Effective Diffusivity Tensors of Point-Like Molecules in ... - Springer Link

Transport in Porous Media 47: 279–293, 2002. c 2002 Kluwer Academic Publishers. Printed in the Netherlands.

279

Effective Diffusivity Tensors of Point-Like Molecules in Anisotropic Porous Media by Monte Carlo Simulation SINH TRINH1, , BRUCE R. LOCKE2 and PEDRO ARCE2,3 1 Center for Materials Research and Technology, The Florida State University, Tallahassee, FL

32310, U.S.A. 2 Department of Chemical Engineering, FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310, U.S.A. 3 Geophysical Fluid Dynamics Institute, GFDI-Florida State University, Tallahassee, FL 32310, U.S.A. (Received: 3 November 1999; in final form: 24 July 2001) Abstract. Monte Carlo simulations of random walks in anisotropic structured media are performed to determine the dependence of effective diffusivities on geometrical properties. The anisotropic media used in this study are periodic systems, which are generated by extending primitive, facecentered, and body-centered unit cells indefinitely in all axial directions. Results of simulations compare well with published experimental data and the calculations by the volume averaging method. In addition, these results suggest that if the 2D media with percolation thresholds subtantially differ from those of 3D, 2D approximations of 3D media are not satisfactory. When percolation thresholds are the same, the effective diffusivity tensors depend solely on the porosity. This fact has been suggested for isotropic media and it seems to hold for anisotropic media. Key words: effective diffusivity tensors, anisotropic media, Monte Carlo simulation.

1. Introduction Many porous materials (either natural, man-made, nonbiological, or biological) are of anisotropic nature, that is, the transport properties such as permeability, diffusivity, and thermal conductivity vary with direction in a specific region in the medium (Bear, 1988). The anisotropy is usually caused by the orientation and shape of asymmetric grains (or obstacles) making up the porous matrix. Determination of the ‘effective’ or macroscopic transport properties of these materials and the relationship to the microscopic structure of the material is very relevant, for example, to the design of new materials for separation and drug delivery processes, to aid in the interpretation of experimental data on diffusivities (Riley et al., 1995), and to improve the diagnostic accuracy in noninvasive techniques currently used in clinical applications such as MRI. Previous efforts have focused on calculation of Author for correspondence

280

SINH TRINH ET AL.

the transport properties of idealized structures by continuum mechanics approaches or by computational methods. There is a need to determine the sensitivity of the calculations for the effective transport properties with regards to variation in structure and to validate the results obtained with a given approach. For example, comparison between calculations based on direct computer simulation methods, such as MC approaches, and those based on macroscopic continuum equations, such as the volume averaging methodologies, have received little attention in previous research efforts. This contribution will present results for direct comparison of Monte Carlo simulations and the volume averaging method. This study is motivated, in part, by recent efforts performed to synthesize new materials where internal structure is tailored for a given application (Rill et al., 1996, 1998). The present analysis will focus on the study of the process of diffusion of point-like molecules in 2D and 3D idealized anisotropic structures. Specifically, the components of the effective diffusivity tensors will be obtained by direct computational methods using Monte Carlo simulations, and for certain cases, also compared with those obtained by volume averaging approaches and with experimental data. The article will be organized in several sections. The section below will provide a brief review of work on transport in anisotropic media. The computational methodology, numerical illustrations and discussion follow. This contribution is a sequel to that of Trinh et al. (1999b) that considered transport in isotropic materials.

2. Overview of Previous Work and Methodologies Nonbiological porous materials constitute a large family with a variety of geological, geophysical, and engineering applications (Toledo, 1992). For example, sedimentary rocks, soils, wood products, and other similar types of porous media found in nature are examples of heterogeneous anisotropic media even at a very small scale (Bear, 1988). Examples of transport and flow in engineering applications include the recovery of oil from underground reservoirs, the motion of pollutants in groundwater flows, and processes related to reaction and separations of products in the synthesis of new chemical and biochemical species and materials (Toledo et al., 1992). Transport in anisotropic porous media is also important in the separation of biomacromolecules in new synthetic materials with an internal anisotropic structure. Materials of this type can be obtained, for example, by using macromolecules and or surfactant assemblies as templates (Rill et al., 1996, 1998). Diffusion in anisotropic media is also important for a number of fundamental biological processes (Basser et al., 1994a, b, 1996; Henkelman et al., 1994) and for several applications in materials science (Johannesson et al., 1996). Recent advances in pulsed field gradient NMR methods allow for rapid and accurate measurement of diffusion tensors in a variety of biological and non-biological materials (Callaghan et al., 1994, 1995). For example, diffusion is highly anisotropic

EFFECTIVE DIFFUSIVITY TENSORS OF POINT-LIKE MOLECULES

281

in many muscles including leg muscle (van Gelderen et al., 1994), fish muscle (Kinsey et al., 1999), and heart muscle (Garrido et al., 1994). Nerve fibers also have highly anisotropic diffusion coefficients (Beaulieu et al., 1994). Transport by diffusion in liquid crystal media has been found to be highly anisotropic. Interest in the development of experimental methods for the study of anisotropic diffusion compliment the motivation behind the present theoretical study where key geometrical parameters that govern diffusion in porous media are investigated for a variety of porous media structures. Analysis of transport in anisotropic media has received less attention in the literature in comparison to that for isotropic materials. There are, of course, some notable exceptions in this regard. For example, Bernasconi (1974) showed that the model based on the use of an effective medium theory works well for networks with anisotropic binary distribution. This approach has been extended by Toledo et al. (1992) to include structures based on any regular lattice regardless of its connectivity. The focus of their efforts was on the determination of ‘effective conductances’ of the network. Monte Carlo simulation has also been used to study effects of anisotropic media on Knudsen diffusivity tensors (Tomadakis and Sotirchos, 1991a, b, 1993). These studies were carried out in porous media constructed to mimic fibrous materials with random and unidirectional orientations. Also, Ochoa-Tapia et al. (1994) presented an analysis of diffusive transport in two-phase media by using the method of volume averaging based on a modification of the unit cell proposed by Chang (1982). Extensions of the method of volume averaging to 3D media were also carried out. These researchers presented a useful review of methodologies used to determine effective transport properties in multiphase systems that included early contributions such as those by Maxwell (1881) and Rayleigh (1892). Applications of the volume averaging approach to obtain transport properties in anisotropic media were also carried out by Saez and Perfetti (1991) and Quintard (1994), and comparisons with experimental measurements (Kim et al., 1987) were presented. The experimental data on effective diffusivities in anisotropic media were collected using media composed of mylar and mica disks. The results of computations by Saez and Perfetti (1991) were based on 2D and 3D ordered media. The results of these computations of the effective diffusivity tensor for anisotropic media show poor agreements with the experimental measurement. By modeling the media with randomly oriented disks, Quintard (1994) demonstrated that it was possible to get the theoretically calculated effective diffusivity tensor closer to the experimentally measured values than those that had been done previously, but no accurate anisotropic diffusivity tensors were obtained. Trinh et al. (1999b) reported that the effective diffusivities of point-like molecules in a variety of isotropic structured media obtained by Monte Carlo simulation agree well with other theoretical predictions and with experimental data. In the present contribution, results using MC simulations for anisotropic media are compared with volume averaging calculations and experimental measurements. The

282

SINH TRINH ET AL.

general methodology of the simulations used to obtain the effective diffusivities has been discussed in detail by Trinh et al. (1999b), and only a brief summary of the modifications necessary for anisotropic media is provided here. 3. Method of Simulations Calculation of the effective diffusivity along a given direction in a specific material structure is based on the following dimensionless equation: De ξ 2 = lim , τ →∞ τ D

(1)

where ξ 2 is the dimensionless mean square displacement of the Brownian particle executing a random walk in the presence of the obstacles. This equation is slightly modified for the case of diffusion in an anisotropic porous medium. Due to the anisotropy of the material, the components of the effective diffusivity tensor are no longer the same for all the directions of transport of the molecule in the media. Hence, Equation (1) is modified to account for anisotropic properties of a given material to yield ξ 2 De,ii = lim i , τ →∞ τ D

(2)

where De,ii /D is the component of the diffusion tensor in the i-direction, and ξi is the mean square displacement in the i principal axis of the diffusion tensor. As in the previous communication (Trinh et al., 1999b) Monte Carlo simulation is used in the present study to obtain the dimensionless mean square displacement, ξi2 . The cross terms in the diffusivity tensor, namely, De,xy , are absent because the principal axes of the diffusivity tensor are also the same as those of the obstacles. The component of the effective diffusivity tensor in the i-direction, De,ii /D, is obtained by allowing Brownian particles to execute random walks on the 2D and 3D lattices. Mean square displacements are determined for random walks of point solutes moving from one unoccupied site to another. The obstacles take up various arrangements of sites in order to form the basic geometrical features of the medium. The basic structure of the medium is the unit cell. Unit cells are constructed with three basic configurations namely, primitive, body-centered and face-centered. The placement of the obstacles at specific locations defines the structure of the unit cell. The obstacles are represented by filling specific lattice sites. The anisotropic porous media are constructed by periodically extending the unit cells indefinitely in all appropriate axial directions (i.e. two directions for 2D and three directions for 3D). In addition, body-centered unit cells are also studied for the cases of 3D media. The obstacles are of rectangular and elliptical geometries for 2D media, and of rectangular parallelepiped and ellipsoid geometries for 3D media.


283

Figure 1. Two-dimensional unit cell with rectangular obstacles. (a) Primitive, (b) facecentered.

Primitive and face-centered unit cells for the 2D media with rectangular and elliptical obstacles are shown in Figure 1 and 2. Face-centered unit cells for the 3D media with circular cylindrical obstacles are shown in Figure 3. Figures 3(a) and 3(b) show the same type of unit cells with different orientations of cylindrical obstacles. For comparison with experimental values of the effective diffusivity tensors Kim et al. (1987), face-centered unit cells (Figure 3(b)) are used in Monte Carlo simulations. For homogeneous periodic porous media, the unit cells are extended indefinitely in all (two perpendicular directions for 2D and three for 3D) directions to mimic the actual media. By definition, porosity of the media is calculated according to the following formula:  n   1  1 − A(C) A(Oi ) 2D   i=1 (3)

= n   1  V (Oi ) 3D   1 − V (C) i=1

where A(O), V (O), A(C), and V (C) are the area and volume of the obstacles, respectively. To obtain the mean square displacement of the Brownian particle for the determination of the components of the diffusivity tensor, Equation (2), the Brownian particle is randomly introduced into the medium until it is successfully placed in an unoccupied region of the unit cell. At time τ = 0, the particle begins to take random jumps to one of the nearest unoccupied sites, and this process is repeated for each time increment. If the selected site for the particle is occupied by an obstacle, then a new site is randomly selected. Otherwise, the particle jumps to the selected site. In either case, the time is incremented by one unit. The process is repeated for sufficiently large time such that variations of the values of the

284

SINH TRINH ET AL.

Figure 2. Two-dimensional unit cell with elliptical obstacles. (a) Primitive, (b) face-centered.

Figure 3. Three-dimensional face-centered unit cell with circular cylindrical obstacles. (a) All cylinders oriented in one direction, (b) axes of cylinders perpendicular to the plane on which the cylinder lies.

effective diffusivity tensor are no longer significant when the time increment is further increased. In these simulations, 50 000 time increments are sufficient to obtain an accurate diffusivity tensor. At every 100 time increments, the location of the particle and the value of τ are recorded and the square of the displacement from the initial location is calculated. The square of the displacements is obtained for 100 Brownian particles and their values are averaged over these particles to obtain the mean square displacements. Five hundred mean square displacements obtained at every 100 time intervals are then fitted with a straight line by using linear regression. The slope of the line thus obtained is the ratio of the component of the effective diffusivity tensor and the molecular diffusion coefficient. The least square fit with 500 points over a period of 50 000 random jumps results in accurate determinations of different components of the diffusivity tensor. The desired accuracy for the diffusivity tensor is established when larger values of τ do not change the values of the diffusivity tensor by more than 10−5 .


285

Figure 4. The y-component effective diffusivity tensor in primitive unit cell with rectangular obstacle for various aspect ratios.

4. Results and Discussion of Simulation The results presented in this section will focus on plots of the components of the effective diffusivity tensor as functions of porosity for various changes in parameters of the unit cell. In Figure 4, 5, and 7, the curves are the results of best fit using a least square technique, the points are obtained by Monte Carlo simulations. The error in the Monte Carlo simulations are deliberately reduced to negligible by taking a large number of samples. Let a and b be the characteristic dimensions of the obstacles, and La and Lb be the corresponding dimensions of the unit cell. For example, a and b are the height and width of a rectangle in the case of rectangular obstacles, and the major and minor axes for the case of elliptical obstacles. Then, the aspect ratio for the obstacles, α is defined as a/b. Figures 4 and 5 (see Figures 1–3 for depiction geometries and labels of coordinate axes) show De,yy /D as functions of porosity for primitive and face-centered unit cells with different geometries and aspect ratios. For porosity greater than 0.6 and aspect ratio of 1.5, De,yy /D for rectangular obstacles is almost the same as that for elliptical obstacles. The difference between the values of De,yy /D for the rectangular obstacles and those for elliptical obstacles are larger for larger

286

SINH TRINH ET AL.

Figure 5. The y-component effective diffusivity tensor in face-centered unit cell with rectangular obstacle for various aspect ratios.

aspect ratios. The differences are more significant near the percolation thresholds. Percolation thresholds are the values of porosity at which the media form isolated pores. The effective diffusivities in porous media made of elliptical obstacles are always less than those made of rectangular obstacles. This characteristic can be explained based on the differences in the geometries of the obstacles. For rectangular and elliptical obstacles with the same values of a and b, the porosities of media consisting of rectangular obstacles is less than those of elliptical obstacles, but the obstruction of the passage of the Brownian particles is the same for both geometries. This characteristic is observed to be true for all unit cells regardless of their dimensionalities. For large α, De,yy /D is nonzero only in a narrow range of porosity. As α increases De,yy /D decreases. For a given porosity, the lesser the aspect ratio, the larger the component of De,yy /D. It can be shown from geometrical arguments that the percolation thresholds are 1 − β/α where β ≡ La /Lb . The percolation thresholds shown in these figures are close to those obtained by geometrical arguments. It follows that the larger the aspect ratio, the larger the percolation threshold. The component, De,yy is nonzero only for porosity greater than the percolation threshold. Thus, the larger the aspect ratio, the greater the changes in De,yy /D


287

with respect to porosity. For large aspect ratios, small changes in the porosity cause large changes in the dimension of the obstacles, which in turn may obstruct the passage of the particle. This observation accounts for the rapid changes in De,yy /D with for large aspect ratio. For large aspect ratio, the media form channels. The formation of channels accounts for the effective diffusivity along the channel being the same as that of the free solution, that is, De,xx = D. The dependence of De,yy /D on the aspect ratios are similar for all unit cells and geometries considered in this study. These simulations show that there exist percolation thresholds for various unit cells, geometries and aspect ratios. The percolation thresholds obtained by simulations agree with the values = 1 − 2β/α which are obtained by geometrical arguments when the obstacles do not overlap, that is, La < 2a and Lb < 2b. For a given aspect ratio, the percolation thresholds corresponding to the face-centered unit cell with rectangular obstacles are less than those of the primitive unit cell with rectangular obstacles. For primitive square unit cells with elliptical obstacles, the percolation thresholds are easily shown to be 1 − πβ/(4α), and for a face-centered unit cell with elliptical obstacles, it is 1 − πβ/(2α) when percolation occurs only in the x-direction, that is, when < 1 − π/8(α/β + β/α). The percolation threshold for both the x- and y-directions is 1 − π/8(α/β + β/α). These values for percolation thresholds along the y direction are close to those obtained by Monte Carlo simulations. The small discrepancies between the percolation thresholds obtained by simulation and those obtained by geometrical methods are due to the effects of discretization. These effects are more pronounced for elliptical obstacles because of the curvature of the ellipses. To reduce these effects, larger unit cells are required, which in turn increase the time of simulations. The method of volume averaging, originally developed by Whitaker (1967), Slattery (1969), and Anderson and Jackson (1967) can also be used to determine effective diffusivities in porous media. By using an approach based on volume averaging, Whitaker and coworkers (Ochoa-Tapia et al., 1994) found the components of the effective diffusivity tensor for a two-phase system consisting of an impermeable matrix and a fluid within the open pores of the media to be given by 1 1 De = I+ α (nαβ f + fnαβ )dA, (4) D V Aαβ 2 where I is the unit tensor, nαβ is the normal unit vector to the interface between the α and β phases, and the f vector field is determined by solving the closure problem in a unit cell. This expression is valid for anisotropic as well as isotropic media. Ochoa and collaborators (Kim et al., 1987), Nozad et al. (1985), and Ryan et al. (1981) have presented extensive analysis and computations of the effective diffusivity in model porous media using the volume averaging method. Equation (4) has been used to determine D e /D for anisotropic media by previous researchers, however, in the present paper Equation (4) will also be used determine the effective diffusivity tensors for some types of media not considered previously by

288

SINH TRINH ET AL.

Figure 6. The y-component effective diffusivity tensor in primitive unit cell with elliptical obstacle for various aspect ratios.

other researchers. Equation (4) has been evaluated in the present study using a finite element code to determine the components of the effective diffusivity tensor for comparison with the MC simulations for the case of face-centered unit cell with rectangular geometry. Figure 6 shows De,yy /D using face-centered unit cell, rectangular obstacles for various values of α. The continuous curves represent the values of the dimensionless effective diffusivity obtained by Monte Carlo simulations and the points were obtained by the method of volume averaging. The percolation threshold increases as the aspect ratio increases. The percolation thresholds agree well with those calculated. The results of Monte Carlo simulation also agree well with those obtained by the method of volume averaging. This comparison is very useful since it indicates that in terms of the diffusion coefficient for certain type of structured and anisotropic media, calculation of the values of the effective diffusivity in the media by averaging the macroscopic continuum equation is very close to direct simulation of Brownian motion. D e /D for face-centered unit cells with circular cylindrical Figure 7 shows D obstacles as illustrated in Figure 3(a). The points represented by the crosses and


289

Figure 7. The effective diffusivity tensor obtained by using the face-centered unit cell illustrated in Figure 3(a).

x’s are the measured effective diffusivity obtained by Kim et al. (1987). All other points shown in the figure were obtained by using Monte Carlo simulations for various aspect ratios. The continuous curves are obtained by least square fitting of the points. Unlike the results obtained for 2D system, the dimensionless effective diffusivity is virtually independent of the aspect ratios. This characteristic seems to suggest that the fact that for systems with substantial differences in percolation thresholds the effective diffusivity tensor for 3D cannot be approximated by that of 2D. For 3D anisotropic media, the effective diffusivity tensor shown in this figure seem to depend solely on the porosity due to the fact that these media of different aspect ratios have the same percolation thresholds. The effects of media on effective diffusivities obtained by Monte Carlo simulation are similar to those D e /D is less than those of the obtained experimentally. The z-components of D x-components. The differences in the effective diffusivity tensors obtained by Monte Carlo simulation and those obtained by experimental measurements are due to the fact that the orientations of the disks in the experimental setup will never align perfectly as those used in the simulations. Monte Carlo simulations were also carried out using media constructed with a different type of face-centered unit cell (Figure 3(b)). The face-centered unit cell

290

SINH TRINH ET AL.

Figure 8. The effective diffusivity tensor obtained using the face-centered unit cell illustrated in Figure 3(b).

is constructed by placing a disk on each of the six faces of the unit cell such that the center of the disk is located at the center of the square and the central axis of the disk coincides with the normal vector of the bounding planes of the unit cell. The obstacles are constructed using circular cylinders with aspect ratios of radius to height ranging from 5 to 8. These cylindrical obstacles are allowed to overlap to mimic mica disks with chips and defects. D e /D obtained from Monte Carlo simulations in the above Figure 8 shows D D e /D tensors obmedium and experimental measurements. The components of D tained using these media show good agreement with those obtained experimentally. The anisotropic diffusivity tensors obtained by Monte Carlo simulations have been favorably compared with the published experimental data based on the models of the media shown in Figure 3(b). Previously, the components of the effective diffusivity tensors calculated by volume averaging did not compare satisfactorily with experimental data. By using more accurate geometrical descriptions of the media in an improved model, Quintard (1994) has shown that it is possible to match the calculated results with experimental data.


291

5. Conclusions Direct simulation of the diffusivity of point-like molecules in a variety of anisotropic media has been performed in this contribution. Monte Carlo simulation has been used for the calculation of the mean square displacement of the molecules in a number of different media. The Einstein relation, with modification for anisotropic media was used to determine the components of the diffusivity tensors for the media under analysis. In addition, the values of the effective diffusivity tensors for certain type of media were obtained by the method of volume averaging. Comparison between the two approaches showed excellent agreement. A model of the media constructed using the unit cell shown in Figure 3(b) was used to obtain the effective diffusivity tensors for comparison with published experimental data, and the agreement of the experimental data with the Monte Carlo simulations in this new medium was very good. For 2D media, the percolation threshold obtained by Monte Carlo simulation, at a given value of the aspect ratio, are consistent with those obtained by using geometrical arguments. For a given porosity, different media structures give rise to different effective diffusivities and demonstrate the fact of the inadequacy of using porosity as the only variable to characterize the effective diffusivity tensor. The use of percolation thresholds to characterize the effective diffusivity has been suggested for isotropic media. These results also suggest that when the percolation thresholds of the two media are quite different, the effective diffusivity tensors for one media cannot be approximated with those for the other. Acknowledgements We gratefully acknowledge support from the National Science Foundation, BES 9521381, the Center for Materials Research and Technology (MARTECH) at Florida State University, and an RTG grant from Institute of Molecular Biophysics (IMB) at Florida State University. We would like to thank Mr M. Benson for his initial contribution to this project, and Dr R. Rill and Dr D. Van Winkle for their helpful comments and discussions. References Anderson, T. B. and Jackson, R.: 1967, A fluid mechanical description of fluidized beds, I & EC Fund. 6, 527–539. Basser, P. J., Mattiello, J. and LeBihan, D.: 1994a, MR diffusion tensor spectroscopy and imaging, Biophys. J. 66, 259. Basser, P. J., Mattiello, J. and LeBihan, D.: 1994b, Estimation of the effective self-diffusion tensor from the NMR spin echo, J. Magnet. Reson. Ser. B 103, 247. Basser, P. J. and Pierpaoli, C.: 1996, Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI, J. Magnet. Reson. Ser. B 111, 209. Bear, J.: 1998, Dynamics of Fluid in Porous Media, Dover, New York. Beaulieu, C. and Allen, P. S.: 1994, Determinants of anisotropic water diffusion in nerves, Magnet. Reson. Med. 31, 394.

292

SINH TRINH ET AL.

Bernasconi, J.: 1974, Conduction in anisotropic disordered systems: effective medium theory, Phys. Rev. B 9, 2612. Callaghan, P. T., Clark, C.J. and Forde, L.C.: 1994, Use of static and dynamic NMR microscopy to investigate the origins of contrast in images of biological tissues, Biophys. Chem. 50, 225. Callaghan, P. T. and Stepisnik, J.: 1995, Spatially-distributed pulsed gradient spin echo NMR using single-wire proximity, Phys. Rev. Lett. 75, 4532. Chang, H.-C.: 1982, Multi-scale analysis of effective transport in periodic heterogeneous media, Chem. Eng. Comm. 15, 83–91. Garrido, L., Wedeen, V. J., Kwong, K. K., Spencer, U. M. and Kantor, H. L.: 1994, Anisotropy of water diffusion in the myocardium of the rat, Circul. Res 74, 789. Henkelman, R. M., Stanisz, G. J., Kim, J. K. and Bronskill, M. J.: 1994, Anisotropy of NMR properties of tissues, Magnet. Reson. Med. 32, 592. Johannesson, H., Furo, I. and Halle, B.: 1996, Orientational order and micelle size in the nematic phase of the cesium pentadecaflurorooctanoate-water system from the anisotropic self-diffusion of water, Phys. Rev. E 53, 4904. Kim, J. H., Ochoa, J. A. and Whitaker, S.: 1987, Diffusion in anisotropic media, Transport in Porous Media 2, 327–356. Kinsey, S. T., Locke, B. R., Penke, B. and Mooreland, T. S.: 1999, Diffusional anisotropy is Induced by subcellular barriers in skeltal muscle, Magnet. Reson. Med. 12, 1–7. Maxwell, J. C.: 1881, A Treatise on Electricity and Magnetism, 2nd edn., Vol. I. Clerdon Press, Oxford, Vol. I, p. 440. Nozad, I., Carbonell, R.G. and Whitaker, S.: 1985, Heat conduction in multiphase systems – I theory and experiment for two-phase systems. Chem. Engng Sci. 40, 843–855. Ochoa-Tapia, J. A., Stroeve, P. and Whitaker, S.: 1994, Diffusive transport in two-phase media: spatially periodic modles and maxwell’s theory for isotropic and anisotropic systems, Chem. Engng Sci. 49, 709–726. Quintard, M.: 1994, Diffusion in isotropic and anisotropic porous systems: three-dimensional calculations, Transport in Porous Media 11, 187–199. Rayleigh, L.: 1892, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Phil. Mag. 34, 481–503. Riley, R. M., Muzzio, F. J., Buettner, H. and Reyes, S. C.: 1995, Diffusion in heterogeneous media: application to immobilized cell systems, AIChE J. 41, 691–700. Rill, R. L., Locke, B. R., Liu, Y., Dharia, J. and Van Winkle, D.: 1996, Protein electrophoresis in polyacrylamide gels with templated pores, Electrophoresis 17, 1304–1312. Rill, R. L., Van Winkle, D. and Locke, B. R.: 1998, Templated pores in hydrogels for improved size selectivity in gel permeation chromatography, Anal. Chem. 70, 2433. Ryan, D., Carbonell, R. G. and Whitaker, S.: 1981, A theory of diffusion and reaction in porous media, AIChE Symp. Ser. 46–62. Saez, A. E. and Perfectti: 1991, Prediction of effective diffusivities in porous media using spatially periodic models, Transport in Porous Media 6, 143–157. Slattery, J. C.: 1969, Single-phase flow through porous media, AIChE. J. 15, 866–872. Toledo, P. G., Davis, H. T. and Scriven, L. E.: Transport properties of anisotropic porous media: effective medium theory, Chem. Eng. Sci. 42, 391. Tomadakis, M. M. and Stratis V. S.: 1991a, Effective Knudsen diffusivities in structures of randomly overlapping fibers, AIChE J. 37, 74–86. Tomadakis, M. M., Stratis V. S.: 1991b, Knudsen diffusivities and properties of structures of unidirectional fibers, AIChE J. 37, 1175–1186. Tomadakis, M. M., Stratis V. S.: 1993, Ordinary, transition, and Knudsen regime diffusion in random capillary structures, Chem. Eng. Sci. 48, 3323–3333. Trinh S., Arce, P. and Locke, B. R.: 1996, Effective diffusivities of point-like molecules in isotropic porous media by Monte Carlo simulation, Transport in Porous Media (in press).


293

van Gelderen, P., DesPres, D., van Zijl, P. C. M. and Moonen, C. T. W.: 1994, Evaluation of restricted diffusion in cylinders. Phosphocreatine in rabbit leg muscle, J. Mag. Reson. Ser. B 103, 255. Whitaker, S.: 1967, Diffusion and dispersion in porous media, AIChE. J. 420–427. Whitaker, S.: 1969, Advances in theory of fluid motion in porous media, I & E. C. Res. 61, 14–28. Yi, Xiaohua, Shing, K. S. and Sahimi, M.: 1996, Molecular simulation of adsorption and diffusion in pillared clays, Chem. Eng. Sci. 51, 3409–3426.