Tortuosity factor for permeant flow through a fractal solid

The theory of finite Markov processes is used to calculate a normalized tor tuosity for a porous solid with a well-defined internal structure character ized by an interconnected network of serial and parallel channels. The mode l introduced is based on the Menger sponge, a symmetric fractal set of dime nsion D=ln 20/ln 3. Numerically exact values of the mean walklength [n] for a permeant diffusing through this system are calculated both in the presen ce and absence of a uniform gradient (bias or external field) acting on the permeant. The ratio of walklengths is then used to define unambiguously a normalized tortuosity for the medium. Different assumptions on the initial spatial distribution of the permeant are investigated and the study is desi gned so that the effects of one or multiple exit sites are quantified. As e xpected, the tortuosity factor is dependent on system size, and quantitativ e results are presented for the first- and second-generation Menger sponge. Our calculations document that for a given system size: (1) the mean passa ge time can change by an order of magnitude depending on the number of exit pores available to the permeant; and (2) restricting the number of exit po res suppresses differences between values of [n] calculated for different i nitial conditions. Finally, we study whether the geometrical self-similarit y of Menger sponge translates into a corresponding scaling of diffusion tim es for different system sizes; we find that there exists a remarkable corre spondence in values calculated for a defined ratio of walklengths for the f irst two generations of this fractal set. (C) 2000 American Institute of Ph ysics. [S0021-9606(00)71322-5].