Validation of mesoscopic theory and its application to computing concentration dependent diffusivities

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.

Quantitative understanding of cell signaling: the importance of membrane organization

Systems biology modeling of signal transduction pathways traditionally employs ordinary differential equations, deterministic models based on the assumptions of spatial homogeneity. However, this can be a poor approximation for certain aspects of signal transduction, especially its initial steps: the cell membrane exhibits significant spatial organization, with diffusion rates approximately two orders of magnitude slower than those in the cytosol. Thus, to unravel the complexities of signaling pathways, quantitative models must consider spatial organization as an important feature of cell signaling. Furthermore, spatial separation limits the number of molecules that can physically interact, requiring stochastic simulation methods that account for individual molecules. Herein, we discuss the need for mathematical models and experiments that appreciate the importance of spatial organization in the membrane.

Continuum mesoscopic framework for multiple interacting species and processes on multiple site types and/or crystallographic planes

While recently derived continuum mesoscopic equations successfully bridge the gap between microscopic and macroscopic physics, so far they have been derived only for simple lattice models. In this paper, general deterministic continuum mesoscopic equations are derived rigorously via nonequilibrium statistical mechanics to account for multiple interacting surface species and multiple processes on multiple site types and/or different crystallographic planes. Adsorption, desorption, reaction, and surface diffusion are modeled. It is demonstrated that contrary to conventional phenomenological continuum models, microscopic physics, such as the interaction potential, determines the final form of the mesoscopic equation. Models of single component diffusion and binary diffusion of interacting particles on single-type site lattice and of single component diffusion on complex microporous materials' lattices consisting of two types of sites are derived, as illustrations of the mesoscopic framework. Simplification of the diffusion mesoscopic model illustrates the relation to phenomenological models, such as the Fickian and Maxwell-Stefan transport models. It is demonstrated that the mesoscopic equations are in good agreement with lattice kinetic Monte Carlo simulations for several prototype examples studied.

Spectral method study of domain coarsening

The self-organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as Ostwald ripening. The late stage of Ostwald ripening is studied here through the use of a mesoscopic model in conjunction with a special configuration that allows for the direct measurement of system characteristics such as domain size. The mesoscopic model is a stochastic partial integrodifferential equation and is studied through the use of recently developed and benchmarked spectral schemes. The size of the growing region is not observed to increase as a power law in this model during the late stages of self-organization in contrast to the power law growth observed in simulations of the earlier stages of self-organization. The results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system.

Multiscale spatial Monte Carlo simulations: Multigriding, computational singular perturbation, and hierarchical stochastic closures

Monte Carlo (MC) simulation of most spatially distributed systems is plagued by several problems, namely, execution of one process at a time, large separation of time scales of various processes, and large length scales. Recently, a coarse-grained Monte Carlo (CGMC) method was introduced that can capture large length scales at reasonable computational times. An inherent assumption in this CGMC method revolves around a mean-field closure invoked in each coarse cell that is inaccurate for short-ranged interactions. Two new approaches are explored to improve upon this closure. The first employs the local quasichemical approximation, which is applicable to first nearest-neighbor interactions. The second, termed multiscale CGMC method, employs singular perturbation ideas on multiple grids to capture the entire cluster probability distribution function via short microscopic MC simulations on small, fine-grid lattices by taking advantage of the time scale separation of multiple processes. Computational strategies for coupling the fast process at small length scales (fine grid) with the slow processes at large length scales (coarse grid) are discussed. Finally, the binomial tau-leap method is combined with the multiscale CGMC method to execute multiple processes over the entire lattice and provide additional computational acceleration. Numerical simulations demonstrate that in the presence of fast diffusion and slow adsorption and desorption processes the two new approaches provide more accurate solutions in comparison to the previously introduced CGMC method.

The role of molecular interactions and interfaces in diffusion: Transport diffusivity and evaluation of the Darken approximation

Kinetic Monte Carlo (KMC) simulations are carried out to directly study diffusion of benzene through thin (37-100 nm) NaX zeolite membranes under a gradient in chemical potential. Nonlinearities in adsorbate loading near the membrane boundaries are shown to arise from the difference in adsorbate density between the zeolite and adjacent fluid phase. Direct extraction of the transport diffusivity from gradient KMC simulations enables testing of the Darken approximation. This rigorous approach reveals limitations of the Darken approximation and, for the first time, the potentially complex nonunique functionality and multiplicity of the transport diffusivity for strongly interacting adsorbates. In the companion paper we explore these nonlinear interfacial effects in the context of permeation through both single-crystal and polycrystalline membranes.

Molecular sieve valves driven by adsorbate-adsorbate interactions: hysteresis in permeation of microporous membranes

A recently derived mesoscopic framework describing activated micropore diffusion is employed to explore system criticality in microporous membranes under nonequilibrium conditions. Rapid exploration of parameter space, possible with this continuum framework, elucidates a novel temperature-induced ignition and extinction of the molecular flux under a macroscopic gradient in pressure (chemical potential). Deviation from equilibrium like phase behavior (i.e., shifting and narrowing of phase envelopes and double hysteresis) derives from asymmetry of the coupled boundaries of the nonequilibrium membrane. We confirm this new phase behavior, akin to "opening" and "closing" of a molecular valve, via gradient kinetic Monte Carlo simulations of thin one-dimensional and three-dimensional systems. The heat of adsorption, strength of adsorbate-adsorbate intermolecular forces, and chemical potential gradient are all shown to control 'valve' actuation, suggesting potential implications in chemical sensing and novel diffusion control.

Lattice density functional theory of molecular diffusion

A density functional theory of diffusion is developed for lattice fluids with molecular flux as a functional of the density distribution. The formalism coincides exactly with the generalized Ono-Kondo density functional theory when there is no gradient of chemical potential, i.e., at equilibrium. Away from equilibrium, it gives Fick's first law in the absence of a potential energy gradient, and it departs from Fickian behavior consistently with the Maxwell-Stefan formulation. The theory is applied to model a nanopore, predicting nonequilibrium phase transitions and the role of surface diffusion in the transport of capillary condensate.