Effect of confinement in nano-porous materials on the solubility of a supercritical gas

By combining Gibbs Ensemble Monte Carlo simulations and density functional theory, we investigate the influence of confinement in a slit-shaped carbon pore on the solubility of a supercritical solute gas in a liquid solvent. In the cases studied here, competing adsorption of the solvent and solute determines whether the solubility is enhanced or suppressed for larger pores. We find that the solubility in the confined system is strongly dependent on pore width, and that molecular packing effects are important for small pore widths. In addition, the solubility decreases on increase in the temperature, as for the bulk mixture, but the rate of decrease is greater in the pore due to a decrease in the partial molar enthalpy of the solute in the pore; this effect becomes greater as pore width is decreased. The solubility is increased on increasing the bulk pressure of the gas in equilibrium with the pore, and obeys Henry's law at lower pressures. However, the Henry constant differs significantly from that for the bulk mixture, and the range of pressure over which Henry's law applies is reduced relative to that for the bulk mixture. The latter observation indicates that solute–solute interactions become more important in the pore than for the bulk at a given bulk pressure. Finally, we note that different authors use different definitions of the solubility in pores, leading to some confusion over the reported phenomenon of ‘oversolubility’. We recommend that solubility be defined as the overall mole fraction of solute in the pores, since it takes into account the increase in density of the solvent in the pores, and avoids ambiguity in the definition of the pore volume.     

Molecular simulation study of triangle-well fluids confined in slit pores

Grand canonical Monte Carlo simulations are used to study the behaviour of triangle-well (TW) fluids with variable well widths confined inside slit pores. The effect of individual factors influencing the properties of confined fluids such as fluid–fluid interactions, pore size and pore wall–fluid interactions are obtained using simulations as it is difficult to experimentally determine the same. An interesting observation of this study is that inside the narrow pore of slit height h* = 5 at the high-pressure condition of P* = 0.8, for the TW fluid with long-range attraction or for the fluid at a low temperature for even a short-range attraction, the density profiles show layering such that there is a sticking tendency of the particles at centre, while there is a depletion of particles near the wall (as the layers at the centre have higher density peak heights than near the walls).     

The phase behaviour of a fluid in a slitlike pore with permeable walls

The confinement of a lattice fluid in a set of slitlike pores separated by semipermeable walls with a finite width has been studied. The walls are modelled by a square-well repulsive potential with a finite height. The thermodynamic properties and the phase behaviour of the system are evaluated by means of Monte Carlo simulations. For some states theoretical calculations have been made using a mean-field-type theory. These investigations confirm previous findings for confined Lennard-Jones fluids, obtained from a density functional approach. For intermediate and low potential barriers that separate the pores, the isotherms exhibit two hysteresis loops and the liquid-vapour coexistence curve divides into two branches describing condensation inside the pore and inside the permeable wall. These two branches are separated by a triple point. At temperatures lower than the triple point temperature, the condensation takes place instantaneously in both the pore and inside the permeable wall. It was found that when the temperature is scaled by the bulk critical temperature, the phase diagram emerging from this simple mean-field treatment is close to the phase diagram obtained from simulation.     

Structural and thermodynamic properties of inhomogeneous fluids in rectangular corrugated nano-pores

Based on the free-energy average method, an area-weighted effective potential is derived for rectangular corrugated nano-pore. With the obtained potential, classical density functional theory is employed to investigate the structural and thermodynamic properties of confined Lennard-Jones fluid in rectangular corrugated slit pores. Firstly, influence of pore geometry on the adsorptive potential is calculated and analyzed. Further, thermodynamic properties including excess adsorption, solvation force, surface free energy and thermodynamic response functions are systematically investigated. It is found that pore geometry can largely modulate the structure of the confined fluids, which in turn influences other thermodynamic properties. In addition, the results show that different geometric elements have different influences on the confined fluids. The work provides an effective route to investigate the effect of roughness on confined fluids. It is expected to shed light on further understanding about interfacial phenomena near rough walls, and then provide useful clues for the design and characterization of novel materials.     

On the solvation force of water-like fluid models with square-well attraction and site–site association in slit-like pores: density functional approach

The solvation force of the water-like fluid models with square-well attraction and site–site chemical association confined to slit-like pores has been explored. Theoretical procedure is based on the application of the density functional approach with mean-field approximation for the attractive interparticle interactions. The chemical association effects are treated by using the first-order thermodynamic perturbation theory of Wertheim. Trends of behaviour of the solvation force are put in correspondence with the distribution of molecules in the pores and with the average density of the adsorbate. Moreover, the distribution of non-bonded species on pore width is described. The influence of the width of the square-well and of the gas–solid attraction is discussed. A comparison of theoretical predictions with computer simulations results for water models in slit-like pores is performed.     

Phase behaviour of a Lennard-Jones fluid in a pore with permeable walls of a finite thickness: a density functional approach

A confinement of a Lennard-Jones fluid in a system of slitlike pores separated by semipermeable walls of a finite width is studied. The walls are modelled by square-well repulsive potential wells. The structure of the confined fluid is investigated by means of a density functional method. For high potential barriers separating the pores, the phase behaviour of the system is similar to that for a single slitlike pore with impenetrable walls. For intermediate and low potential barriers the system shows different phase behaviour. Within some temperature range the isotherms exhibit two hysteresis loops, which characterize the condensation of the fluid in different parts of the system, namely in the pore and inside the semipermeable walls. The systems characterized by low and intermediate potential barriers exhibit the triple point, such that at temperatures below that triple point the condensation instantaneously takes place in both the pore and inside the permeable wall.     

Fluids absorbed in structured pores

A model of fluids absorbed in a pore with walls patterned with parallel channels is used to demonstrate some of the unity that can be proved to hold between the statistical mechanics of fluids absorbed in structured pores and of fluids adsorbed at unstructured walls and at edges/wedges where walls meet. In particular, the work done to reversibly shear a corrugated pore immersed in liquid is related to the difference in the density profile structure of liquid adsorbed near the edges of the channels. When the channel dimensions are mesoscopic or macroscopic but the minimum pore width is microscopic, statistical mechanics generates remarkable links between the surface tension of planar wall-fluid interfaces or, more generally, the solvation free energy of a planar pore, and the density profile at the sides of a channel wall in the vicinity of edges and wedges.     

Phase transitions of fluids confined in porous silicon: A differential calorimetry investigation

The phase transitions of non-polar organic fluids and of water, confined in the pores of porous silicon samples, were investigated by Differential Scanning Calorimetry (DSC). Two types of PS samples (p^- and p⁺ type) with different pore size and morphology were used (with spherical pores with a radius of about 1.5 nm and cylindrical shape with a radius of about 4 nm respectively). The DSC results clearly show that the smaller the pores are, the larger is the decrease in the transition temperature. Moreover, a larger hysteresis between melting and freezing is observed for p⁺ type than for p^- type samples. A critical review of the thermodynamical properties of small particles and confined fluids is presented and used to interpret and discuss our DSC results. The effects of the chemical dissolution as well as the influence of anodization time are presented, showing that thick p⁺ type porous silicon layers are non-homogeneous. The DSC technique which was used for the first time to investigate fluids confined in porous silicon, enables us to deduce original information, such as the pore size distribution, the decrease in the freezing temperature of confined water, and the thickness of non-freezing liquid layer at the pore wall surface. Received: 11 May 1998 / Revised and Accepted: 29 July 1998     

Computer simulation investigation of diffusion selectivity in graphite slit pores

Equilibrium molecular dynamics simulations are reported of oxygen and nitrogen molecules confined in graphite slit pores. Self- and collective diffusion coefficients have been calculated as a function of pore width, temperature and density for each pure component in the pore space. The aim of this study was to elucidate the mechanism by which oxygen and nitrogen are kinetically separated when air is passed over an adsorbent bed consisting of molecular sieving carbon in the commercial production of oxygen. It was found that a critical pore width exists for each species at which there is a sharp drop in the rate of diffusion (both self- and collective diffusion) of each fluid. The critical pore width is one for which the individual molecules are prevented from rotating freely about one of their axes. The greater length of a nitrogen molecule means that the critical pore width is higher for this species than for oxygen. Consequently, oxygen molecules diffuse substantially faster than nitrogen molecules in the vicinity of the nitrogen critical pore width. From an analysis of correlation functions and their corresponding power spectra it is shown that the restricted rotations, which occur at or below the critical pore width, cause a decoupling of translational and rotational modes, with the net result being a lowering of translational diffusion. The nitrogen critical pore width lies within the range of the mean pore size of most commercial molecular sieving carbons, and so this mechanism may help to explain the high oxygen selectivities reported in the literature.     

受限于对称性破缺狭缝间氢键流体的相平衡研究

利用密度泛函理论并结合改进的基本度量理论研究了受限于对称性破缺狭缝间氢键流体的相平衡. 首先根据氢键流体的吸附-脱附等温线及相应巨势获得不同条件下氢键流体的相图. 进一步讨论了氢键作用、狭缝间距、狭缝与流体分子间相互作用及对称性破缺程度等因素对氢键流体相平衡的影响. 结果表明, 由于狭缝与流体分子及流体分子间的相互作用存在竞争, 使得受限于对称性破缺条件下的氢键流体呈现更为复杂的相态特征.     

Demixing transitions in a binary Gaussian-core fluid confined in narrow slit-like pores

Using a mean-field density functional approach we investigate phase separation transition in a binary mixture of Gaussian-core molecules confined in narrow slit-like pores. We consider pores with repulsive and attractive walls. In the case of fluid confinement in pores with repulsive and non-selective attracting walls, no phase separation in the confined fluid, prior to the bulk separation transition, was observed. However, in the case of pores with the walls selectively attracting fluid particles, we reveal that the separation transition may take place as a two-step process. During the first step the composition change occurs within a few layers adjacent to the pore walls, whereas in the second step, it takes place in the pore interior.     

Phase transitions and ordering of confined dipolar fluids

We apply a modified mean-field density functional theory to determine the phase behavior of Stockmayer fluids in slit-like pores formed by two walls with identical substrate potentials. Based on the Carnahan-Starling equation of state, a fundamental-measure theory is employed to incorporate the effects of short-ranged hard-sphere-like correlations while the long-ranged contributions to the fluid interaction potential are treated perturbatively. The liquid-vapor, ferromagnetic-liquid-vapor, and ferromagnetic-liquid-isotropic-liquid first-order phase separations are investigated. The local orientational structure of the anisotropic and inhomogeneous ferromagnetic liquid phase is also studied. We discuss how the phase diagrams are shifted and distorted upon varying the pore width.     

Adsorption isotherm of a Lennard-Jones nitrogen in a carbon slitlike pore

A density functional perturbation approximation based both on second-order perturbation theory and on the pore average density has been proposed to study the adsorption hysteresis of nitrogen in a carbon slit pore. The main advantage of the present approximation is that it is computationally much simpler than the original density functional approximation based on the second-order perturbation theory of liquids, and can be applied to several model fluids confined in a strong external field in order to study their structural and thermodynamic properties. The calculated adsorption hysteresis for the confined Lennard-Jones nitrogen is in very good agreement with computer simulation, even if its accuracy slightly deteriorates for the desorption branch. The calculated equilibrium particle density distributions also compare well with computer simulations, and are better than those of a density functional theory based on the so-called mean-field approximation.     

平行板间方势阱偶极流体的结构性质研究

在密度泛函理论(DFT)框架下, 应用改进的基本度量理论(MFMT)表达硬球作用对自由能泛函的贡献, 根据统计力学理论结合加权密度近似(WDA)表达偶极作用对自由能泛函的贡献,得到了方势阱偶极流体在平行板间的密度分布表达式, 计算了偶极流体在两平行板间的密度分布, 并探讨了方势阱深度和宽度对体系密度分布的影响. 此外, 通过体系密度分布, 进一步分析了方势阱宽度和深度以及板间尺度与溶剂化力的关系.     

平行板间方势阱偶极流体的结构性质研究

在密度泛函理论(DFT)框架下, 应用改进的基本度量理论(MFMT)表达硬球作用对自由能泛函的贡献, 根据统计力学理论结合加权密度近似(WDA)表达偶极作用对自由能泛函的贡献,得到了方势阱偶极流体在平行板间的密度分布表达式, 计算了偶极流体在两平行板间的密度分布, 并探讨了方势阱深度和宽度对体系密度分布的影响. 此外, 通过体系密度分布, 进一步分析了方势阱宽度和深度以及板间尺度与溶剂化力的关系.     

Effect of surface modification on the pore condensation of fluids: experimental results and density functional theory

The physisorption and pore condensation of a polar fluid (CHF₃) in a series of MCM-41 type mesoporous silica materials with native and chemically modified pore walls has been studied over the temperature range 168–293 K, corresponding to reduced temperatures T/T_c in the range 0.56-0.98, where T_c is the critical temperature of the fluid. Chemical modification of the pore walls by attachment of Si(CH₃)₃ groups causes a shift in pore condensation to higher relative pressures p/p0. This effect is most pronounced for materials with narrow pores (2.9 nm) at low temperatures. In the theoretical part of the work density functional theory based on a simple cubic lattice model of the confined fluid has been used to analyse the combined effect of a reduced pore width and weaker fluid-wall interaction caused by the surface coating. For realistic values of the model parameters it is found that the effect of the lower pore width is outweighed by the opposing effect of the lower fluid-wall interactions. The weaker temperature dependence of the pore existence curve observed experimentally for the surface modified materials can be traced back to a crossover from a two-step to a single-step process of pore filling predicted by the model.     

Reentrant filling transitions in Lennard-Jones fluids confined in nanoscopic slit-like pores

The reentrant first-order pore filling transition in a Lennard-Jones (LJ) fluid confined between the parallel (100) planes of the face centered cubic (fcc) crystal is studied by means of Monte Carlo simulations in the canonical as well as grand canonical ensembles. Slit-like pores of the width allowing for the formation of only three adsorbate layers are used. It is demonstrated that the reentrant first-order pore filling transition, associated with the condensation of the middle layer, appears only when the height of the potential barrier between adjacent sites, V_D, fulfills a condition V_D∈[V_D,min,V_D,max]. The lower limit of V_D (V_D,min) is primarily determined by the stability of commensurate monolayers formed at both pore walls during the first step of adsorbate condensation, while the upper limit of V_D (V_D,max) depends on the stability of the commensurate three-layer structure. It is also shown that both the misfit between the size of adsorbed atom and the surface lattice as well as the pore width have a great influence on the phase behavior of confined fluids. Moreover, the effects of the phase shift between the confining lattices on the reentrant first-order filling transition are discussed.     

On the freezing and structure of hard spheres under spherical confinement

The equation of state and the structure of hard spheres confined in spherical pores have been investigated via molecular dynamics for different pore radii ranging from 5.0 to 10.0?σ, where σ is the particle diameter. The hard boundary is chosen to capture the pure geometric effect of spherical confinement. A discontinuity in the equation of state was observed, indicating the onset of a freezing-like phase transition, which was similar to that of the bulk hard-sphere fluids. The behaviour of confined particles resembles that of the bulk with increase in the pore size, while its deviation from the bulk is found to be larger at the solid-like phase. For the pore radius below 5.0, FCC-like crystal clusters are not formed in spherically confined hard spheres.     

Nanopore wall effect on surface tension of methane

Local pressure is known to be anisotropic across the interfaces separating fluids in equilibrium. Tangential pressure profiles show characteristic negative peaks as a result of surface tension forces parallel to the interface. Nearby attractive forces parallel to the interface are larger than the repulsive forces and, hence, constitute the surface tension. In this work, using molecular dynamics simulations of methane inside nano-scale pores, we show this surface tension behaviour could be significantly influenced by confinement effects. The layering structure, characterised by damped oscillations in local liquid density and tangential pressures, extends deep into the pore and can be a few nanometers thick. The surface tension is measured numerically using local pressures across the interface. Results show that the tension is smaller under confinement and becomes a variable in small pores, mainly controlled by the thickness of the liquid density layering (or liquid saturation) and the pore width. If the liquid saturation inside the pore is high enough, the vapour–liquid interface is not interfered by the pore wall and the surface tension remains the same as the bulk values. The results are important for understanding phase change and multi-phase transport phenomena in nanoporous materials.     

Size selectivity of narrow pores

The selectivity of micropores and ion channels is examined for simple pore topologies within the framework of density functional theory of highly confined fluids. In an infinite cylindrical pore purely steric (excluded volume) effects are shown to lead to strong, nontrivial size selectivity, which is highly sensitive to the pore radius. A crude modeling of electrostatic effects does not alter the relative absorbance of Na+ and K+ ions in a significant way.