Numerical estimation of effective diffusion coefficients for charged porous materials based on micro-scale analyses
Introduction
For many low permeability engineering materials such as clays and reactive barriers, landfill covers, concrete walls, cement pastes, ceramics and biological tissues, diffusion is the main transport mechanism of solutes [5], [6], [7], [8], [9], [10]. In these materials diffusion takes place in the fluid-saturated pore space. The solute flux equation for the fluid phase (of uncharged porous materials) can be described by Fick’s first law [11], [12]:where is the self-diffusion coefficient of the ith solute and denotes the solute concentration gradient. The above equation can be applied to the fluid phase of porous materials to obtain, via upscaling methods, the solute flux at the macro-scale. Applying the volume averaging method to Fick’s first law, one can obtain the macro-scale (cm–m range) flux vector (for more details, see [1]):where is the effective diffusion coefficient, n is the porosity of the porous material, is the macroscopic flux vector and denotes the average concentration gradient. For electrically uncharged porous media, the effective diffusion coefficient can be found as [13]:where Df,i is the self-diffusion coefficient or the diffusion coefficient of the ith ion in pure water and τ is a second order (symmetric) tensor that is called tortuosity. Note that Eq. (2) has been derived assuming that there is no adsorption and chemical reactions between the fluid phase and solid phase; and that for isotropic porous media, the tortuosity tensor can simply be described by a scalar value (tortuosity factor), i.e., τ = τ1, where 1 is the second order unity tensor.
To find the effective diffusion coefficient, the two parameters on the right hand side of Eq. (3) should be obtained first. Self-diffusion coefficients of solutes can be measured directly from laboratory experiments but tortuosity is thought to be a physical characteristic of the pore scale of the material that is related to the ratio of average pathlines (or streamlines) through the porous media (i.e., the tortuous length) to the straight length or thickness of the porous media through which diffusion takes place. The concept of tortuosity for uncharged porous media is based on irregularity and random setting of particles and as a result, the local variation of tortuosity is related to variation of the pore morphology [11], [14], [15], [16].
Different expressions for the tortuosity factor have been suggested by different authors [1], [17]. These multiple ‘definitions’ of tortuosity have lead to misinterpretations of the parameter throughout the years. The tortuosity factor was first suggested by Carman [14] as the square of the ratio between the length of the tortuous pathline le and the thickness of porous media l [14], [15], [17], [18] (Fig. 1). Nevertheless, some other researchers have defined the tortuosity factor as the inverse of that [1], [17]:
Both apparently different definitions give the same final results when used appropriately in Eqs. (2), (3). However, tortuosity has been also defined as only the ratio of pathline le to the thickness of porous membrane l (i.e., without squaring this ratio), which is not correct. The reader should be aware of these ‘definitions’ of tortuosity. We will see that Eq. (4) should in fact prevail in the context of this article.
Even though there are expressions to estimate the tortuosity factor, the complex nature of porous media and somehow erratic movement of solutes within their pores, make it very hard to calculate the actual length of pathlines le, and closed form solutions were only limited to simple geometries [19], [20]. The underlying assumption in all the tortuosity models and above tortuosity equations is that ion diffusion occurs in uncharged porous media. However, electrically charged porous materials are quite common in engineering applications and it is not clear if the above formulations for tortuosity are applicable in charged porous media.
Studies have shown that for charged porous materials, the distribution of cations and anions are different depending on the distribution of electric charges. Thus, one could expect to obtain a different tortuosity value using traditional tortuosity equations (e.g. Eq. (4)). Recent studies have estimated the Donnan potential based on micro-scale thermodynamic equilibrium between ions in solution [22]. In a further extension of the model, non-equilibrium ion transport has been added. Pivonka et al. [3] derived a consistently up-scaled generalized Poisson–Nernst–Planck (PNP) equation that can model electro-diffusive transport [3]. It has also been shown that this generalized PNP system of governing equations is a generalization of Fick’s law that takes into account the effect of electrical charges in porous media together with electro-static interactions of ions. These equations have been applied to calculate effective diffusion coefficients for straight cylindrical pores containing surface charges. It has been shown that the effective diffusion coefficients of anions and cations can be up to 20–30% increased in the case of charged porous materials. However, in these studies the property of tortuosity in charged porous materials has not been investigated (due to the simplifying assumption of straight pores, τ = 1) [3]. In this paper we address an important question, whether in charged porous materials the tortuosity factor is (like in uncharged materials) an intrinsic property which only depends on the pore morphology or whether it depends on the surface charge and the background electrolyte concentration. In order to do this, we apply a numerical approach to estimate the tortuosity factor, which is based on the relationship between the PNP equations at the micro-scale and the generalized PNP equations at the macro-scale.
Different pore geometries for thin membranes are numerically solved here for charged and uncharged porous materials and the results for tortuosity factors are compared with the ones available from the literature.
Section snippets
Multi-ionic transport governing equations
The Poisson–Nernst–Planck system of equations can be used to describe the multi-ion electro-diffusion transport in charged porous media. These equations may be referred to as the micro-scale PNP equations. For engineering applications it is necessary to know about macroscopic parameters of the porous media rather than micro-scale characterisation. The traditional way to learn about these parameters is performing experimental tests. However, derivation of macro-scale PNP equation using the
Methodology
The most straight forward way to estimate the value of tortuosity is to apply the micro-scale PNP equations to the entire material domain and then compute respective macro-scale quantities encountered in the generalized PNP equations (for both uncharged and charged porous materials). One drawback of this approach is that only relatively thin porous materials (i.e., membranes) can be investigated given the high computational costs. Using volume averaging, when all parameters in Eqs. (12), (13)
Results and discussion
The multi-ionic transport through uncharged and charged porous media is modelled for different pore geometries to show the consistency of the numerical approach for estimating the tortuosity.
The micro-scale models are applied to: (a) 2D Omega-shape channel, (b) 2D circular particles, (c) 2D rectangular particles, (d) 2D z-shaped channel and (e) 2D slits. For all these cases, a corresponding 1D macro-scale model can be formulated requiring the effective ion diffusion coefficients as input
Conclusions
From the presented numerical results the following conclusions can be drawn:
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For thin porous membranes it is possible to estimate tortuosity values via a numerical method for both electrically charged and uncharged porous materials, by applying the micro-scale equations to the whole domain and then using the macro-scale equations to estimate tortuosity.
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The so computed values for τ are in good agreement with the ones from ‘direct measurements’ of the tortuous ion path length le and the thickness l
Acknowledgments
The authors are grateful to the Australian Research Council (ARC) for Discovery Project funding provided. Partial support was also provided by the Geotechnical Group at Melbourne University. We thank the reviewers for valuable suggestions.
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