Diffusion and reaction in heterogeneous media: Pore size distribution, relaxation times, and mean survival time

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Abstract
Diffusion and reaction in heterogeneous media plays an important role in a variety of processes arising in the physical and biological sciences. The determination of the relaxation times Tn (n=1,2,...) and the mean survival time τ is considered for diffusion and reaction among partially absorbing traps with dimensionless surface rate constant κ̄. The limits κ̄=∞ and κ̄=0 correspond to the diffusion-controlled case (i.e., perfect absorbers) and reaction-controlled case (i.e., perfect reflectors), respectively. Rigorous lower bounds on the principal (or largest) relaxation time T1 and mean survival time τ for arbitrary κ̄ are derived in terms of the pore size distribution P(δ). Here P(δ)dδ is the probability that a randomly chosen point in the pore region lies at a distance δ and δ+dδ from the nearest point on the pore–trap interface. The aforementioned moments and hence the bounds on T1 and τ are evaluated for distributions of interpenetrable spherical traps. The length scales 〈δ〉 and 〈δ2〉1/2, under certain conditions, can yield useful information about the times T1 and τ, underscoring the importance of experimentally measuring or theoretically determining the pore size distribution P(δ). Moreover, rigorous relations between the relaxation times Tn and the mean survival time are proved. One states that τ is a certain weighted sum over the Tn, while another bounds τ from above and below in terms of the principal relaxation time T1. Consequences of these relationships are examined for diffusion interior and exterior to distributions of spheres. Finally, we note the connection between the times T1 and τ and the fluid permeability for flow through porous media, in light of a previously proved theorem, and nuclear magnetic resonance (NMR) relaxation in fluid-saturated porous media.