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A calculation procedure of the surface tension (ST) of vapor–liquid menisci in porous bodies based on the microscopic lattice gas model (LGM) is presented. Ideas of this model are used to describe a two-level structural model of mesoporous bodies and to calculate adsorbate distributions in a given structure of an adsorbent. The supramolecular level of the model is presented as a granule/grain of a porous material with the set distribution function of connected pores of different sizes and types. At the molecular level, a LGM ensures an equally accurate description of adsorbate molecular distributions in the coexisting vapor and liquid phases and at their interface. ST is calculated through a local excess of the free energy that is generated by vapor–liquid interfaces inside the pores. Calculated STs characterize the spatial distribution of mobile phases in porous systems. The capability to obtain local information about ST, which reflects the size and energy features of porous systems, significantly exceeds the capabilities of the thermodynamic approach.

Three types of two-phase interfaces (vapor–liquid, solid–vapor, and solid–liquid) are considered in the “vapor and liquid meniscus” system inside a slitlike pore with rough walls. A unified description of these interface surfaces is given based on the lattice gas model that enables an equally accurate calculation of molecular distributions in heterogeneous distributed models of transition regions of interfaces. Undeformable pore walls generate an external field affecting the molecular distribution and forming adsorption films due to the adsorbate–adsorbent interaction potential. Surface tensions (STs) are calculated from the excess of the free energy of the interface (according to Gibbs) for each three types of two-phase interfaces. The ST for the solid–liquid interface corresponds to the surface passing through the contacting phases, i.e., via bonds between the adsorbate and the adsorbent. The state of coexisting “vapor in a pore” and “liquid in a pore” phases must satisfy the equality of chemical potentials, excluding the appearance of metastable states. Distinctions introduced by the wall roughness are mainly observed for narrow pores and they decrease as the pore width increases. The wall roughness changes the critical parameters of fluid stratification into liquid and vapor phases. The calculated ST values are compared with similar values for smooth walls of a slitlike pore. It is found that solid–liquid and solid–vapor STs are close to each other (their difference is smaller on a rough surface than on the smooth one) and are larger (in their absolute value) than the liquid–vapor ST in the center of the pore by approximately an order of magnitude. Local vapor–liquid ST values nonmonotonically change when removed from the wall.

An adsorbate can simultaneously affect the external and internal surfaces of an adsorbent, changing the volume of a sample with a fixed amount of the adsorbent. This process is discussed from the standpoints of mechanics of continuous media, thermodynamics, and molecular models. The microscopic theory of solids distinguishes between external and internal deformations, which makes it possible to associate mechanical characteristics with adsorption ones. Based on the lattice gas model, the principles of two-level structural models are formulated for deformable porous bodies. They enable the self-consistent description of changes in their volumes and adsorption isotherms as a function of the adsorbate external pressure at a fixed temperature. For the simple presentation of the calculation scheme, it is accepted that the atomic sizes of a solid and components of the adsorbate mobile phase are comparable. The molecular level reflects the volume of molecules and their lateral interaction in the quasi-chemical approximation. The supramolecular level of the model is presented as a granule/grain of a porous material with the set distribution function of different (in size and type) pores connected with each other. A relationship between internal deformations and the nonequilibrium state of a solid is shown. A calculation procedure for local mechanical modules with regard to internal and external deformations characterizing the mechanical properties of solids is exemplified by the compression modulus. The procedure takes into account the effect of the nonequilibrium of an adsorbent on adsorbent isotherms and its volume. The approach proposed can be also applied to related systems in which the adsorption of active mobile solutions by highly dispersed compounds changes their volume.

A unified description of three types of two-phase interfaces (vapor–liquid, solid–vapor, and solid–liquid) at a vapor–liquid meniscus inside a slit-like pore are considered on the basis of the lattice gas model. This approach allows equally accurate calculations of molecular distributions in heterogeneous distributed models of transitional regions at all interfaces. It is assumed that pore walls cannot be deformed, and they create an external field for a stratifying fluid. Adsorption films form at solid–mobile phase interfaces due to the potential of adsorbate–adsorbent interaction. The state of the coexisting vapor-in-a-pore and liquid-in-a-pore phases satisfy the equality of the chemical potential that excludes the appearance of metastable states. Conditions for distinguishing regions of the system that lie beyond solid–liquid–vapor three-phase contact are discussed. A procedure is discussed for introducing a contact angle into a liquid–vapor–solid pore wall system through molecular distributions of the adsorbate in a slit-like pore. Dependences of the width of the considered interfaces and the contact angle as a function of the pore width and pore wall potential are found.

The effect of adsorption on the characteristics of equilibrium rough surfaces of vapor–liquid and vapor–solid interfaces at a temperature close to the melting point is investigated. A discrete model of a dense phase—the lattice gas model, which includes direct correlations of all neighboring components of the system in a quasi-chemical approximation—is applied in the calculations. For simplicity, the model implies commensurability of the component size and the considered vibrations of particles through the effective contributions of the lateral interaction parameters for both adsorbent atoms and adsorbate molecules. The equilibrium profiles of adsorbent A and adsorbate B inside the transition area are calculated (provided that the adsorbent’s bulk phase remains unchanged) for physical adsorption and chemisorption. The effect of adsorption on the surface tension of the dense phase is examined. The probability of fluctuation processes of a new phase formation determined by the surface roughness of the pure adsorbent A, taking into account the adsorption of B particles on it, is estimated.

The fulfillment of the self-consistency condition in the description of adsorption and desorption the rates of dissociating vapor molecules on the rough surface of an adsorbent (equilibrium and tempered from an equilibrium state) is considered. The adsorption process causes the reconstruction of the equilibrium near-surface region of the adsorbent, but does not change the state of the tempered surface. The adsorption–desorption rates are calculated taking into account the interaction of the nearest neighbors in the quasi-chemical approximation. Four models are considered for the description of a nonuniform surface: the initial averaged multilayer model with different types of adsorption sites (and Henry constants), a model containing all the sites available for adsorption, and three types of its averaging. The first type of averaging is associated with the transition to a single-layer nonuniform surface containing different types of adsorption sites with different Henry constants, the second is associated with the transition to a set of effective uniform monolayers in the multilayer transition region, and the third is associated with the transition to a single-layer effective uniform surface. The condition of self-consistency is fulfilled in the last two types of averaging and is violated in the first type of averaging, as well as in the averaged model taking into account the differences between the types of sites. It is found that the state of a nonuniform surface (equilibrium or tempered) does not affect the self-consistency of the description of the rates of adsorption and desorption in the latter two types of models. The influence of taking into account the effects of correlation of the interaction of species on the condition of self-consistency is shown. The neglect of the effects of correlation of interacting species leads to the violation of the self-consistency condition in all models.

The work analyzes the energy characteristics of rough interfaces with and without taking into account adsorption. For simplicity, the theory of the lattice gas model is exemplified by taking into account a lateral interaction between the nearest components of the system. The behavior of the surface free and internal energies of the interface between the rough solid and the adsorbed gas is considered. The effect of the non-equilibrium states of interfaces on the energy characteristics of the surface are analyzed depending on the formation temperature of a flat rough surface. The difference between the excess surface energy at the solid–mobile phase interface and the non-equilibrium analogue of the surface tension of the same system, which transforms into the surface tension in the complete equilibrium state, is discussed. The effect of adsorption on the energy characteristics of the flat rough surface and the position of the equimolar surface is explored, depending on the gas pressure during physical adsorption and chemisorption. The means of calculating the fluctuation estimates of the formation probability of a rough solid boundary from the metastable state with a flat boundary is formulated. The size dependences of the formation probability of a circular region with a rough solid boundary on its curvature radius are obtained.

Ways of satisfying the condition for the self-consistent description of adsorption and desorption rates of non-dissociated molecules on a rough aerosol surface are considered. The rates are calculated with allowance for the interaction between nearest neighbors in the quasi-chemical approximation. Four types of models are analyzed. One describes the initial distributed heterogeneous surface, while the other three are variants averaged over the number of monolayers (with a transition to a single-layered heterogeneous surface) and different types of adsorption centers with different Henry constants (with a transition to an effective homogeneous monolayer in a multilayered transition region), or over the number of monolayers and types of adsorption centers (with a transition to the single-layered homogeneous surface) simultaneously. Types of models are distinguished for which the self-consistency condition is strictly satisfied and for which it is violated. Observed discrepancies are explained. The dependence of the degree of discrepancy (if there is one) on the density of the gas in the system, the energy of interaction of the activated complex, and the radius of adsorbent particles is established.

An approach to calculating the effects of density fluctuations is generalized with allowance for the movement of molecules in small condensed multicomponent heterogeneous systems (e.g., droplets, microcrystals, and adsorption on microcrystal faces). The collective character of statistical sums of vibrational, rotational, and translational motions of separate molecules in dense phases is expressed in dependences of these statistical sums on the local configurations of neighboring molecules. Using the quasi-chemical approximation with allowance for interparticle interactions reflecting the effects of direct short-range correlations, a relationship is derived between the statistical sum of a system consisting of a multicomponent mixture of molecules and the statistical sums of individual components. A structure is formulated for isotherm equations associating the density of mixture components and their chemical potentials in a thermostat, and for pair distribution functions.

The position of the inflection point in polylayer adsorption isotherms, which has been traditionally used to determine the specific surface of many disperse systems, is analyzed. The inflection point positions versus the molecular parameters of the adsorption system and the degree of surface nonuniformity are considered. The surface heterogeneity is exemplified by the droplet roughness formed on rapid cooling. The vapor–liquid-droplet interface creates the thermal heterogeneity of the surface on which gas-phase molecules are adsorbed after cooling. Conditions for using the inflection point to estimate the specific surface of silica samples during argon and nitrogen adsorption are discussed, along with distinctions between the monolayer capacity found from the isotherm and complete filling of the monolayer itself determined by the degree of filling surface centers.

An approach is developed for studying polymolecular adsorption on the modeled rough surface of a small aerosol obtained from a liquid droplet on its rapid cooling. A way of estimating the specific surface of adsorbent droplets with rough surfaces is proposed, and the temperature and size dependences of the specific surface are established. Isotherms of N2 and Ar polymolecular adsorption on a heterogeneous surface of small spherical particles of SiO2 are derived. The possibility of using this approach to describe an experiment is demonstrated. Comparison to the experimental isotherms reveals agreement with isotherms of argon and nitrogen on silica surfaces, with an error of up to 4.5%.

An approach is formulated that theoretically describes the structure of a rough surface of small aerosol particles obtained from a liquid droplet upon its rapid cooling. The problem consists of two stages. In the first stage, a concentration profile of the droplet–vapor transition region is calculated. In the second stage, local fractions of vacant sites and their pairs are found on the basis of this profile, and the rough structure of a frozen droplet surface transitioning to the solid state is calculated. Model parameters are the temperature of the initial droplet and those of the lateral interaction between droplet atoms. Information on vacant sites inside the region of transition allows us to identify adsorption centers and estimate the monolayer capacity, compared to that of the total space of the region of transition. The approach is oriented toward calculating adsorption isotherms on real surfaces.

A microscopic theory of adsorption, based on a discrete continuum lattice gas model for noninert (including deformable) adsorbents that change their lattice parameters during adsorption, is presented. Cases of the complete and partial equilibrium states of the adsorbent are considered. In the former, the adsorbent consists of coexisting solid and vapor phases of adsorbent components, and the adsorbate is a mobile component of the vapor phase with an arbitrary density (up to that of the liquid adsorbate phase). The adsorptive transitioning to the bound state changes the state of the near-surface region of the adsorbent. In the latter, there are no equilibrium components of the adsorbent between the solid and vapor phases. The adsorbent state is shown to be determined by its prehistory, rather than set by chemical potentials of vapor of its components. Relations between the microscopic theory and thermodynamic interpretations are discussed: (1) adsorption on an open surface, (2) two-dimensional stratification of the adsorbate mobile phase on an open homogeneous surface, (3) small microcrystals in vacuum and the gas phase, and (4) adsorption in porous systems.

A procedure for analyzing the formation processes of two-dimensional droplets of an adsorbate on a rigid adsorbent support is considered. The molecular theory is based on data on the potential functions between adsorbent atoms and adsorbate molecules. Interactions between nearest neighbors are considered in the quasi-chemical approximation. The internal motions of adsorbent atoms and adsorbate molecules are ignored. Problems of describing the formation of droplets on heterogeneous adsorbents are associated with calculations for binodals (illustrated with the simplest example of two different homogeneous crystal faces) due to the choice of methods for calculating linear tension and the structural model of the region of the liquid–vapor transition. The dependence of the characteristics of droplets in the layered structural model on the method for determining the reference lines of the tension is shown for their metastable and equilibrium states. It is found that for a number of structural parameters, the thermodynamic determination of the line of tensions of metastable droplets can result in nonmonotonic dependences of the linear tension on their radii. The characteristics of two-dimensional liquid–vapor interfaces are compared for two structural models: coordination sphere and layered. It is found that the coordination sphere model allows the exclusion of the structural parameter of the layered model, but both models need refinement at small radii.

A molecular statistical theory for calculating the linear tension of small multicomponent droplets in two-dimensional adsorption systems is developed. The theory describes discrete distributions of molecules in space (on a scale comparable to molecular size) and continuous distributions of molecules (at short distances inside cells) in their translational and vibrational motions. Pair intermolecular interaction potentials (the Mie type potential) in several coordination spheres are considered. For simplicity, it is assumed that distinctions in the sizes of mixture components are slight and comparable to the sizes of adsorbent adsorption centers. Expressions for the pressure tensor components inside small droplets on the heterogeneous surface of an adsorbent are obtained, allowing calculations of the thermodynamic characteristics of a vapor–fluid interface, including linear tension. Problems in refining the molecular theory are discussed: describing the properties of small droplets using a coordination model of their structure, considering the effect an adsorbate has on the state of a near-surface adsorbent region, and the surface heterogeneity factor in the conditions for the formation of droplets.

The size dependence of the linear tension of round two-dimensional equilibrium drops in the vapor phase on a homogeneous surface of an adsorbent is studied at the pressure of saturated two-dimensional vapor. The calculations are based on the lattice gas model in a quasi-chemical approximation with allowance for the correlation effects of the nearest interacting molecules. Methods for calculating linear tension using the equimolecular reference line are considered. Temperature dependences of the linear tension are studied for metastable and equilibrium drops. It is found that the differences between the thermodynamic properties of two types of drops are slight over a wide range of variation in drop radii.

The effect of internal motions of an adsorbate on the local characteristics of adsorption and layering phase diagrams are studied for structurally heterogeneous surfaces of slit-like pores. A molecular model describing adsorbate distributions inside slit-like pores, which is based on discrete distribution functions (lattice gas model), is used for the calculation. Molecular distributions are calculated by the Lennard-Jones potential (12–6) in a quasi-chemical approximation reflecting the effects of direct correlations of interacting particles and for the combined interaction of an adsorbate with walls in the average potential approximation (9–3) and the short-range Lennard-Jones potential for structurally heterogeneous surface areas. The conclusion is made that internal motions reflect the vibrational motion of molecules within a modified quasi-dimer model and a displacement of the adsorbate during its translational motion inside cells. It was found that the taking into account of internal motions decreases the critical temperature of adsorbate layering in slit-like pores.

The molecular theory of adsorbate melting in slit pores or near planar open surfaces is generalized to the case of arbitrary pores with a random character of the spatial distribution of adsorption centers (a fully distributed model). The states of liquid and crystalline adsorbates with regard to the contributions from vibrational motions are described using a unified molecular approach based on discrete distribution functions (the lattice gas model). Equations for the chemical potential of the adsorbate in a defect crystal and a vapor-liquid system are derived with allowance for their vibrational motion within the modified quasi-dimeric model. The concentration profile of a substance at the planar interface between two solid phases or a solid-liquid interface and inside a slit pore is calculated. Molecular distributions are calculated in a quasi-chemical approximation reflecting the effects of direct correlations of interacting particles via the Lennard–Jones potential. The transition to the averaged interaction potential of an adsorbate with structurally heterogeneous walls of slit pores, the transition to an averaged description of the lattice parameter of the system, and the problem of estimating the vibrational spectrum of the adsorption system are discussed.

The constraints of classical thermodynamics on analyzing nanosized objects are considered. It is shown that it is essential to separate the thermodynamic concepts of a cluster and a drop. Using the lattice gas model, a molecular statistical theory of small multicomponent mixtures is developed that considers inter-particle interactions in several coordination spheres and the effects of size equilibrium fluctuations. Kinetic equations describing the particle redistribution dynamics inside small bodies and the problems of allowing for size fluctuations in them are discussed. The principles of the molecular statistical theory are compared to results from applying molecular dynamics to modeling the states of small clusters and the surface states of metal atoms during their adsorption and diffusion on graphene.

The phase state of adsorbate in porous systems is considered, in which the walls are characterized by interaction potential with adsorbate: from strong attraction to absence of attraction. The equations are based on discrete distribution functions (the lattice gas model) taking into account the contribution of the vibration motions of adsorbate. The vibration motion of molecules is taken into account in the modified quasi-dimer Mie model. Molecular distributions are calculated in the quasi-chemical approximation reflecting the effects of direct correlations of interacting particles using the Lennard-Jones potential. Conditions of express estimates of vibration frequencies are discussed in which vibration frequencies weakly affect molecular distributions in the case of strong adsorption. It is found that consideration of vibrations results in a shift in adsorption isotherms in the coordinates of chemical potential-density to higher values of vapor pressure in the range of high fillings. This is related to the necessity of additional vapor “compression” for its transition into the adsorbed state characterized by higher kinetic energy.

A molecular model of adsorbate melting in slit-like pores [1] is used to calculate adsorption isotherms with regard to the contribution from vibrational motions of the adsorbate. The equations are based on discrete distribution functions (the lattice gas model). Molecular distributions are calculated in a quasi-chemical approximation reflecting the effects of direct correlations of interacting particles using the Lennard-Jones potential. The vibrational motion of molecules is taken into account using a modified quasi-dimer Mie model. It is shown that considering vibrations shifts the adsorption isotherms in the chemical potential-density coordinates to higher vapor pressures in the region of high filling. This is due to the need for additional compression of the vapor to transfer it into the adsorbed state with increased kinetic energy.

Problems that arise in refining calculations of the equilibrium thermodynamic functions of an adsorbate located near an adsorbent’s surface or inside slit-like pores when its collective vibrational motions are considered are discussed. A technique is proposed for calculating collective vibrations for a small number of vacancies in a defective heterogeneous adsorbate from changes in the number of degrees of freedom and the local frequencies in the frequency distribution function in an ideal bulk crystal.

Molecular principles of a theory of adsorbate melting near the open surfaces of adsorbents in the frozen state or in slit-shaped pores are discussed. The states of liquid and crystalline adsorbates are described in terms of a single molecular approach (the lattice gas model). The crystalline state is described using the concept of quasi-average distributions, for which the degeneracy of the density distribution function in the plane of the adsorbent is eliminated. Equations for the chemical potential of the adsorbate in defective crystals and vapor-liquid systems are derived with allowance for their vibrational motion, making it possible to calculate the concentration profile of the substance at the planar interface between two solid phases, between a solid and a liquid, and inside a slit-shaped pore.

The correctness of using the macroscopic Gibbs phase rule in microheterogeneous systems is considered. Results from using molecular theory are discussed for a single-component substance in describing delamination and fusion on open surfaces and in the pores of a solid, along with the interfaces between three states of a substance in the bulk phase. It is established that the form of adsorption system phase diagrams is determined by the surface potential, depending on the composition and structure of heterogeneous surfaces, and is characterized by multiple first-order phase transitions. It is concluded that the thermodynamic approach to heterogeneous systems makes any microscale description of the real interface distribution of molecules less accurate, and all quasithermodynamic theories are insufficiently precise for describing small systems (adsorption systems and interfaces).

We investigate the effect of restricting the area of planes of microcrystals and equilibrium density fluctuations in components of binary mixtures on partial isotherms of the adsorption of binary mixtures of molecules and the rate of a surface reaction of the Langmuir-Hinshelwood type. Adsorption of components of mixture is considered in a large canonical assembly, and the rate of an elementary step is calculated in kinetic regime. The value of a section of the surface on a plane contains a number of adsorption centers in the range of 10 to 105. The effect of the structure of a heterogeneous surface on the rate of the considered reaction is studied. The effect of the density fluctuations of adsorbed molecules on partial adsorption isotherms and fluctuations in the rate of reaction on heterogeneous surfaces is discussed. It is shown that the greatest effect of density fluctuations on the rate of a step is observed at low fillings of each plane of a particle and at the almost complete filling of a plane.

Equations for the isotherms of adsorption of pure substances on small inhomogeneous crystalline particles with taking into account density fluctuations and intermolecular interactions in the quasi-chemical approximation were obtained for the first time.

The effect of the finiteness of face areas and equilibrium fluctuations on the adsorption isotherms of binary molecular mixtures and the rate of surface processes occurring on nanosized particles is studied. The adsorption process is considered in the grand canonical ensemble; the rates of elementary stages are calculated in the kinetic mode. The surface region on the face is found to contain a number of adsorption centers ranging from 10 to 105. The effect of density fluctuations of adsorbed molecules on the partial adsorption isotherms and rate fluctuations is discussed. A calculation procedure for density fluctuations in heterogeneous microparticles with different faces and different binding energies of molecules with the surface is considered. It is shown that the greatest effect of density fluctuations is manifested at a low occupancy of each face of the particle.

Principles of the fluctuation theory of microheterogeneous systems are considered based on the example of adsorption on a heterogeneous surface of microcrystals and spherical liquid droplets in the vapor phase. Equations that describe the equilibrium distributions of molecules are derived within the lattice gas model with allowance for lateral interactions in the mean-field approximation. This approximation does not take into account the correlation effects of interacting particles. We discuss the structure of entropy and energy contributions in the derived equations and the relationship between the molecular chemical potential and the expansion pressure, which is analogous to the equation of the state, as well as the molecular interpretation of the concept of pressure in an ensemble of small particles and its relation to the mechanical interpretation of the surface tension.

Fundamentals of the equilibrium fluctuation theory of adsorption of molecules on microcrystalline particles taking into account the atomic structure of nonuniform surfaces were developed. The importance of taking into account the discrete character of adsorption centers in constructing adsorption isotherm equations was demonstrated. This changes the type of the mathematical apparatus in the search for a maximum distribution function term in a grand canonical ensemble: instead of differential derivatives, symmetrized difference derivatives should be used. The fluctuation theory of adsorption was generalized to ideal multicomponent mixtures of molecules. Adsorption isotherm equations for multicomponent mixtures on uniform and nonuniform surfaces taking into account limited sizes of the surface of various microcrystal faces and fluctuation contributions were obtained. The equations describe the influence of equilibrium fluctuations on adsorption isotherms for adsorbent particles of all sizes, from nanometric to macroscopic. An analysis of the equations showed that the influence of fluctuations was strongest at low coverages for each microcrystal face. The simplest case of taking fluctuations into account in the presence of contributions of lateral interactions in the mean field approximation was discussed.

The effect of an inert supercritical (SC) component of a gas mixture on the equilibrium adsorption conditions was investigated theoretically. The adsorption isotherms were calculated within the framework of the lattice-gas model taking into account lateral interactions of the nearest neighbors in the quasichemical approximation. The physical adsorption and chemisorption isotherms were calculated at specified chemical potentials of the main and supercritical components and at specified mole fractions of the supercritical component. The binary phase diagrams SC component-atoms of the solid characterizing the solubility conditions of SC molecules in the solids are considered.

The molecular theory of adsorption is used to estimate the upper bound of the validity of the Kelvin equation for calculating the saturated vapor pressure in meso- and macroporous adsorbents. The values of saturated vapor pressure are compared according to the theoretical adsorption isotherms obtained in terms of the lattice gas model and the Kelvin equation for atoms of argon and molecules of nitrogen and tetrachloride carbon. As the temperature increases, the agreement between the two methods improves. It is shown that the validity of the results obtained using the Kelvin equation for problems of absorption porometry is even limited for pores as small as 50 nm.

The question of the role of equilibrium fluctuations in the adsorption theory and kinetics of surface processes occurring on the particles of the nanometer size range is discussed. Differences are put forward that need to be introduced to the fluctuation theory of surface processes on microparticles and that generalize Hill’s approach to describing the thermodynamic properties of small systems. We show the importance of allowing for the discrete character of adsorption centers on the surfaces and their heterogeneity when describing adsorption isotherms and the rates of adsorption processes.

The discussed results yielded from development of the absolute reaction rate theory for surface processes, which was initiated by works of M. I. Temkin on uniform and nonuniform solid surfaces. His works also studied the interaction effect of adsorbed particles to the kinetics of surface reactions and discussed several adsorption sites blocked by adsorbed particles, kinetic complications caused by low mobility of adsorbed particles, and effect of high pressures in gas phase. At the present time, the theory of surface processes provides simultaneous account of all the major physical factors: nonuniform surface, interaction of adsorbed particles, size of particles, limited mobility of reagents, and rearrangement of surface. This review analyzes the role of self-consistent description of elementary reaction rates and equilibrium reaction system, which was emphasized by M.I. Temkin. The self-consistency requirement for dense surface phases, which realize cooperative behavior of reagents, indicates to the restrictions of the earlier derived kinetic equations.

The micro-hydrodynamic method is applid to the calculation of the molecular transport in narrow channels in case of capillary condensation, at the flow anisotropy resulted from the potential of the wall surface and/or of boundary vapor and fluid phases. The mechanisms of molecular transport in the one-phase and two-phase fluid flows as a dependence of fluid density and adsorption potential of channel walls are discussed.

The properties of the van der Waals loops for isotherms of the adsorbate located in the slit-shaped and cylindrical pores of different width were analyzed. The adsorbate molecules are modeled by spherical Lennard-Jones type particles. The calculation was based on the latticegas model in the quasichemical approximation for intermolecular interactions. The accuracy of calculations of the adsorption hysteresis loops increases due to the use of distributed models, which reflect the spatial inhomogeneity of the distribution of molecules along the normal to the pore wall. The effect of the adsorbate-adsorbent interaction potential and pore width on the pattern of adsorption isotherms is considered. Taking into account the inhomogeneity of the spatial distribution of molecules changes the course of the spinodal sections of the adsorption isotherms.

Stratification of molecules in inhomogeneous systems can cause the appearance of multiplicity of critical points on vapor-liquid stratification curves. Such systems are open inhomogeneous surfaces and porous solids. The adsorption of molecules on them is determined by the joint action of the surface potential and intermolecular interactions. The joint influence of these potentials forms subregions of the system as a whole in which first-order phase transitions occur. The number and parameters of critical points depend on the composition and structure of inhomogeneous systems. The largest possible number of critical points is related to the presence of macroscopic regions of various types. If regions comprising adsorption centers of various types have locally ordered structures, the system contains one critical point with modified critical parameters, similarly to homogeneous systems. If the influence of the surface potential prevails over intermolecular interaction, the system does not contain critical points.

The possibility of determining the function characterizing the connectedness of pores of different types and sizes from the experimental adsorption isotherms is discussed. It is shown that the presence of joints of pores manifests itself by two indications in phase diagrams and adsorption isotherms that contain hysteresis loops. The first indication is related to the appearance of an additional phase transition in the phase diagram and/or a density jump in the isotherm. The second indication is related to different numbers of density jumps in the desorption and adsorption branches of adsorption isotherms. The character of the behavior of the adsorption branch of the isotherm in the case of the presence of regions of joints in a porous system is established.

The conditions of layering of adsorbate molecules in porous systems with characteristic sizes of from 1 to 50–100 nm are discussed. The porous systems contain both very narrow pores, in which interaction potentials of pore walls overlap, and comparatively broad pores without overlapping of surface potentials. Three pore size intervals are distinguished. In the first interval, no adsorbate layering occurs, the second interval is characterized by capillary condensation with critical parameters different from their volume values, and, in the third interval, capillary condensation conditions are almost the same as in the volume adsorbtive phase. Criteria of the characteristic pore sizes of different geometries are formulated; the criteria correspond to small volumes in which first-order phase transitions are absent. The boundary between the first and second pore size regions is observed experimentally as the disappearance/appearance of hysteresis loops in adsorption-desorption isotherms measured under strictly equilibrium conditions as the size of pores decreases/increases. A nonuniform distribution of the surface potential is shown to be responsible for the multiplicity of local regions in porous media with their own vapor-liquid coexisting phases. The spinodal transitions in adsorption-desorption in pores can occur between various local regions. An analysis is performed in terms of the lattice gas model with short-range Lennard-Jones interaction of adsorbate molecules with each other and adsorbent walls.

The gas and liquid spinodal branches for an adsorbate located in narrow slit-shaped, cylindrical, and spherocylindrical pores were calculated. The adsorbate is modeled by Lennard-Jones spherical particles. The calculation was based on the lattice gas model taking into account the intermolecular interactions of nearest neighbors in the quasichemical approximation. The density-temperature diagrams for the gas and liquid spinodal branches in the pores are similar to the equilibrium vapor-liquid phase diagrams: they have a common critical point; the dense-phase branches are shifted to lower pore fillings, while the rarefied-phase branches are shifted toward higher pore fillings. The width of adsorption-desorption hysteresis loop in the adsorption isotherms for Lennard-Jones particles was analyzed as a function of the pore size and the interaction potential of the adsorbate with the pore walls. The effect of pore wall roughness and the accuracy of isotherm calculation on the width of the adsorption-desorption hysteresis loop in cylindrical pores is discussed

Adsorption isotherms are predicted for spherical adsorbates in cylindrical channels of MCM-41 mesoporous materials over a wide range of temperatures by using the “fragment method”. This prediction shows that an equilibrium capillary condensation is impossible for pores with diameters smaller than 2.5 nm. The adsorbate distribution in relatively large pore channels was described by the quasi-chemical approximation (QCA) that takes into account direct pair correlations between interacting molecules. In order to improve the lattice–gas model in the vicinity of the critical point, a calibration function that takes into account information from the fragment method, was introduced into the QCA equations. The influence of the size factor of pores on argon adsorption isotherms was demonstrated.

The adsorption of rod-like molecules in slit-shaped pores was considered within the frame-work of the lattice-gas model. This model is applicable over a broad range of fluid concentrations (from the gaseous to the liquid state) and temperatures (including the critical region). In the calculation of the local distributions of mixture components in the equilibrium states, lateral interactions are taken into account. The equations of the model reflect the strong anisotropy of the distribution of mixture components along the normal to the pore wall surface and ordering of the rods along various directions.

The concentration dependences of the label transport and shear viscosity coefficients for rod-like molecules in slit-shaped pores were studied. The calculations were carried out using the lattice gas model, which describes a broad range of fluid concentrations (from the gaseous to the liquid state) and temperatures (including the critical region). In the calculation of the local distributions of mixture components in the equilibrium states, lateral interactions were taken into account. The translational and rotational motions of molecules were described in terms of the transition state theory for nonideal reaction systems, which took into account the influence of neighboring molecules on the height of the activation barrier. The model equations reflect the pronounced anisotropy of the distribution of system components along the normal to the pore wall surface and ordering effects of molecules along various directions.

Expressions for calculating the thermal velocities of the motion of rodlike molecules in slitlike pores were obtained within the framework of a lattice gas model valid over a wide range of fluid densities (from rarefied gases to liquids) and temperatures, including the critical region. The translational and rotational motion of molecules was described using the transition-state theory for nonideal reaction systems, which takes into account the effect of the neighboring molecules on the activation barrier height. The local distributions of the components of the mixture under equilibrium conditions were calculated by describing lateral interactions in the approximation of isolated contacts. The equations of the model reflect the fact that the distributions of the components in the direction perpendicular to the pore walls (due to the effect of adsorption forces) and along the pore axis (if capillary condensation occurs) exhibit a strong anisotropy.

A unified approach for simulation of the vapour–liquid coexisting curves on phase diagrams and hysteresis phenomena for simple adsorbates in various type porous adsorbents was developed on the basis of the lattice–gas model. The ideas of this model in complex porous systems were used twice: for a construction of the over-molecular structure for high-dispersed bodies and for a description of the molecular distribution within the complex porous space. The supramolecular structure of these systems is modelled by sections with a simple regular geometry (slit-like, cylindrical and spherical) with additional inclusion of junctions between different pore sections. At the description of the adsorbate distribution, the lateral interactions between the fluidic molecules were included through the quasi-chemical approximation (QCA) that takes into account direct pair correlations between interacting molecules. The knowledge of phase diagrams allows improving a description of the adsorption hysteresis phenomena. The lattice–gas model enables explaining the experimental data for MCM-41 like sample in which the hysteresis is absent, when the pore size is narrowed more than ∼4 nm.

A self-consistent approach to the calculation of equilibrium and transport characteristics of inert gases and liquids in complex narrow-pore systems based on the lattice-gas model is proposed. A supramolecular structure for fine-grained solids was constructed and the adsorbate distribution within the pore volume is described. The supramolecular structure is simulated using slit-shaped, cylindrical, spherical, and globular segments. Additionally, junctions of pore systems with different structures are included, and the heterogeneity of their walls and the presence of structural defects in the pore segments are taken into account. The distributions of molecules are described in the quasi-chemical approximation to take into account intermolecular interactions using calibration functions to correct this approximation in the near-critical area. Expressions for local and integrated flow transfer coefficients are constructed, in particular, self-diffusion, shear viscosity and heat conductivity. The contributions of the near-wall areas and the core parts of pores to the general form of phase diagrams, the effect of the pore size on the conditions of capillary condensation, and the role of surface mobility of molecules are discussed.

A globular model for silica-gel systems has been proposed in order to perform calculations for adsorption in porous systems on basis of the lattice–gas model. The globular model operates with fixed size spherical particles. These particles form a nonregular structure, which can be constructed by removing of a portion of globules from the regular three-dimensional cubic structure. Types of phase diagrams for the vapour–liquid transition of an adsorbate in porous globular adsorbents with regular and nonregular structures have been considered. The calculations for three types of adsorbates: nitrogen, argon and a model system with low binding energy of an adsorbent were performed. A size of globules was changed from 7 up to 40 lattice parameter of the model. The phase diagrams of the model have several domes which correspond to surface, over-surface, and inner part of pore volumes.

A microhydrodynamic approach is formulated to describe the molecular flows in micro- and mesoporous systems, including the region of capillary phenomena, where one needs a single set of equations characterizing the flows of both dense gases and liquids. The set of equations of the transfer of dense fluids in narrow pores is closed by using the simplest molecular model, specifically, the lattice gas model, which describes the behavior of fluids over wide ranges of concentrations (from the gas to the liquid state) and temperatures, including the critical region. This model takes into account the volume of molecules and the intermolecular interactions in a quasi-chemical approximation, which retains short-range order correlations. The model reflects the strong anisotropy of the molecular distributions along a normal to the pore wall (because of the adsorption forces) and along the pore axis if there is capillary condensation. The derived set of equations is a set of finite-difference equations in increments of coordinates (instead of derivatives). The dissipative coefficients have a nonlocal anisotropic character. The equations of the transfer in near-wall cells and the prospects of the microhydrodynamic approach are discussed.

The calibration functions based on an information on exact solutions were used for the uniform and nonuniform systems to enhance the accuracy of phase diagram calculations using approximate cluster methods. This approach was used for the description of adsorption on uniform surfaces (two-dimensional system) and in porous sorbents (three-dimensional system). The quasichemical approximation and accurate information on the regions of critical points in uniform lattices were used. The current state of phase diagram simulations in porous systems is discussed.

The phase diagrams describing condensation of adsorbate in micro- and mesoporous adsorbents having slit-shaped and cylindrical pores whose size varied from 1 to 20 monolayers were constructed. The study was performed using the lattice-gas model in the quasichemical approximation to take into account the intermolecular interactions. The phase diagrams for various values of the potential arising from different types of adsorbate--adsorbent interaction were analyzed for adsorption of helium, neon, methane, and carbon tetrachloride in graphite pores. Other adsorption systems are considered and the relationship between the pressure and temperature of adsorbate condensation is discussed. A nonmonotonic variation of the critical densities for pore widths from 3 to 10 molecular diameters was found. The pattern of this variation depends on the ratio of the energy of lateral interactions of the adsorbate molecules to the energy of interaction of the adsorbate molecules with pore walls. The critical temperature decreases monotonically with a decrease in the pore width. The stronger the adsorbate interaction with the pore walls, the greater the decrease in the critical temperature.