Modeling molecular transport in slit pores

We examine the transport of methane in microporous carbon by performing equilibrium and nonequilibrium molecular dynamics simulations over a range of pore sizes, densities, and temperatures. We interpret these simulation results using two models of the transport process. At low densities, we consider a molecular flow model, in which intermolecular interactions are neglected, and find excellent agreement between transport diffusion coefficients determined from simulation, and those predicted by the model. Simulation results indicate that the model can be applied up to fluid densities of the order to 0.1-1 nm(-3). Above these densities, we consider a slip flow model, combining hydrodynamic theory with a slip condition at the solid-fluid interface. As the diffusion coefficient at low densities can be accurately determined by the molecular flow model, we also consider a model where the slip condition is supplied by the molecular flow model. We find that both density-dependent models provide a useful means of estimating the transport coefficient that compares well with simulation.     

Some considerations concerning the technical use of narrow pores

Carrier transport and diffusion through narrow pores (single-file diffusion) are basic transport mechanisms that have been proposed to explain the specific passive permeation properties of biological membranes. Unlike carrier diffusion, the concept of single-file diffusion has been scarcely considered in connection with technical purposes. Taking into account, however, that extensive studies have been made in recent years on artificial lipid membranes with single-file pores and that cylindrical pores with a very narrow diameter distribution can be produced in thin plates by etching nuclear tracks, it seems useful to discuss more carefully the specific properties and conceivable applications of the single-file pore. As properties of obvious technical interest one finds high permselectivity and high transport rate. In addition, the voltage-dependent block of narrow pores provides an intriguing possibility of ionic flux control. A therapeutic system functioning as an artificial suprarenal gland is briefly outlined as an example.     

Anomalous transport in molecularly confined spaces

We develop a novel theory to predict the density dependence of the diffusivity of simple fluids in a molecularly sized nanopore with diffusely reflecting walls, incorporating nearest neighbor intermolecular interactions within the framework of the recent oscillator model of low density transport arising from this laboratory. It is shown that when the pore width is about two molecular diameters, at sufficiently high densities these interactions lead to a repulsive inner core, as a result of which the diffusing molecules undergo more frequent reflections at the wall. This leads to a reduction in diffusivity with increase in density, which is consistent with molecular dynamics simulation results, and contrasts with the behavior in larger pores where the transport coefficient has previously been shown to increase with increase in density due to viscouslike intermolecular interactions. At low densities the behavior is opposite, with the inner core becoming more attractive with increase in density, which can lead to an increase in diffusivity. The theory consistently explains molecular dynamics simulation results when the inhomogeneous pair distribution function of moving particles in the pore is axially periodic, suggesting concerted motion of neighboring molecules. It is also shown that a potential of mean force concept is inadequate for describing the influence of intermolecular interactions on transport.     

Modeling the selective partitioning of cations into negatively charged nanopores in water

Partitioning and transport of water and small solutes into and through nanopores are important to a variety of chemical and biological processes and applications. Here we study water structure in negatively charged model cylindrical [carbon nanotube (CNT)-like] nanopores, as well as the partitioning of positive ions of increasing size (Na+, K+, and Cs+) into the pore interior using extensive molecular dynamics simulations. Despite the simplicity of the simulation system-containing a short CNT-like nanopore in water carrying a uniformly distributed charge of qpore=-ne surrounded by n (=0,...,8) cations, making the overall system charge neutral-the results provide new and useful insights on both the pore hydration and ion partitioning. For n=0, that is, for a neutral nanopore, water molecules partition into the pore and form single-file hydrogen-bonded wire spanning the pore length. With increasing n, water molecules enter the pore from both ends with preferred orientations, resulting in a mutual repulsion between oriented water molecules at the pore center and creating a cavity-like low density region at the center. For low negative charge densities on the pore, the driving force for partitioning of positive ions into the pore is weak, and no partitioning is observed. Increasing the pore charge gradually leads to partitioning of positive ions into the pore. Interestingly, over a range of intermediate negative charge densities, nanopores display both thermodynamic as well as kinetic selectivity toward partitioning of the larger K+ and Cs+ ions into their interior over the smaller Na+ ions. Specifically, the driving force is in the order K+>Cs+>Na+, and K+ and Cs+ ions enter the pore much more rapidly than Na+ ions. At higher charge densities, the driving force for partitioning increases for all cations-it is highest for K+ ions-and becomes similar for Na+ and Cs+ ions. The variation of thermodynamic driving force and the average partitioning time with the pore charge density together suggest the presence of free energy barriers in the partitioning process. We discuss the role of ion hydration in the bulk and in the pore interior as well as of the pore hydration in determining the barrier heights for ion partitioning and the observed thermodynamic and kinetic selectivities.     

Modeling of submicrometer aerosol penetration through sintered granular membrane filters

We present a deep-bed aerosol filtration model that can be used to estimate the efficiency of sintered granular membrane filters in the region of the most penetrating particle size. In this region the capture of submicrometer aerosols, much smaller than the filter pore size, takes place mainly via Brownian diffusion and direct interception acting in synergy. By modeling the disordered sintered grain packing of such filters as a simple cubic lattice, and mapping the corresponding 3D connected pore volume onto a discrete cylindrical pore network, the efficiency of a granular filter can be estimated, using new analytical results for the efficiency of cylindrical pores. This model for aerosol penetration in sintered granular filters includes flow slip and the kinetics of particle capture by the pore surface. With a unique choice for two parameters, namely the structural tortuosity and effective kinetic coefficient of particle adsorption, this semiempirical model can account for the experimental efficiency of a new class of "high-efficiency particulate air" ceramic membrane filters as a function of particle size over a wide range of filter thickness and texture (pore size and porosity) and operating conditions (face velocity).     

A density functional theory for Lennard-Jones fluids in cylindrical pores and its applications to adsorption of nitrogen on MCM-41 materials

A density functional theory (DFT) constructed from the modified fundamental-measure theory and the modified Benedict-Webb-Rubin equation of state is presented. The Helmholtz free energy functional due to attractive interaction is expressed as a functional of attractive weighted-density in which the weight function is a mean-field-like type. An obvious advantage of the present theory is that it reproduces accurate bulk properties such as chemical potential, bulk pressure, vapor-liquid interfacial tension, and so forth when compared with molecular simulations and experiments with the same set of molecular parameters. Capabilities of the present DFT are demonstrated by its applicability to adsorption of argon and nitrogen on, respectively, a model cylindrical pore and mesoporous MCM-41 materials. Comparison of the theoretical results of argon in the model cylindrical pore with those from the newly published molecular simulations indicates that the present DFT predicts accurate average densities in the pore, slightly overestimates the pore pressure, and correctly describes the effect of the fluid-pore wall interaction on average densities and pressures in the pore. Application to adsorption of nitrogen on MCM-41 at 77.4 K shows that the present DFT predicts density profiles and adsorption isotherms in good agreement with those from molecular simulations and experiments. In contrast, the hysteresis loop of adsorption calculated from the mean-field theory shifts toward the low pressure region because a low bulk saturated pressure is produced from the mean-field equation of state. The present DFT offers a good way to describe the adsorption isotherms of porous materials as a function of temperature and pressure.     

Filling and emptying transitions in cylindrical channels: a density functional approach

The authors use density functional theory in a square gradient approximation to investigate capillary condensation and evaporation in cylindrical channels of finite lengths. The model allows them to systematically explore the effect of the pore's length, width, and surface fields on the location of the transition between "empty" (vapor-filled) and "full" (liquid-filled) states. In general, their results indicate that decreasing the length of the channel drastically reduces the range of pore widths where a transition between liquidlike and vaporlike configurations may be observed. For the wide pores, the transition occurs at very low pressures where the liquid is no longer stable, while for the narrow pores, the transition is hindered by the solid-fluid interactions that favor the vapor phase in lyophobic pores. For the limited range of sizes where the transition can occur, the authors' results confirm the existence of two competing minima that may explain the density oscillations observed in a computer simulation of nanochannels. Comparisons between these results with those generated using a phenomenological model based on the capillary approximation indicate that this simplified approach yields fairly good predictions for medium size pores. However, the capillary approach fails to properly describe the properties of the very small and very large pores.     

Molecular modeling of surfactant covered oil-water interfaces: Dynamics, microstructure, and barrier for mass transport

Mass transport across surfactant-covered oil-water interfaces of microemulsions plays an important role in numerous applications. In the current work, we use coarse-grained molecular dynamics simulations to investigate model systems containing flat hexadecane-water interfaces covered by monolayers of nonionic surfactants of various lengths. Several properties of the surfactant monolayers relevant to the mass transport are considered, including the monolayer microstructure, dynamics, and a free energy barrier to the solute transport. It is observed that the dominant contribution of a surfactant monolayer to the free energy barrier is a steric repulsion caused by a local density increase inside the monolayer. The local densities, and hence the free energy barriers, are larger for monolayers composed of longer surfactants. Since it is likely that the solute transport mechanism involves a sequence of jumps between short-lived pores within a monolayer, we perform a detailed analysis of structure, size, and lifetime of these pores. We demonstrate that the pore statistics is consistent with predictions of percolation theory and apply this theory to identify the characteristic length scale of the monolayer microstructure. The obtained pore structures are sensitive to minute changes of surfactant configurations occurring on the picosecond time scale. To reduce this sensitivity, the pores are averaged over short time intervals. The optimal duration of these time intervals is estimated from analysis of dynamics of pores with diameters comparable to or exceeding the characteristic percolation length scale. The developed approach allows one to filter out transient events of the pore dynamics and to focus on events leading to substantial changes of the monolayer microstructure.     

Structure and rheological behavior of highly charged colloidal particles in a cylindrical pore I. Effect of pore size

In this work we performed nonequilibrium Brownian dynamics (NEBD) computer simulations of highly charged colloidal particles in diluted suspension under a parabolic flow in cylindrical pores. The influence of charged and neutral cylindrical pores on the structure and rheology of suspensions is analyzed. A shear-induced disorder-order-disorder-like transition was monitored for low shear rates and small pore diameters. We calculate the concentration profiles, axial distribution functions, and axial-angular pair correlation functions to determine the structural properties at steady state for a constant shear flow for different pore sizes and flow strengths. Similar behavior has been observed in a planar narrow channel in the case of charged interacting colloidal particles (M.A. Valdez, O. Manero, J. Colloid Interface Sci. 190 (1997) 81). The mobility of the particles in the radial direction decreases rapidly with the flow and becomes practically frozen. The flow exhibits non-Newtonian shear thinning behavior due to interparticle interactions and particle-wall interaction; the apparent viscosity is lower as the pore diameter decreases, giving rise to an apparent slip in the colloidal suspension. The calculated slip velocity was higher than that obtained in a rectangular slit under shear flow.     

Single-file water in nanopores

Water molecules confined to pores with sub-nanometre diameters form single-file hydrogen-bonded chains. In such nanoscale confinement, water has unusual physical properties that are exploited in biology and hold promise for a wide range of biomimetic and nanotechnological applications. The latter can be realized by carbon and boron nitride nanotubes which confine water in a relatively non-specific way and lend themselves to the study of intrinsic properties of single-file water. As a consequence of strong water-water hydrogen bonds, many characteristics of single-file water are conserved in biological and synthetic pores despite differences in their atomistic structures. Charge transport and orientational order in water chains depend sensitively on and are mainly determined by electrostatic effects. Thus, mimicking functions of biological pores with apolar pores and corresponding external fields gives insight into the structure-function relation of biological pores and allows the development of technical applications beyond the molecular devices found in living systems. In this Perspective, we revisit results for single-file water in apolar pores, and examine the similarities and the differences between these simple systems and water in more complex pores.     

Theoretical effects of macromolecule concentration and charge on membrane rejection coefficients

A hydrodynamic model for the convection of rigid, spherical solutes through cylindrical pores, which includes both steric and electrostatic interactions between pairs of solute particles and between solutes and the pore wall, has been developed to examine the effects of solute concentration and charge on solute rejection by membrane pores during ultrafiltration. Calculations have been performed for a wide range of charge conditions and the results are presented in terms of the membrane rejection coefficient at infinite dilution and a correction factor which accounts for the first-order effects of concentration. For pores and solutes of like charge, the rejection coefficient is predicted to decrease with increasing feed concentration or ionic strength.     

Approximation of contacts for calculating the velocity of the thermal motion of rodlike molecules in narrow slitlike pores

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.     

Equilibrium adsorption in cylindrical mesopores: a modified Broekhoff and de Boer theory versus density functional theory

This paper presents a thermodynamic analysis of capillary condensation phenomena in cylindrical pores. Here, we modified the Broekhoff and de Boer (BdB) model for cylindrical pores accounting for the effect of the pore radius on the potential exerted by the pore walls. The new approach incorporates the recently published standard nitrogen and argon adsorption isotherm on nonporous silica LiChrospher Si-1000. The developed model is tested against the nonlocal density functional theory (NLDFT), and the criterion for this comparison is the condensation/evaporation pressure versus the pore diameter. The quantitative agreement between the NLDFT and the refined version of the BdB theory is ascertained for pores larger than 2 nm. The modified BdB theory was applied to the experimental adsorption branch of adsorption isotherms of a number of MCM-41 samples to determine their pore size distributions (PSDs). It was found that the PSDs determined with the new BdB approach coincide with those determined with the NLDFT (also using the experimental adsorption branch). As opposed to the NLDFT, the modified BdB theory is very simple in its utilization and therefore can be used as a convenient tool to obtain PSDs of all mesoporous solids from the analysis of the adsorption branch of adsorption isotherms of any subcritical fluids.     

Concentration dependence of the distribution coefficient for macromolecules in porous media

Partitioning of macromolecules between pore and bulk solutions directly affects both equilibrium and transport processes such as exclusion chromatography and movement of solutes through porous media. Because of interactions between macromolecules and the pore wall, the variation of the macromolecule activity with concentration is different inside the pore than in bulk solution. This difference causes a concentration dependence of the distribution coefficient, as reported in experiments involving exclusion chromatography. In order to explain this effect, we develop a model for a concentration-dependent distribution which explicitly accounts for a coupling between pore–macromolecule and macromolecule–macromolecule interactions. Predictions using this model are reported for the case of rigid spherical macromolecules in both cylindrical and slit pores, including both steric (hard sphere–hard wall) and long-range (screened electrostatic) interactions. An important result is the existence of a general correlation between the first order concentration effect and measurable properties of the macromolecule and porous medium.     

Modeling of Adsorption in Finite Cylindrical Pores by Means of Density Functional Theory

Adsorption of argon at its boiling point in finite cylindrical pores is considered by means of the non-local density functional theory (NLDFT) with a reference to MCM-41 silica. The NLDFT was adjusted to amorphous solids, which allowed us to quantitatively describe argon adsorption isotherm on nonporous reference silica in the entire bulk pressure range. In contrast to the conventional NLDFT technique, application of the model to cylindrical pores does not show any layering before the phase transition in conformity with experimental data. The finite pore is modeled as a cylindrical cavity bounded from its mouth by an infinite flat surface perpendicular to the pore axis. The adsorption of argon in pores of 4 and 5 nm diameters is analyzed in canonical and grand canonical ensembles using a two-dimensional version of NLDFT, which accounts for the radial and longitudinal fluid density distributions. The simulation results did not show any unusual features associated with accounting for the outer surface and support the conclusions obtained from the classical analysis of capillary condensation and evaporation. That is, the spontaneous condensation occurs at the vapor-like spinodal point, which is the upper limit of mechanical stability of the liquid-like film wetting the pore wall, while the evaporation occurs via a mechanism of receding of the semispherical meniscus from the pore mouth and the complete evaporation of the core occurs at the equilibrium transition pressure. Visualization of the pore filling and empting in the form of contour lines is presented.     

Simple models of adsorption in nanotubes

We present two very simple models of adsorption in cylindrical pores. It is assumed that a layer-by-layer mechanism occurs similarly to that in the BET theory. The major assumption is that in the pores having an adsorption space with cylindrical geometry, the surface area of the upper surface (in comparison with the bottom surface) should be diminished in proportion to the radii of a cylinder. Two cases are considered: the adsorbate-adsorbate interactions are neglected or they are taken into account according to the lattice model developed by Fowler and Guggenheim. It is shown that the data simulated by Ohba and Kaneko for adsorption of nitrogen in the internal space of carbon nanotubes are successfully described by our models. On the basis of the fitted data we show that the relation between the monolayer capacity in cylindrical pores and on flat surfaces is in excellent agreement with the equation developed recently by Salmas and Androutsopoulos. Moreover, our models are verified for two sets of experimental data reported by Kaneko et al. We obtain excellent agreement between the values of the pore diameters calculated by us and suggested by these authors (from HRTEM, the GCMC simulations, and the IDBdB model). It is concluded that proposed simple and fast models can be applied as a first approximation to the estimation of the internal nanotube diameters if they do not exceed ca. 5 nm and are slightly dispersed.     

Catalytic conversion reactions mediated by single-file diffusion in linear nanopores: hydrodynamic versus stochastic behavior

We analyze the spatiotemporal behavior of species concentrations in a diffusion-mediated conversion reaction which occurs at catalytic sites within linear pores of nanometer diameter. Diffusion within the pores is subject to a strict single-file (no passing) constraint. Both transient and steady-state behavior is precisely characterized by kinetic Monte Carlo simulations of a spatially discrete lattice-gas model for this reaction-diffusion process considering various distributions of catalytic sites. Exact hierarchical master equations can also be developed for this model. Their analysis, after application of mean-field type truncation approximations, produces discrete reaction-diffusion type equations (mf-RDE). For slowly varying concentrations, we further develop coarse-grained continuum hydrodynamic reaction-diffusion equations (h-RDE) incorporating a precise treatment of single-file diffusion in this multispecies system. The h-RDE successfully describe nontrivial aspects of transient behavior, in contrast to the mf-RDE, and also correctly capture unreactive steady-state behavior in the pore interior. However, steady-state reactivity, which is localized near the pore ends when those regions are catalytic, is controlled by fluctuations not incorporated into the hydrodynamic treatment. The mf-RDE partly capture these fluctuation effects, but cannot describe scaling behavior of the reactivity.     

Molecular Mechanisms Causing Anomalously High Thermal Expansion of Nanoconfined Water

Anomalously high thermal expansion is measured in water confined in nanoscale pores in amorphous silica and the molecular mechanisms are identified by molecular dynamics (MD) simulations using an accurate dissociative water potential. The experimentally measured coefficient of thermal expansion (CTE) of nanoconfined water increases as pore dimension decreases. The simulations match this behavior for water confined in 30 Å and 70 Å pores in silica. The cause of the high expansion is associated with the structure and increased CTE of a region of water ～6 Å thick adjacent to the silica. The structure of water in the first 3 Å of this interface is templated by the atomically rough silica surface, while the water in the second 3 Å just beyond the atomically rough silica surface sits in an asymmetric potential well and displays a high density, with a structure comparable to bulk water at higher pressure.