From thermodynamic cell models to partitioning cellular automata for diffusion in zeolites. II. Static and dynamic properties

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.

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.

Thermodynamics of the one-dimensional parallel Kawasaki model: exact solution and mean-field approximations

The adsorption isotherm for the recently proposed parallel Kawasaki (PK) lattice-gas model [Phys. Rev. E 88, 062144 (2013)] is calculated exactly in one dimension. To do so, a third-order difference equation for the grand-canonical partition function is derived and solved analytically. In the present version of the PK model, the attraction and repulsion effects between two neighboring particles and between a particle and a neighboring empty site are ruled, respectively, by the dimensionless parameters ϕ and θ. We discuss the inflections induced in the isotherms by situations of high repulsion, the role played by finite lattice sizes in the emergence of substeps, and the adequacy of the two most widely used mean-field approximations in lattice gases, namely, the Bragg-Williams and the Bethe-Peierls approximations.

Synchronous equilibrium model for the diffusion of mutually exclusive particles in a heterogeneous lattice of adsorption sites

Through straight synchronization and proper manipulation of a sequential Monte Carlo glass-forming rule introduced by Fröbose and Jäckle [J. Stat. Phys. 42, 551 (1986)], we constructed a synchronous, non-glass-forming rule for diffusion of mutually exclusive particles in a lattice of adsorption sites. The rule satisfies detailed balance in the presence of both homogeneous and heterogeneous adsorption energies. Our model differs from the usual lattice-gas cellular automata diffusion rules in that the mutual exclusion holds on the lattice sites rather than on the channels which connect neighboring sites, and from the mass-conserving cellular automata rules in the use of a no-partitioning scheme. The first aim of this work is to show that, although some prescriptions in the synchronous rule are introduced just to allow that both detailed balance and mutual exclusion can coexist with synchronicity, the diffusion process produced by the rule is not anomalous so that the rule can be regarded as a diffusion model. We then compare the diffusion isotherms of several test systems with the ones obtained by means of sequential Monte Carlo simulations of Arrhenius jumps of particles on a lattice. Finally, we apply the rule to the case of a (100) fcc model surface and estimate the amount of time correlation in the migration process, and show that the synchronous rule produces higher correlations and slightly lower diffusivity than the sequential Monte Carlo counterpart.

Chemical potential evaluation in NVT lattice-gas simulations

The discrete nature of the partition function of a lattice-gas system can be exploited to build an efficient strategy for the evaluation of the chemical potential of a periodic lattice-gas with arbitrarily ranged interactions during a simulation in the canonical ensemble, with the need of no additional sampling as it were required instead by the Widom insertion/deletion approach. The present method is based on the main concepts of the small system grand ensemble [for details, see G. Soto-Campos, D. S. Corti, and H. Reiss, J. Chem. Phys. 108, 2563 (1998)], whose key idea is to study the properties of a sublattice (called small system) and of its complementary (the reservoir sublattice) as they were two separated subsystems. The accuracy of the measured chemical potential can be further improved by artificially "restoring" the missing connections among the reservoir sublattice sites located at the boundaries with the small system. We first illustrate the theory and then we compare μVT with NVT simulation results on several test systems.

The central cell model: a mesoscopic hopping model for the study of the displacement autocorrelation function

On the mesoscale, the molecular motion in a microporous material can be represented as a sequence of hops between different pore locations and from one pore to the other. On the same scale, the memory effects in the motion of a tagged particle are embedded in the displacement autocorrelation function (DACF), the discrete counterpart of the velocity autocorrelation function (VACF). In this paper, a mesoscopic hopping model, based on a lattice-gas automata dynamics, is presented for the coarse-grained modeling of the DACF in a microporous material under conditions of thermodynamic equilibrium. In our model, that we will refer to as central cell model, the motion of one tagged particle is mimicked through probabilistic hops from one location to the other in a small lattice of cells where all the other particles are indistinguishable; the cells closest to the one containing the tagged particle are simulated explicitly in the canonical ensemble, whereas the border cells are treated as mean-field cells in the grand-canonical ensemble. In the present paper, numerical simulation of the central cell model are shown to provide the same results as a traditional lattice-gas simulation. Along with this a mean-field theory of self-diffusion which incorporates time correlations is discussed.