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

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 T_n (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 T_n and the mean survival time are proved. One states that τ is a certain weighted sum over the T_n, 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.