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

Phase Behavior and Capillary Condensation Hysteresis of Carbon Dioxide in Mesopores

Carbon dioxide adsorption on micro- and mesoporous carbonaceous materials in a wide range of temperatures and pressures is of great importance for the problems of gas separations, greenhouse gas capture and sequestration, enhanced hydrocarbon recovery from shales and coals, as well as for the characterization of nanoporous materials using CO₂ as a molecular probe. We investigate the influence of temperature on CO₂ adsorption focusing on the capillary condensation and hysteresis phenomena. We present experimental data on the adsorption of CO₂ on CMK-3, ordered carbon with mesopores of ∼5-6 nm, at various temperatures (185-273 K) and pressures (up to 35 bars). Using Monte Carlo (MC) simulations in the grand canonical and mesocanonical ensembles, we attempt to predict the transition from reversible capillary condensation to hysteretic adsorption-desorption cycles that is experimentally observed with the decrease of temperature. We show that although the desorption at all temperatures occurs at the conditions of pore vapor-liquid equilibrium, the capillary condensation is a nucleation-driven process associated with an effective energy barrier of ∼43 kT, specific to the sample used in this work. This barrier can be overcome at the equilibrium conditions in the region of reversible condensation at temperatures higher than 240 K. At lower temperatures, the regime of developing hysteresis is observed with progressively widening hysteresis loops. The position of capillary condensation transition is estimated using the pressure dependence of the energy barrier calculated by the thermodynamic integration of the van der Waals-type continuous canonical isotherm simulated with the gauge cell MC method. These findings lay the foundation for developing kernels of CO₂ adsorption and desorption isotherm for calculating the pore size distribution in the entire range of micropore and mesopore sizes from one high-pressure experimental isotherm.

Chemical potential calculations in non-homogeneous liquids

The numerical computation of chemical potential in dense non-homogeneous fluids is a key problem in the study of confined fluid thermodynamics. To this day, several methods have been proposed; however, there is still need for a robust technique, capable of obtaining accurate estimates at large average densities. A widely established technique is the Widom insertion method, which computes the chemical potential by sampling the energy of insertion of a test particle. Non-homogeneity is accounted for by assigning a density dependent weight to the insertion points. However, in dense systems, the poor sampling of the insertion energy is a source of inefficiency, hampering a reliable convergence. We have recently presented a new technique for the chemical potential calculation in homogeneous fluids. This novel method enhances the sampling of the insertion energy via well-tempered metadynamics, reaching accurate estimates at very large densities. In this paper, we extend the technique to the case of non-homogeneous fluids. The method is successfully tested on a confined Lennard-Jones fluid. In particular, we show that, thanks to the improved sampling, our technique does not suffer from a systematic error that affects the classic Widom method for non-homogeneous fluids, providing a precise and accurate result.

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

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

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

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

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.

Determination of phase equilibria in confined systems by open pore cell Monte Carlo method

We present a modification of the molecular dynamics simulation method with a unit pore cell with imaginary gas phase [M. Miyahara, T. Yoshioka, and M. Okazaki, J. Chem. Phys. 106, 8124 (1997)] designed for determination of phase equilibria in nanopores. This new method is based on a Monte Carlo technique and it combines the pore cell, opened to the imaginary gas phase (open pore cell), with a gas cell to measure the equilibrium chemical potential of the confined system. The most striking feature of our new method is that the confined system is steadily led to a thermodynamically stable state by forming concave menisci in the open pore cell. This feature of the open pore cell makes it possible to obtain the equilibrium chemical potential with only a single simulation run, unlike existing simulation methods, which need a number of additional runs. We apply the method to evaluate the equilibrium chemical potentials of confined nitrogen in carbon slit pores and silica cylindrical pores at 77 K, and show that the results are in good agreement with those obtained by two conventional thermodynamic integration methods. Moreover, we also show that the proposed method can be particularly useful for determining vapor-liquid and vapor-solid coexistence curves and the triple point of the confined system.

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.

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.

Binary mixture adsorbed in a slit pore: Field-induced population inversion near the bulk instability

The recently proposed method for modulating through an external field the composition of a binary fluid mixture adsorbed in a slit pore is discussed. The population inversion near the bulk (demixing) instability is first analyzed in the case of a symmetric mixture of nonadditive hard spheres, without field. It is next investigated for a mixture comprising dipolar particles subject to an external field. The influence of several factors on the adsorption curves including bulk composition, pore width, field direction, polarizability versus permanent dipoles, and temperature on this field induced population inversion is shown by Monte Carlo simulation.

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.

Studies of capillary phase transitions of methane in metal-organic frameworks by gauge cell Monte Carlo simulation

Capillary phase transitions of CH(4) confined in a series of metal-organic frameworks (MOFs) were investigated in this work using gauge cell Monte Carlo simulations. The results show that capillary phase transitions can occur in MOFs, and the effects of temperature, pore size, and adsorption energy are very significant. Furthermore, this work shows the confinement can induce a shift in critical point for fluids confined in MOFs, leading to a decrease in critical temperature and an increase in critical density. The critical point shift is more obvious for MOFs with small pore size and large adsorption energy.

Solvation forces between silica bodies in supercritical carbon dioxide

We report Monte Carlo simulations of the solvation pressure between two planar surfaces, which represent the interface of spherical silica nanoparticles in supercritical carbon dioxide. Carbon dioxide (CO2) was modeled as an atomistic dumbbell or a spherical Lennard-Jones particle. The interaction between CO2 molecules and silica surfaces was characterized by the standard Steele potential with energetic heterogeneities representing the hydrogen bonds. The parameters for the solid-fluid interaction potentials were obtained by fitting our simulations to the experimental isotherms of CO2 sorption on mesoporous siliceous materials. We studied the dependence of the solvation force on the distance between planar silica surfaces at T = 318 K, at equilibrium bulk pressures p(bulk) ranging from 69 to 200 atm. At 69 atm, we observed a long-range attraction between the two surfaces, and it vanished when the pressure was increased to 102 and then 200 atm. The results obtained with different fluid models were consistent with each other. According to our observations, energetic heterogeneities of the surface have negligible influence on the solvation pressure. Using the Derjaguin approximation, we calculated the solvation forces between spherical silica nanoparticles in supercritical CO2 from the solvation pressures between the planar surfaces.

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.

Adsorption equilibria of hydrocarbons in the structure of the reforming catalyst: Molecular modeling

Molecular modeling was used to analyze the phenomena involved in the sorption of hydrocarbons by the Pt/Al2O3 reforming catalyst. The interactions between the atoms of the catalyst structure and the molecules of the model reforming compounds were described in terms of the universal forcefield. Making use of the GCMC algorithms, the adsorption isotherms for the reagents in the catalytic system and the temperature dependence of the Henry constant were determined. The research has produced the following major findings: the amount of the hydrocarbon molecules adsorbed rises with increasing pressure and decreasing temperature, and the adsorption isotherm for toluene has a characteristic plot as compared to the isotherms of the other hydrocarbons studied. Mass cloud analysis has revealed a favorable effect of platinum on adsorption in the catalyst model.

Phase Transitions and Criticality in Small Systems: Vapor−Liquid Transition in Nanoscale Spherical Cavities

Phase transformations in fluids confined to nanoscale pores, which demonstrate characteristic signatures of first-order phase transitions, have been extensively documented in experiments and molecular simulations. They are characterized by a pronounced hysteresis, which disappears above a certain temperature. A rigorous interpretation of these observations represents a fundamental problem from the point of view of statistical mechanics. Nanoscale systems are essentially small, finite volume systems, in which the concept of the thermodynamic limit is no longer valid, and the statistical ensembles are not equivalent. Here, we present a rigorous approach to the description and molecular simulations of phase transitions and criticality in small confined systems, as illustrated by the example of vapor-liquid transition (capillary condensation) in spherical cavities. The method is based on the analysis of the canonical ensemble isotherms, which can be generated by the gauge cell Monte Carlo simulation method. The method allows one to define the critical temperature of phase transition, conditions of phase equilibrium, limits of stability of metastable states, and nucleation barriers, which determine hysteretic phase transformations.