Transp Porous Med (2011) 88:193–203 DOI 10.1007/s11242-011-9734-9
Simple Expression for the Tortuosity of Porous Media L. Pisani
Received: 29 May 2009 / Accepted: 19 January 2011 / Published online: 12 February 2011 © Springer Science+Business Media B.V. 2011
Abstract In this article, we derive a simple expression for the tortuosity of porous media as a function of porosity and of a single parameter characterizing the shape of the porous medium components. Following its value, a very large range of porous materials is described, from non-tortuous to high tortuosity ones with percolation limits. The proposed relation is compared with a widely used expression derived from percolation theory, and its predictive power is demonstrated through comparison with numerical simulations of diffusion phenomena. Application to the tortuosity of hydrated polymeric membranes is shown. Keywords
Tortuosity · Porosity · Porous media · Diffusion
1 Introduction Transport in porous media is a very important issue in a large number of science branches, including geophysics, chemical engineering and material science. In addition to the specific interactions which may occur between the fluid phase and the solid medium, all the transport phenomena must take into account the decrease of the volume available to fluid transport due to the presence of the solid medium and an increase of the path that the fluid must walk to cross the tortuous medium. These two effects can be described at a macroscopic level using the porosity (ε) and the tortuosity (τ ) parameters and by rescaling the transport coefficients according to D=
ε Dvoid τ
(1)
where Dvoid is the corresponding parameter in the void space. In this article, ε represents the fraction of free volume, whilst τ is the reciprocal of the average ratio between the straight
L. Pisani (B) Center for Advanced Studies, Research and Development in Sardinia (CRS4), Parco Scientifico e Tecnologico, POLARIS, Edificio 1, 09010 Pula, CA, Italy e-mail:
[email protected]
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distance connecting two points and the actual path length inside the porous medium. The ratio f = ε/τ is known as the structure factor.1 The geometrical definition of tortuosity given above, although largely used in the literature, is not universally accepted. Discussion of the tortuosity concept, however, lays beyond the scope of this article, and we address the interested reader to the following basic papers Carman (1937), Carman (1956), Kubik (1986), Adler (1992) and Cieszko (2009). Whilst the porosity of a medium can be easily derived from its weight and density, the tortuosity of porous media in general depends on the pore volume fraction, shape and connectivity. However, for some classes of materials, theoretical (or phenomenological) relations exist expressing the tortuosity as a function of the porosity only (Koponen et al. 1997; Mackie and Meares 1955; Matyka et al. 2008; Prager 1960; Shen and Chen 2007; Weissberg 1963). The best known amongst these expressions is the Bruggeman relation (Meredith and Tobias 1962) τ = ε −0.5 ,
(2)
which has been proven to be a good approximation for a large class of porous materials. Such a relation is valid only when the pore phase is connected. Actually, pore closure phenomena occur, which, at porosities lower than a characteristic percolation threshold (ε < ε0 ) can interrupt mass (and charge) passing through the pores. Percolation theory provides a conductivity expression that is valid for ε close to the percolation threshold (Stauffer and Aharony 1994) D ∝ (ε − ε0 )t and can be normalized to obtain the correct limit for ε close to unity ε − ε0 t f = ε/τ = . (3) 1 − ε0 Equation 3 generalizes the Bruggeman relation Eq. 2, which is obtained for ε0 = 0 and t = 1.5, and is able to describe a large number of transport phenomena but has a serious drawback being a function of two parameters, ε0 and t, which are known only for simple model systems. In this study, we propose a simple geometrical approach to derive the tortuosity as a function of the porosity. It will result in an expression as simple as Eq. 3 but with a single parameter with a well-defined physical meaning.
2 Methods 2.1 Model The model describes the porous media as a series of solid objects of given shape and dimensions. In particular, we suppose to know the volume V of the objects, their cross section σ , and their “radius” r , namely, the average distance necessary to by-pass them. While V and σ have a well-defined quantitative geometrical meaning, the definition of r is qualitative, since “the average distance necessary to bypass the obstacle” depends not only on the obstacle shape but also on the transport phenomenon. However, in the result section, 1 When direct interactions of the fluid phase with the porous solid surface are not negligible, Eq. 1 is no longer valid, and the dependence of the transport parameters on the porous structure characteristics becomes more complex. For simplicity, in this article, the analysis is limited to diffusion–conduction phenomena for which Eq. 1 hold.
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we show two possible strategies to use the model in a predictive quantitative way. Notice also that both σ and r depend on the orientation of the object with respect to the direction of motion so that non isotropic media can be described as well. For a sphere of radius R, we have V = 4π R 3 /3, σ = π R 2 , r ≈ R
(4)
These objects are randomly ranged in space. By defining Vvoid as the average void volume around each object, the system porosity can be expressed as ε = Vvoid / (V + Vvoid ) .
(5)
The tortuosity, defined above as the ratio between effective path length and straight distance in the direction of motion through the porous medium, can be expressed in terms of the geometry of the scattering objects. The path of the moving particles through the medium can be represented as a succession of free rides between the obstacles in the direction of motion, alternated with deviations to bypass the obstacles. From this “microscopic” point of view, ¯ i.e. the average distance between one object and sketched in Fig. 1, the straight distance is d, the nearest ones in the direction of motion, whilst the effective path length can be expressed as the sum of d¯ and δ, i.e. the average additional path necessary to bypass the obstacle ¯ τ = d¯ + δ /d. (6) The average distance d¯ can be easily expressed in terms of object volumes and cross section; as illustrated in Fig. 2, we can write ¯ V + Vvoid = d dS = σ d. (7) σ
When the density of solid objects is low, and they did not touch each other, we have δ = r.
(8)
However, at higher densities, whilst bypassing an object, other objects can obstruct the passage, and δ becomes larger than r . In other words, the “path necessary to bypass the object” becomes tortuous as well. Thus, we can write δ = r τ.
(9)
Equation 9 is only approximate as it uses the “macroscopic” tortuosity parameter inside a microscopic relation; furthermore, δ is orthogonal to the direction of motion for which τ is defined. However, such correction is necessary to describe dense media, whilst the simpler relation Eq. 8, leads to a tortuosity expression that is valid only at very high porosity. From Eq. 6 and 9 we get ¯ d¯ − r ). τ = d/(
(10)
Finally, by combining Eq. 10 with Eqs. 5 and 7 we obtain τ = [1 − α (1 − ε)]−1 .
(11)
We have here introduced the shape factor α = r σ/V
(12)
which will be seen to characterize the shape of the obstacles. For spheres, from Eq. 4, we get α ≈ 0.75.
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Fig. 1 Scheme of the model system δ
d δ
Fig. 2 Scheme illustrating the relation between d, V , Vvoid and σ
By following the same procedure with Eq. 8 instead of Eq. 9, we get τ = 1 + α (1 − ε).
(13)
which we expect to hold in the high porosity limit only. Equation 13 is the linear expansion of Eq. 11 for ε ≈ 1. Using Eq. 11, we obtain f = ε/τ = αε2 + (1 − α)ε.
(14)
In Fig. 3, the structure factors of Eqs. 3 and 14 are compared. Equation 14 represents parabolas passing through the (0,0) and (1,1) points, whilst Eq. 3 represents t-power curves with origin in (ε0 ,0) and passing through (1,1). The two functions coincide in the linear limit (ε0 = 0, t = 1 in Eq. 3 and α = 0 in Eq. 14 and in the quadratic one (ε0 = 0, t = 2 in Eq. 3 and α = 1 in Eq. 14. These two cases are represented in Fig. 3 by the curves A and D. In Fig. 3, it can be observed that the Bruggeman expression Eq. 2 (dashed line) is comprised between α = 0.5 and α = 0.75 (curves B and C); the latter value corresponding to a media composed of non-overlapping spheres explains why Eq. 2 is valid for a large range of porous materials. For α > 1, the structure factor in Eq. 14 is positive only for ε larger than a threshold value ε0 , with ε0 = (α − 1)/α. From a geometrical point of view (see Eqs. 7 and 12), α > 1 corresponds to r > d¯ (Vvoid = 0). In such conditions, for ε small enough, the distance r necessary to by-pass the object becomes larger than the mean free path d, and a new obstacle is met before the previous one is passed, thus impeding the transport through the medium. This condition describes a medium with percolation limits; in Fig. 3, we see that expression Eq. 14 with α = 1.5 (curve E) corresponds quite well to a typical structure factor of a medium with a percolation threshold of 0.32 (dotted line).
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Simple Expression for the Tortuosity of Porous Media Fig. 3 Comparison between the structure factors of percolation theory and of the present study
1
197
A B C D E
0.8
ε/τ
0.6
0.4
α=0 α=.5 α=.75 α=1 α=1.5 ε0=0,t=1.5 ε0=.32,t=1.5
A B C D E
0.2
0 0
0.2
0.4
0.6
0.8
1
ε
This behaviour origins by the use of Eqs. 9 and 11, whilst when using Eqs. 8 and 13 the structure factor is always positive and it is not possible to describe media with percolation limits. We shall also observe that the discussion above is valid in the limits of validity of Eq. 9 and could fail for highly anisotropic media. One of the advantages of the present approach is to describe the tortuosity in terms of a shape factor defined by Eq. 12. However, evaluation of the shape factor is not trivial; the first problem is that α depends on the parameter r , which is only qualitatively defined above. Another important limitation is that, especially at low porosity, the definition of the scattering objects can be difficult and non-univocal, and may change when changing the system density. 2.2 Numerical simulations In order to illustrate these aspects and validate the theory, a series of numerical experiments has been performed. The simulated phenomenon is steady-state diffusion. The model equations are N = −Dvoid ∇c
(15)
∇·N =0
(16)
where c represents a scalar quantity, N is the corresponding flux per unit area and Dvoid a diffusion coefficient. Equation 15 represents diffusion inside a void space, and Eq. 16 represents conservation in steady-state conditions. Such equations represent a large range of diffusion phenomena including mass, charge and heat diffusion under concentration, potential and temperature gradients, respectively. The porous media is represented by partially filling the simulation volume with solid objects and imposing zero normal flux on their surface. Equations 15 and 16 are explicitly solved by finite differences, on a Cartesian 3D grid. On the x and y boundaries, periodic conditions are implemented, whilst at the bottom and upper z boundaries, z=0 and z=1, the value of the scalar c is fixed to the values of 1 and 0, respectively. At a macroscopic level, the flux equations can be written as N s = −D∇c, ∇ · N s = 0,
(17) (18)
where N s is the average flux per unit surface, and D the effective diffusion coefficient.
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By making use of Eq. 1, Eq. 17 can be rewritten as N s = −(ε/τ )Dvoid ∇c
(19)
Equations 18 and 19, with the above boundary conditions, have a trivial analytical solution c = 1 − z,
Nzs = (ε/τ )Dvoid
(20)
The structure factor, therefore, can be numerically estimated from the computed flux, as 1 Nz , (21) (ε/τ ) = S Dvoid S
where S is a surface orthogonal to the z axis. Since the porosity is known, tortuosity and shape factor can be easily derived from Eq. 21. It can be useful to stress that the numerical experiment outlined in this section implies choices which are made only to make the numerical computations feasible on a PC and does not correspond to limit of validity of the tortuosity expression. For example, here we simulate a diffusion process whilst the expression of tortuosity is general and should be valid for other phenomena as well; we use periodic boundary conditions, which is a common procedure to close the simulation box, but this does not imply that the expression validity is limited to periodic media. Finally, the procedure of deriving the tortuosity from Eq. 21 is not as rigorous as directly calculating the length of the flow-lines (Matyka et al. 2008), but is simpler and should lead to equivalent results.
3 Results 3.1 Numerical model results In a first numerical experiment, the shape factor is estimated by putting a single-solid object near the centre of the simulation volume, following the procedure outlined in the previous section. In Fig. 4, the shape factor of spheres, cubes (with a face orthogonal to the flow) and elongated parallelepipeds (with the long side orthogonal to the flow and four times larger than the short ones) is shown for different object scales. In the abscissa, the diameter of spheres, side of cubes and short side of parallelepipeds is reported. The spheres of diameter d are represented by filling the grid units whose centre is distant less than d from a given point. Such representation is very rough when d is a few grid units but becomes more accurate for larger d. By looking at Fig. 4, it can be noticed that, by increasing the object dimensions, the shape factors decrease towards an asymptotic value. This behaviour is not a physical effect but only a grid-resolution artefact, which can be explained by considering that when the object dimensions are of few grid units, the flow around them is not adequately resolved. The asymptotic convergence of the spheres takes longer since the sphere representation converges slowly to a real sphere. The white, grey and black symbols are obtained using computational domains of 50 × 50 × 50, 100 × 100 × 100 and 150 × 150 × 150 grid units, respectively. It can be seen that when the object dimensions are comparable with those of the computational domain, problems may arise due to interactions with the periodic images. By extrapolating the numerical shape factors, we get values of 0.97, 0.62 and 0.50 for parallelepipeds, cubes and spheres, respectively. The shape factor of infinitely long parallelepipeds
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Simple Expression for the Tortuosity of Porous Media Fig. 4 Numerical evaluation of the shape factors of some characteristic objects
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1.4
1.2
parallelepipeds
α
1
0.8
cubes 0.6
spheres 0.4
0
10
20
30
40
d (grid units)
Table 1 Dimensions and object characteristics for the numerical simulations α
Objects
Dimensions (grid units)
Domain dim. (grid units)
Spheres
d = 12
100 × 100 × 100
0.64
Cubes
6×6×6
50 × 50 × 50
0.73
Large parallelepipeds
6 × 6 × 24
100 × 100 × 100
1.07
Small parallelepipeds
3 × 3 × 12
50 × 50 × 50
1.21
orthogonal to the flow has been estimated through a two-dimensional simulation and is worth 1.16. It is worth noticing that the shape factor “measured” for the sphere is quite smaller than the value of 0.75 estimated at the end of section 2. This can be explained by considering that the streamlines of the diffusion flow turn smoothly around the spheres instead of deviate abruptly as sketched in Fig. 1, thus reducing the distance r necessary to by-pass the obstacle. Such behaviour can explain also the difference between the shape factor of spheres and cubes. In the second numerical experiment, the porous media is represented by randomly filling the simulation volume with solid objects, until the desired porosity is reached. The dimensions of the objects and of the computational domain have been chosen on the basis of the results of Fig. 4 and are reported in Table 1. Since the number of objects inside the simulation box is not very large, preliminary calculations have been performed to estimate the stochastic variability of the tortuosity and its possible dependence on the box dimensions. In Fig. 5, the calculated tortuosity of a solid medium composed by 6 × 6 × 6 cubes and with a porosity of 0.7 is reported for ten different random solid positions and two different domain dimensions. It appears that, by doubling the dimension of the simulation box, the tortuosity stochastic variability decreases significantly, whilst its average value stay almost unchanged. Therefore, for each configuration, three calculations, with different random solid positions, are performed to estimate the average flux. The parallelepipeds are randomly oriented with the long side along one of the two Cartesian directions orthogonal to the flow. During the filling procedure, the objects are allowed to overlap by, at most, one grid unit. Thus, cubes and spheres mimic a porous media obtained by assembling and pressing powders, whilst parallelepipeds mimic a medium made by fibres (as, for example, carbon paper). By looking, through cluster analysis, if the large majority of
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Fig. 5 Stochastic variability of the computed tortuosity for two different sizes of the computational domain tortuosity
1.36
1.34
1.32
1.3 0
100
50
150
side of cubic domain (grid units) 3
spheres cubes large parallel. small parallel.
2.5
tortuosity
Fig. 6 Tortuosity versus porosity relation for porous media made by partially overlapping objects. The symbols represent the numerical simulation results, and the lines the analytical prediction of Eq. 11
2
1.5
1 0.4
0.5
0.6
0.7
0.8
0.9
1
porosity
the objects belongs to a single cluster of interconnected particles, it is established which of the model solids has a percolating solid structure. In Fig. 6, the tortuosity values obtained by the above procedure (symbols) are shown as a function of porosity and compared with the theoretical relation Eq. 11 (curves). The theoretical curves are obtained using the values of alpha corresponding to the chosen object dimension, measured in the previous experiment and reported in Fig. 4 and Table 1. Black symbols are used when the solid medium has a percolating solid structure whilst otherwise white symbols are used. It is seen that Eq. 11 provides a quite good prediction of the medium tortuosity even when the solid objects are strongly interconnected and partially overlapping. As foreseeable, when cubes and spheres overlap, they partially loose their original shape and the shape factor slightly increases leading to a tortuosity a bit larger than predicted. Nevertheless the prediction remains quite satisfactory, and the difference in tortuosity between the two different shapes is well preserved. On the other hand, the prediction made for parallelepipeds is very good and does not degrade with their overlapping. 3.2 Semi-empirical model results Some porous media, however, have topologies which cannot be easily defined as an ensemble of objects. This may happen, for example, when porosity and pores shape depend on the fluid–solid interaction. An example of such kind of porous media with important
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Simple Expression for the Tortuosity of Porous Media 0.4
Dow Dow Nafion Speek A α=0.60 B α=0.85 C α=1.35
0.3
ε/τ
Fig. 7 Structure factor for the conductivity of polymeric membranes. Experimental data (symbols) are compared with this study theory (full lines) and with percolation theory (dashed lines). Experimental data, denoted by filled squares, are from Zawodzinsky et al. (1993). Experimental data, denoted by open squares, are from Edmondson et al. (2000). All other experimental data are from Kreuer et al. (2004)
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0.2
A B C
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
ε
technological application is provided by hydrated polymeric membranes. In particular, some special polymers with strong acid groups attached, when hydrated, can form membranes with very good proton conductivity. Also polymeric membranes are made by assembling similar solid particles (the polymers). However, their interconnecting structure is very complex and flexible and may change with the amount of absorbed fluid, so that the building blocks are not easily recognisable. In the literature, these systems are largely studied, and the tortuosity effects on conductivity are mostly described using Eq. 3 with the parameter ε0 determined by empirical fit and t either used as a fitted parameter or arbitrarily fixed to 1.5 (Thampan et al. 2000; Fimrite et al. 2005; Pisani et al. 2008). Here, we show that Eq. 14 can be used to describe the membrane morphology by fitting the single parameter α and using its geometrical meaning to interpret the fitting results. The structure factor of hydrated polymeric membranes can be evaluated by dividing the measured proton conductivity by a model “bulk conductivity” as described, for example, in (Pisani et al. 2008; Pisani 2009). In Fig. 7, the structure factors of three different membranes are represented as a function of ε, which, in this case, represents the volume fraction of liquid phase inside the polymer membrane. It should be noted that, at low porosity values, the relative error is quite high and at porosity around or larger than 0.5, the model bulk conductivity becomes too high since one of the model assumptions (homogeneous proton distribution) is not true anymore (Pisani et al. 2008). Nevertheless, a quite good fit has been obtained using Eq. 14. It can be noted that the α value obtained by fitting the Speek membrane structure factor is much larger than the ones obtained for Nafion and Dow indicating a difference in the morphology of the pore structure. This result can be interpreted by considering that in Nafion and Dow membranes, the hydrophilic acid groups are appended to the flexible lateral polymer chains, which can move and arrange to minimize the water–hydrophobic polymer contact, thus forming quasi spherical agglomerates. On the contrary, the sulphonic groups are attached to the “body” of the Speek polymer, so that, when hydrated, the linear structure of the polymer is partially preserved. A difference in the shape factor, therefore, is largely justified. A similar agreement can be obtained by fitting the polymer structure factors using Eq. 3 (Pisani et al. 2008) (dashed lines in Fig. 7), but at the price of using two fitting parameters, and without gaining insight in the pore morphology. Finally, the advantage of measuring α instead of directly measuring the tortuosity appears clearly in Fig. 7, where one single value of α is used to describe the behaviour of each
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membrane, whilst each of the 45 represented points have its own tortuosity. Physically, the above procedure works when the polymer shape factor is preserved in different hydration conditions, whilst porosity and tortuosity change.
4 Conclusions The definition of a geometrical–physical parameter r as “the average distance necessary to bypass the obstacle” allows a trivial resolution of the tortuosity problem without any other approximation and leads to a very simple expression of the tortuosity. This result is, by itself, an important achievement, since the resulting expression is general (does not depend on the particular phenomenon or on the presence or not of percolation limits) and provides a very clear geometrical interpretation. The evaluation of r , or, equivalently, of α, is complicated by the fact that it depends both on the shape of the object and on the transport phenomena. Another important limitation is that, especially at low porosity, the definition of the scattering objects can be difficult and non-univocal, and may change when changing the system density. However, two strategies for possible applications of the present model are shown. The first one is to measure α, by considering a single-isolated object. This has been done numerically here for diffusion phenomena and spherical, cubic and elongated parallelepipeds objects. Use of the measured α value to predict the tortuosity of solids made by stochastic partial overlapping of the considered objects has shown a very good agreement. This is a second important achievement, since a large class of porous media, including agglomerates of powders or fibres, has this kind of structure. The second strategy is to use the model as a semi-empirical one, and fit the α parameter to experimental data. This has been done here to model the conductivity of polymeric membranes at different levels of hydration. The model has been able to describe membranes both with and without percolation limits and in the whole range of hydration. Comparison of this model with the most used one in literature shows that a similar agreement is obtained but with the advantages of using a single fitting parameter instead of two, and of gaining insight into the membrane structure. As mentioned in the introduction, Eqs. 1 and 3 are valid only when the solid–fluid interactions are negligible, as in most of the diffusion phenomena. The concept of tortuosity and its geometrical definition used in this study, however, are commonly used to evaluate other transport properties as, for example, the Darcy permeability (Matyka et al. 2008). Application of the present model to the permeability of porous media is currently under investigation. Acknowledgements This study has been carried out with the financial contribution of the Sardinia Regional Authorities and of the Italian Ministry for University and Research (MIUR), NUME project (http//www. progetto-nume.it). The author is also indebted to Cesare Pisani for helpful discussions.
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