Dynamics of capillary condensation in lattice gas models of confined fluids: A comparison of dynamic mean field theory with dynamic Monte Carlo simulations

How 2D Nanoflakes Improve Transport in Mixed Matrix Membranes: Insights from a Simple Lattice Model and Dynamic Mean Field Theory

Mixed matrix membranes (MMMs), incorporating graphene and graphene oxide structural fragments, have emerged as promising materials for challenging gas separation processes. What remains unclear is the actual molecular mechanism responsible for the enhanced permeability and perm-selectivity of these materials. With the fully atomistic models still unable to handle the required time and length scales, here, we employ a simple qualitative model based on the lattice representation of the physical system and dynamic mean field theory. We demonstrate that the performance enhancement results from the flux-regularization impact of the 2D nanoflakes and that this effect sensitively depends on the orientation of the nanoflakes and the properties of the interface between the nanoflakes and the polymer.

Connecting dynamic pore filling mechanisms with equilibrium and out of equilibrium configurations of fluids in nanopores

In the present study, using dynamic mean field theory complemented by grand canonical molecular dynamics simulations, we investigate the extent to which the density distributions encountered during the dynamics of capillary condensation are related to those distributions at equilibrium or metastable equilibrium in a system at fixed average density (canonical ensemble). We find that the states encountered can be categorized as out of equilibrium or quasi-equilibrium based on the magnitude of the driving force for mass transfer. More specifically, in open-ended slit pores, pore filling via double bridging is an out of equilibrium process, induced by the dynamics of the system, while pore filling by single bridge formation is connected to a series of configurations that are equilibrium configurations in the canonical ensemble and that cannot be observed experimentally by a standard adsorption process, corresponding to the grand canonical ensemble. Likewise, in closed cap slits, the formation of a liquid bridge near the pore opening and its subsequent growth while the initially detached meniscus from the capped end remains immobilized are out of equilibrium processes that occur at large driving forces. On the other hand, at small driving forces, there is a continuous acceleration of the detached meniscus from the capped end, which is associated with complete reversibility in the limit of an infinitesimally small driving force.

Machine-Learned Free Energy Surfaces for Capillary Condensation and Evaporation in Mesopores

Using molecular simulations, we study the processes of capillary condensation and capillary evaporation in model mesopores. To determine the phase transition pathway, as well as the corresponding free energy profile, we carry out enhanced sampling molecular simulations using entropy as a reaction coordinate to map the onset of order during the condensation process and of disorder during the evaporation process. The structural analysis shows the role played by intermediate states, characterized by the onset of capillary liquid bridges and bubbles. We also analyze the dependence of the free energy barrier on the pore width. Furthermore, we propose a method to build a machine learning model for the prediction of the free energy surfaces underlying capillary phase transition processes in mesopores.

Application of the dynamic mean field theory to fluid transport in slit pores

We explore the applicability of the lattice model and dynamic mean field theory as a computationally efficient tool to study transport across heterogeneous porous media, such as mixed matrix membranes. As a starting point and to establish some basic definitions of properties analogous to those in the off-lattice systems, we consider transport across simple models of porous materials represented by a slit pore in a chemical potential gradient. Using this simple model, we investigate the distribution of density and flux under steady state conditions, define the permeability across the system, and explore how this property depends on the length of the pore and the solid-fluid interactions. Among other effects, we observe that the flux in the system goes through a maximum as the solid-fluid interaction is varied from weak to strong. This effect is dominated by the behavior of the fluid near the walls and is also confirmed by off-lattice molecular dynamics simulations. We further extend this study to explore transport across heterogeneous slit pore channels composed of two solids with different values of solid-fluid interaction strengths. We demonstrate that the lattice models and dynamic mean field theory provide a useful framework to pose questions on the accuracy and applicability of the classical theories of transport across heterogeneous porous systems.

Nucleation of Capillary Bridges and Bubbles in Nanoconfined CO2

Using molecular simulation, we examine the capillary condensation and the capillary evaporation of CO₂ in cylindrical nanopores. More specifically, we employ the recently developed μV T-S method to determine the microscopic mechanism associated with these processes and the corresponding free energy profiles. We calculate the free energy barrier for capillary condensation and identify that the key step consists in the nucleation of a liquid bridge of a critical size. Similarly, the free energy maximum found for the capillary evaporation process is found to correspond to the nucleation of a vapor bubble of a critical size. In addition, we assess the impact of the strength of the wall-fluid on the height of the free energy barrier and on the critical size of liquid bridges (condensation process) and vapor bubbles (evaporation process). We observe that the height of the free energy barrier increases with the strength of the wall-fluid interactions. Finally, we build a theoretical model, based on capillary theory, to rationalize our findings. In particular, the simulation results reveal a linear scaling of the free energy barrier with the critical size, in excellent agreement with the theoretical predictions.

Water Filling and Emptying Kinetics in Two-Dimensional Hexagonal Mesoporous Silica of the Same Pore Diameter but Different Pore Lengths

The effect of pore length on the water filling and emptying rates was studied using mesoporous silica (MPS) with same pore diameter but different pore lengths. The pore diameter of the synthesized MPS was ∼8 nm, whereas the average pore lengths were 460, 1,770, and 4000 nm. The gravimetric method was employed to record the time course of the adsorbed mass of water in MPS at 298 K and 1 atm. In both the filling and emptying processes, the relaxation curves (time course of adsorbed mass of water per unit mass of sample) were not significantly related to the pore length. This independence of the initial adsorption and desorption rates on the pore length suggests that the surface of the MPS aggregates is the bottleneck in the overall adsorption and desorption processes and that the initial mass flux in each nanopore is inversely proportional to the pore length. Furthermore, because the relaxation times to reach the equilibrium state were independent of the pore length, the mass flux of water uptake, release, and transport probably increase with an increase in the pore length during the entire adsorption and desorption processes. A transport model to describe these phenomena was proposed.

A comparison of dynamic mean field theory and grand canonical molecular dynamics for the dynamics of pore filling and capillary condensation of fluids in mesopores

We use results from grand canonical molecular dynamics (GCMD) to test the predictions from dynamic mean field theory (DMFT) for the pore filling and capillary condensation mechanisms of a fluid confined in slit shaped mesopores. The theory predicts that capillary condensation occurs by a nucleation process in which a liquid bridge forms between the two walls, and the pore is filled via the growth of this bridge. For longer pores, multiple bridging is seen. These mechanisms are confirmed by the molecular dynamics simulations. The primary difference between the theory and simulations lies in the role of fluctuations. DMFT predicts a single nucleation time and location, while in GCMD (and in nature) a distribution of nucleation times and locations is seen.

A single-walker approach for studying quasi-nonergodic systems

The jump-walking Monte-Carlo algorithm is revisited and updated to study the equilibrium properties of systems exhibiting quasi-nonergodicity. It is designed for a single processing thread as opposed to currently predominant algorithms for large parallel processing systems. The updated algorithm is tested on the Ising model and applied to the lattice-gas model for sorption in aerogel at low temperatures, when dynamics of the system is critically slowed down. It is demonstrated that the updated jump-walking simulations are able to produce equilibrium isotherms which are typically hidden by the hysteresis effect characteristic of the standard single-flip simulations.

Modeling the influence of side stream and ink bottle structures on adsorption/desorption dynamics of fluids in long pores

We apply dynamic mean field theory to study relaxation dynamics for lattice models of fluids confined in linear pores with side streams and with ink bottle structures. Our results show several mechanisms for how the pore structure affects the dynamics, and these are amplified in longer pores. An important conclusion of this work is that features such as side streams and ink bottle segments can substantially slow the equilibration of fluids confined in long pore systems where the pore lengths can be more than 100 micrometers, such as in porous silicon. This may make it difficult to properly equilibrate these systems for states close to those where the pores should be completely filled with liquids. The presence of trapped bubbles in the system may change the desorption characteristics of the system and the shape of the hysteresis loops.

Influence of system size on the properties of a fluid adsorbed in a nanopore: Physical manifestations and methodological consequences

Hysteresis and discontinuities in the isotherms of a fluid adsorbed in a nanopore in general hamper the determination of equilibrium thermodynamic properties, even in computer simulations. A way around this has been to consider both a reservoir of small size and a pore of small extent in order to restrict the fluctuations of density and approach a classical van der Waals loop. We assess this suggestion by thoroughly studying through Monte Carlo simulations and density functional theory the influence of system size on the equilibrium configurations of the adsorbed fluid and on the resulting isotherms. We stress the importance of pore-symmetry-breaking states that even for modest pore sizes lead to discontinuous isotherms and we discuss the physical relevance of these states and the methodological consequences for computing thermodynamic quantities.

Filling dynamics of closed end nanocapillaries

We have studied the filling dynamics of model capillaries using dynamic mean field theory for a confined lattice gas and Kawasaki dynamics simulations. We have found two different scenarios for filling of capped nanocapillaries from the vapor phase. As compared to channels with macroscopic width, in which the filling process occurs by the detachment of the meniscus from the cap, in mesoscopic channels there is an alternative mechanism associated with the spontaneous condensation of the liquid close to the pore opening and its subsequent growth toward the closed pore end. We show that these two scenarios have totally different filling dynamics, providing an additional mechanism for slow capillary condensation kinetics in nanoscopic objects.

Dynamic mean field theory for lattice gas models of fluid mixtures confined in mesoporous materials

We present the extension of dynamic mean field theory (DMFT) for fluids in porous materials (Monson, P. A. J. Chem. Phys. 2008, 128, 084701) to the case of mixtures. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable equilibrium states for fluids in pores after a change in the bulk pressure or composition. It is especially useful for studying systems where there are capillary condensation or evaporation transitions. Nucleation processes associated with these transitions are emergent features of the theory and can be visualized via the time dependence of the density distribution and composition distribution in the system. For mixtures an important component of the dynamics is relaxation of the composition distribution in the system, especially in the neighborhood of vapor-liquid interfaces. We consider two different types of mixtures, modeling hydrocarbon adsorption in carbon-like slit pores. We first present results on bulk phase equilibria of the mixtures and then the equilibrium (stable/metastable) behavior of these mixtures in a finite slit pore and an inkbottle pore. We then use DMFT to describe the evolution of the density and composition in the pore in the approach to equilibrium after changing the state of the bulk fluid via composition or pressure changes.