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 tau is considered for diffusion and reaction among partially absorbing traps with dimensionless surface rate constant kappaBAR. The limits kappaBAR = infinity and kappaBAR = 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 tau for arbitrary kappaBAR are derived in terms of the pore size distribution P(delta). Here P(delta)d-delta is the probability that a randomly chosen point in the pore region lies at a distance delta and delta + d-delta from the nearest point on the pore-trap interface. The aforementioned moments and hence the bounds on T1 and tau are evaluated for distributions of interpenetrable spherical traps. The length scales <delta> and <delta-2>1/2, under certain conditions, can yield useful information about the times T1 and tau, underscoring the importance of experimentally measuring or theoretically determining the pore size distribution P(delta). Moreover, rigorous relations between the relaxation times T(n) and the mean survival time are proved. One states that tau is a certain weighted sum over the T(n), while another bounds tau 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 tau 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.