Optimal Flux Properties of Fractal Respiratory Membranes

Peter Pfeifer

Center for Nonlinear Studies
Los Alamos National Laboratory
and
Department of Physics
University of Missouri

When particles diffuse from a source to a convoluted surface and  cross the surface irreversibly (membrane, heat exchanger, electrode, catalyst), the screened diffusion field in remote fjords produces a strongly nonlinear flux across the surface as function of bulk diffusion constant D and surface  permeability W. This leads to three well-defined regimes---complete screening, partial screening, and no screening---similar to a phase transition. The regime of partial screening is of particular interest because it is sensitive to the surface geometry:  For a self-similar surface with fractal dimension  D_f, the flux is proportional to D(W/D)^{1/(D_f-1)}; for a  self-affine surface (steep regime) with q-th order roughness exponent H_q, the flux  is proportional to D(W/D)^{H_1}^{1,\,2}.  The two power laws, when applied to the oxygen flux across a macroscopic respiratory membrane, predict that D_ f=3 and H_1=0, each operating near the transition from partial to no screening, are optimal architectures with respect to four seemingly incompatible criteria:  maximum flux, maximum insensitivity to loss of permeability, no waste of surface area, and  no waste of permeability.  The architectures are realized as lung in mammals and  gill in fish, respectively.  The two power laws also shed light on the evolutionary path that led to the two architectures.

P. Pfeifer and P.J. Hagerty, in: Fractals and Chaos in  Chemical Engineering, eds. M. Giona and G. Biardi (World Scientific, Singapore, 1997), p. 151-164.

S. Gheorghiu and P. Pfeifer, Phys. Rev. Lett. 85, 3894 (2000).