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Substantial progress has been made in elucidating the impact of a poroelastic medium, such as a deformable solid containing a saturating fluid, on the transport behavior of the fluid.  This coupling of fluid transport and material deformation is relevant for many kinetic, structural, and geomechanical applications including, for example, flow through gels, soil consolidation, and earth subsidence associated with oil recovery.  Our program comprises two interrelated efforts.  First, recognizing that the elastic coupling between the solid and the fluid can lead to complex transport kinetics in an anisotropic solid, we have developed a framework that naturally incorporates stress-driven diffusion in a two-phase system.  More specifically, we have constructed and validated a simplified phase-field model of fluid imbibition in a weakly poroelastic solid.  In this context, phase-field models represent the overdamped dynamics of order parameters that follow from a free-energy functional and that can be identified with the solid and fluid phases.  With this model, we investigated the motion of fluid fronts as a function of fluid self-strain and the anisotropic character of the medium.  It was found that the rate of fluid imbibition depends on the strength of the crystalline anisotropy as well as the mutual orientation of the crystallographic axes and the propagating fluid front.  Perhaps the most interesting result of this work was the finding that fluid transport may be spatially nonlocal (i.e., non-Fickian) in an anisotropic medium as the corresponding diffusive flux depends on the location of other fluid elements throughout the system.  The results of this work are summarized in a recent publication in the Journal of Applied Physics.

In our second, complementary effort, we have examined tracer diffusion in the presence of obstacles having a well-defined mobility.  As hindered diffusion occurs in a wide variety of systems, we have chosen to focus on the prototype of tracer diffusion in a binary alloy.  In particular, we have employed Monte Carlo simulation to describe the nonequilibrium diffusive behavior of the components of a two-dimensional lattice gas comprising A and B atoms wherein one of the species (B) interacts with randomly-distributed line defects to create equilibrium  atmospheres at late times.  Various kinetic assumptions and defect densities are explored to highlight the role of B-atom mobility and defect interaction strength on the transport behavior of the A atoms.  From the calculated instantaneous diffusivity, several diffusive regimes are then identified and related to evolving segregation profiles and, in particular, to the free area available for diffusion. 

More generally, the system considered here can be viewed as an illustration of coupled diffusion/relaxation behavior in which diffusional pathways evolve with time due to an interaction between a diffusant and its environment.  For example, in systems where a diffusant penetrates a high viscosity medium, the diffusion equation may be generalized by the addition of a memory term to describe the slow structural relaxation of the solvent.  In this case, the coupling between diffusional processes and structural relaxation arises due to the swelling of the solvent as solute particles penetrate the medium, and different diffusional  regimes are predicted here as well.  Thus, it may be possible to construct a similar generalized diffusion equation for tracers in our system in which the tracer/obstacle coupling is subsumed into a frequency-dependent chemical potential.  The results of this work are summarized in a manuscript that has been submitted for publication.

The next phase of our program involves the incorporation of realistic pore structures in the phase-field model described above.  The current model includes porosity only in a spatially coarse-grained representation, and so it is of interest to employ a plausible distribution of pore sizes, shapes, and interconnectivity.  It is expected that this approach will permit us to make a closer connection with other models of poroelastic transport.  Finally, we may also wish to perform complementary molecular dynamics simulations of diffusion in a deformable porous medium.  For practical purposes, we might model fluid flow in, for example, two narrow, nearby channels embedded in an elastic solid, the latter comprising atoms that interact via a prescribed interatomic potential.  It will be of interest to quantify the extent to which fluid flow in one channel affects flow in the nearby channel as a function of the compressibility of the solid, the channel geometry, the density of the fluid, etc.