Volume 149, number 1 PHYSICS LETFERS A 10 September 1990 Non-Fickian diffusive flow through disordered membranes S.F. Burlatsky, G.S. Oshanin Institute ofChemical Physics, USSR Academy ofSciences, Kosygin Street 4, 117334 Moscow V-334, USSR and A.!. Chernoutsan Department ofPhysics, Gubkin’s Gas and Oil Institute, Leninsky Prospekt 65, 117296 Moscow, USSR Received 19 June 1990; accepted for publication 29 June 1990 Communicated by V.M. Agranovich The diffusive flow through an inhomogeneous membrane containing repulsive impurities is investigated. We show that fluctua- tions in the impurity distribution lead to an anomalous exponential dependence ofthe permeability on the membrane thickness in the limit of small L. Such a behavior is governed by the random impurity-free channels that span the membrane. For larger L the dependence of the permeability x on L changes from an exponential to a stretched exponential law. In the large L limit the permeabilityx approaches a constant mean-field value. 1. Introduction the other hand, in numerical simulations (see ref. [11], and references therein) and experimental The study of diffusive processes in disordered me- studies of porous media some new effects were ob- dia attracts considerable attention [1—3]due to a served which are actually connected with fluctua- wide variety of applications in physics, chemical tions of pore sizes. In particular, it was shown in ref. physics and biology. A partial list of such applica- [12] that the steady flux of particles through 2D po- tions includes separation or catalytic processes in rous media exponentially vanishes with particle ra- zeolites or pillared clays, reverse osmosis membrane dius ri,, J—~ exp ( — r~/z), where z is the mean size of separation, diffusion-controlled fusion of excitations pores. An analogous dependence upon particle ra- in porous films, polyelectrolyte gels, etc. dius was predicted in ref. [13] for the steady flux An essential place is taken by the problem of dif- through a barrier membrane containing randomly fusive transport through disordered membranes. The distributed hard-core obstacles. Besides, there it has membrane is usually modeled as a flat object with been shown that the steady flux has an anomalous one dimension fixed (membrane thickness L) and dependence upon the membrane thickness L: two other dimensions tending to infinity in the ther- ,~. ~. modynamic limit. It is natural to expect that in such exp~ —J ~ mesoscopic objects the spatial fluctuations would play where f depends on the geometric characteristics of an essential role. However, most of the available the- the obstacles. oretical models of transport through porous media In the present Letter we investigate the steady [4—71, synthetic and biological barrier membranes permeability X(L) of a three-dimensional barrier (see refs. [8—10],and references therein) describe membrane containing randomly distributed repul- only mean-field properties and neglect effects con- sive impurities as a function of the membrane thick- nected with fluctuations in the distributions of po- ness L. The membrane’s diffusive permeability is de- res, obstacles or impurities of different nature. On fined as 0375-96011901$ 03.50 C 1990 — Elsevier Science Publishers B.V. (North-Holland) 47