Computer Simulation of Isothermal Mass Transport in Graphite Slit Pores

Abstract

Results are presented from a simulation study of the mass transport of oxygen and nitrogen through graphite slit pores. The work is motivated by an attempt to understand the molecular origins of the kinetic selectivity displayed when air is separated into its major components using pressure swing adsorption. A combination of non-equilibrium molecular dynamics (NEMD), equilibrium molecular dynamics (EMD) and grand canonical Monte Carlo methods has been employed in our study to extract the maximum information. Transport diffusivities, self-diffusivities, permeabilities and Darken thermodynamic factors have been calculated as a function of pore width and temperature for pure component oxygen and nitrogen. In addition, new EMD simulation data for an 80:20 mixture of nitrogen and oxygen is reported, including a direct calculation of the Stefan-Maxwell coefficients. The results are discussed in terms of the oxygen selectivity and the possible mechanisms, which increase or decrease this quantity.

We find that the pore width behaviour of the diffusion coefficients consists of three distinct regimes: a regime at larger pore widths in which single component diffusion coefficients are largely independent of pore width, an optimum pore width at which both diffusivities increase substantially but the slit pore is selective towards nitrogen, and a regime at very low pore widths at which the diffusivities decrease sharply, but the slits are selective towards oxygen. The mechanism behind each of these regimes is discussed in terms of “entropic” effects and potential barrier heights.

We have also found that permeability selectivity is substantially reduced in a mixture of the two gases with a composition similar to that of air. Cross diffusion coefficients in the mixture have been calculated and shown to be non-negligible.     

Transport diffusivities of fluids in nanopores by non-equilibrium molecular dynamics simulation

We present a method to study fluid transport through nanoporous materials using highly efficient non-equilibrium molecular dynamics simulations. A steady flow is induced by applying an external field to the fluid particles within a small slab of the simulation cell. The external field generates a density gradient between both sides of the porous material, which in turn triggers a convective flux through the porous medium. The heat dissipated by the fluid flow is released by a Gaussian thermostat applied to the wall particles. This method is effective for studying diffusivities in a slit pore as well as more natural, complex wall geometries. The dependence of the diffusive flux on the external field sheds light on the transport diffusivities and allows a direct calculation of effective diffusivities. Both pore and fluid particle interactions are represented by coarse-grained molecular models in order to present a proof-of-concept and to retain computational efficiency in the simulations. The application of the method is demonstrated in two different scenarios, namely the effective mass transport through a slit pore and the calculation of the effective self-diffusion through this system. The method allows for a distinction between diffusive and convective contributions of the mass transport.     

Mass and Energy Transport Through Slit Pores: Application to Planar Poiseuille Flow

Abstract

We develop a simple, efficient and general statistical mechanical technique for calculating the pressure tensor and the heat flux vector in atomic fluids. The method is applied to the case of planar Poiseuille flow through a narrow slit pore and the results indicate that our technique is accurate and relatively efficient. A second method to calculate shear stress is derived from the momentum continuity equation. This mesoscopic method again is seen to be accurate with good computational efficiency.

We also find that the commonly used approximation to the Irving-Kirkwood expression for the heat flux and the pressure tensor (where the Irving-Kirkwood O_ij operator is set equal to unity-the so-called IK1 approximation), leads to incorrect results for highly inhomogeneous fluids. In such cases the pressure tensor and heat flux vector display spurious oscillations.

We calculate the spatially dependent viscosity across a narrow pore and find that it exhibits real but weak oscillations, a consequence of oscillations in the number density. Finally we point out that if the heat flux vector is coupled to the gradient of the square of the strain rate tensor such an effect will only affect the shape of the temperature profile. For planar Poiseuille flow, the temperature profile should deviate from the classical quartic form and include an additional quadratic component. The actual magnitude and shape of the heat flux vector remain exactly as they would if such a coupling did not exist.     

Revising the Concept of Pore Hierarchy for Ionic Transport in Carbon Materials for Supercapacitors

Rapid motion of electrolyte ions is a crucial requirement to ensure the fast charging/discharging and the high power densities of supercapacitor devices. This motion is primarily determined by the pore size and connectivity of the used porous carbon electrodes. Here, the diffusion characteristics of each individual electrolyte component, that is, anion, cation, and solvent confined to model carbons with uniform and well‐defined pore sizes are quantified. As a result, the contributions of micropores, mesopores, and hierarchical pore architectures to the overall transport of adsorbed mobile species are rationalized. Unexpectedly, it is observed that the presence of a network of mesopores, in addition to smaller micropores—the concept widely used in heterogeneous catalysis to promote diffusion of sorbates—does not necessarily enhance ionic transport in carbon materials. The observed phenomenon is explained by the stripping off the surrounding solvent shell from the electrolyte ions entering the micropores of the hierarchical material, and the resulting enrichment of solvent molecules preferably in the mesopores. It is believed that the presented findings serve to provide fundamental understanding of the mechanisms of electrolyte diffusion in carbon materials and depict a quantitative platform for the future designing of supercapacitor electrodes on a rational basis.     

Atomistic Simulations of Rare Gas Transport through Breathable Single-wall Nanotubes with Constrictions and Knees

We present the results of molecular dynamics (MD) computer simulations of rare gas diffusion through breathable nanotubes with pentagon–heptagon pair defects resulting in constrictions and knees. Diffusion involves interrupted high speed “choppy” motion with intermittent reversal in velocity direction. Single atoms exhibit a spiral-like path, in contrast to atoms traveling in groups. Considerable resistance to flow appears to reside in the upstream section of the nanotube where density gradients are small, prior to the constriction. Subsequently, considerable density gradients are present and speeds increase, becoming greatest at the tube exit. For the nanotubes examined, Kr and Xe diffusion was too hindered to provide reliable results. Diffusion of He through the nanotubes with knees occurs in a single-file fashion nearly along the center of the tube and the knee has no detectable effect on the diffusion kinetics. Transport diffusion coefficients are in the order of 10^-4–10^-2?cm²/s.     

Simulation of Diffusive Transport of Radionuclides in the Soil of a Radioactive Waste Disposal Site

The diffusive transport of ¹³⁷Cs, ⁹⁰Sr, and ⁶⁰Co in the clay of a radioactive waste disposal site at PINSTECH was studied to assess the safety of the underlying permeable zone against the release of these radionuclides from buried waste containers in the clay. Diffusion coefficients of these radionuclides were estimated by reservoir to sediment diffusion method via their stable counterparts in a laboratory experiment. A curve-fitting procedure was applied on the measured concentration-time profiles of the reservoir using the one-dimensional solute transport equation with a nonlinear least squares technique. Distribution coefficients were determined in laboratory batch experiments. Diffusive transport simulations were performed with the estimated values of diffusion coefficients and distribution coefficients using the one-dimensional solute transport equation describing Fickian diffusion, equilibrium adsorption, and radioactive decay. The transport simulation results showed that ¹³⁷Cs, ⁹⁰Sr, and ⁶⁰Co will transport distances of 4.33, 3.77, and 1.51 meters, respectively, in the clay before their activity concentrations will drop to clearance levels set by the International Atomic Energy Agency (IAEA), below which the waste is treated as non-radioactive. This showed that concentrations more than clearance levels will not be able to transport to the permeable zone at a minimum depth of seven meters from the ground surface if the waste containers are disposed in a trench below which a clay layer with a thickness of 4.33 meters or more exists.     

A comparison of model linear chain molecules with constrained and flexible bond lengths under planar Couette and extensional flows

We compare directly under flow two commonly used coarse grained models of linear polymers, namely the flexible finitely extensible nonlinear elastic (FENE) chain, and the freely jointed tangent sphere chain, otherwise known as the freely jointed chain. The comparison is based on viscometric, structural and dynamical properties. We use non-equilibrium molecular dynamics to simulate steady-state systems under planar Couette flow and planar extensional flow. Computed properties include shear and elongational viscosities, normal stresses, radius of gyration and end-to-end distances, order parameters, alignment angles and spin angular velocities. In all computed properties we observe very little difference between the two molecular models. Therefore, the choice of either model is suitable, though there is a computational advantage in the use of the FENE model.     

Periodic boundary conditions for the simulation of uniaxial extensional flow of arbitrary duration

It is very common with molecular dynamics and other simulation techniques to apply Lees–Edwards periodic boundary conditions (PBCs) for the simulation of shear flow. However, the behaviour of a complex liquid can be quite different under extensional flow. Simple deformation of a simulation cell and its periodic images only allows for simulations of these flows with short duration. For the simulation of planar extensional flow, it was recognised that the PBCs of Kraynik and Reinelt [Int. J. Multiphase Flow 1992;18:1045] could be used to perform simulations of this flow with arbitrary duration. However, a very common extensional flow in industrial applications and experiment is uniaxial extensional flow. Kraynik and Reinelt found that their method could not be directly generalised to this flow because of the lack of a lattice which reproduces itself during uniaxial extension. PBCs are presented in this article, which avoid this problem by finding a lattice which is compatible with the flow, finding the reduced basis to the lattice at all times and using this basis when calculating the position and separation of particles. Using these new PBCs, we perform nonequilibrium molecular dynamics simulations of a simple liquid and show that the technique gives results which agree with those from simulations using simply deforming PBCs.     

Strain‐Based In Situ Study of Anion and Cation Insertion into Porous Carbon Electrodes with Different Pore Sizes

The expansion of porous carbon electrodes in a room temperature ionic liquid (RTIL) is studied using in situ atomic force microscopy (AFM). The effect of carbon surface area and pore size/pore size distribution on the observed strain profile and ion kinetics is examined. Additionally, the influence of the potential scan rate on the strain response is investigated. By analyzing the strain data at various potential scan rates, information on ion kinetics in the different carbon materials is obtained. Molecular dynamics (MD) simulations are performed to compare with and provide molecular insights into the experimental results; this is the first MD work investigating the pressure exerted on porous electrodes under applied potential in a RTIL electrolyte. Using MD, the pressure exerted on the pore wall is calculated as a function of potential/charge for both a micropore (1.2 nm) and a mesopore (7.0 nm). The shape of the calculated pressure profile matches closely with the strain profiles observed experimentally.     

Homogeneous non-equilibrium molecular dynamics simulations of viscous flow: techniques and applications

We provide a review of the literature for non-equilibrium molecular dynamics (NEMD) simulations of homogeneous fluids. Our review focuses on techniques for simulations of shear and elongational flows in viscous fluids and covers the formulation and application of NEMD algorithms for atomic and molecular fluids. We provide a set of expositions that can be effectively used as guidelines to formulate the relevant equations of motion, periodic boundary conditions and thermostats. We also provide a survey of applications in a convenient tabular form as an aid to researchers who wish to use NEMD to study transport phenomena.     

Fabrication and Visualization of Capillary Bridges in Slit Pore Geometry

A procedure for creating and imaging capillary bridges in slit-pore geometry is presented. High aspect ratio hydrophobic pillars are fabricated and functionalized to render their top surfaces hydrophilic. The combination of a physical feature (the pillar) with a chemical boundary (the hydrophilic film on the top of the pillar) provides both a physical and chemical heterogeneity that pins the triple contact line, a necessary feature to create stable long but narrow capillary bridges. The substrates with the pillars are attached to glass slides and secured into custom holders. The holders are then mounted onto four axis microstages and positioned such that the pillars are parallel and facing each other. The capillary bridges are formed by introducing a fluid in the gap between the two substrates once the separation between the facing pillars has been reduced to a few hundred micrometers. The custom microstage is then employed to vary the height of the capillary bridge. A CCD camera is positioned to image either the length or the width of the capillary bridge to characterize the morphology of the fluid interface. Pillars with widths down to 250 µm and lengths up to 70 mm were fabricated with this method, leading to capillary bridges with aspect ratios (length/width) of over 100¹.     

Diffusive transport in Stokeslet flow and its application to plankton ecology

In this paper we consider the advective/diffusive transport of a solute near a hovering zooplankter. We approximate the fluid flow with that of a Stokeslet, corresponding to the plankter exerting a point force on the water, and assume that the plankter acts as a point source for the transported solute, located at the same point as the force. We find an analytical expression in closed form for the steady-state concentration of the solute. We also discuss the situation where the plankter performs Brownian motion. Finally we apply the results to the courtship of the marine copepod Pseudocalanus elongatus, where the male performs a mating dance below the hovering female. For this situation, our model supports the hypothesis that the mating dance is guided by the plume of a signalling pheromone.     

Conformational dynamics of the mitochondrial ADP/ATP carrier: a simulation study

The mitochondrial ADP/ATP carrier is a six helix bundle membrane transport protein, which couples the exit of ATP from the mitochondrial matrix to the entry of ADP. Extended (4×20 ns) molecular dynamics simulations of the carrier, in the presence and absence of bound inhibitor (carboxyatractyloside), have been used to explore the conformational dynamics of the protein in a lipid bilayer environment, in the presence and absence of the carboxyatractyloside inhibitor. The dynamic flexibility (measured as conformational drift and fluctuations) of the protein is reduced in the presence of bound inhibitor. Proline residues in transmembrane helices H1, H3 and H5 appear to form dynamic hinges. Fluctuations in inter-helix salt bridges are also observed over the time course of the simulations. Inhibitor-protein and lipid-protein interactions have been characterised in some detail. Overall, the simulations support a transport mechanism in which flexibility about the proline hinges enables a transition between a ‘closed’ and an ‘open’ pore-like state of the carrier protein.     

Molecular Dynamics Studies of Light Hydrocarbons Diffusion in Zeolites

Molecular dynamics simulation techniques have been used to study the diffusion of methane, ethane, propane and i-butane into the zeolite ZSM-5. From the trajectories, the mean-square displacements were obtained and the diffusion coefficients determined using Einstein's diffusion equation. The results, when compared to the available experimental data, indicate that the simulations can provide a realistic representation of the microscopic process of diffusion into the zeolite pores. This revised version was published online in June 2006 with corrections to the Cover Date.     

Molecular Dynamics Simulation of Self Diffusion in Microclusters

Abstract

Molecular dynamics simulation is carried out to study the mechanism of self diffusion which is characteristic of solid-like microclusters. A two-dimensional system with the Lennard-Jones potential is employed and the temperatures near the triple point of the two-dimensional bulk system are adopted for the simulation. The results show that: a) microclusters consist of two regions, i.e., solid-like cores and liquid-like surface regions, b) the size dependence of the diffusion coefficient for microclusters is weak, and the value of the solid-like core region is not much different from that of the bulk liquid, c) the activation energy of diffusion for microclusters is twenty to thirty times larger than that for the bulk liquid, d) the diffusion mechanism in the solid-like region involves the collective motion of small domains containing ten to twenty atoms which results in the formation of low density regions, sometimes even vacancy clusters, between them, and atoms in the low density regions change their positions to cause diffusion.     

Structural and dynamic properties of liquid alkali metals: molecular dynamics

Molecular dynamics simulations in the canonical ensemble have been performed to calculate structural and transport properties of liquid alkali metals. The alkali metals considered in this work were Na, K, Rb and Cs. Two Yukawa-type potentials were employed as the interaction law among particles. This function is written in terms of two adjustment parameters which make possible tuning the softness of the potential core and the range of the attractive part. The radial distribution functions and transport properties such as self-diffusion and shear viscosity, were calculated in a thermodynamic state near the melting point. The radial distribution function calculated for each alkali metal was compared with previous simulation results where a more elaborated potential was employed. From this comparison we found an excellent agreement. Our results for transport properties were also compared with the available experimental data and a good agreement was found.     

Molecular dynamics simulation of O2 diffusion in polydimethylsiloxane (PDMS) and end-linked PDMS networks

Molecular dynamics simulations are used to compute diffusion coefficients for O₂ molecules in polydimethylsiloxane (PDMS) and end-linked PDMS networks. The PDMS chains and penetrants are modelled using a hybrid interatomic potential which treats the Si and O atoms along the chain backbone explicitly while coarse-graining the methyl side groups and penetrants. In PDMS models with different molecular weights, diffusivity of the O₂ penetrants is found to modestly decrease with an increase in chain length. To match typical experimental conditions, the end-linked PDMS networks are constructed with a PDMS to crosslinking (CL) molecule mass ratio of 5:1 or 10:1, demanding that the number of CL molecules exceeds the number of PDMS chains in each model. Despite end-linking, the presence of non-bonded CL molecules promotes increased O₂ diffusivity in comparison with uncrosslinked PDMS. Temperature dependence is captured using the Williams–Landel–Ferry equation.