Modeling Cage-to-Cage Dynamics of Adsorbates at Arbitrary Loadings with Dynamically Corrected Transition-State Theory

Storage and Diffusion of Carbon Dioxide in the Metal Organic Framework MOF-5─A Semi-empirical Molecular Dynamics Study

Metal-organic frameworks (MOFs) have attracted increasing attention due to their high porosity for exceptional gas storage applications. MOF-5 belongs to the family of isoreticular MOFs (IRMOFs) and consists of Zn4O6+ clusters linked by 1,4-benzenedicarboxylate. Due to the large number of atoms in the unit cell, molecular dynamics simulation based on density functional theory has proved to be too demanding, while force field models are often inadequate to model complex host-guest interactions. To overcome this limitation, an alternative semi-empirical approach using a set of approximations and extensive parametrization of interactions called density functional tight binding (DFTB) was applied in this work to study CO2 in the MOF-5 host. Calculations of pristine MOF-5 yield very good agreement with experimental data in terms of X-ray diffraction patterns as well as mechanical properties, such as the negative thermal expansion coefficient and the bulk modulus. In addition, different loadings of CO2 were introduced, and the associated self-diffusion coefficients and activation energies were investigated. The results show very good agreement with those of other experimental and theoretical investigations. This study provides detailed insights into the capability of semi-empirical DFTB-based molecular dynamics simulations of these challenging guest@host systems. Based on the comparison of the guest-guest pair distributions observed inside the MOF host and the corresponding gas-phase reference, a liquid-like structure of CO2 can be deduced upon storage in the host material.

Quasiequilibrium multistate cellular automata

In our effort to tackle the problem of letting nontrivial interactions, thermodynamic equilibrium, and full synchronicity coexist, and in the hope of reviving interest in cellular automata as promising tools for the quantitative, large-scale investigation of multiparticle systems, we built a fully synchronous cellular automaton rule for the simulation of occupancy-based lattice systems with multistate cells and neighboring interactions. The core of this rule, which constitutes an actual synchronous sampling scheme, is a negotiation stage; it produces cell occupancy distributions in very good agreement with their sequential Monte Carlo counterparts, and it satisfies a cellwise detailed balance principle thanks to the use of "mixed" intermediate states that allow for the computation of locally averaged acceptance probabilities. We took a square lattice (but the rule itself is not bound by dimensionality) as a basis for comparison with sequential Monte Carlo for showing that this synchronous rule leads to quasiequilibrium; the fulfillment of cellwise detailed balance is shown through results obtained for a small one-dimensional system, where the transition matrix could be computed exactly.

Scaling-Up Simulations of Diffusion in Microporous Materials

We introduce and demonstrate the coarse-graining of static and dynamical properties of host-guest systems constituted by methane in two different microporous materials. The reference systems are mapped to occupancy-based pore-scale lattice models. Each coarse-grained model is equipped with an appropriate coarse-grained potential and a local dynamical operator, which represents the probability of interpore molecular jumps between different cages. Coarse-grained thermodynamics and dynamics are both defined based on small-scale atomistic simulations of the reference systems. We considered two host materials: the widely studied ITQ-29 zeolite and the LTA-zeolite-templated carbon, which was recently theorized. Our method allows for representing with satisfactory accuracy and a considerably reduced computational effort the reference systems while providing new interesting physical insights in terms of static and diffusive properties.

Local free energies for the coarse-graining of adsorption phenomena: The interacting pair approximation

We investigate the coarse-graining of host-guest systems under the perspective of the local distribution of pore occupancies, along with the physical meaning and actual computability of the coarse-interaction terms. We show that the widely accepted approach, in which the contributions to the free energy given by the molecules located in two neighboring pores are estimated through Monte Carlo simulations where the two pores are kept separated from the rest of the system, leads to inaccurate results at high sorbate densities. In the coarse-graining strategy that we propose, which is based on the Bethe-Peierls approximation, density-independent interaction terms are instead computed according to local effective potentials that take into account the correlations between the pore pair and its surroundings by means of mean-field correction terms without the need for simulating the pore pair separately. Use of the interaction parameters obtained this way allows the coarse-grained system to reproduce more closely the equilibrium properties of the original one. Results are shown for lattice-gases where the local free energy can be computed exactly and for a system of Lennard-Jones particles under the effect of a static confining field.

Diffusion of interacting particles in discrete geometries: Equilibrium and dynamical properties

We expand on a recent study of a lattice model of interacting particles [Phys. Rev. Lett. 111, 110601 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.110601]. The adsorption isotherm and equilibrium fluctuations in particle number are discussed as a function of the interaction. Their behavior is similar to that of interacting particles in porous materials. Different expressions for the particle jump rates are derived from transition-state theory. Which expression should be used depends on the strength of the interparticle interactions. Analytical expressions for the self- and transport diffusion are derived when correlations, caused by memory effects in the environment, are neglected. The diffusive behavior is studied numerically with kinetic Monte Carlo (kMC) simulations, which reproduces the diffusion including correlations. The effect of correlations is studied by comparing the analytical expressions with the kMC simulations. It is found that the Maxwell-Stefan diffusion can exceed the self-diffusion. To our knowledge, this is the first time this is observed. The diffusive behavior in one-dimensional and higher-dimensional systems is qualitatively the same, with the effect of correlations decreasing for increasing dimension. The length dependence of both the self- and transport diffusion is studied for one-dimensional systems. For long lengths the self-diffusion shows a 1/L dependence. Finally, we discuss when agreement with experiments and simulations can be expected. The assumption that particles in different cavities do not interact is expected to hold quantitatively at low and medium particle concentrations if the particles are not strongly interacting.

Diffusion in tight confinement: A lattice-gas cellular automaton approach. II. Transport properties

In this second paper the authors study the transport properties of the lattice-gas cellular automaton presented in Paper I [J. Chem. Phys. 126, 194709 (2007)] to model adsorption and dynamics of particles in a lattice of confining cells. Their work shows how a surprisingly simple parallel rule applied to a static network of cells joined by links set in space and time can generate a wide range of dynamical behaviors. In their model the cells are the elementary constituent objects of the network. They are a portion of space structured in sites which are energetically different. Each cell can accommodate a given maximum number of particles, and each pair of neighboring cells can exchange at most one particle at a time. The predictions of the model are in qualitative agreement with both experimental observations and molecular dynamics simulation results.

Molecular simulation of loading-dependent diffusion in nanoporous materials using extended dynamically corrected transition state theory

A dynamically corrected transition state theory method is presented that is capable of computing quantitatively the self-diffusivity of adsorbed molecules in confined systems at nonzero loading. This extension to traditional transition state theory is free of additional assumptions and yields a diffusivity identical to that obtained by conventional molecular-dynamics simulations. While molecular-dynamics calculations are limited to relatively fast diffusing molecules, our approach extends the range of accessible time scales significantly beyond currently available methods. We show results for methane, ethane, and propane in LTL- and LTA-type zeolites over a wide range of temperatures and loadings, and demonstrate the extensibility of the method to mixtures.

Loading dependence of the diffusion coefficient of methane in nanoporous materials

In this work, we use molecular simulations to study the loading dependence of the self-and collective diffusion coefficients of methane in various zeolite structures. To arrive at a microscopic interpretation of the loading dependence, we interpret the diffusion behavior in terms of hopping rates over a free-energy barrier. These free-energy barriers are computed directly from a molecular simulation. We show that these free-energy profiles are a convenient starting point to explain a particular loading dependence of the diffusion coefficient. On the basis of these observations, we present a classification of zeolite structures for the diffusion of methane as a function of loading: three-dimensional cagelike structures, one-dimensional channels, and intersecting channels. Structures in each of these classes have their loading dependence of the free-energy profiles in common. An important conclusion of this work is that diffusion in nanoporous materials can never be described by one single effect so that we need to distinguish different loading regimes to describe the diffusion over the entire loading range.

Influence of Cation Na/Ca Ratio on Adsorption in LTA 5A: A Systematic Molecular Simulation Study of Alkane Chain Length

Recent adsorption isotherms of n-alkanes on Ca,Na-LTA-type zeolite afford development of a force field describing the interactions between calcium and n-alkanes in configurational-bias Monte Carlo simulations. The force field of Calero et al. (J. Am. Chem. Soc. 2004, 126, 11377-11386) is able to accurately describe the adsorption properties of linear alkanes in the sodium form of FAU-type zeolites. Here, we extend upon this type of force field by including calcium-type ions. The force field was fitted to reproduce the calcium and sodium cations positions on LTA 5A and the experimental adsorption properties of n-alkanes over all range of temperatures and pressures. This opens up a vast amount of experimental data on LTA 5A, both on adsorption and diffusion. Furthermore, evaluation of half a century of reported n-alkane adsorption data on LTA-type zeolites indicates that there are many inconsistencies between the various data sets, possibly as a result of (i) undisclosed calcium and sodium contents, (ii) less than perfect drying of the hygroscopic zeolite, and (iii) coadsorption of contaminants such as vacuum grease. Having obtained our force field, and confirmed its reliability on predictions outside the calibration set, we apply the force field on two "open" problems: (a) the heats of adsorption and Henry coefficient as a function of chain length and (b) the effect of cations in LTA-type zeolites. The molecular simulations shed new light on previous experimental findings, and we provide rationalizations on the molecular level that can be generalized to the class of cage/window-type nanoporous materials.

Dynamically Corrected Transition State Theory Calculations of Self-Diffusion in Anisotropic Nanoporous Materials

We apply the dynamically corrected transition state theory to confinements with complex structures. This method is able to compute self-diffusion coefficients for adsorbate-adsorbent systems far beyond the time scales accessible to molecular dynamics. Two example cage/window-type confinements are examined: ethane in ERI- and CHA-type zeolites. In ERI-type zeolites, each hop in the z direction is preceded by a hop in xy direction and diffusion is anisotropic. The lattice for CHA-type zeolite is a rhombohedral Bravais lattice, and diffusion can be considered isotropic in practice. The anisotropic behavior of ERI-type cages reverses with loading, i.e., at low loading the diffusion in the z direction is two times faster than in the xy direction, while for higher loadings this changes to a z diffusivity that is more than two times slower. At low loading the diffusion is impeded by the eight-ring windows, i.e., the exits out of the cage to the next, but at higher loadings the barrier is formed by the center of the cages.

Understanding Diffusion in Confined Systems: Methane in a ZK4 Molecular Sieve. A Molecular Dynamics Simulation Study

The equilibrium probability distribution of N methane molecules adsorbed in the interior of n alpha cages of the ZK4 zeolite, the all-silica analogue of zeolite A, is modeled by a modified hypergeometric distribution where the effects of mutual exclusion between particles are extracted from long molecular dynamics simulations. The trajectories are then analyzed in terms of time-correlation functions for the fluctuations in the occupation number of the alpha cages. The analysis digs out the correlations induced by the spatial distribution of the adsorbed molecules coupled with a migration mechanism where a molecule can pass from one alpha cage to another, one-by-one. These correlations lead to cooperative motion, which manifests itself as a nonexponential decay of the correlators. Our results suggest ways of developing improved lattice approaches that may be useful for studying diffusion in much larger systems and for a much longer observation time.

Self-Diffusion and Transport Diffusion of Light Gases in Metal-Organic Framework Materials Assessed Using Molecular Dynamics Simulations

Metal-organic framework (MOF) materials pose an interesting alternative to more traditional nanoporous materials for a variety of separation processes. Separation processes involving nanoporous materials can be controlled by either adsorption equilibrium, diffusive transport rates, or a combination of these factors. Adsorption equilibrium has been studied for a variety of gases in MOFs, but almost nothing is currently known about molecular diffusion rates in MOFs. We have used equilibrium molecular dynamics (MD) to probe the self-diffusion and transport diffusion of a number of small gas species in several MOFs as a function of pore loading at room temperature. Specifically, we have studied Ar, CH4, CO2, N2, and H2 diffusion in MOF-5. The diffusion of Ar in MOF-2, MOF-3, and Cu-BTC has been assessed in a similar manner. Our results greatly expand the range of MOFs for which data describing molecular diffusion is available. We discuss the prospects for exploiting molecular transport properties in MOFs in practical separation processes and the future role of MD simulations in screening families of MOFs for these processes.

Coarse-grained nonequilibrium approach to the molecular modeling of permeation through microporous membranes

We present a modeling technique that combines a statistical-mechanical coarse-graining scheme with a nonequilibrium molecular simulation algorithm to provide an efficient simulation of steady-state permeation across a microporous material. The coarse-graining scheme is based on the mapping of an atomistic model to a lattice using multidimensional free-energy and transition-state calculations. The nonequilibrium simulation algorithm is a stochastic, lattice version of the recently developed atomistic dual-control-volume grand canonical molecular dynamics. We demonstrate the approach on a model of methane permeating through a bulk portion of siliceous zeolite ZK4 at 300 K under imposed fugacity differences. We predict the coarse-grained (cage-level) density profiles and observe the development of nonlinearities as the magnitude of the fugacity difference is increased. From the net flux of methane we also predict a mean permeability coefficient under the various conditions. The simulation results are obtained over time scales on the order of microseconds and length scales on the order of dozens of nanometers.