Difusividad Efectiva

JOURNAL OF CHEMICAL PHYSICS

VOLUME 119, NUMBER 14

8 OCTOBER 2003

COMMUNICATIONS Effective diffusivity in periodic porous materials
Alexander M. Berezhkovskiia)
Center for Information Technology, National Institutes of Health, Bethesda, Maryland 20892

Vladimir Yu. Zitserman
Thermophysical Center, Institute for High Temperatures, Russian Academy of Sciences, Ul. Izhorskaia 13/19,127412 Moscow, Russia

Stanislav Y. Shvartsmanb)
Department of Chemical Engineering and Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544

Received 28 April 2003; accepted 13 August 2003 Diffusion of a solute in a periodic porous solid is analyzed. An expression for the effective diffusion coefficient is derived for a solute diffusing in a porousmedium formed by a simple cubic lattice of spherical cavities connected by narrow tubes. This expression shows how the effective diffusion coefficient depends on microgeometry of the porous material. Generalizations to nonspherical cavities, other lattices, and nonequal diffusion coefficients in the cavities and in the tubes are discussed. © 2003 American Institute of Physics. DOI: 10.1063/1.1615758Motivated by recent successes in the design of periodic porous materials,1 we study solute diffusion in a solventfilled periodic array of identical cavities connected by narrow tubes and derive an expression for the effective diffusivity. The wide range of architectures in a new class of manmade porous materials promises a large number of applications in chemistry and biology. One of thepotential applications is in controlled drug release: By regulating the microstructure of a porous material one can tune the molecular diffusivity of a drug and, hence, the rate of its efflux from the encapsulating matrix.2,3 Recently, we have derived the effective diffusivity for the case of periodic arrays of touching spherical cavities.4 Our analysis was based on the recent results for the escape ofBrownian particles through the entropic barrier.5 Here, we combine the general approach presented in Ref. 4 with the analysis of the particle translocation through a narrow tube connecting two reservoirs.6 This enables the derivation of the expression for the effective diffusivity, D e f f , in a porous medium formed by spherical cavities arranged in a lattice and connected by tubes. The derivedstructure/property relationship shows how D e f f depends on the microgeometry of the medium. Straightforward generalizations of our analysis to other lattices, nonspherical cavities, and nonequal diffusion coefficients in the cavities and in the tubes are discussed at the end of the paper. Consider a particle diffusing in a porous medium formed by a simple cubic lattice of spherical cavities ofradius R connected by tubes of length L and radius a, which is much
a

smaller than R. The particle diffusion constant in the unconstrained solvent is D 0 . The effective diffusivity characterizes diffusion of a particle on times when its mean square displacement is much larger than the square of the lattice period, (2R L) 2 . To derive an expression for D e f f one has to find the mean squaredisplacement by solving the diffusion equation in real geometry. To perform this calculation one has to solve the diffusion equation in the cavity and in the tube and match the two solutions at the tube entrance. This is an extremely complicated task, which cannot be carried out. One can find a discussion of different approximate approaches to the problem in Ref. 7. While the problem cannot be solvedanalytically in the general case, an expression for D e f f can be derived when a R. In this case, diffusion can be replaced by a lattice random walk between the neighboring sites, which coincide with the centers of the spheres. Indeed, when a R, the time required for a particle to find an entrance to a tube is much larger than the characteristic relaxation time of the distribution in the cavity to...