Restricted diffusion within a pore space is commonly described in terms of the apparent diffusion coefficient, Dapp, which can be measured experimentally using NMR. The time dependence of Dapp/D0 can be used to infer certain features of the pore-space morphology, making NMR diffusion measurements an invaluable tool for structural characterisation of porous media.
Calculation of Dapp from known pore-space geometry has traditionally been based on “geometric” considerations,1,2 whereby the diffusion propagator is found by solving the diffusion equation subject to the appropriate boundary conditions, and then translated into Dapp(Δ). This is technically challenging for all but the simplest geometries, and no general analytic solution is known for a pore space with arbitrary boundaries. Short-Δ asymptotic analysis yields an analytic relationship between Dapp/D0 and the surface-to-volume ratio of the pore space. In the long-Δ limit, a similar relationship exists between Dapp/D0 and tortuosity. The latter is typically represented as an expansion in terms of 1/Δ and 1/Δ3/2; however, no comprehensive justification exists for the form of this expansion.
We present a novel method for the calculation of Dapp(Δ) based on replacing geometric boundary conditions with an analytic potential representing the interaction between diffusing particles and the pore-space walls, and applying Langevin dynamics (LD) to model the diffusion of the particles under the assumption of local Boltzmann equilibrium. For proof-of-principle, we considered a classic model system, a stack of parallel semipermeable barriers.3 We obtained an analytic solution for Dapp(Δ) in the long-Δ limit and compared it with the results of numerical LD simulations. The analytic solution was consistent with the numerical LD results, demonstrating the viability of the LD-based approach for semi-analytic evaluation of Dapp in the long-Δ limit for a pore-space with a periodic but otherwise arbitrary geometry. We discuss possible extension of the LD approach to 2D and 3D cases.