Computer simulation of diffusion within and through membranes using nonequilibrium molecular dynamics

A novel nonequilibrium molecular dynamics (NEMD) method introduced in 1994 and its recent application to investigations of the transport properties of gases and dense fluids within strongly inhomogeneous pore structures are reviewed. In this technique molecular simulations are conducted under realistic nonequilibrium (experimental) conditions thus enabling direct insight into the underlying microscopic processes taking place during transport within pores. The case studies reviewed in this paper establish the versatility and scope of the NEMD technique and also demonstrate its significant advantages over prior molecular simulation procedures as a tool to assist in the design and tailoring of novel nanopore systems.     

Water vapor/propylene sorption and diffusion behavior in PVA-P(AA-AMPS) blend membranes by GCMC and MD simulation

Detailed atomistic structures of blend membranes (poly vinyl alcohol (PVA)/(acrylic acid-co-2-acrylamido-2-methylpropylsulfonic acid) (P(AA-AMPS)) were constructed to investigate the sorption and diffusion behavior of gas molecules (water and propylene) in the membranes. Interaction and miscibility between PVA and P(AA-AMPS) were calculated, and it was found that strong intermolecular interaction resulted in good miscibility of PVA and P(AA-AMPS) in the blend. The polymer chains mobility and free volume properties of the blend membranes were characterized. The sorption quantities and sorption sites of water and propylene in the blend membranes were calculated using Grand Canonical Monte Carlo (GCMC) method. The diffusion coefficients of water in the blend membranes were calculated by molecular dynamics (MD) simulation. The simulated results of the membrane structure (chain mobility, free volume properties), the sorption quantities and diffusion coefficients of water/propylene in the blend membranes showed the identical changing trends as the experimental results. Hopefully, this study could offer qualitative insight into the mass transport phenomena within the blend membranes.     

Molecular simulation on penetrants diffusion at the interface region of organic-inorganic hybrid membranes

Detailed atomistic structures of blend (class I) and chemical tethered (class II) poly vinyl alcohol (PVA)-silica hybrid membranes were constructed to investigate the diffusion behavior of small molecules at the interfacial region. Stronger interaction between PVA and silica, as well as more appropriate interfacial morphology which was evaluated by interchain distance, chain mobility and free volume were found in class II hybrid membranes due to the presence of covalent bonds. The effects of interfacial morphology on diffusion behavior of small molecules including benzene and cyclohexane were tentatively explored, and found that diffusion coefficients were closely related to fraction of free volume (FFV). Subsequently, the diffusion selectivity was calculated by the ratio of FFV probed by benzene and cyclohexane molecules. Hopefully, this study will offer some important qualitative insight into the transport phenomena within organic-inorganic hybrid membranes.     

Hydrogen adsorption capacity of vanadium oxide nanotube from pure and mixture gas environment through molecular simulation

The effects of temperature and pressure on the adsorption capacity of vanadium oxide nanotube (VONT) were investigated by Monte Carlo (MC) simulation. Hydrogen adsorption was an increasing function of pressure and in about 50 MPa VONT showed the maximum total hydrogen capacity of 5.17, 4.67 and 4.56 wt%. at 250, 275 and 300 K, respectively. On increasing the temperature to room temperature, 75% decrease in the initial adsorption capacity was observed.

Appraising adsorption of the same number of hydrogen molecules from pure and hydrogen/nitrogen mixture at 300 K indicated that under 10 MPa, changes in adsorption capacities were inconsequential after N₂ addition to the environment.     

Atomistic insight into the structure and diffusion properties of pollucite glass-ceramics

Molecular dynamics simulations have been performed with an aim to better understand the structure and ion diffusion properties of pollucite glass-ceramics with alkali metal cations dopants. The minor-radius alkali metal cations influence the resistance of volume deformation but have little effect on that of shear deformation. Compared to pollucite, the bond length and bond angle distributions of glass exhibit a broadening characteristic, and the alkali metal cation type would not affect the local environment of Si/Al. It is found that the self-diffusion coefficient of Cs cations in glass phase is lower than the ones of minor-radius alkali metal cations and is reduced as Na/Li is introduced. In comparison, the self-diffusion coefficient of Cs in pollucite is near zero, and the migration energy barrier of Cs cation obtained from metadynamics simulation reaches up to 124.0 kJ/mol, indicating that pollucite is a kind of excellent immobilization candidate material for radioactive Cs.     

Multi-component lattice gas diffusion

In this paper, diffusional transport of multi-component mixtures within the framework of a three-dimensional lattice gas is studied using dynamic Monte Carlo simulations. The mobile species, instantaneously hopping from one site to another, are assumed to have no mutual interactions, other than the usual ‘hard core’ interactions. Most strikingly, percolation phenomena occur for multi-component mixtures with significant differences in mobility. These greatly reduce the flux of the mobile component and cause failure of the standard macroscopic theories, including, e.g., the Maxwell-Stefan theory. Furthermore, we demonstrate that the well-known correlation effect disappears for systems in which gradients in the vacancy concentration are present. For systems in which co-operative displacements of two or more molecules are allowed to occur the effect of correlation between successive jumps vanishes, while the plot of the mobility versus occupancy shows a maximum. This intricate relation between mobility and occupancy again complicates the use of standard theories for describing mass transport.     

Asymptotically exact driving force approximation for intraparticle mass transfer rate in diffusion and adsorption processes: nonlinear isotherm systems with macropore diffusion control

An asymptotically exact nonlinear driving force model of intra-particle mass-transfer rate for nonlinear isotherm systems with macropore diffusion control is presented. The obtained expression is compared with the solutions of the Fickian diffusion and adsorption model and excellent accuracy over the entire time (fractional uptake) domain is demonstrated. In the case of an irreversible isotherm the model reduces to the equations resulting from the shrinking core model a fact that guarantees its accuracy for highly nonlinear systems. The high accuracy of the model is further demonstrated by comparison with experimental data under various operational conditions.     

Molecular simulations of gas transport in nitrile rubber and styrene butadiene rubber

Nitrile rubber (NBR, 39:61 wt% of acrylonitrile:butadiene) and styrene butadiene rubber (SBR, 50:50 wt% of styrene:butadiene) matrices have been equilibrated by molecular dynamics (MD) simulations. Transition-state approach is used to calculate the diffusion and solubility coefficients of small penetrants in these matrices, indicating quite low values in NBR and reasonable agreement with experimental results. MD simulations have been performed to analyze water diffusion in these matrices. Aggregation of water molecules is observed in the hydrophobic matrix SBR. MD simulations with fictitious nonpolar water molecules inhibit aggregation and lead to enhanced diffusion in SBR. In NBR there is a slight increase in diffusion for fictitious water molecules. The lower diffusion constants in NBR result from slower local relaxation of the matrix due to tighter intermolecular packing and higher cohesive energy density. The free volume distribution that affects solubility coefficients is not a major determining factor for the diffusion coefficients in these matrices.     

Molecular dynamics studies on the structures of polymer electrolyte membranes and diffusion mechanism of protons and small molecules

We performed a series of molecular dynamics (MD) simulations on Nafion^® membranes containing various quantities of H₂O and CH₃OH. The simulations afforded diverse nanoscale phase-separated structures, such as clusters, channels, and cluster–channels. The calculated cluster–channel structure qualitatively agrees with the experimental results of X-ray diffraction studies. We also investigated the diffusion mechanisms for H₂O, protons, CH₃OH, H₂, and O₂ in these membranes. To reproduce the hopping transfer of protons, we employed a semi-classical MD approach using the empirical valence bond method. The estimated diffusion coefficients of H₂O and proton in the membranes significantly depended on the H₂O content, and these values showed qualitatively good agreement with the experimental results. The diffusion coefficient of proton in H₂O-rich membranes was much larger than that of H₂O, and the proton mainly formed H₅O₂⁺ complex. Furthermore, the simulation results indicate that the majority of CH₃OH permeates through the H₂O clusters, and the majority of H₂ and O₂ permeates through the hydrophobic region of the membrane.     

A pore network model for diffusion in nanoporous carbons: Validation by molecular dynamics simulation

A hybrid molecular dynamics simulation/pore network model (MD/PNM) approach is developed for predicting diffusion in nanoporous carbons. This approach is computationally fast, and related to the structure of the real material. The PNM takes into account both the geometrical (a distribution of pore sizes) and topological (the pore network connectivity) characteristics of nanoporous carbons, which are obtained by analysing adsorption data. The effective diffusion coefficient is calculated by taking the transport diffusion coefficients in single slit-shaped model pores from MD simulation and then computing the effective value over the PNM. The reliability of this approach is evaluated by comparing the results of the PNM analysis with a more rigorous, but much slower, simulation applied to a realistic model material, the virtual porous carbon (VPC). We obtain good agreement between the diffusion coefficients for the PNM and the VPC, indicating the reliability of the hybrid MD/PNM method and it can be used in industry for materials design.     

Effects of carbonisation on pore evolution and gas permeation properties of carbon membranes from Kapton polyimide

Carbon membranes were prepared by carbonisation of Kapton^® polyimide at different temperatures under vacuum and nitrogen flow. Pore structure development of the membranes during carbonisation was studied. Carbonisation temperature was critical in the modification of membrane structure. At the same temperature, the carbon membranes fabricated under nitrogen atmosphere had higher gas permeances than those fabricated under vacuum. During heat treatment, the value of d-spacing for the carbon membranes decreased with increasing temperature, however, vacuum and nitrogen atmosphere had different influences on the changes in the d-spacing. CO₂ adsorption showed that the carbon membranes prepared at 1273 K under vacuum had the highest micropore volume whilst the membranes prepared at 1073 K under vacuum had the highest characteristic adsorption energy. N₂ adsorption showed that the samples obtained at 873 K under vacuum had the highest nitrogen uptake. Mesopores were deemed to be connected through micropores and narrow channels between meso- and/or micropores were supposedly present. The micropores predominantly controlled the transport properties of the carbon membranes. The membrane samples obtained at 1173 K under vacuum yielded ideal separation factors of 558.27, 60.87, 19.69 and 138.53 for He/N₂, CO₂/N₂, O₂/N₂ and CO₂/CH₄, respectively, with permeances of 7.26, 0.79, 0.26, 0.13 and 0.006 mol/(m² s Pa) for He, CO₂, O₂, N₂ and CH₄, respectively.     

Adsorption and diffusion of light alkanes on nanoporous faujasite catalysts investigated by molecular dynamics simulations

Molecular dynamics simulations were performed for ethane, propane, and n-butane in siliceous faujasite for different numbers of molecules per unit cell (loadings) at 300 K. Both the adsorbed molecules and the zeolite framework were modeled as flexible entities. A new semiempirical analytical potential function for the systems was constructed. From the mean-square displacement of the molecules, self-diffusion coefficients of 18.7 × 10⁻⁵, 13.3 × 10⁻⁵, and 4.3 × 10⁻⁵ cm²/s were calculated for ethane, propane, and n-butane, respectively at a loading of 8 molecules/unit cell. They compare well with experimental values from pulsed-field gradient NMR measurements (10 × 10⁻⁵, 9 × 10⁻⁵, and 6 × 10⁻⁵ cm²/s, respectively). Besides depending on the size of the hydrocarbon, the heats of adsorption and self-diffusion coefficients also strongly depend on the loading of adsorbate molecules. The results suggest that the new intermolecular force field can reasonably describe the adsorption and diffusion behavior of ethane, propane, and n-butane in faujasite zeolite.     

Atomistic molecular dynamics simulations of gas diffusion and solubility in rubbery amorphous hydrocarbon polymers

Diffusion coefficients D of H₂, He, O₂, N₂, and CO₂ in different rubbery amorphous polymeric matrices were estimated by atomistic molecular dynamics simulations at 298 K using the Einstein relationship, and compared with the relevant experimental values, where available. The simulated diffusion coefficients D of all the gases in all polymers considered almost regularly decreased with increasing molecular gas volumes and increasing polymer glass transition temperature. Further, solubility coefficients and heats of solution were obtained for all gases from Grand Canonical Monte Carlo simulations, which were also used to calculate sorption isotherms. In general, there is a good agreement between experimental and simulated values of diffusion and solubility coefficients for all gases considered.     

A hierarchical approach to the molecular modeling of diffusion and adsorption at nonzero loading in microporous materials

A new hierarchical approach is presented for the modeling of small molecules at nonzero concentrations in microporous materials. This approach is complementary to other methods recently appearing in the literature; it is targeted for systems with pores that are well defined, large enough to host multiple molecules, and energetically uncorrugated in the interior. Statistical mechanical partition functions are calculated on molecular-level models and used as input to coarse-grained models, to predict both adsorption isotherms and self-diffusion coefficients. Certain physically reasonable simplifying approximations are employed to make the partition functions tractable. The approach is demonstrated on the model system of methane in siliceous zeolite ZK4 at , and the results are judged in comparison to those from traditional grand canonical Monte Carlo and molecular dynamics simulations. The adsorption isotherm is predicted to a high degree of accuracy across a large pressure range. The predicted trends in the self-diffusion coefficient are in qualitative agreement with the molecular dynamics results, but there is some quantitative disagreement at the lowest and highest adsorbate loadings.     

Molecular dynamics simulation of oxygen diffusion in dry and water-containing poly(vinyl alcohol)

The kinetics and mechanisms of diffusion of oxygen and water in dry and water-containing amorphous syndiotactic poly(vinyl alcohol) were studied at 502 K and normal pressure by molecular dynamics simulation. Penetrant molecule trajectories were obtained in a system with 600 repeating units of poly(vinyl alcohol) and 0, 40 (2.6 wt%) and 80 (5.2 wt%) water molecules. Under dry conditions, oxygen molecules jumped in a cage-like fashion. The oxygen molecule diffused in a liquid-like fashion while water diffusion was cage-like in the system with 5.2 wt% water. The hydrogen bond lifetimes among the water molecules were significantly shorter than those formed between water and the polymer and between different polymer segments. The hydrogen bond lifetimes among all species were, within experimental error, unaffected by the content of water, even though the oxygen diffusivity increased exponentially and the water diffusivity increased to some extent with increasing water content. It seemed that the diffusivity was sensitive primarily to the decrease in concentration of polymer-polymer hydrogen bonds, which followed from the increase in water content. This finding was consonant with the analysis of the oxygen molecule motion relative to the nearest polymer backbone, which revealed that it jumped preferentially along the polymer chain and towards the backbone. This behavior was more pronounced when the dynamics were analyzed over longer distances (5 Å) and it was less pronounced in the water-rich systems. The simulations indicated that water clustering was absent and consequently that water was homogeneously distributed in the polymer systems.     

Non-equilibrium molecular dynamics simulation of gas separation in a microporous carbon membrane

We have carried out non-equilibrium molecular dynamics simulations of gas separation in a “selective surface flow” membrane. The gas mixture studied is hydrogen/methane, which is relevant to hydrogen purification in refineries. The simulations give insight into the separation mechanism, which is based on the transport of the more strongly adsorbing species (methane) in a dense layer near the pore wall, with the less strongly adsorbed species (hydrogen) diffusing through a less dense region close to the centre of the pore. Good agreement is obtained with experimental selectivity data. This work is also relevant to the study of the combined effects of adsorption and diffusion in microporous carbon adsorbents.     

Molecular dynamics simulation of diffusion of O2 and CO2 in blends of amorphous poly(ethylene terephthalate) and related polyesters

The diffusion of small molecules through polymers is important in many areas of polymer science, such as gas barrier and separation membrane materials, polymeric foams, and in the processing and properties of polymers. Molecular dynamics simulation techniques have been applied to study the diffusion of oxygen and carbon dioxide as small molecule penetrants in models polyester blends of bulk amorphous poly(ethylene terephthalate) and related aromatic polyesters. A bulk amorphous configuration with periodic boundary conditions was generated into a unit cell whose dimensions were determined for each of the simulated polyester blends in the cell having the experimental density. The diffusion coefficients for O₂ and CO₂ were determined via NVE molecular dynamics simulations using the Dreiding 2.21 molecular mechanics force field over a range of temperatures (300, 500 and 600 K) using up to 40 ns simulation time. We have focussed on the influence of the temperature, polymer dynamics, density and free volume distribution on the diffusion properties. Correlation of diffusion coefficients with free volume distribution was found.     

Molecular dynamics simulation of diffusion of O2 and CO2 in amorphous poly(ethylene terephthalate) and related aromatic polyesters

The diffusion of small molecules through polymers is important in many areas of polymer science, such as gas barrier and separation membrane materials, polymeric foams, and in the processing and properties of polymers. Molecular simulation techniques have been applied to study the diffusion of oxygen and dioxide of carbon as small molecule penetrants in models of bulk amorphous poly(ethylene terephthalate) and related aromatic polyesters. A bulk amorphous configuration with periodic boundary conditions is generated into a unit cell whose dimensions are determined for each of the simulated aromatic polyesters in the cell to have the experimental density. The aim for this research is to explore and investigate the diffusion of gases through bulk amorphous poly(ethylene terephthalate) and related aromatic polyesters. The diffusion coefficients for O₂ and CO₂ were determined via NVE molecular dynamics simulations using the Dreiding 2.21 molecular mechanics force field over a range of temperatures (300, 500 and 600 K) using up to 30 ns simulation time. We have focussed on the influence of the temperature, polymer dynamics, number of aromatic rings, ortho-, meta-, para-isomers, density and free volume distribution on the diffusion properties. Correlation of diffusion coefficients with free volume, temperature, number of aromatic rings, ortho-, meta- and para-isomers was found.