Quantifying Pore Width Effects on Diffusivity via a Novel 3D Stochastic Approach with Input from Atomistic Molecular Dynamics Simulations

Diffusion and Gas Flow Dynamics in Partially Saturated Smectites

Clays and clay rocks are considered good natural and engineered barriers for deep geological disposal of nuclear waste worldwide. Metal corrosion and organic waste degradation in underground repositories generate significant amounts of gas that should be able to migrate through the multibarrier system to avoid potential pressure buildup, which could be compromising the integrity of the barriers and host rocks. The gas is expected to accumulate in larger pores and eventually form an interconnected network. Under such conditions, the migration of gas molecules takes place both in pore water films and gas-filled macropores. Therefore, mass fluxes depend on the distribution of gas molecules between the water-rich and gas-rich phases and their mobility in both compartments. Classical molecular dynamics (MD) simulations were employed to investigate the mobilities of He, H₂, CO₂, Ar, and CH₄ in a Na-montmorillonite mesopore as a function of the degree of saturation, as well as evaluate the hydrodynamic behavior of the pore fluid in partially saturated clays. The diffusivity of the gas molecules was determined by observing the asymptotic behavior of the mean square displacement in the gas-rich phase and at the gas-water interface. The partition coefficient and Gibbs free energy were analyzed to investigate the transfer of gas molecules between the gas-rich and water-rich phases by observing the molecular trajectories as they cross the vapor-liquid interface. The results revealed that the diffusion coefficient in the gas phase increased with increasing gas-filled pore width and converged asymptotically toward the diffusion coefficient in the bulk state. It could be shown that the diffusion coefficient of gas molecules dissolved in the water films remained constant as long as the interacting water surface was in the bulk-liquid-like phase. This behavior changes in very thin water films. It was observed that the partitioning coefficient of gas molecules at the solid-liquid interface is nearly the same as that in the bulk-liquid-like phase. Partitioning is observed to be strongly dependent on the temperature and gas molecular weights. In the second part of the study, nonequilibrium molecular dynamics (NEMD) simulations were performed to investigate the mobility of gases in pressure-driven decoupled gas-phase dynamics (DGPD) and coupled gas and water phase dynamics (CGWPD) in a partially saturated Na-montmorillonite slit mesopore. The dynamic viscosity of the gas phase was calculated from NEMD simulations and indicated that the viscosity of the gas phase was almost the same in both methods (DGPD and CGWPD). The average slip length for gas molecules at the gas-water interface was also calculated, revealing that the slip-free boundary condition assumed in continuum models is generally invalid for microfluidics and that a slip boundary condition exists at the microscale for specific surface interactions. Finally, a Bosanquet-type equation was developed to predict the diffusion coefficient and dynamic viscosity of gas as a function of the average pore width, gas mean-free path, geometric factor, and thickness of the adsorbed water film.

Exact distributed kinetic Monte Carlo simulations for on-lattice chemical kinetics: lessons learnt from medium- and large-scale benchmarks

Kinetic Monte Carlo (KMC) simulations have been instrumental in multiscale catalysis studies, enabling the elucidation of the complex dynamics of heterogeneous catalysts and the prediction of macroscopic performance metrics, such as activity and selectivity. However, the accessible length- and time-scales have been a limiting factor in such simulations. For instance, handling lattices containing millions of sites with 'traditional' sequential KMC implementations is prohibitive owing to large memory requirements and long simulation times. We have recently established an approach for exact, distributed, lattice-based simulations of catalytic kinetics which couples the Time-Warp algorithm with the Graph-Theoretical KMC framework, enabling the handling of complex adsorbate lateral interactions and reaction events within large lattices. In this work, we develop a lattice-based variant of the Brusselator system, a prototype chemical oscillator pioneered by Prigogine and Lefever in the late 60s, to benchmark and demonstrate our approach. This system can form spiral wave patterns, which would be computationally intractable with sequential KMC, while our distributed KMC approach can simulate such patterns 15 and 36 times faster with 625 and 1600 processors, respectively. The medium- and large-scale benchmarks thus conducted, demonstrate the robustness of the approach, and reveal computational bottlenecks that could be targeted in further development efforts. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Prediction of Adsorption and Diffusion of Shale Gas in Composite Pores Consisting of Kaolinite and Kerogen using Molecular Simulation

Natural gas production from shale formations is one of the most recent and fast growing developments in the oil and gas industry. The accurate prediction of the adsorption and transport of shale gas is essential for estimating shale gas production capacity and improving existing extractions. To realistically represent heterogeneous shale formations, a composite pore model was built from a kaolinite slit mesopore hosting a kerogen matrix. Moreover, empty slabs (2, 3, and 4 nm) were added between the kerogen matrix and siloxane surface of kaolinite. Using Grand-Canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations, the adsorption and diffusion of pure methane, pure ethane, and a shale gas mixture were computed at various high pressures (100, 150, and 250 atm) and temperature of 298.15 K. The addition of an inner slit pore was found to significantly increase the excess adsorption of methane, as a pure component and in the shale gas mixture. The saturation of the composite pore with methane was observed to be at a higher pressure compared to ethane. The excess adsorption of carbon dioxide was not largely affected by pressure, and the local number density profile showed its strong affinity to kerogen micropores and the hydroxylated gibbsite surface under all conditions and pore widths. Lateral diffusion coefficients were found to increase with increasing the width of the empty slab inside the composite pore. Statistical errors of diffusion coefficients were found to be large for the case of shale gas components present at low composition. A larger composite pore configuration was created to investigate the diffusion of methane in different regions of the composite pore. The calculated diffusion coefficients and mean residence times were found to be indicative of the different adsorption mechanisms occurring inside the pore.

Machine Learning Predictions of Simulated Self-Diffusion Coefficients for Bulk and Confined Pure Liquids

Diffusion properties of bulk fluids have been predicted using empirical expressions and machine learning (ML) models, suggesting that predictions of diffusion also should be possible for fluids in confined environments. The ability to quickly and accurately predict diffusion in porous materials would enable new discoveries and spur development in relevant technologies such as separations, catalysis, batteries, and subsurface applications. In this work, we apply artificial neural network (ANN) models to predict the simulated self-diffusion coefficients of real liquids in both bulk and pore environments. The training data sets were generated from molecular dynamics (MD) simulations of Lennard-Jones particles representing a diverse set of 14 molecules ranging from ammonia to dodecane over a range of liquid pressures and temperatures. Planar, cylindrical, and hexagonal pore models consisted of walls composed of carbon atoms. Our simple model for these liquids was primarily used to generate ANN training data, but the simulated self-diffusion coefficients of bulk liquids show excellent agreement with experimental diffusion coefficients. ANN models based on simple descriptors accurately reproduced the MD diffusion data for both bulk and confined liquids, including the trend of increased mobility in large pores relative to the corresponding bulk liquid.

Multiscale modeling of solute diffusion in triblock copolymer membranes

We develop a multiscale simulation model for diffusion of solutes through porous triblock copolymer membranes. The approach combines two techniques: self-consistent field theory (SCFT) to predict the structure of the self-assembled, solvated membrane and on-lattice kinetic Monte Carlo (kMC) simulations to model diffusion of solutes. Solvation is simulated in SCFT by constraining the glassy membrane matrix while relaxing the brush-like membrane pore coating against the solvent. The kMC simulations capture the resulting solute spatial distribution and concentration-dependent local diffusivity in the polymer-coated pores; we parameterize the latter using particle-based simulations. We apply our approach to simulate solute diffusion through nonequilibrium morphologies of a model triblock copolymer, and we correlate diffusivity with structural descriptors of the morphologies. We also compare the model's predictions to alternative approaches based on simple lattice random walks and find our multiscale model to be more robust and systematic to parameterize. Our multiscale modeling approach is general and can be readily extended in the future to other chemistries, morphologies, and models for the local solute diffusivity and interactions with the membrane.

Quantifying the Effect of Spatial Confinement on the Diffusion and Nuclear Magnetic Resonance Relaxation of Water/Hydrocarbon Mixtures: A Molecular Dynamics Simulation Study

We implement molecular dynamics (MD) simulations to explore the relaxation mechanisms involved in the description of translational diffusion and rotational dynamics in water/hydrocarbon interfaces. The analysis of density profiles, self-diffusion coefficients, and nuclear magnetic resonance (NMR) relaxation properties as a function of the confinement layer width and type of hydrocarbons improves the understanding of confined water properties at water/oil interfaces. Density profile fluctuations reveal the presence of water-oil interactions close to the interface. MD results show that self-diffusion coefficients and NMR relaxation times of planar and cylindrical water/oil interfaces are strongly influenced by layer thickness and geometry. Shorter (between 20 and 60%) self-diffusion coefficients and ¹H NMR relaxation times were obtained for water/n-pentane, water/n-decane, and water/n-hexadecane systems than bulk diffusion coefficients. An increase in Larmor frequency from 2.3 MHz to 400 MHz shows that longitudinal relaxation time (T₁) of confined oil has slightly larger differences at higher frequencies than the transverse relaxation time (T₂). At 400 MHz, n-alkanes (n-pentane, n-decane, and n-hexadecane) exhibit longer relaxation times than at smaller frequency values (2.3 and 22 MHz). Analysis of spin-spin and spin-lattice times provides relevant information about inter- and intramolecular relaxation mechanisms of water and oil as a function of geometry and width of the interface layer. These MD results suggest that the strength of confinement and geometry play a vital role in the diffusion and NMR relaxation properties of water/oil interfaces.

Homogenization of Continuum-Scale Transport Properties from Molecular Dynamics Simulations: An Application to Aqueous-Phase Methane Diffusion in Silicate Channels

Silica-based materials including zeolites are commonly used for wide-ranging applications including separations and catalysis. Substrate transport rates in these materials can significantly influence the efficiency of such applications. Two factors that contribute to transport rates include (1) the porosity of the silicate matrix and (2) nonbonding interactions between the diffusing species and the silicate surface. These contributions generally emerge from disparate length scales, namely, "microscopic" (roughly nanometer-scale) and "macroscopic" (roughly micron-scale), respectively. Here, we develop a simulation framework to estimate the simultaneous impact of these factors on methane mass transport in silicate channels. Specifically, we develop a model of methane transport using homogenization theory to obtain transport parameters valid at length scales of hundreds to thousands of nanometers. These parameters implicitly reflect interactions taking place at fractions of a nanometer. The inputs to the homogenization analysis are data from extensive molecular dynamics simulations that incorporate atomistic-scale interactions, processed to yield local diffusion coefficients and mean force potentials. With this model, we demonstrate how nuances in silicate hydration and silica/methane interactions impact methane diffusion rates in silicate materials, including the effects of silicate surface chemistry such as the presence of silanol groups. The molecular dynamics simulations indicate that methane diffusivity at the silica surface is lower than the bulk-like rates observed at the center of channels of sufficient width. However, potentials of mean force generally evidence attractive methane/silica interactions that enhance diffusion overall. By simultaneously accounting for both of these effects, we show that the effective diffusion coefficient for the nanoporous silicate can be approximately double the value of estimates assuming fully bulk-like behavior in the channel. This study therefore demonstrates the importance of determining diffusion coefficients and potentials of mean force at an atomistic level when estimating transport properties in bulk materials. Importantly, we provide a simple homogenization framework to incorporate these molecular-scale attributes into bulk material transport estimates. This hybrid homogenization/molecular dynamics approach will be of general use for describing small-molecule transport in materials with detailed molecular interactions.

A Novel Modeling Approach to Stochastically Evaluate the Impact of Pore Network Geometry, Chemistry and Topology on Fluid Transport

Fine-grained sandstones, siltstones, and shales have become increasingly important to satisfy the ever-growing global energy demands. Of particular current interest are shale rocks, which are mudstones made up of organic and inorganic constituents of varying pore sizes. These materials exhibit high heterogeneity, low porosity, varying chemical composition and low pore connectivity. Due to the complexity and the importance of such materials, many experimental, theoretical and computational efforts have attempted to quantify the impact of rock features on fluids diffusivity and ultimately on permeability. In this study, we introduce a stochastic kinetic Monte Carlo approach developed to simulate fluid transport. The features of this approach allow us to discuss the applicability of 2D vs 3D models for the calculation of transport properties. It is found that a successful model should consider realistic 3D pore networks consisting of pore bodies that communicate via pore throats, which however requires a prohibitive amount of computational resources. To overcome current limitations, we present a rigorous protocol to stochastically generate synthetic 3D pore networks in which pore features can be isolated and varied systematically and individually. These synthetic networks do not correspond to real sample scenarios but are crucial to achieve a systematic evaluation of the pore features on the transport properties. Using this protocol, we quantify the contribution of the pore network’s connectivity, porosity, mineralogy, and pore throat width distribution on the diffusivity of supercritical methane. A sensitivity analysis is conducted to rank the significance of the various network features on methane diffusivity. Connectivity is found to be the most important descriptor, followed by pore throat width distribution and porosity. Based on such insights, recommendations are provided on possible technological approaches to enhance fluid transport through shale rocks and equally complex pore networks. The purpose of this work is to identify the significance of various pore network characteristics using a stochastic KMC algorithm to simulate the transport of fluids. Our findings could be relevant for applications that make use of porous media, ranging from catalysis to radioactive waste management, and from environmental remediation to shale gas production.

Comparison of Queueing Data-Structures for Kinetic Monte Carlo Simulations of Heterogeneous Catalysts

On-lattice kinetic Monte Carlo (KMC) is a computational method used to simulate (among others) physicochemical processes on catalytic surfaces. The KMC algorithm propagates the system through discrete configurations by selecting (with the use of random numbers) the next elementary process to be simulated, e.g., adsorption, desorption, diffusion, or reaction. An implementation of such a selection procedure is the first-reaction method in which all realizable elementary processes are identified and assigned a random occurrence time based on their rate constant. The next event to be executed will then be the one with the minimum interarrival time. Thus, a fast and efficient algorithm for selecting the most imminent process and performing all of the necessary updates on the list of realizable processes post execution is of great importance. In the current work, we implement five data-structures to handle the elementary process queue during a KMC run: an unsorted list, a binary heap, a pairing heap, a one-way skip list, and finally, a novel two-way skip list with a mapping array specialized for KMC simulations. We also investigate the effect of compiler optimizations on the performance of these data-structures on three benchmark models, capturing CO oxidation, a simplified water gas shift mechanism, and a temperature-programmed desorption run. Excluding the least efficient and impractical for large-problems unsorted list, we observe a 3× speedup of the binary or pairing heaps (most efficient) compared to the one-way skip list (least efficient). Compiler optimizations deliver a speedup of up to 1.8×. These benchmarks provide valuable insight into the importance of, often-overlooked, implementation-related aspects of KMC simulations, such as the queueing data-structures. Our results could be particularly useful in guiding the choice of data-structures and algorithms that would minimize the computational cost of large-scale simulations.

A Method for Obtaining Liquid-Solid Adsorption Rates from Molecular Dynamics Simulations: Applied to Methanol on Pt(111) in H2O

Adsorption is an important step in heterogeneous catalysis as it predetermines how many reactant molecules can participate in a surface reaction per unit time. While the rate of adsorption processes is well studied in gas-solid adsorption in both theory and experiment, such rates are still not well studied for liquid-solid adsorption. This is partly because the ever-changing configurations of liquid-phase solvent molecules impede the ability to study a molecule approaching a surface from a liquid phase by either experiment or theory. In this work, we develop a method using molecular dynamics (MD) simulations to study the rate of adsorption in liquid-solid adsorption processes. Specifically, we use MD to model the diffusion of a methanol molecule in aqueous solvent and its adsorption to a Pt(111) surface. We find that by approximating the solute motion as following the same displacement rates as a random walk model, the adsorbed and non-adsorbed states of the methanol molecule near the Pt(111) surface can be discerned and quantified. In particular, this methodology enables extracting a sticking coefficient and a macroscopically relatable adsorption rate. This method can be applied to arbitrary types of reactants and surfaces, as well as different liquid environments, thus providing a general tool for predicting quantitative adsorption rates of liquid-solid adsorption systems.