The effects of diffusion mechanism and void structure on transport rates and tortuosity factors in complex porous structures

Tracer diffusion simulations within random porous structures show that tortuosity factors are independent of diffusion mechanism for all practical void fractions when an equivalent Knudsen diffusivity is correctly defined. Previous studies concluded that tortuosity factors, a geometric property of the void space as defined, increase with increasing Knudsen number, K_n, a measure of the relative number of molecule-surface and intermolecular collisions. The model porous structures in this study consist of random-loose packings of spheres overlapped to achieve a given void fraction and to accurately reflect the void space in practical porous solids. Effective diffusivities were estimated using tracer or flux-based Monte Carlo methods for Knudsen numbers of 10⁻³-10¹⁰; the two methods lead to similar diffusivities for void fractions of 0.06-0.42. Tortuosity factors estimated using the number-averaged distance between collisions, 〈l_p〉, for the characteristic void length scale increased with increasing Knudsen number, even though simulations in infinite cylinders confirmed the accuracy of the Bosanquet equation for all values of K_n. These unexpected changes in a geometric property of the void space become most apparent near the percolation void fraction (∼0.04). For example, the Knudsen tortuosity factor defined in this manner is 1.8 times larger than in the bulk regime for a solid with 0.10 void fraction. Even at high void fractions (∼0.42), the two extreme values of tortuosity factor differ by a factor of ∼1.4. These apparent effects of diffusion mechanism on tortuosities reflect the inaccurate use of number-averaged chord lengths when tracer reflections from random obstacles obey the Knudsen cosine law for diffuse reflection. A corrected length scale, first proposed by Derjaguin, leads to tortuosity factors independent of K_n for void fractions above 0.20; tortuosities differ by only 18% and 4% between Knudsen and bulk regimes even for void fractions of 0.10 and 0.15, respectively. The residual differences at void fractions below 0.10 arise from the increasingly serial nature of the remaining voids. Thus, a long-standing inconsistency between the defined geometric nature of tortuosity factors and their inexplicable dependence on diffusion mechanism is essentially resolved. In practice, these simulations allow the consistent and accurate use of tortuosity factors determined at any value of K_n for all diffusion regimes; they also prescribe, rigorously for void fractions above 0.15 and empirically for lower void fractions, the length scale relevant to diffusion in the Knudsen and transition diffusion regimes.