We explore the applicability of lattice model and dynamic mean field theory (DMFT) 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 behaviour 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.

This work offers a computationally efficient theoretical framework to investigate transport phenomena in complex systems such as mixed matrix membranes (MMMs). It is demonstrated here that a wide variety of heterogeneous, disordered geometries can be constructed using lattice models. These geometries combined with the dynamic mean field theory (DMFT) provide useful insights on the distribution of density and flux in the structures. From this work, the DMFT emerges as a theoretical playground to explore transport phenomena in MMMs under a variety of conditions and probe some of the assumptions involved in the commonly used macroscopic models. As a case study, a comparison of the predictions of the DMFT is considered with several classical macroscopic theories. In the preliminary observations, the study points to a much greater impact of the pore blocking effects on the overall transport of the composite system in comparison with the macroscopic models.

Heterogeneous nucleation is the dominant form of liquid-to-solid transition in nature. Although molecular simulations are most uniquely suited to studying nucleation, the waiting time to observe even a single nucleation event can easily exceed the current computational capabilities. Therefore, there exists an imminent need for methods that enable computationally fast and feasible studies of heterogeneous nucleation. Seeding is a technique that has proven to be successful at dramatically expanding the range of computationally accessible nucleation rates in simulation studies of homogeneous crystal nucleation. In this article, we introduce a new seeding method for heterogeneous nucleation called Rigid Seeding (RSeeds). Crystalline seeds are treated as pseudorigid bodies and simulated on a surface with metastable liquid above its melting temperature. This allows the seeds to adapt to the surface and identify favorable seed–surface configurations, which is necessary for reliable predictions of crystal polymorphs that form and the corresponding heterogeneous nucleation rates. We demonstrate and validate RSeeds for heterogeneous ice nucleation on a flexible self-assembled monolayer surface, a mineral surface based on kaolinite, and two model surfaces. RSeeds predicts the correct ice polymorph, exposed crystal plane, and rotation on the surface. RSeeds is semiquantitative and can be used to estimate the critical nucleus size and nucleation rate when combined with classical nucleation theory. We demonstrate that RSeeds can be used to evaluate nucleation rates spanning many orders of magnitude.

The Robeson bound is a theoretical limit that applies to kinetics-driven membrane separations of gas mixtures. However, this bound does not apply to sorption-driven membrane processes such as CO$_\text{2}$/N$_\text{2}$ separation, which lacks a theoretical explanation. As a result, we are uncertain about the factors that control the limiting behavior of sorption-driven separations. To address this issue, we employed a simple lattice model and dynamic mean field theory to examine the transport properties of disordered model structures, isolating sorption effects from purely kinetic effects. Our findings indicate that transport effects play a crucial role in sorption-driven processes, and perm-selectivity is consistently lower than sorption selectivity, which is an unattainable limit. We used basic geometric fragments of the structure to explain how transport effects emerge and manifest themselves in sorption-driven processes.