Numerical estimation of effective diffusion coefficients for charged porous materials based on micro-scale analyses

Abstract

In order to describe diffusive transport of solutes through a porous material, estimation of effective diffusion coefficients is required. It has been shown theoretically that in the case of uncharged porous materials the effective diffusion coefficient of solutes is a function of the pore morphology of the material and can be described by the tortuosity (tensor) (Bear, 1988 [1]). Given detailed information of the pore geometry at the micro-scale the tortuosity of different materials can be accurately estimated using homogenization procedures. However, many engineering materials (e.g., clays and shales) are characterized by electrical surface charges on particles of the porous material which strongly affect the (diffusive) transport properties of ions. For these type of materials, estimation of effective diffusion coefficients have been mostly based on phenomenological equations with no link to underlying micro-scale properties of these charged materials although a few recent studies have used alternative methods to obtain the diffusion parameters (Jougnot et al., 2009; Pivonka et al., 2009; Revil and Linde, 2006 [2], [3], [4]). In this paper we employ a recently proposed up-scaled Poisson–Nernst–Planck type of equation (PNP) and its micro-scale counterpart to estimate effective ion diffusion coefficients in thin charged membranes. We investigate a variety of different pore geometries together with different surface charges on particles. Here, we show that independent of the charges on particles, a (generalized) tortuosity factor can be identified as function of the pore morphology only using the new PNP model. On the other hand, all electro-static interactions of ions and charges on particles can consistently be captured by the ratio of average concentration to effective intrinsic concentration in the macroscopic PNP equations. Using this formulation allows to consistently take into account electrochemical interactions of ions and charges on particles and so excludes any ambiguity generally encountered in phenomenological equations.

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]:

$${\mathbf{j}}_i=-D_{f\text{,}i}\nabla c_i$$

where $D_{f\text{,}i}$ is the self-diffusion coefficient of the ith solute and $\nabla c_i$ 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]):

$$\overline{{\mathbf{j}}_i}=-n{\mathbf{D}}_{\mathit{eff}\text{,}i}\nabla {\overline{c}}_i$$

where ${\mathbf{D}}_{\mathit{eff}\text{,}i}$ is the effective diffusion coefficient, n is the porosity of the porous material, $\overline{{\mathbf{j}}_i}$ is the macroscopic flux vector and $\nabla {\overline{c}}_i$ denotes the average concentration gradient. For electrically uncharged porous media, the effective diffusion coefficient can be found as [13]:

$${\mathbf{D}}_{\mathit{eff}\text{,}i}=\mathit{\tau }{D}_{f\text{,}i}$$

where D_f,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 l_e 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]:

$$\tau ={\left(\frac{l}{l_e}\right)}^2⩽1$$

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 l_e 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 l_e, 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.