(174b) Dynamic Mean Field Theory with Hydrodynamic Interactions: Theoretical Development and Applications in the Study of Fluid Transport in Porous Adsorbents

Understanding the dynamics and equilibrium behavior of inhomogeneous fluids is of great importance in many fields of science and engineering, including gas and liquid separations in porous adsorbents and membranes, heterogeneous catalysis, enhanced oil recovery, microfluidics, etc. In these examples the structure of the fluids can become quite rich, especially if the fluid is transported through confined or complex geometries. In these examples, the thermodynamic view of a fluid under confinement can be quite different from what is observed in the bulk, while metastability as well as mass transfer limitations to equilibration can play an important role in this case [1]. Such challenging issues are tackled using molecular simulation methods and classical Density Functional Theory (DFT). The latter method can in principle describe accurately the thermodynamics of fluids confined in porous structures of arbitrary complexity. Furthermore, a dynamic version of the theory has been developed focusing primarily on the study of spinodal decomposition of a single fluid. Unfortunately, implementation of DFT in two or three dimensional systems is extremely expensive computationally even for equilibrium studies.

Coarse-grained methods based on lattice fluid models offer an attractive alternative for such cases, as has been demonstrated by several research groups over the last decade. Furthermore, a dynamic version the theory has been developed (Dynamic Mean Field Theory, DMFT) to describe the relaxation processes of fluids confined in porous materials [2]. The theory is based on a lattice fluid model and can be viewed as an approximation to Kawasaki dynamics, a process where the molecules hop between nearest neighbour sites under the action of transition probabilities such as those in the Metropolis Monte Carlo method, constrained by the restriction of single occupancy. DMFT reproduces the thermodynamic description from mean field DFT (MFT) in the appropriate limit so that it can describe both the dynamics and the thermodynamics of the system.

A limitation of DDFT and its lattice analogue, DMFT, is that when viewed within the context of the theory of transport phenomena it is essentially a continuity equation for the fluid density and produces overdamped dynamics, which characterize the interaction among colloidal particles immersed in a solvent acting as a heat bath. Thus the DDFT method is strictly valid for colloidal fluids, where there is friction between the colloidal particles and the solvent, so that one does not have to account for momentum currents and the velocity field is fully determined by the density field. For the case of molecular fluids on the other hand, there is no solvent present so the momentum of the particles is no longer damped by the friction with the solvent. In this case equilibration is realized primarily due to the action of viscosity and a momentum conservation equation must be included and be coupled with the continuity equation, so that the momentum current is treated on equal footing with the density field.

In this work we present a recently developed DMFT model that includes hydrodynamic interactions [3]. The new model satisfies simultaneously continuity and momentum equations under the assumptions of constant dynamic or kinematic viscosity and small velocities and/or density gradients. We first present applications of the theory to spinodal decomposition of subcritical temperatures for a lattice gas model in MFT. We find that the DMFT/HI model correctly describes the transition from diffusive phase separation at short times to hydrodynamic behaviour at long times. In addition, in common with DMFT, it provides a theory of transport that is fully consistent with the theory of the thermodynamics from density functional theory.

Accordingly, we apply the DMFT/HI model to study the transport of fluids in porous media. The model is extended to account for frictional effects due to the presence of the internal pore surface. We propose a simple formalism based on the concept of adding a local frictional force at the pore-solid interface. The frictional force is determined from the oscillator model employing the low density diffusion theory proposed by Bhatia and Nicholson [4]. The DMFT/HI model has been applied to study steady state and dynamic transport of a lattice fluid in a 2D slit pore at various degrees of wetting. Steady state simulations provide density and velocity profiles that are in qualitative agreement with results from rigorous Non-Equilibrium Molecular Dynamics simulations. Dynamic adsorption-desorption experiments reveal that hydrodynamic interactions result in faster uptake rates compared to the standard DMF model, and can have a major impact in the transport mechanism of the fluid at strong wetting conditions. In the latter case, the formation of the monolayer is slower than pore condensation during both adsorption and desorption. On the other hand, for the case of weak wetting conditions, the effect of hydrodynamic interactions does not alter the basic characteristics of the transport mechanism and the standard DMFT can still give a good qualitative picture of the transport process.

 

References

[1] P. A. Monson, Microporous and Mesoporous Materials 160, 47 (2012).

[2] P. A. Monson, Journal of Chemical Physics 128, 084701 (2008).

[3] E. S. Kikkinides and P. A. Monson, Journal of Chemical Physics 142 (2015).

[4] S. K. Bhatia and D. Nicholson, AICHE Journal 52, 29 (2006).