfaces: a gauge cell monte carlo study in the lattice model

By combining the gauge cell method and lattice model, we study the surface phase transition and adsorption behaviors of surfactants on a solid surface. Two different cases are considered in this work: macrophase transition and adsorption in a single-phase region. For the case of macrophase transition, where two phases coexist, we investigate the shape and size of the critical nuclei and determine the height of the nucleation barrier. It is found that the nucleation depends on the bulk surfactant concentration. Our simulations show that there exist a critical temperature and critical adsorption energy, below which the transition from low-affinity adsorption to the bilayer structure shows the characteristic of a typical first-order phase transition. Such a surface phase transition in the adsorption isotherm is featured by a hysteresis loop. The hysteresis loop becomes narrower at higher temperature and weaker adsorption energy and finally disappears at the critical value. For the case where no macrophase transition occurs, we study the adsorption isotherm and microphase separation in a single-phase region. The simulation results indicate that the adsorption isotherm in adsorption processes is divided into four regions in a log-log plot, being in agreement with experimental observations. In this work, the four regions are called the low-affinity adsorption region, the hemimicelle region, the morphological transition region, and the plateau region. Simulation results reveal that in the second region the adsorbed monomers aggregate and nucleate hemimicelles, while adsorption in the third region is accompanied by morphological transitions.

Adsorption of fluids in a pore with chemical heterogeneities: the cooperative effect

In this work, we study the cooperative adsorption of fluids in a heterogeneous pore, in which the pore walls are composed of homogeneous substrates with chemical groups (CGs) decorating them. The adsorption caused by the homogeneous substrates alone and that by CGs do not add up to the overall adsorption, indicating the existence of a cooperative effect. The cooperative effect is the source of cooperative adsorption, and is characterized in this work by the ratio of the overall adsorption to the sum of adsorption by the substrate only and that by CGs. It is found that the cooperative adsorption does not depend monotonically on the substrate or the CGs. Two different origins of the cooperative adsorption play different roles depending on which one dominates the overall adsorption. Our simulations reveal that, when the homogeneous substrate dominates the overall adsorption, weakening of the attractive fluid-substrate interaction or alternatively strengthening of the fluid-CGs interaction leads to a stronger cooperative effect and enhances the cooperative adsorption. However, when CGs dominate the overall adsorption, weakening of the attractive fluid-CG interaction or strengthening the fluid-substrate interaction results in strong cooperative adsorption. In order to investigate the effects of the distribution of CGs on cooperative adsorption, a design-test method is generalized and used in this work. Simulation results show that the overall adsorption can be significantly affected by the CG distribution.

Symmetry breaking in binary mixtures in closed nanoslits

The symmetry breaking (SB) of the fluid density distribution (FDD) in closed nanoslits between two identical parallel solid walls described by Berim and Ruckenstein [J. Chem. Phys. 128, 024704 (2008)] for a single component fluid is examined for binary mixtures on the basis of a nonlocal canonical ensemble density functional theory. As in Monte Carlo simulations, the periodicity of the FDD in one of the lateral (parallel to the wall surfaces) directions, denoted as the x direction, was assumed. In the other lateral direction, y direction, the FDD was considered to be uniform. The molecules of the two components have different diameters and their Lennard-Jones interaction potentials have different energy parameters. It was found that depending on the average fluid density in the slit and mixture composition, SB can occur for both or none of the components but never for only one of them. In the direction perpendicular to the walls (h direction), the FDDs of both components can be asymmetrical about the middle plane between walls. In the x direction, the SB occurs as bumps and bridges enriched in one of the components, whereas the composition of the mixture between them is enriched in the other component. The dependence of the SB states on the length Lx of the FDD period at fixed average densities of the two components was examined for Lx in the range from 10 to 120 molecular diameters of the smaller size component. It was shown that for large Lx, the stable state of the system corresponds to a bridge. Because the free energy of that state decreases monotonically with increasing Lx, one can conclude that the real period is very large (infinite) and that a single bridge exists in the slit.

Two-dimensional symmetry breaking of fluid density distribution in closed nanoslits

Stable and metastable fluid density distributions (FDDs) in a closed nanoslit between two identical parallel solid walls have been identified on the basis of a nonlocal canonical ensemble density functional theory. Similar to Monte Carlo simulations, periodicity of the FDD in one of the lateral (parallel to the walls surfaces) directions, denoted as the x direction, was assumed. In the other lateral direction, y direction, the FDD was considered uniform. It was found that depending on the average fluid density in the slit, both uniform as well as nonuniform FDDs in the x direction can occur. The uniform FDDs are either symmetric or asymmetric about the middle plane between walls; the latter FDD being the consequence of a symmetry breaking across the slit. The nonuniform FDDs in the x direction occur either in the form of a bump on a thin liquid film covering the walls or as a liquid bridge between those walls and provide symmetry breaking in the x direction. For small and large average densities, the stable state is uniform in the x direction and is symmetric about the middle plane between walls. In the intermediate range of the average density and depending on the length L(x) of the FDD period, the stable state can be represented either by a FDD, which is uniform in the x direction and asymmetric about the middle of the slit (small values of L(x)), or by a bump- and bridgelike FDD for intermediate and large values of L(x), respectively. These results are in agreement with the Monte Carlo simulations performed earlier by other authors. Because the free energy of the stable state decreases monotonically with increasing L(x), one can conclude that the real period is very large (infinite) and that for the values of the parameters employed, a single bridge of finite length over the entire slit is generated.

Distribution of carbon nanotube sizes from adsorption measurements and computer simulation

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

Mechanism of adsorption in cylindrical nanopores: The roles of fluctuations and correlations in stabilizing the adsorbed phase

The mechanism of the adsorption in nanometric cylindrical pores has been studied using grand canonical Monte Carlo simulations. The results have been analyzed from the point of view of microscopic correlations. It has been shown that the correlations between the energy components and between the energy and the number of adsorbed particles provide crucial information concerning the microscopic mechanism of the formation of adsorbed layers. Typical susceptibility functions have been calculated. They give the relations between the statistical correlations and the stability of the adsorbed system in different stages of adsorption. The numerical calculations have been carried out for Kr atoms adsorbed in an MCM-41 model porous material with pores of diameter d = 4 nm. The smooth-wall model as well as the model of a wall with micropores have been discussed.

Capillary condensation in cylindrical nanopores

Using grand canonical Monte Carlo simulations, we have explored the phenomenon of capillary condensation (CC) of Ar at the triple temperature inside infinitely long, cylindrical pores. Pores of radius R = 1 nm, 1.7 nm, and 2.5 nm have been investigated using a gas-surface interaction potential parametrized by the well depth D of the gas on a planar surface made of the same material as that comprising the porous host. For strongly attractive situations--i.e., large D--one or more (depending on R) Ar layers adsorb successively before liquid fills the pore. For very small values of D, in contrast, negligible adsorption occurs at any pressure P below saturated vapor pressure P0; above saturation, there eventually occurs a threshold value of P at which the coverage jumps from empty to full, nearly discontinuously. Hysteresis is found to occur in the simulation data whenever abrupt CC occurs--i.e., for R > or = 1.7 nm--and for small D when R = 1 nm. Then, the pore-emptying branch of the adsorption isotherm exhibits larger coverage than the pore-filling branch, as is known from many experiments and simulation studies. The relation between CC and wetting on planar surfaces is discussed in terms of a threshold value of D, which is about one-half of the value found for the wetting threshold on a planar surface. This finding is consistent with a simple thermodynamic model of the wetting transition developed previously.

Hysteresis and scanning behavior of mesoporous molecular sieves

Sorption hysteresis is a widely studied phenomenon whose predicted behavior is well documented and researched. On the other hand, there is much less known about the region that lies between sorption isotherms, believed to be a metastable region. Scanning curves are a way of understanding the mechanism of hysteresis and a tool for hysteresis model validation. Scanning curves were produced for mesoporous materials: SBA-15 and MCM-41 for N(2) sorption at 77 K and Ar sorption at 87 K. A limited set of different scanning behaviors is identified. Like most hysteresis theories, it was found that a single model for scanning behavior cannot be extended to all materials under the same or different experimental conditions. Two behaviors are consistent with recent theories and simulations; however, several are not. The implications as to the characterization of pore dimensions and structure are discussed.

A simulation method for the calculation of chemical potentials in small, inhomogeneous, and dense systems

We present a modification of the gauge cell Monte Carlo simulation method [A. V. Neimark and A. Vishnyakov, Phys. Rev. E 62, 4611 (2000)] designed for chemical potential calculations in small confined inhomogeneous systems. To measure the chemical potential, the system under study is set in chemical equilibrium with the gauge cell, which represents a finite volume reservoir of ideal particles. The system and the gauge cell are immersed into the thermal bath of a given temperature. The size of the gauge cell controls the level of density fluctuations in the system. The chemical potential is rigorously calculated from the equilibrium distribution of particles between the system cell and the gauge cell and does not depend on the gauge cell size. This scheme, which we call a mesoscopic canonical ensemble, bridges the gap between the canonical and the grand canonical ensembles, which are known to be inconsistent for small systems. The ideal gas gauge cell method is illustrated with Monte Carlo simulations of Lennard-Jones fluid confined to spherical pores of different sizes. Special attention is paid to the case of extreme confinement of several molecular diameters in cross section where the inconsistency between the canonical ensemble and the grand canonical ensemble is most pronounced. For sufficiently large systems, the chemical potential can be reliably determined from the mean density in the gauge cell as it was implied in the original gauge cell method. The method is applied to study the transition from supercritical adsorption to subcritical capillary condensation, which is observed in nanoporous materials as the pore size increases.

Phase coexistence in heterogeneous porous media: A new extension to Gibbs ensemble Monte Carlo simulation method

The effect of confinement on phase behavior of simple fluids is still an area of intensive research. In between experiment and theory, molecular simulation is a powerful tool to study the effect of confinement in realistic porous materials, containing some disorder. Previous simulation works aiming at establishing the phase diagram of a confined Lennard-Jones-type fluid, concentrated on simple pore geometries (slits or cylinders). The development of the Gibbs ensemble Monte Carlo technique by Panagiotopoulos [Mol. Phys. 61, 813 (1987)], greatly favored the study of such simple geometries for two reasons. First, the technique is very efficient to calculate the phase diagram, since each run (at a given temperature) converges directly to an equilibrium between a gaslike and a liquidlike phase. Second, due to volume exchange procedure between the two phases, at least one invariant direction of space is required for applicability of this method, which is the case for slits or cylinders. Generally, the introduction of some disorder in such simple pores breaks the initial invariance in one of the space directions and prevents to work in the Gibbs ensemble. The simulation techniques for such disordered systems are numerous (grand canonical Monte Carlo, molecular dynamics, histogram reweighting, N-P-T+test method, Gibbs-Duhem integration procedure, etc.). However, the Gibbs ensemble technique, which gives directly the coexistence between phases, was never generalized to such systems. In this work, we focus on two weakly disordered pores for which a modified Gibbs ensemble Monte Carlo technique can be applied. One of the pores is geometrically undulated, whereas the second is cylindrical but presents a chemical variation which gives rise to a modulation of the wall potential. In the first case almost no change in the phase diagram is observed, whereas in the second strong modifications are reported.

Vapor-to-droplet transition in a Lennard-Jones fluid: simulation study of nucleation barriers using the ghost field method

We report a comprehensive Monte Carlo (MC) simulation study of the vapor-to-droplet transition in Lennard-Jones fluid confined to a spherical container with repulsive walls, which is a case study system to investigate homogeneous nucleation. The focus is made on the application of a modified version of the ghost field method (Vishnyakov, A.; Neimark, A. V. J. Chem. Phys. 2003, 119, 9755) to calculate the nucleation barrier. This method allows one to build up a continuous trajectory of equilibrium states stabilized by the ghost field potential, which connects a reference droplet with a reference vapor state. Two computation schemes are employed for free energy calculations, direct thermodynamic integration along the constructed trajectory and umbrella sampling. The nucleation barriers and the size dependence of the surface tension are reported for droplets containing from 260 to 2000 molecules. The MC simulation study is complemented by a review of the simulation methods applied to computing the nucleation barriers and a detailed analysis of the vapor-to-droplet transition by means of the classical nucleation theory.

Thermodynamic Characterization of Fluids Confined in Heterogeneous Pores by Monte Carlo Simulations in the Grand Canonical and the Isobaric−Isothermal Ensembles

Materials presenting nanoscale porosity are able to condense gases in their structure. This "capillary condensation" phenomenon has been studied for more than one century. Theoretical models help to understand experimental results but fail in explaining all experimental features. Most of the time, the difficulties in making quantitative or even qualitative predictions are due to the geometric complexity of the porous materials, such as large pore size distribution, chemical heterogeneities, or pore interconnections. Numerical calculations (lattice gas models or molecular simulations) are of considerable interest to calculate the adsorption properties of a fluid confined in a porous model with characteristic sizes up to several tens of nanometers. For instance, the grand canonical Monte Carlo method allows one to compute the average amount of fluid adsorbed in the porous model as a function of the temperature and the chemical potential of the fluid. However, the grand potential, necessary for a complete characterization of the system, is not a direct output of the algorithm. It is shown in this paper that the use of the isobaric-isothermal (NPT) ensemble allows one to circumvent this problem; that is, it is possible to get in one single Monte Carlo run the absolute grand potential for any given thermodynamic state of the fluid. A simplified thermodynamic integration scheme is then used to evaluate the grand potential over the whole isotherm branch passing through this initially given point. Since the usual NPT technique is a priori limited to homogeneous pores, it is proposed, for the first time, to generalize this procedure to a pore presenting a chemical heterogeneity along its axis. The new method gives the same results as the previous for homogeneous pores and allows new predictions for chemically heterogeneous pores. Comparison with the full integration scheme shows that the proposed direct calculation is faster since it avoids multiple Monte Carlo runs and more precise because it avoids the possible cumulative errors of the integration procedure.

Grand Potential, Helmholtz Free Energy, and Entropy Calculation in Heterogeneous Cylindrical Pores by the Grand Canonical Monte Carlo Simulation Method

The adsorption of fluids in porous media is still an open area of research, since no model is able to explain all experimental features. The difficulties rise from the complexity of the real porous materials which present surface heterogeneities, large pore size distributions, and complex networks of interconnected pores. In parallel to experimental efforts trying to produce more ordered porous materials, theoreticians try to introduce more disorder in their models, with the help of molecular simulation for instance. This grand canonical Monte Carlo simulation study concentrates on the adsorption of a simple Lennard-Jones fluid in three porous substrates, to compare the effect of purely geometric heterogeneity (spatial deformation of the external potential) as opposed to purely chemical heterogeneity (amplitude variations of the external potential). This separation is unrealistic, since geometric fluctuations of a real pore diameter along its axis generally induce variations in the amplitude of the external potential created by the pore. However it enables one to compare both effects. In this paper, a thermodynamic integration scheme is applied to a complete set of adsorption/desorption isotherms. The grand potential, free energy, and entropy are calculated, which allows one to discuss the features of the phase diagrams. It is shown that a purely geometric deformation (undulation) of the external potential does not affect the thermodynamic characteristics of the confined fluid. On the other hand, amplitude modulation of the external potential (chemical heterogeneity) strongly distorts the phase diagram. This heterogeneity is actually able to stabilize a "bridgelike" phase which corresponds to an accumulation of molecules in the most attractive region of the pore.

Nucleation in cylindrical capillaries

We use a local density functional theory in the square gradient approximation to explore the properties of critical nuclei for the liquid-vapor transition of van der Waals fluids in cylindrical capillaries. The proposed model allows us to investigate the effect of pore size, surface field, and supersaturation on the behavior of the system. Our calculations predict the existence of at least three different pathways for the nucleation of droplets and bubbles in these confined fluids: axisymmetric annular bumps and lenses, and asymmetric droplets. The morphological transition between these different structures is driven by the existence of states of zero compressibility in the capillary. We show that the classical capillarity theory provides surprisingly accurate predictions for the work of formation of critical nuclei in cylindrical pores when line tension contributions to the free energy are taken into account.