A Two-Scale Computational Model of pH-Sensitive Expansive Porous Media

Short abstract

We propose a new two-scale model to compute the swelling pressure in colloidal systems with microstructure sensitive to pH changes from an outer bulk fluid in thermodynamic equilibrium with the electrolyte solution in the nanopores. The model is based on establishing the microscopic pore scale governing equations for a biphasic porous medium composed of surface charged macromolecules saturated by the aqueous electrolyte solution containing four monovalent ions $({\mathit{Na}}^{+},{\mathit{Cl}}^{-},H^{+},{\mathit{OH}}^{-})$. Ion exchange reactions occur at the surface of the particles leading to a pH-dependent surface charge density, giving rise to a nonlinear Neumann condition for the Poisson–Boltzmann problem for the electric double layer potential. The homogenization procedure, based on formal matched asymptotic expansions, is applied to up-scale the pore-scale model to the macroscale. Modified forms of Terzaghi's effective stress principle and mass balance of the solid phase, including a disjoining stress tensor and electrochemical compressibility, are rigorously derived from the upscaling procedure. New constitutive laws are constructed for these quantities incorporating the pH-dependency. The two-scale model is discretized by the finite element method and applied to numerically simulate a free swelling experiment induced by chemical stimulation of the external bulk solution.

1. Introduction

In the past few decades there has been an increasing interest in the comprehensive understanding of the response of swelling porous media under various loading conditions induced by electro-chemo-mechanical-thermal stimulations. Applications are widespread in nature and in modern technologies involving diverse fields such as: soil science, hydrogeology, geotechnical and petroleum engineering, chemical and mechanical sciences, colloid chemistry, pharmaceutical and life sciences, biomechanics, clinical, and medical fields.

Historically, owing to its critical role in the quality of groundwater, swelling of clay-rich formations, particularly montmorillonites, during moisture imbibition and sorption of ionic species has been widely reported in the literature [1]. The exposure of swelling soils to free polar fluids induces stresses which can be very troublesome to foundations, leading to the failure of buildings, bridges, highways, and runways. Upon inundation structures founded on collapsible and expansive soils are subject to severe damage ranging from minor cracking to irreparable displacements of footings, which may reduce the stability of land slopes [2,3]. In the petroleum industry swelling/contraction of clay-rich rocks such as shales, which are strongly dependent on the water-based drilling mud concentrations, has been responsible for most of the stability problems of drilled boreholes [4]. Swelling can also be explored for beneficial purposes. For instance bentonitic based compacted expansive clays play a critical role in the safety assessments of their capacity to act as a host rock of high-level long lived radioactive waste disposal sites. In addition, they act as a geochemical filter for environmental protection to inhibit the migration of contaminants from hazardous wastes in sanitary landfills [5].

Swelling polymers have numerous technological applications in the development of smart drug delivery substrates, in contact lenses, and in many biological and biomedical devices [6]. In particular, the osmotic induced swelling of cross-linked ionized hydrogels characterized by a stable microstructure [7] make them very attractive as sensor devices [8,9].

In biomedical technology, articular cartilage reinforced by collagen fibers illustrates the enormous complexity inherent to the modeling of electrically charged swollen soft tissues. The tissue fabric consists of a multiphasic hydrated mixture mainly composed of proteoglycan, collagen, and water [10]. The proteoglycans are negatively charged biomacromolecules, which play a crucial role in the load-bearing capacity of the cartilage by dictating the magnitude of the shear and compressive modulus and determining the ability to develop prestresses in the articulating joints and damping the dynamic forces in the human body. Thus, it is imperative that any macroscopic model describing the complex electro-chemo-mechanical interactions inherent to swelling systems is capable of capturing their complex response to different types of stimulation.

Expansive materials have a multiphasic porous microstructure in common, which is composed of a charged solid matrix, identified as a mixture of macromolecules (polymers, active clay particles, and proteoglycans) and an interstitial fluid, which is either adsorbed to the macromolecules in the form of a thin film (or electrolyte solution) or in a bulk state where the electroneutrality condition is fulfilled pointwisely [11]. The electrochemical properties of macromolecules/water interfaces have been intensively studied and have a tremendous impact on the volume, stiffness, strength, conformational properties, and the permeability of reactive porous media [12,13]. Electrically charged interfaces in colloidal system are very sensitive to several mechanisms of different natures, such as electro-osmosis, chemico-osmosis, streaming current, streaming potentials, and electric currents, which may induce severe alterations in the ionic structure and charge distribution in the medium [14].

The mechanisms underlying the swelling/shrinking phenomena have been widely studied by many researchers in an effort to reach a fundamental approach to establish the dominant swelling mechanisms. The difficulty in accomplishing this task lies is the high sensitivity of the medium when the thickness of the aqueous solution attains a value of 10 nanometers or less. Experimental evidence indicates that for interstices smaller than $50\AA$ swelling is due primarily to hydration forces and diffuse double layer forces are believed to be too weak to explain the anomalous behavior of the adsorbed water (see Low and coworkers [15,16], Derjaguin and coworkers [17], and Israelachvili and coworkers [18]). Conversely, since hydration stresses operate in a short-range fashion, their influence rapidly decays and, consequently, for long-range interactions swelling is dominated by electrostatic effects. In this range, the adsorbed fluid is viewed as an aqueous electrolyte solution consisting of water and entirely or partially dissociated electrolytes whose concentrations are governed by the conventional Gouy–Chapman theory of a diffuse double layer, which assumes point charge ions embedded in a dielectric continuum distributed according to Boltzmann statistics [19–21].

Derjaguin and coworkers [17] described the lyophilic interaction between the adsorbed fluid and solid surface in terms of a disjoining pressure, defined as the excess in the normal fluid pressure relative to the surrounding bulk phase. The electrostatic component of this repulsive force results from the overlapping between adjacent ionic double layers [21]. The averaged counterpart of the disjoining pressure is the swelling pressure, defined as the overburden pressure excess relative to the bulk phase pressure of the outer solution that must be applied to a saturated mixture of particles and fluid in equilibrium with the bulk water in order to keep the layers from moving apart [15].

During the past two decades a significant amount of fundamental research has been undertaken for swelling systems based on purely macroscopic phenomenological approaches (see, e.g., Ref. [22]) and also by using mixture theory for multiphase multi-ionic species porous media ([23–25]) which is also based on the framework of thermodynamics of irreversible processes and Onsager's reciprocity relations (see, e.g., [26–28]). Despite their success, limited accomplishments have been achieved toward the incorporation of the morphology and local electrical double layer (EDL) properties in the macroscopic model and very little information has been available to identify some of the macroscopic electrokinetic coefficients with the local EDL properties [29]. In order to incorporate this information, it becomes essential to develop up-scaling methods [30], wherein the effective swelling medium behavior is rigorously constructed from the propagation of information available at the pore-scale to the macroscale. In homogenization-based approaches, the electrostatic component of the swelling pressure naturally appears in the modified Terzaghi's effective stress principle with the magnitude dictated by the profile of the local EDL potential which satisfies the Poisson–Boltzmann problem [30].

Despite the enormous improvement achieved in the development of constitutive theories for the effective parameters based on the homogenization of the nanopore-scale description, previous models were developed for electrolyte solutions composed of two fully dissociated monovalent ions, ${\mathrm{Na}}^{+}$ and ${\mathrm{Cl}}^{-}$, and a constant $p\mathrm{H}=7$ with the constant surface charge solely dictated by the isomorphous substitution reactions. Consequently, the influence of partially dissociated ions within the aqueous solution such as ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$, along with chemical reactions between the ${\mathrm{H}}^{+}$ ions and surface charged groups lying on the particle surface leading to pH-dependent surface charge were neglected (see, e.g., Refs. [31,32]). In contrast to the sodium chloride, acids and bases have a profound influence on the magnitude of the surface charge through protonation/deprotonation and ion exchange reactions or surface complexation mechanisms taking place at particle-edges. Such reactions are strongly pH-dependent (see, e.g., Refs. [31,32]).

A first attempt at incorporating pH-effects in the homogenization-based approach has been pursued by Lemaire et al. [33]. Subsequently, in a series of papers (Lima et al. [34–37]) have extended the homogenization based approach of Moyne and Murad [30] to incorporate pH effects in a non-deforming kaolinite clay and applied the theory to numerically simulate an electro-osmosis experiment. The results obtained highlighted the paramount role of the acidification/alkalinization phenomena on the effectiveness of the electrokinetic remediation processes (see Ref. [35] for details).

The aim of this contribution is the development of a two–scale constitutive model for the swelling pressure in charged hydrated porous media at equilibrium with an outer bulk solution incorporating a pH-dependency. To accomplish this task, we proceed within a formal homogenization procedure applied to a nanoscopic portrait of electrically charged macromolecules saturated by an aqueous electrolyte solution composed of four ionic species $({\mathrm{Na}}^{+},{\mathrm{H}}^{+},{\mathrm{Cl}}^{-},{\mathrm{OH}}^{-})$. Throughout the manuscript we develop the formal homogenization procedure aiming at constructing macroscopic constitutive laws for the effective electrochemical parameters in terms of pH, salinity, and nanoporosity.

The effective swelling medium behavior appears to be strongly dependent on the local profile of the EDL potential in the nano-pores governed by the Poisson–Boltzmann equation supplemented by a nonlinear Neumann boundary condition enforced by the pH-dependent surface charge. Such a local problem is discretized by the finite-element method and numerically solved in a domain composed of two parallel particles associated with a stratified microstructure to obtain the local EDL potential profile and further reconstruct the macroscopic constitutive law for the swelling pressure. Finally, the two-scale model is applied to numerically simulate a free-swelling experiment with expansion induced by gradual alkalinization of the outer bulk solution.

2. Microscopic Modeling

We begin by presenting the microscopic model describing the electro-chemo-mechanical coupled phenomena at the nanopores. At the local scale, consider the domain occupied by a biphasic solid/fluid system. The solid phase consists of elastic macromolecules electrically charged at the surface, whereas the nanopores are filled by an aqueous completely dissociated NaCl solution (see Fig. 1). In contrast, water molecules are partially dissociated with proton and hydroxile concentrations constrained by the equilibrium constant of the dissociation reaction. The monovalent ions are treated as point charges embedded in a continuum dielectric solvent so that the hydration/water dipole effects and ion finite size effects are thoroughly neglected.

Fig. 1.

Sketch of the microscopic domains of the model

Ion exchange reactions occur between the macromolecules and the electrolyte solution leading to a varying surface charge density which is neutralized by counter ions to fulfill global electroneutrality. Isomorphous substitution processes, which are typical of montmorillonite clays, provide a constant surface charge implying swelling solely dictated by salinity and particle distance. In addition, new physics stems from the ion exchange reactions at the particle surface which entails a dependence of the surface charge on the pH, giving rise to a nonlinear Neumann boundary condition for the Poisson–Boltzmann problem [34].

2.1. Microscopic Electrochemical Model.

We begin by presenting the electrochemical model for the local ionic concentrations, electric potential, and surface charge at the pore-scale. The well-known behavior of ions in solution is described by the ionization theory, which explains that some molecules can partially or completely dissociate in an aqueous solvent. For example, water molecules are normally subjected to partial dissociation due to a weak phenomenon called autoionization, which is represented by the equilibrium reaction

$${\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{OH}}^{-}$$ (2.1)

In contrast, sodium chloride ($\mathrm{NaCl}$) in water is considered a completely dissociated strong electrolyte ruled by the reaction

$$\mathrm{NaCl}\to {\mathrm{Na}}^{+}+{\mathrm{Cl}}^{-}$$

The degree of dissociation due to hydrolysis is dictated by the ionic product of water $K_W$, which is defined by the product between the ${\mathrm{H}}^{+}$ and ${\mathrm{OH}}^{-}$ molar concentrations

$$K_W:=C_{\mathrm{H}{b}^{+}}{C}_{\mathrm{OH}{b}^{-}}={10}^{-14}(\mathrm{mol}/\mathrm{l}){}^2$$ (2.2)

where $C_{\mathit{ib}}$ $(i={\mathrm{Na}}^{+},{\mathrm{H}}^{+},{\mathrm{Cl}}^{-},{\mathrm{OH}}^{-})$ denotes the molar concentration of each ionic species in the bulk fluid. Denoting $C_b$ as the total concentration of cations (or anions) in the bulk solution the electroneutrality condition enforces the constraint [11]

$$C_b=C_{\mathrm{Na}{b}^{+}}+C_{\mathrm{H}{b}^{+}}=C_{\mathrm{Cl}{b}^{-}}+C_{\mathrm{OH}{b}^{-}}$$ (2.3)

Protonation/deprotonation and surface complexation reactions occur at the particle surface. Denoting $\mathit{MH}$ as the protonated structural groups associated with the primary mineral ($M^{-}$), the hydrogen adsorption/desorption phenomenon is described by the reaction

$$\mathit{MH}\rightleftharpoons M^{-}+{\mathrm{H}}^{+}(\mathrm{aq})$$ (2.4)

In addition, the charged deprotonated sites are neutralized by the ${\mathrm{Na}}^{+}$ ions in the form

$$M^{-}+{\mathrm{Na}}^{+}(\mathrm{aq})\rightleftharpoons M\mathrm{Na}$$ (2.5)

Note that the combination of the preceding two mechanisms gives rise to a coupled exchange reaction. Denoting ${\Gamma }_{\max }$ as the surface density of the particles sites which measures the total number of sites available for adsorption per unit area and ${\gamma }_j$ $(j=M^{-},\mathit{MH},M\mathrm{Na})$, the surface concentration of each reagent/product in Eqs. (2.4) and (2.5) we have the constraint

$${\Gamma }_{\max }:={\gamma }_{M^{-}}+{\gamma }_{\mathit{MH}}+{\gamma }_{M\mathrm{Na}}$$ (2.6)

For cation exchange surfaces, the maximum surface density is given by ${\Gamma }_{\max }:=\mathrm{CEC}/A_s$, where $\mathrm{CEC}$ is the cation exchange capacity and $A_s$ is the specific surface area [38]. Denoting ${\overline{C}}_j:={\gamma }_j/{\Gamma }_{\max }$ $(j=M^{-},\mathit{MH},M\mathrm{Na})$ as the corresponding dimensionless surface concentration and $C_{\mathrm{H}{0}^{+}}$ and $C_{\mathrm{Na}{0}^{+}}$ the cation molar concentrations at the solid surface, the equilibrium constants associated with Eqs. (2.4) and (2.5) are defined by

$$K_1:=\frac{C_{\mathrm{H}{0}^{+}}{\overline{C}}_{M^{-}}}{{\overline{C}}_{\mathit{MH}}}=\frac{C_{\mathrm{H}{0}^{+}}{\gamma }_{M^{-}}}{{\gamma }_{\mathit{MH}}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}{K}_2:=\frac{{\overline{C}}_{M\mathrm{Na}}}{C_{\mathrm{Na}{0}^{+}}{\overline{C}}_{M^{-}}}=\frac{{\gamma }_{M\mathrm{Na}}}{C_{\mathrm{Na}{0}^{+}}{\gamma }_{M^{-}}}$$ (2.7)

For monovalent species we define the surface charge density $\sigma$ by the product between the surface concentration of the charged species ${\gamma }_{M^{-}}$ and Faraday's constant $F$. Using Eqs. (2.6) and (2.7) we obtain the representation

$$\sigma :=-F{\gamma }_{M^{-}}=-F\frac{\mathrm{CEC}}{A_s}\left\{\frac{K_1}{K_1+C_{\mathrm{H}{0}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{0}^{+}}}\right\}$$ (2.8)

In addition to information on the parameters ($\mathrm{CEC}$, $A_s$, $K_1$, $K_2$) the complete characterization of the surface charge requires knowledge of the cation concentrations at the particle surface ($C_{\mathrm{H}{0}^{+}}$,$C_{\mathrm{Na}{0}^{+}}$), which are strongly dependent on the local profile of the EDL potential $\Psi$. Denoting ${\Omega }_f$ as the subdomain occupied by the electrolyte solution, $E$ as the electric field, and $(\tilde{\epsilon },{\tilde{\epsilon }}_0)$ as the pair of the vacuum permittivity and relative dielectric constant of the solvent, the Gauss–Poisson equation reads as [39]

$$\begin{array}{c}{\epsilon \tilde{\epsilon }}_0\nabla ·E=qE=-\nabla \Psi \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\Omega }_f\end{array}$$ (2.9)

In the preceding model $q$ denotes the net volumetric charge density defined as

$$q:=F(C_{{\mathrm{Na}}^{+}}+C_{{\mathrm{H}}^{+}}-C_{{\mathrm{Cl}}^{-}}-C_{{\mathrm{OH}}^{-}})$$ (2.10)

where $C_i$ $(i={\mathrm{Na}}^{+},{\mathrm{Cl}}^{-},{\mathrm{H}}^{+},{\mathrm{OH}}^{-})$ is the ionic concentration in the electrolyte solution, which is related to the ionic concentration in the bulk fluid $C_{\mathit{ib}}$ through the Boltzmann distributions [19]

$$C_{i^{\pm }}=C_{\mathit{ib}}\exp (\mp \frac{F\Psi }{\mathit{RT}})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(i={\mathrm{Na}}^{+},{\mathrm{H}}^{+},{\mathrm{Cl}}^{-},{\mathrm{OH}}^{-})$$ (2.11)

with $R$ being the ideal gas constant and $T$ denoting the temperature. Using Eqs. (2.3) and (2.11) in Eq. (2.10), we obtain

$$q=-2{\mathit{FC}}_b\sinh \left(\frac{F\Psi }{\mathit{RT}}\right)$$ (2.12)

By combining Eqs. (2.9) and (2.12), the local EDL distribution is ruled by the Poisson–Boltzmann equation

$$\Delta \Psi =\frac{2{\mathit{FC}}_b}{{\tilde{\epsilon }\tilde{\epsilon }}_0}\sinh \left(\frac{F\Psi }{\mathit{RT}}\right)$$ (2.13)

supplemented by the Neumann condition at the particle/fluid interface

$${\tilde{\epsilon }\tilde{\epsilon }}_{0}E·\mathbf{n}=-\sigma \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{on}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\Gamma }_{\mathit{fs}}$$ (2.14)

with n being the unitary outward normal. The compatibility condition between $q$ and $\sigma$, which is typical of the Neumann problem, stems from the global electroneutrality condition

$${\int }_{{\Omega }_f}\mathit{qd}{\Omega }_f={\tilde{\epsilon }\tilde{\epsilon }}_0{\int }_{{\Omega }_f}\nabla ·\mathit{Ed}{\Omega }_f={\tilde{\epsilon }\tilde{\epsilon }}_0{\int }_{\Gamma }E·\mathbf{n}d\Gamma =-{\int }_{{\Gamma }_{\mathit{fs}}}\sigma d\Gamma$$ (2.15)

To close the local electrochemical model, for a given input of bulk concentrations ($C_{\mathrm{Na}{b}^{+}}$, $C_{\mathrm{H}{b}^{+}}$) it remains to completely characterize the surface charge $\sigma$. Recalling the pair of cation concentrations at the particle surface ($C_{\mathrm{Na}{0}^{+}}$,$C_{\mathrm{H}{0}^{+}}$), from the Boltzmann distribution we have

$$C_{i0}=C_{\mathit{ib}}\exp (-\frac{F\zeta }{\mathit{RT}})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{with}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}i={\mathrm{H}}^{+},{\mathrm{Na}}^{+}$$ (2.16)

where $\zeta :=\Psi |{}_{{\Gamma }_{\mathit{fs}}}$ is the so-called ζ-potential which quantifies the value of the EDL potential at the particle surface. Using Eq. (2.16) in Eq. (2.8) we obtain

$$\sigma =-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}}{A_s}\left[\frac{K_1}{K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\exp (-F\zeta /\mathit{RT})}\right]$$ (2.17)

It is worth noting that the dependence of the surface charge on the ζ potential brings additional nonlinearity to the Poisson–Boltzmann problem.

We are now ready to formulate our microscopic electrochemical model. Denoting $\zeta :=\Psi |{}_{{\Gamma }_{\mathit{fs}}}$ as the $\zeta$-potential and given the parameters ($F$, $R$, $T$, $K_W$, $K_1$, $K_2$, $\mathrm{CEC}$, $A_s$, $\tilde{\epsilon }$, ${\tilde{\epsilon }}_0$) along with the cationic concentrations in the bulk fluid $(C_{\mathrm{Na}{b}^{+}},C_{\mathrm{H}{b}^{+}})$, we find the EDL potential $\Psi$ satisfying

$$\{\begin{array}{cc}\Delta \Psi =\frac{2{\mathit{FC}}_b}{{\tilde{\epsilon }\tilde{\epsilon }}_0}\sinh \left(\frac{F\Psi }{\mathit{RT}}\right)& \mathrm{in}\hspace{1em}{\Omega }_f\\ \nabla \Psi ·\mathbf{n}=-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}\hspace{1em}{K}_1}{{\tilde{\epsilon }\tilde{\epsilon }}_0{A}_s[K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\exp (-\frac{F\Psi }{\mathit{RT}})]}& \mathrm{on}\hspace{1em}{\Gamma }_{\mathit{fs}}\end{array}$$ (2.18)

with

$$C_b=C_{\mathrm{Na}{b}^{+}}+C_{\mathrm{H}{b}^{+}}=C_{\mathrm{Cl}{b}^{-}}+C_{\mathrm{OH}{b}^{-}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{C}_{\mathrm{OH}{b}^{-}}=K_W{C}_{\mathrm{H}{b}^{+}}^{-1}$$

2.2. Disjoining Pressure.

Given the local profile of the EDL potential satisfying Eq. (2.18), one can explore the local equilibrium condition in the fluid in order to construct the constitutive law for the disjoining pressure. Under thermodynamic equilibrium, the pressure gradient in the electrolyte solution is counterbalanced by a body force of the Coulomb type, given by the product between the net charge density $q$ and the electric field $E$ [40]. We then have

$$-\nabla p+q\mathit{E}=-\nabla p-q\nabla \Psi =0$$ (2.19)

where $p$ is the thermodynamic pressure in the electrolyte solution. To define the osmotic pressure we follow the procedure of Moyne and Murad [30], which consists of inserting the Boltzmann distribution for the net charge density in Eq. (2.12) in the preceding result to obtain

$$\nabla p+q\nabla \Psi =\nabla (p+{\int }_0^{\Psi }q(\Psi \prime )d\Psi \prime )=\nabla [p-2{\mathit{RTC}}_b(\cosh \left(\frac{F\Psi }{\mathit{RT}}\right)-1)]=0$$ (2.20)

Hence, defining the constant bulk phase pressure of the outer solution

$$p_b:=p-2{\mathit{RTC}}_b[\cosh \left(\frac{F\Psi }{\mathit{RT}}\right)-1]$$

the classical osmotic pressure governed by the van't Hoff relation reads

$$\pi :=p-p_b=\mathit{RT}(C_{{\mathrm{Na}}^{+}}+C_{{\mathrm{H}}^{+}}+C_{{\mathrm{Cl}}^{-}}+C_{{\mathrm{OH}}^{-}}-2C_b)=2{\mathit{RTC}}_b[\cosh \left(\frac{F\Psi }{\mathit{RT}}\right)-1]$$ (2.21)

where Eqs. (2.3) and (2.11) have been used. The equilibrium in the fluid can also be rephrased in terms of the Cauchy stress tensor

$$\nabla ·{\mathbf{\sigma }}_f=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{with}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\mathbf{\sigma }}_f:=-p\mathit{I}+{\mathbf{\tau }}_{\mathit{M}}=-(p_b+\pi )\mathit{I}+{\mathbf{\tau }}_{\mathit{M}}$$ (2.22)

where the Maxwell stress tensor is defined by [38]

$${\mathbf{\tau }}_{\mathit{M}}:=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0}{2}(2E\otimes E-E^2\mathit{I})$$ (2.23)

with the symbol $\otimes$ denoting the classical tensorial product between vectors. The overall excess in stress in the electrolyte solution relative to the bulk phase pressure is nothing but the disjoining stress tensor, which incorporates the combined effects of the osmotic pressure and Maxwell stress tensor [41]

$${\mathbf{\Pi }}_d:=\pi \mathit{I}-{\mathbf{\tau }}_{\mathit{M}}$$ (2.24)

Using the definition in Eq. (2.22) yields

$${\mathbf{\sigma }}_f:=-p_b\mathit{I}-{\mathbf{\Pi }}_d$$

2.3. Elasticity of the Macromolecules.

Let ${\Omega }_s$ be the microscopic domain occupied by the solid particles. Denoting $(\mathit{u},{\mathbf{\sigma }}_s)$ as the microscopic displacement and stress tensor of the solid phase, assuming a linear elastic and isotropic solid, the deformation is governed by the classical linear elasticity problem

$$\nabla ·{\mathbf{\sigma }}_s=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\hspace{1em}{\Omega }_s$$ (2.25)

$${\mathbf{\sigma }}_s={\mathit{c}}_s\mathcal{E}\left(\mathit{u}\right)$$ (2.26)

where ${\mathit{c}}_s$ is the fourth order elastic tensor given by $c_{\mathit{ijkl}}={\lambda }_s{\delta }_{\mathit{ij}}{\delta }_{\mathit{kl}}+{\mu }_s({\delta }_{\mathit{ik}}{\delta }_{\mathit{jl}}+{\delta }_{\mathit{il}}{\delta }_{\mathit{jk}})$ with ${\lambda }_s=E_s\nu /(1+\nu )(1-2\nu )$ and ${\mu }_s=E_s(1+\nu )/2$ as the Lame constants, ($E_s$,$\nu$) as Young's modulus and Poisson's ratio, and ${\delta }_{\mathit{ij}}$ being the Kronecker δ symbol.

Continuity of the traction vector at the particle/fluid interface yields

$${\mathbf{\sigma }}_s\mathbf{n}={\mathbf{\sigma }}_f\mathbf{n}$$ (2.27)

2.4. Mass Balance of the Solid Phase.

Denoting ${\rho }_s$ and ${\rho }_{s0}$ as the densities of the solid particles in the current and reference configurations, under the small strain assumption within the solid particles the mass balance of the solid phase reads

$${\rho }_s={\rho }_{s0}(1-\nabla ·\mathit{u})$$ (2.28)

2.5. Summary of the Microscopic Model.

The quasi–static microscopic model consists in: given the set of constants $(F$, $R$, $T$, ${\tilde{\epsilon }}_0$, $\tilde{\epsilon }$, ${\rho }_{s0}$, $E_s$, $\nu$, $A_s$, $K_W)$, the parameters $(\mathrm{CEC}$, $K_1$, $K_2)$ depending on the salinity, the pressure, and cation concentrations in the outer bulk solution $(p_b,C_{\mathrm{Na}{b}^{+}},C_{\mathrm{H}{b}^{+}})$ and $C_b:=C_{{\mathit{Nab}}^{+}}+C_{\mathrm{H}{b}^{+}}$, find the microscopic fields $(\Psi ,\pi ,E,{\mathbf{\sigma }}_f$, ${\mathbf{\tau }}_{\mathit{M}},{\mathbf{\Pi }}_d)$ and $({\rho }_s,\mathit{u},{\mathbf{\sigma }}_s)$ satisfying

$$\{\begin{array}{c}\Delta \Psi =\frac{2{\mathit{FC}}_b}{{\tilde{\epsilon }\tilde{\epsilon }}_0}\sinh \left(\frac{F\Psi }{\mathit{RT}}\right)\\ \nabla ·{\mathbf{\sigma }}_f=0\\ {\mathbf{\sigma }}_f=-(p_b+\pi )\mathit{I}+{\mathbf{\tau }}_{\mathit{M}}\\ {\mathbf{\Pi }}_d=\pi \mathit{I}-{\mathbf{\tau }}_{\mathit{M}}\\ {\mathbf{\tau }}_{\mathit{M}}=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0}{2}(2E\otimes E-E^2\mathit{I})\\ E=-\nabla \Psi \\ \pi =2{\mathit{RTC}}_b[\cosh \left(\frac{F\Psi }{\mathit{RT}}\right)-1]\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\hspace{1em}{\Omega }_f$$ (2.29)

and

$$\{\begin{array}{c}\nabla ·{\mathbf{\sigma }}_s=0\\ {\mathbf{\sigma }}_s={\mathit{c}}_s\mathcal{E}\left(\mathit{u}\right)\\ {\rho }_s={\rho }_{s0}(1-\nabla ·\mathit{u})\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\hspace{1em}{\Omega }_s$$ (2.30)

along with the interface conditions

$$\{\begin{array}{c}\nabla \Psi ·\mathbf{n}=\frac{\sigma }{{\tilde{\epsilon }\tilde{\epsilon }}_0}\\ {\mathbf{\sigma }}_f\mathbf{n}={\mathbf{\sigma }}_s\mathbf{n}\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{on}\mathrm{\hspace{1em}}{\Gamma }_{\mathit{fs}}$$ (2.31)

with the surface charge density given by

$$\sigma =-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}}{A_s}\left[\frac{K_1}{K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\exp (-\frac{F\zeta }{\mathit{RT}})}\right]$$ (2.32)

After solving for the primary unknowns, the quantities ($p$,$q$,$C_i$) ($i={\mathrm{Na}}^{+}$, ${\mathrm{H}}^{+}$, ${\mathrm{Cl}}^{-}$, ${\mathrm{OH}}^{-}$) can be computed within the postprocessing approach using Eqs. (2.21), (2.12) and (2.11).

3. Homogenization

In this section, we apply the homogenization procedure [42] to upscale the microscopic model in Eqs. (2.29)–(2.32) to the macroscale. Consider the overall microscopic domain $\Omega ={\Omega }_f\hspace{1em}\cup \hspace{1em}{\Omega }_s$ equipped with a periodic structure. Under the scale-separation assumption, introduce the perturbation parameter $\varepsilon :=\mathit{l}/\mathit{L}$ with $l$ and $L$ the microscopic and macroscopic characteristic lengths, respectively. The family of perturbed models, referred to herein as $\varepsilon$-models, consist of properly scaled equations posed in the perturbed domain ${\Omega }^{\varepsilon }$, which are considered to be the union of disjoint fluid and solid subdomains ${\Omega }_f^{\varepsilon }$ and ${\Omega }_s^{\varepsilon }$ with a common interface ${\Gamma }_{\mathit{fs}}^{\varepsilon }$. The perturbed domain ${\Omega }^{\varepsilon }$ is periodically reconstructed from the repetition of a micro cell $Y^{\varepsilon }$ and the subdomains ${\Omega }_f^{\varepsilon }$ and ${\Omega }_s^{\varepsilon }$ along with the interface ${\Gamma }_{\mathit{fs}}^{\varepsilon }$ given by the union of cell subdomains $Y_f^{\varepsilon }$ and $Y_s^{\varepsilon }$ and the cell inner boundary $\partial Y_{\mathit{fs}}^{\varepsilon }$, respectively. Each cell is congruent to a standard unitary parallelepiped period $Y$ (with outer boundary $\partial Y$) composed of subdomains $Y_f$ and $Y_s$ sharing a common inner boundary $\partial Y_{\mathit{fs}}$. Following the standard homogenization procedure we introduce two coordinate systems x and y with x the macroscopic (slow) coordinate and y the microscopic (fast) coordinate satisfying the constraint $\mathit{y}=\mathit{x}/\varepsilon$. The starting point $\varepsilon =1$ corresponds to our microscopic model. The goal is to investigate the asymptotic homogenized behavior of the swelling medium as $\varepsilon \to 0$.

3.1. Dimensionless Form of the Governing Equations.

To capture the correct physics in the upscaling process the coefficients of the microscopic model shall be properly scaled in terms of powers of $\varepsilon$. This is accomplished by pursuing the procedure proposed by Auriault [43], which consists of rephrasing the microscopic model in dimensionless form and estimating the dimensionless numbers involved in powers of $\varepsilon$. Here, we adopt such a procedure to properly scale the terms in the Poisson–Boltzmann problem. By normalizing the unknowns with respect to the reference values (denoted by the subscript “ref”) and designating the dimensionless variables with the superscript “*,” we have

$${\Psi }^{*}=\frac{\Psi }{{\Psi }_{\mathrm{ref}}},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{E}^{*}=\frac{E}{E_{\mathrm{ref}}},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\sigma }^{*}=\frac{\sigma }{{\sigma }_{\mathrm{ref}}},{\mathit{x}}^{*}=\frac{\mathit{x}}{L},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\nabla =L{\nabla }^{*},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathbf{\Delta }=L^2{\Delta }^{*}$$ (3.1)

In terms of the preceding dimensionless variables, the Poisson– Boltzmann problem can be rephrased as

$$\{\begin{array}{c}{B}_1{\Delta }^{*}{\Psi }^{*}=\sinh \left(B_2{\Psi }^{*}\right)\\ E^{*}=-B_3\nabla {\Psi }^{*}\\ B_4{\nabla }^{*}{\Psi }^{*}·\mathbf{n}={\sigma }^{*}\end{array}$$ (3.2)

with

$$\begin{array}{cccc}{B}_1:=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0{\Psi }_{\mathrm{ref}}}{2{\mathit{FC}}_b{L}^2},& B_2:=\frac{F{\Psi }_{\mathrm{ref}}}{\mathit{RT}},& B_3:=\frac{{\Psi }_{\mathrm{ref}}}{{\mathit{LE}}_{\mathrm{ref}}},& B_4:=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0{\Psi }_{\mathrm{ref}}}{L{\sigma }_{\mathrm{ref}}}\end{array}$$ (3.3)

In order to estimate the quantities in the preceding equations, we begin by identifying the microscopic length with the Debye length $l:=L_{\mathrm{D}}:=({\tilde{\epsilon }\tilde{\epsilon }}_0\mathit{RT}/2F^2{C}_b){}^{1/2}$, which represents the characteristic thickness of the electric double layer. By choosing ${\Psi }_{\mathrm{ref}}$ of $\mathfrak{O}(\mathit{RT}/F)$, the reference electric field is characterized by the ratio between ${\Psi }_{\mathrm{ref}}$ and the microscopic length ${\Psi }_{\mathrm{ref}}={\mathit{lE}}_{\mathrm{ref}}$. Furthermore, the first term in the boundary condition (2.31) suggests the choice of ${\sigma }_{\mathrm{ref}}={\tilde{\epsilon }\tilde{\epsilon }}_0{E}_{\mathrm{ref}}$. Since the net volumetric charge $q$ is counterbalanced by the surface charge density $\sigma$ due to electroneutrality (Eq. (2.15)), we choose ${\sigma }_{\mathrm{ref}}=2{\mathit{FC}}_{b}l$. By using these reference values in Eq. (3.3), we obtain the estimates

$$B_1={\left(\frac{l}{L}\right)}^2=\mathfrak{O}\left({\varepsilon }^2\right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{B}_2=\mathfrak{O}(1),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{B}_3=B_4=\left(\frac{l}{L}\right)=\mathfrak{O}\left(\varepsilon \right)$$

By inserting the aforementioned estimates in Eq. (3.2), the rescaled Poisson–Boltzmann equation reads as

$$\{\begin{array}{c}{\varepsilon }^2\Delta {\Psi }^{\varepsilon }=\frac{2{\mathit{FC}}_b}{{\tilde{\epsilon }\tilde{\epsilon }}_0}\sinh \left(\frac{F{\Psi }^{\varepsilon }}{\mathit{RT}}\right)\\ E^{\varepsilon }=-\varepsilon \nabla {\Psi }^{\varepsilon }\\ \varepsilon \nabla {\Psi }^{\varepsilon }·\mathbf{n}=\frac{\sigma }{{\tilde{\epsilon }\tilde{\epsilon }}_0}\end{array}$$ (3.4)

3.2. Matched Asymptotic Expansions.

To upscale the microscopic model, we adopt the methodology based on two-scale asymptotic expansions [42]. The usual procedure consists of postulating an expansion for the unknowns in the form

$$f^{\varepsilon }(\mathit{x},\mathit{y})=\sum _{k=0}^{\infty }{\varepsilon }^k{f}^k(\mathit{x},\mathit{y})=f^0(\mathit{x},\mathit{y})+\varepsilon f^1(\mathit{x},\mathit{y})+{\varepsilon }^2{f}^2(\mathit{x},\mathit{y})+\dots$$ (3.5)

with the coefficients $f^k=f^k(\mathit{x},\mathit{y})$ $(k=0,1,2...)$, $Y$-periodic. Adopting the macroscopic point of view, the differential operators are represented in the form

$$\begin{array}{c}\nabla f^{\varepsilon }(\mathit{x},\mathit{y})={\nabla }_{\mathit{x}}\hspace{1em}{f}^{\varepsilon }(\mathit{x},\mathit{y})+\frac{1}{\varepsilon }{\nabla }_{\mathit{y}}\hspace{1em}{f}^{\varepsilon }(\mathit{x},\mathit{y})\end{array}$$ (3.6)

Inserting the ansatz (Eq. (3.5)) in our rescaled microscopic governing equations and collecting the successive powers of $\varepsilon$, we obtain equations at different orders. In the fluid domain, we have

$${\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_f^0=0$$ (3.7)

$${\mathbf{\Delta }}_{\mathit{yy}}{\Psi }^0=\frac{2{\mathit{FC}}_b}{{\tilde{\epsilon }\tilde{\epsilon }}_0}\sinh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)$$ (3.8)

$${\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_f^0+{\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_f^1=0$$ (3.9)

$${\pi }^0=2{\mathit{RTC}}_b[\cosh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)-1]$$ (3.10)

$$E^0=-{\nabla }_{\mathit{y}}{\Psi }^0$$ (3.11)

$${\mathbf{\tau }}_M^0=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0}{2}(2E^0\otimes E^0-E^{02}\mathit{I})$$ (3.12)

$${\mathbf{\sigma }}_f^0=-p_b\mathit{I}-{\mathbf{\Pi }}_d^0$$ (3.13)

$${\mathbf{\Pi }}_d^0={\pi }^0\mathit{I}-{\mathbf{\tau }}_M^0$$ (3.14)

For the solid particles, we obtain

$${\nabla }_{\mathit{y}}\left[{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}^0\right)\right]=0$$ (3.15)

$${\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_s^0=0$$ (3.16)

$${\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_s^0+{\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_s^1=0$$ (3.17)

$${\rho }_s^0={\rho }_{s_0}[1-({\nabla }_{\mathit{x}}·{\mathit{u}}^0+{\nabla }_{\mathit{y}}·{\mathit{u}}^1)]$$ (3.18)

$${\mathbf{\sigma }}_s^0={\mathit{c}}_s[{\mathcal{E}}_{\mathit{x}}\left({\mathit{u}}^0\right)+{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}^1\right)]$$ (3.19)

whereas the successive orders of the interface conditions read as

$${\nabla }_{\mathit{y}}{\Psi }^0·\mathbf{n}=\frac{{\sigma }^0}{{\tilde{\epsilon }\tilde{\epsilon }}_0}$$ (3.20)

$${\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}^0\right)·\mathbf{n}=0$$ (3.21)

$${\mathbf{\sigma }}_s^0\mathbf{n}={\mathbf{\sigma }}_f^0\mathbf{n}$$ (3.22)

$${\mathbf{\sigma }}_s^1\mathbf{n}={\mathbf{\sigma }}_f^1\mathbf{n}$$ (3.23)

with the $\mathfrak{O}({\varepsilon }^0)$-surface charge density given by

$${\sigma }^0=-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}\hspace{1em}{K}_1}{A_s[K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\exp (-\frac{F{\zeta }^0}{\mathit{RT}})]}$$ (3.24)

3.3. Nonoscillatory Displacement.

By invoking Eq. (3.15) together with the homogeneous Neumann condition Eq. (3.21) gives the local rigid motion ${\mathit{u}}^0(\mathit{x},\mathit{y})={\mathit{u}}^0\left(\mathit{x}\right)$.

3.4. Local Poisson–Boltzmann Equation.

The local nonlinear problem for the EDL potential at $\mathfrak{O}({\varepsilon }^0)$ stems from Eq. (3.8) along with the boundary conditions in Eqs. (3.28) and (3.24)

$$\{\begin{array}{cc}{\mathbf{\Delta }}_{\mathit{y}}{\Psi }^0=\frac{2F}{{\tilde{\epsilon }\tilde{\epsilon }}_0}(C_{\mathrm{N}{\mathit{ab}}^{+}}+C_{\mathrm{H}{b}^{+}})\sinh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)& \text{in}\hspace{1em}\mathbf{Y}\\ {\nabla }_{\mathit{y}}{\Psi }^0·\mathbf{n}=\frac{{\sigma }^0}{{\tilde{\epsilon }\tilde{\epsilon }}_0}& \text{on}\mathrm{\hspace{1em}}\partial {\mathbf{Y}}_{\mathit{fs}}\\ {\sigma }^0=-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}\hspace{1em}{K}_1}{A_s[K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\mathit{exp}(-F{\zeta }^0/\mathit{RT})]}& \end{array}$$ (3.25)

constrained by the global electroneutrality condition

$${\int }_{Y_f}{q}^0{\mathit{dY}}_f=-\frac{2{\mathit{FC}}_b}{\left|Y\right|}{\int }_{Y_f}\sinh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right){\mathit{dY}}_f=-\frac{1}{\left|Y\right|}{\int }_{\partial Y_{\mathit{sf}}}{\sigma }^0\partial Y$$ (3.26)

It is worth noting that the Poisson–Boltzmann problem “shrinks” in the upscaling procedure and does not survive at the macroscale. The local distribution ${\Psi }^0={\Psi }^0(y)$ will be subsequently used to build-up the effective constitutive laws of the macroscopic parameters. The novelty here is the additional nonlinear dependence of ${\sigma }^0$ on ${\Psi }^0$ and pH, which will be subsequently be transferred to the effective medium behavior through the upscaling process.

3.5. Closure.

To derive the closure problem for the fluctuating displacement of the macromolecules we combine Eqs. (3.16) and (3.19) with the boundary condition in Eq. (3.22). Recalling that ${\mathit{u}}^0(\mathit{x},\mathit{y})={\mathit{u}}^0(\mathit{x})$ and using Eq. (3.13), we obtain the local problem

$$\begin{array}{c}{\nabla }_{\mathit{y}}·\left[{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}^1\right)\right]=0\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\hspace{1em}{Y}_s{\mathit{c}}_s[{\mathcal{E}}_{\mathit{x}}\left({\mathit{u}}^0\right)+{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}^1\right)]\mathbf{n}=-[{\mathit{p}}_{\mathit{b}}\mathit{I}+{\mathbf{\Pi }}_{\mathit{d}}^0]\mathbf{n}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{on}\hspace{1em}\partial {\mathit{Y}}_{\mathit{fs}}\end{array}$$ (3.27)

By comparing the preceding result with the one derived from the classical homogenization of Biot's equations of poroelasticity derived by Auriault and Sanchez-Palencia [44] or Terada et al. [45], the novelty is the appearance of the disjoining tensor ${\mathbf{\Pi }}_d^0$ in the boundary condition, which incorporates the electrochemical effects.

By linearity, the solution of Eq. (3.27) can be written in the form

$${\mathit{u}}^1(\mathit{x},\mathit{y})=\mathbf{\zeta }\left(\mathit{y}\right)p_b^0\left(\mathit{x}\right)+\mathbf{\xi }\left(\mathit{y}\right){\mathcal{E}}_x\left({\mathit{u}}^0\left(\mathit{x}\right)\right)+{\mathit{u}}_{\pi }^1(\mathit{x},\mathit{y})+\stackrel{\wedge }{\mathit{u}}\left(\mathit{x}\right)$$ (3.28)

The canonical cell problems for the unknowns ($\mathbf{\zeta }$,$\mathbf{\xi }$) read as (see Refs. [30,41] for details)

$$\{\begin{array}{c}{\nabla }_{\mathit{y}}·\left[{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\zeta }\right)\right]=0\\ {\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\zeta }\right)\mathbf{n}=-\mathit{I}\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\{\begin{array}{cc}{\nabla }_{\mathit{y}}·\left[{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\xi }\right)\right]=0& \mathrm{in}\hspace{1em}{Y}_s\\ {\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\xi }\right)\mathbf{n}=-{\mathit{c}}_s\mathit{II}\mathbf{nn}& \mathrm{on}\mathrm{\hspace{1em}}\partial Y_{\mathit{sf}}\end{array}$$ (3.29)

with $\mathit{II}$ being the unity fourth-order tensor. The appearance of the disjoining tensor ${\mathbf{\Pi }}_d^0$ in the traction boundary condition in Eq. (3.27) gives rise to the electrochemical component ${\mathit{u}}_{\pi }^1$ in Eq. (3.28) satisfying the local problem

$$\{\begin{array}{cc}{\nabla }_{\mathit{y}}·\left[{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)\right]=0& \mathrm{in}\mathrm{\hspace{1em}}{Y}_s\\ {\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)\mathbf{n}=-{\mathbf{\Pi }}_d^0\mathbf{n}& \mathrm{on}\hspace{1em}\partial Y_{\mathit{sf}}\end{array}$$ (3.30)

The electrochemical stress ${\mathbf{\sigma }}_{\Pi }:={\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}({\mathit{u}}_{\pi }^1)$, which stems from the traction boundary condition due to the disjoining pressure (depending on ${\Psi }^0$ and pH), is a key variable since it incorporates the local mechanical coupling induced by the pH changes in the external bulk solution.

3.6. Macroscopic Effective Stress Principle.

Defining the average and intrinsic average operators

$$〈·〉:=\frac{1}{\left|Y\right|}{\int }_{Y_i}(·)\mathit{dY}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}〈·〉{}^i:=\frac{1}{\left|Y_i\right|}{\int }_{Y_i}(·){\mathit{dY}}_i,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}i=(s,f)$$ (3.31)

by averaging Eqs. (3.9) and (3.17) after adding and using the divergence theorem along with the periodicity assumption and the traction boundary condition in Eq. (3.23), yields (recall that $\mathbf{n}$ was chosen outward to $Y_f$)

$$〈{\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_f^0+{\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_s^0〉=-〈{\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_f^1+{\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_s^1〉=-\frac{1}{\left|Y\right|}{\int }_{\partial Y_{\mathit{fs}}}({\mathbf{\sigma }}_f^1-{\mathbf{\sigma }}_s^1)\mathbf{n}d\Gamma =0$$ (3.32)

Defining the overall stress tensor of the mixture ${\mathbf{\sigma }}_T^0$ as

$${\mathbf{\sigma }}_T^0:=〈{\mathbf{\sigma }}_f^0〉+〈{\mathbf{\sigma }}_s^0〉$$ (3.33)

the overall equilibrium reads

$${\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_T^0=0$$ (3.34)

Furthermore, using the closure (3.28) in Eq. (3.19) yields

$${\mathbf{\sigma }}_s^0=[{\mathit{c}}_s\mathit{II}+{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\xi }\right)]{\mathcal{E}}_{\mathit{x}}\left({\mathit{u}}^0\right)-{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\zeta }\right)p_b-{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)$$ (3.35)

in which, when combined with Eqs. (3.13) and (3.33), after averaging, gives the macroscopic effective stress principle

$${\mathbf{\sigma }}_T^0=-\mathbf{\alpha }{p}_b+{\mathit{C}}_s{\mathcal{E}}_{\mathit{x}}\left({\mathit{u}}^0\right)-{\mathbf{\Pi }}^0$$ (3.36)

where the macroscopic coefficients ($\mathbf{\alpha }$, ${\mathit{C}}_s$, ${\mathbf{\Pi }}^0$) are given by

$$\begin{array}{c}\mathbf{\alpha }={\phi }_f\mathit{I}-〈{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\zeta }\right)〉\\ {\mathit{C}}_s=〈{\mathit{c}}_s(\mathit{II}+{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\xi }\right))〉\\ {\mathbf{\Pi }}^0=〈{\mathbf{\Pi }}_d^0〉+{\phi }_s{\mathbf{\Pi }}_S^0\\ {\mathbf{\Pi }}_S^0:=-〈{\mathbf{\sigma }}_{\Pi }〉=-{〈{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)〉}^s\end{array}$$ (3.37)

with ${\phi }_i:=\left|Y_i\right|/\left|Y\right|$ $(i=s,f)$ being the volume fractions of the solid and fluid phase, respectively. In the preceding representation of the modified Terzaghi's decomposition, $\mathbf{\alpha }$ is the Biot–Willis coefficient, ${\mathit{C}}_s$ is the macroscopic elastic modulus (fourth-rank tensor) and ${\mathbf{\Pi }}^0$ is the macroscopic electrochemical tensor. By invoking the classical variational analysis presented by Auriault and Sanchez-Palencia [44], one can show the result $〈{\mathit{c}}_s{\mathbf{\varepsilon }}_{\mathit{y}}(\mathbf{\zeta })〉=〈{\nabla }_{\mathit{y}}·\mathbf{\xi }(\mathit{y})〉$. The macroscopic tensor ${\mathbf{\Pi }}_S^0$, given by the intrinsic average of the local coupling electrochemical stress, corresponds to the component of ${\mathbf{\Pi }}^0$ directly responsible for the expansion of the solid matrix and shall henceforth be referred to as the swelling stress tensor [41,46].

The homogenized result (Eqs. (3.36) and (3.37)), together with the cell problem in Eq. (3.30) for ${\mathit{u}}_{\pi }^1$ and the relations in Eqs. (3.10), (3.12), and (3.14) for the disjoining pressure requiring the solution of the local Poisson–Boltzmann (Eq. (3.15) with the Neumann condition for ${\sigma }^0$ (dependent on $C_{\mathrm{H}{b}^{+}}$) provides the two-scale dependency of swelling with the pH of the bulk solution.

3.7. Particle Density.

In order to derive the macroscopic constitutive law for particle density, we use the closure in Eq. (3.28) in Eq. (3.18) to obtain

$$\begin{array}{c}{\rho }_s^0={\rho }_{s0}[1-({\nabla }_{\mathit{x}}·{\mathit{u}}^0+p_b{\nabla }_{\mathit{y}}·\mathbf{\zeta }\left(\mathit{y}\right)+{\nabla }_{\mathit{y}}·\mathbf{\xi }\left(\mathit{y}\right):{\mathbf{\varepsilon }}_x\left({\mathit{u}}^0\left(\mathit{x}\right)\right)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+{\nabla }_{\mathit{y}}·{\mathit{u}}_{\pi }^1(\mathit{x},\mathit{y}))](3.38)\end{array}$$ (3.38)

where $A:B=\mathrm{tr}({\mathit{AB}}^T)$ denotes the classical inner product between tensors. Without loss of generality, assume the absence of fluctuations at the reference configuration so that ${\rho }_{s0}={\rho }_{s0}(\mathit{x})$. Applying the intrinsic average operator the effective equation for the particle density is given by

$$\frac{〈{\rho }_s^0〉{}^s-{\rho }_{s0}}{{\rho }_{s0}}=-\mathbf{\tau }:{\mathcal{E}}_x\left({\mathit{u}}^0\left(\mathit{x}\right)\right)+p_b\beta +\gamma$$ (3.39)

where

$$\begin{array}{c}\mathbf{\tau }=({\phi }_s\mathit{I}+〈{\nabla }_{\mathit{y}}·\mathbf{\xi }\left(\mathit{y}\right)〉{}^s)=\mathit{I}-({\phi }_f\mathit{I}-〈{\mathit{c}}_s{\mathbf{\varepsilon }}_{\mathit{y}}\left(\mathbf{\zeta }\right)〉)=\mathit{I}-\mathbf{\alpha }\beta =-{〈{\nabla }_{\mathit{y}}·\mathbf{\zeta }\left(\mathit{y}\right)〉}^s\gamma =-{〈{\nabla }_{\mathit{y}}·{\mathit{u}}_{\pi }^1(\mathit{x},\mathit{y})〉}^s\end{array}$$ (3.40)

The pair of coefficients $(\beta ,\mathbf{\tau })$, which are solely dependent on cell geometry, are classical poroelastic coefficients [47]. The parameter $\beta$ is the particle compressibility given by the inverse of the local bulk modulus $\beta =K_{s0}^{-1}$. The coefficient $\mathbf{\tau }=\mathit{I}-\mathbf{\alpha }$ consists of a tensorial generalization of the ratio between the bulk modulus of the matrix and the solid. The novelty here is the additional electrochemical compressibility $\gamma$ which from the closure (Eq. (3.30)) quantifies changes in the particle volume due to the effects of the disjoining pressure. Since ${\mathbf{\Pi }}_d^0$ varies with the fast coordinate $\mathit{y}$ (with the exception of microstructures characterized by parallel particles [46]), unlike the other linear poroelastic parameters, $\gamma$ is a nonlinear function of pH and salinity. We remark upon the highly innovative issue elucidated herein regarding the construction of the two-scale dependency of the electrochemical compressibility on the pH.

3.8. Closure Law for the Volume Fraction.

To close the macroscopic system it remains for us to establish a complementary equation for the porosity. This can be accomplished by considering the Lagrangian form of the macroscopic mass balance of the solid phase. Denoting $\left|Y_0\right|$ and $\left|Y_{s0}\right|$ as the volumes of the cell and occupied by the particles in the reference configuration, respectively, and ${\phi }_{s0}=\left|Y_{s0}\right|/\left|Y_0\right|$ as the corresponding volume fraction, the integral form of the solid phase mass balance reads as

$$\frac{1}{\left|Y_0\right|}{\int }_{Y_{s0}}{\rho }_{s0}\mathit{dy}=\frac{1}{\left|Y_0\right|}{\int }_{Y_s}{\rho }_s^0\mathit{dy}$$

In terms of the intrinsic volume averaging operator (the second term in Eq. (3.31)) the preceding result can be rewritten as (recall that ${\rho }_{s0}={\rho }_{s0}(\mathit{x})$)

$${\phi }_{s0}{\rho }_{s0}=\frac{\left|Y_s\right|}{\left|Y_0\right|}{〈{\rho }_s^0〉}^s=\frac{{\phi }_s\left|Y\right|}{\left|Y_0\right|}{〈{\rho }_s^0〉}^s$$ (3.41)

In terms of the macroscopic Jacobian $J=\left|Y\right|/\left|Y_0\right|$, which governs volume changes relative to the reference configuration, the preceding result can be rephrased as

$${\rho }_{s_0}{\phi }_{s0}={〈{\rho }_s^0〉}^s{\phi }_{s}J$$ (3.42)

which corresponds to the form postulated by Coussy [47]. Using the macroscopic closure relation $\mathit{ln}J={\nabla }_x·{\mathit{u}}^0$, we have

$${〈{\rho }_s^0〉}^s{\phi }_s\exp ({\nabla }_{\mathit{x}}·{\mathit{u}}^0)={\rho }_{s_0}{\phi }_{s0}$$ (3.43)

which, when combined with Eq. (3.39), yields

$$[1+(\mathit{I}-\mathbf{\alpha }):{\mathcal{E}}_x\left({\mathit{u}}^0\left(\mathit{x}\right)\right)+p_b\beta +\gamma ]{\phi }_s={\phi }_{s_0}\exp (-{\nabla }_{\mathit{x}}·{\mathit{u}}^0)$$

The preceding result brings new insight into the role of the electrochemical compressibility $\gamma$ in the macroscopic mass balance of the solid phase.

3.9. Summary of the Two-Scale Model.

We are now ready to formulate our two-scale model. Let ${\Omega }_M$ be the macroscopic domain occupied by an expansive material saturated by an aqueous electrolyte solution containing four monovalent ions $({\mathrm{Na}}^{+}{,\mathrm{H}}^{+},{\mathrm{Cl}}^{-},{\mathrm{OH}}^{-})$. Given the set of constants $({\rho }_{s0}$, ${\phi }_{s0}$, ${\mathit{c}}_s$, $F$, $R$, $T$, ${\tilde{\epsilon }}_0$, $\tilde{\epsilon }$, $A_s)$, the salinity dependent parameters $(\mathrm{CEC}$, $K_1$, $K_2)$, the characteristic functions $(\mathbf{\zeta }$, $\mathbf{\xi })$ solution of Eq. (3.29), and the input data in the bulk fluid $(C_{\mathrm{Na}{b}^{+}}$, $C_{\mathrm{H}{b}^{+}}$, $p_b)$, the two-scale problem consists of finding the macroscopic unknowns $({\mathbf{\sigma }}_T^0,{\mathit{u}}^0,〈{\rho }_s^0〉,{\phi }_s)$ satisfying

$$\{\begin{array}{c}{\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_T^0=0\\ {\mathbf{\sigma }}_T^0=-\mathbf{\alpha }{p}_b+{\mathit{C}}_s{\mathcal{E}}_{\mathit{x}}\left({\mathit{u}}^0\right)-{\mathbf{\Pi }}^0\\ {⟨{\rho }_s^0⟩}^s={\rho }_{s0}[1+(\mathit{I}-\mathbf{\alpha }):{\mathcal{E}}_x({\mathit{u}}^0)+p_b\beta +\gamma ]\\ [1+(\mathit{I}-\mathbf{\alpha }):{\mathcal{E}}_x\left({\mathit{u}}^0\left(\mathit{x}\right)\right)+p_b\beta +\gamma ]{\phi }_s={\phi }_{s_0}\mathit{exp}(-{\nabla }_{\mathit{x}}·{\mathit{u}}^0)\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{in}\mathrm{\hspace{1em}}{\Omega }_M$$ (3.44)

where the effective parameters $(\mathbf{\alpha },{\mathit{C}}_s,{\mathbf{\Pi }}^0,\mathbf{\tau },\beta$,$\gamma )$ are given by

$$\{\begin{array}{c}\mathbf{\alpha }=(1-{\phi }_s)\mathit{I}-〈{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\zeta }\right)〉\\ {\mathit{C}}_s=〈{\mathit{c}}_s(\mathit{II}+{\mathcal{E}}_{\mathit{y}}\left(\mathbf{\xi }\right))〉\\ \beta ={〈{\nabla }_{\mathit{y}}·\mathbf{\zeta }\left(\mathit{y}\right)〉}^s\\ \gamma ={〈{\nabla }_{\mathit{y}}·{\mathit{u}}_{\pi }^1(\mathit{x},\mathit{y})〉}^s\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}$$ (3.45)

and

$$\{\begin{array}{c}{\mathbf{\Pi }}^0=⟨{\mathbf{\Pi }}_d^0⟩+{\phi }_s{\mathbf{\Pi }}_S^0\\ ⟨{\mathbf{\Pi }}_d^0⟩=⟨{\pi }^0⟩\mathit{I}-⟨{\mathbf{\tau }}_M^0⟩\\ {\mathbf{\Pi }}_S^0=-{⟨{\mathbf{\sigma }}_{\Pi }⟩}^s=-{⟨{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)⟩}^s\\ {\pi }^0=2{\mathit{RTC}}_b[\cosh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)-1]\\ {\mathbf{\tau }}_M^0=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0}{2}(2E^0\otimes E^0-E^{0^2}\mathit{I})\end{array}$$ (3.46)

together with

$$\{\begin{array}{cc}{\mathbf{\Delta }}_{\mathit{y}}{\Psi }^0=\frac{2F}{{\tilde{\epsilon }\tilde{\epsilon }}_0}(C_{\mathrm{Na}{b}^{+}}+C_{\mathrm{H}{b}^{+}})\sinh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)& \mathrm{in}\hspace{1em}{\mathbf{Y}}_f\\ {\nabla }_{\mathit{y}}{\Psi }^0·\mathbf{n}=\frac{{\sigma }^0}{{\tilde{\epsilon }\tilde{\epsilon }}_0}& \mathrm{on}\hspace{1em}\partial {\mathbf{Y}}_{\mathit{fs}}\\ {\sigma }^0=-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}\hspace{1em}{K}_1}{A_s[K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\exp (-F\Psi |{}_{\Gamma }/\mathit{RT})]}\end{array}$$ (3.47)

and

$$\{\begin{array}{c}{\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_{\Pi }=0\\ {\mathbf{\sigma }}_{\Pi }={\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\hspace{1em}{\mathbf{Y}}_s\\ {\mathbf{\sigma }}_{\Pi }\mathbf{n}=-{\mathbf{\Pi }}_d^0\mathbf{n}\hspace{1em}\hspace{1em}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{on}\hspace{1em}\partial {\mathbf{Y}}_{\mathit{sf}}\end{array}$$ (3.48)

In the preceding representations, the purely mechanical coefficients $(\mathbf{\alpha },{\mathit{C}}_s,\beta )$ solely depend on the cell geometry, whereas the electrochemical parameters $({\mathbf{\Pi }}^0$, $\gamma )$ also depend on the pH and salinity of the bulk water.

The model developed herein generalizes the previous twoscale models [30,41,46] for swelling media by incorporating the dependence of the electrochemical parameters on the pH. The new microphysics has been incorporated through the nonlinear Neumann condition for the Poisson–Boltzmann equation which entails pH-dependence at the pore-scale. The propagation of such dependency to the macroscale through the homogenization procedure gives rise to the new pH-dependent constitutive laws for the effective parameters.

4. Reduced Cases

In the following, we consider particular cases of the general two–scale model.

4.1. Microscopic Incompressibility.

We begin by considering the case where the local volumetric deformation of the particles is small compared to the volume change of the matrix. Under this local incompressibility assumption, the local characteristic functions satisfy the constraints ${\nabla }_{\mathit{y}}·\mathbf{\zeta }={\nabla }_{\mathit{y}}·{\mathit{u}}_{\pi }^1=0$ and ${\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}(\mathbf{\zeta })={\nabla }_{\mathit{y}}·\mathbf{\xi }=-\mathit{I}$, which yields $〈{\nabla }_{\mathit{y}}·\mathbf{\xi }(\mathit{y})〉=〈{\mathit{c}}_s{\mathcal{E}}_{\mathit{y}}(\mathbf{\tau })〉=-{\phi }_s\mathit{I}$ and $\mathbf{\alpha }=\mathit{I}$ and $\beta =\gamma =\mathbf{\tau }=0$, along with ${〈{\rho }_s^0〉}^s={\rho }_{s0}$. Hence the macroscopic governing Eq. (3.44) reduces to

$$\{\begin{array}{c}{\nabla }_{\mathit{x}}·{\mathbf{\sigma }}_T^0=0\\ {\mathbf{\sigma }}_T^0=-p_b\mathit{I}+{\mathit{C}}_s{\mathcal{E}}_{\mathit{x}}\left({\mathit{u}}^0\right)-{\mathbf{\Pi }}^0\\ {\phi }_s={\phi }_{s_0}\exp (-{\nabla }_{\mathit{x}}·{\mathit{u}}^0)\end{array}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\mathrm{\hspace{1em}}{\Omega }_M$$ (4.1)

whereas the elastic problem (Eq. (3.48)) for ${\mathit{u}}_{\pi }^1$ becomes an incompressibility elasticity problem

$$\{\begin{array}{c}{\nabla }_{\mathit{y}}·{\mathbf{\sigma }}_{\Pi }=0\\ {\mathbf{\sigma }}_{\Pi }:=-{\stackrel{\wedge }{p}}_s\mathit{I}+2{\mu }_s{\mathcal{E}}_{\mathit{y}}\left({\mathit{u}}_{\pi }^1\right)\\ {\nabla }_{\mathit{y}}·{\mathit{u}}_{\pi }^1=0& \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{in}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\mathbf{Y}}_s\\ {\mathbf{\sigma }}_{\Pi }\mathbf{n}=-{\mathbf{\Pi }}_d^0& \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{on}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\partial {\mathbf{Y}}_{\mathit{fs}}\end{array}$$ (4.2)

with ${\mathbf{\Pi }}_S^0=-〈{\mathbf{\sigma }}_{\Pi }〉$ and ${\stackrel{\wedge }{p}}_s$ denoting a nonconstitutive pressure in the solid particles.

4.2. Stratified Microstructure.

Hereafter we consider a particular form of microstructure composed of parallel particles of face-to-face contact subject to the traction induced by the disjoining stress and counterbalanced by the elastic force ${\mathit{c}}_s{\mathcal{E}}_{\mathit{x}}({\mathit{u}}^0)$, which acts to mitigate swelling in a parallel plane where the solid matrix is connected. In Fig. 2 we depict a parallel particle arrangement with each particle of thickness $\delta$ is separated by each other by a distance $2\mathit{H}$. In this idealized portrait $(x,y)$ denotes the pair of one-dimensional macroscopic and microscopic coordinates in the direction orthogonal to the particle surface, respectively, and the components of (${\mathbf{\Pi }}^0$, ${\mathbf{\Pi }}_d^0$, ${\mathbf{\Pi }}_S^0$, ${\mathbf{\tau }}_M^0$, $E$, ${\mathit{u}}^0$, ${\mathit{u}}^1$, ${\mathit{C}}_s$) normal to the particle are denoted without bold face. Let (${\sigma }_y^0$) be the component of the total stress and $E_s$ is the Young's modulus in the $y$-direction. The effective equations in Eq. (4.1) reduce to the following one-dimensional system

$$\{\begin{array}{c}\frac{d{\sigma }_y^0}{\mathit{dx}}=0\\ {\sigma }_y^0=-p_b+E_s\frac{{\mathit{du}}^0}{\mathit{dx}}-{\Pi }^0\\ {\phi }_s={\phi }_{s_0}\exp (-\frac{{\mathit{du}}^0}{\mathit{dx}})\end{array}$$ (4.3)

where the one-dimensional problems for $({\Pi }^0,{\Pi }_S^0)$ reduce to

$$\{\begin{array}{c}{\Pi }^0=〈{\Pi }_d^0〉+{\phi }_s{\Pi }_S^0\\ {\Pi }_S^0=-{〈{\sigma }_{\Pi }〉}^s\end{array}$$ (4.4)

with

$$\begin{array}{c}{\mathbf{\tau }}_M^0=\frac{{\tilde{\epsilon }\tilde{\epsilon }}_0}{2}{\left(\frac{d{\Psi }^0}{\mathit{dy}}\right)}^2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\Pi }_d^0={\pi }^0-{\mathbf{\tau }}_M^0\end{array}$$ (4.5)

Fig. 2.

Stratified microstructure of parallel particles

The local electric potential distribution ${\Psi }^0={\Psi }^0(y)$ is governed by the one-dimensional version of the Poisson–Boltzmann problem

$$\{\begin{array}{cc}\frac{d^2{\Psi }^0}{{\mathit{dy}}^2}=\frac{2{\mathit{FC}}_b}{{\tilde{\epsilon }\tilde{\epsilon }}_0}\sinh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)& \mathrm{for}\hspace{1em}\hspace{1em}y\in (0,H)\\ \frac{d{\Psi }^0}{\mathit{dy}}=0& \mathrm{at}\hspace{1em}\hspace{1em}y=0\\ \frac{d{\Psi }^0}{\mathit{dy}}=\frac{{\sigma }^0}{{\tilde{\epsilon }\tilde{\epsilon }}_0}& \mathrm{at}\hspace{1em}\hspace{1em}y=H\\ {\sigma }^0=-\frac{F\mathrm{\hspace{1em}}\mathrm{CEC}\hspace{1em}{K}_1}{A_s[K_1+(C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}})\exp (-F{\zeta }^0/\mathit{RT})]}\end{array}$$ (4.6)

and the local one-dimensional incompressible elasticity problem (Eq. (4.2)) for the pair $({\mathit{u}}_{\pi }^1,{\sigma }_{\Pi })$ reduces to

$$\{\begin{array}{c}\frac{d{\sigma }_{\Pi }}{\mathit{dy}}=0\\ {\sigma }_{\Pi }=-{\tilde{p}}_s+{\mu }_s\frac{{\mathit{du}}_{\pi }^1}{\mathit{dy}}\\ \frac{{\mathit{du}}_{\pi }^1}{\mathit{dy}}=0& \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{for}\hspace{1em}y\in [H,H+\delta /2]\\ {\sigma }_{\Pi }=-{\Pi }_d^0& \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{at}\hspace{1em}y=H\end{array}$$ (4.7)

In the stratified arrangement, the reduced representation for the swelling pressure is obtained by multiplying the first term in the Poisson–Boltzmann problem (Eq. (4.6)) by $d{\Psi }^0/\mathit{dy}$ and integrating from $y=0$ to an arbitrary point “$y$.” Using the boundary conditions in the second term in Eq. (4.6), the normal component of the Maxwell tensor in the first term in Eq. (4.5) is given by

$${\tau }_M^0=2{\mathit{FC}}_b{\int }_{{\Psi }_{\star }^0}^{{\Psi }^0}\sinh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)d\Psi =2{\mathit{RTC}}_b[\cosh \left(\frac{F{\Psi }^0}{\mathit{RT}}\right)-\cosh \left(\frac{F{\Psi }_{\star }^0}{\mathit{RT}}\right)]$$

where ${\Psi }_{\star }^0$ denotes the value of ${\Psi }^0$ at the line of symmetry $y=0$.

By inserting the representation (Eq. (4.5)) along with the van't Hoff relation for the osmotic pressure, Eq. (3.10) gives the constant value for the disjoining pressure

$${\Pi }_d^0=2{\mathit{RTC}}_b[\cosh \left(\frac{F{\Psi }_{\star }^0}{\mathit{RT}}\right)-1]$$ (4.8)

In addition, the solution of the incompressible elasticity problem (Eq. (4.7)) with the constant traction boundary condition gives ${\sigma }_{\Pi }=-{\Pi }_d^0$ and furnishes the simplified averaged result

$${\Pi }_S^0={〈{\Pi }_d^0〉}^s={\Pi }_d^0$$ (4.9)

which reproduces the classical conjecture that the swelling pressure is nothing but the disjoining pressure in the stratified arrangement of face-to-face particles.

Finally, combining the first term in Eq. (4.4) and Eq. (4.9) we also obtain for the electrochemical tensor

$${\Pi }^0={\phi }_f{\Pi }_d^0+{\phi }_s{\Pi }_d^0={\Pi }_d^0={\Pi }_S^0$$ (4.10)

It is worth noting that $({\tau }_M^0,{\Pi }_d^0,{\Pi }^0)$ also exhibit a tangential component to the particle surface giving rise to a surface tension (see Ref. [46] for details).

5. Numerical Modeling

In what follows we illustrate the potential of the two–scale model in reconstructing new pH-dependent constitutive laws for the swelling pressure. To this end, one needs to solve the Poisson–Boltzmann problem, numerically supplemented by the nonlinear boundary condition in Eq. (3.47). For simplicity, the problem is presented in dimensionless form. Denoting ${\Psi }_{*}=F{\Psi }^0/\mathit{RT}$ as the dimensionless electric potential and $L_{\mathrm{D}}:=(\varepsilon {\varepsilon }_0\mathit{RT}/2F^2{C}_b){}^{1/2}$ as the Debye length, we rephrase Eq. (3.47) in the form

$$\{\begin{array}{cc}{\mathbf{\Delta }}_{\mathit{y}}{\Psi }_{*}={\left(\frac{1}{L_{\mathrm{D}}}\right)}^2\sinh \left({\Psi }_{*}\right)& \mathrm{in}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\mathbf{Y}}_f\\ {\nabla }_{\mathit{y}}{\Psi }_{*}·\mathbf{n}=\frac{F{\sigma }^0}{\mathit{RT}{\tilde{\epsilon }\tilde{\epsilon }}_0}& \mathrm{on}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\partial {\mathbf{Y}}_{\mathit{fs}}\end{array}$$ (5.1)

We present the variational formulation of Eq. (5.1), followed by a linearization procedure within the Newton–Raphson method and spatial discretization by the Galerkin method. To this end, begin by introducing the appropriate function spaces. Let $L^2({\mathbf{Y}}_f)$ be the space of square integrable scalar-valued functions defined in ${\mathbf{Y}}_f$ equipped with the usual inner products

$$(f,g):={\int }_{Y_f}\mathit{fgd}\mathit{y}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{(f,g)}_{\Gamma }:={\int }_{\partial Y_{\mathit{fs}}}\mathit{fgd}\Gamma$$ (5.2)

Furthermore, let $U:=H^1({\mathbf{Y}}_f)$ be the subspace of $L^2({\mathbf{Y}}_f)$ of functions with derivative $\partial f$ in $L^2({\mathbf{Y}}_f)$. The weak form of the nonlinear Poisson–Boltzmann problem consists of: Find ${\Psi }_{*}\in U$ such that

$$({\nabla }_{\mathit{y}}{\Psi }_{*},{\nabla }_{\mathit{y}}w)+L_D^{-2}(\sinh {\Psi }_{*},w)=f({\Psi }_{*},w),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\forall w\in U$$ (5.3)

with

$$f({\Psi }_{*},w)=\frac{F}{\mathit{RT}{\tilde{\epsilon }\tilde{\epsilon }}_0}{\int }_{\partial Y_{\mathit{fs}}}{\sigma }^0({\Psi }_{*})\mathit{wd}\Gamma$$ (5.4)

Application of the Newton–Raphson method to Eq. (5.3) gives the linearized problem: Given ${\Psi }_{*}^k$, the solution of Eq. (5.3) at a $K\mathrm{th}$-iteration, find the subsequent approximation ${\Psi }_{*}^{k+1}$ satisfying

$$B({\Psi }_{*}^{k+1},w)=F({\Psi }_{*}^k,w),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\forall w\in U\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}k=1,2,...,n$$ (5.5)

where

$$\begin{array}{c}B({\Psi }_{*}^{k+1},w)=({\nabla }_{\mathit{y}}{\Psi }_{*}^{k+1},{\nabla }_{\mathit{y}}w)+L_D^{-2}(\cosh \left({\Psi }_{*}^k\right){\Psi }_{*}^{k+1},w)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-\mathit{AB}{(\frac{{\Psi }_{*}^{k+1}\exp (-{\zeta }_{*}^k)}{{(1+B\exp (-{\zeta }_{*}^k))}^2},w)}_{\Gamma }F({\Psi }_{*}^k,w)=L_D^{-2}(\cosh \left({\Psi }_{*}^k\right)({\Psi }_{*}^k-\sinh ({\Psi }_{*}^k),w)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-A{(\frac{1+B\exp (-{\zeta }_{*}^k)(1-{\Psi }_{*}^k)}{{(1+B\exp (-{\zeta }_{*}^k))}^2},w)}_{\Gamma }\mathrm{with}{\zeta }_{*}:=\frac{F\zeta }{\mathit{RT}};\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}A:=-\frac{F^2\mathit{CTC}}{\mathit{RT}{\tilde{\epsilon }\tilde{\epsilon }}_0{A}_s};\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}B:=\frac{C_{\mathrm{H}{b}^{+}}+K_1{K}_2{C}_{\mathrm{Na}{b}^{+}}}{K_1}\end{array}$$ (5.6)

Denoting $U_h$ as a finite-dimensional subspace of $U$ containing piecewise linear or bilinear polynomials on triangles or quadrilaterals of a partition of ${\mathbf{Y}}_f$, the Galerkin approximation of Eq. (5.5) reads as

$$B({\Psi }_{*h}^{k+1},w_h)=F({\Psi }_{*h}^k,w_h)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\forall w_h\in U_h\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{and}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}k=1,2,...,n$$ (5.7)

6. Macroscopic Constitutive Laws for a Cation Exchange Resin

The two-scale model is now applied to numerically reconstruct the macroscopic constitutive law for the swelling pressure in a stratified arrangement of parallel particles. For the determination of the microscopic parameters ($\mathrm{CEC}$, $K_1$, $K_2$, $A_s$), we consider a particular colloidal system characterized by a cation exchange resin (Purolite C104E (as shown in Fig. 3); see Ref. [48]).

Fig. 3.

Portrait of dry and saturated swollen Purolite C104E cation exchange resin

The microstructure of such an expansive medium is given by a polymeric chain predominantly composed of carboxylic functional groups $(-{\mathit{RCOO}}^{-})$ with $R$ denoting the polyacrilic acid and divinylbenzene groups lying in the resin matrix (see, e.g., Ref. [49]). The particular forms of the cation exchange reaction, Eqs. (2.4) and (2.5) (with $M^{-}={\mathit{RCOO}}^{-}$), reads as

$$\begin{array}{c}\mathit{RCOO}\mathrm{H}\rightleftharpoons {\mathit{RCOO}}^{-}+{\mathrm{H}}^{+}(\mathrm{aq}),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{K}_1:=\frac{C_{\mathrm{H}{0}^{+}}{\overline{C}}_{{\mathit{RCOO}}^{-}}}{{\overline{C}}_{\mathit{RCOO}\mathrm{H}}}{\mathit{RCOO}}^{-}+{\mathrm{Na}}^{+}(\mathrm{aq})\rightleftharpoons \mathit{RCOO}\mathrm{Na},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{K}_2:=\frac{{\overline{C}}_{\mathit{RCOO}\mathrm{Na}}}{C_{\mathrm{Na}{0}^{+}}{\overline{C}}_{{\mathit{RCOO}}^{-}}}\end{array}$$

In Ref. [50] Ponce reported the results obtained for the cation exchange capacity measured by batch potentiometric titrations of the carboxylic resin $C104E$ Purolite and blank solution [51,52]. In a concise description of the experiment, a matrix of Purolite and $\mathrm{NaCl}$ blank solution is gradually mixed with a base ($\mathrm{NaOH}$) and the pH is measured at equilibrium for each concentration of $\mathrm{NaCl}$ and $\mathrm{NaOH}$. The experimental results are depicted in Fig. 4.

Fig. 4.

Batch titration curve of the C104E resin for three NaCl concentrations

Following the procedure described in Helfferich [52], from the potentiometric titrations, the cation exchange capacity can be computed from the maximum value of the milliequivalent of ${\mathrm{H}}^{+}$ released in the titration experiment. For each salinity, such a value is computed from the difference between the milliequivalent of base added in the resin and blank solution (for a fixed pH) in the form

$$\mathrm{CEC}=\max {(\left[{\mathrm{NaOH}}_R\right]-\left[{\mathrm{NaOH}}_B\right])}_{p\mathrm{H}}$$

where the notation $\left[{\mathrm{NaOH}}_R\right]$ and $\left[{\mathrm{NaOH}}_B\right]$ represents the milliequivalents of titrant added during the titration of the resin and the blank solution, respectively.

The data of the titration experiment can also be used to compute the dissociation constants $K_1$ and $K_2$ using the Hyperquad software [53-55]. In Table 1 the values of $(K_1,K_2,\mathrm{CEC})$ are displayed as a function of salinity. For the specific surface area of the resin, we adopt the experimental value $A_s=900{\mathrm{\hspace{1em}}\mathrm{m}}^2/\mathrm{g}$, which is reported in the literature for similar carboxylic resins [56-58].

Table 1.

Equilibrium parameters obtained for the Purolite C104E ion exchange resin

[NaCl]	K 1 (M)	K 2 (${\mathrm{M}}^{-1}$)	CEC (m Eq/g)
1	9.$8084\times {10}^{-6}$	0.6987	4.712
0.1	6.$4669\times {10}^{-6}$	5.1963	4.516
0.01	6.$7390\times {10}^{-7}$	0.8482	4.410

The previously discussed experimental values of the microscopic parameters complete the microscopic information for the numerical upscaling procedure. By exploring the numerical simulations of the Poisson–Boltzmann problem at the pore-scale and averaging to the macroscale, we obtain the constitutive law for the swelling pressure in terms of the pH, salinity, and particle separation H. Since the macroscopic response is strongly dependent on the local profiles of the EDL potential and electric field, in what follows we begin by depicting the local behavior of these quantities parametrized by the pH and salinity. The simulations were carried out using linear elements. In Fig. 5, we display the local profile of ${\Psi }_{*}$ along with the dimensionless electric field $E_{*}=\mathit{Hd}{\Psi }_{*}/\mathit{dx}$ for $C_b={10}^{-3}\mathrm{M}$ in the acid regime of $p\mathrm{H}=3$. The magnitude of ${\Psi }_{*}$ is higher next to the particle $(y=0)$ and attains a minimum in the middle of the interlayer spacing where the electric field vanishes.

Fig. 5.

Electric potential and electric field distributions (${\mathit{E}}_{*}={H\nabla }_{\mathit{y}}{\mathbf{\Psi }}_{*}$) for NaCl concentration 0.001 M and pH = 3

The increase in pH tends to amplify the surface charge in the fourth term in Eq. (4.6), enforcing more strongly the Neumann condition for the Poisson–Boltzmann problem, leading to an increase in the magnitude of the EDL potential (see Fig. 6). In Figs. 7 and 8, we depict the dependence of the surface charge with pH for two different salinities and particle separation, respectively. One may clearly observe a high pH-dependency in the acid range $(2\le p\mathrm{H}\le 6)$. Conversely, in the alkaline range $p\mathrm{H}>6$, since the concentration of ${\mathrm{H}}^{+}$ is low, the constitutive law of the surface charge $\sigma$ (the fourth term in Eq. 4.6) is dominated by the term involving the ${\mathrm{Na}}^{+}$ concentration leading to a strong dependence of $\sigma$ with salinity. More precisely, the decrease in salinity and particle separation tends to amplify the magnitude of the surface charge, as we may observe in Figs. 7 and 8.

Fig. 6.

Electric potential and electric field distributions (${\mathit{E}}_{*}={H\nabla }_{\mathit{y}}{\mathbf{\Psi }}_{*}$) for NaCl concentration 0.001 M and pH = 11

Fig. 7.

Charge density as a function of pH for H = 1 nm and two salinities

Fig. 8.

Charge density as a function of pH for NaCl concentration 0.1 M and two thicknesses

Similar behavior is observed for the $\zeta$-potential (see Figs. 9 and 10) since the increase in $\sigma$ with pH more strongly enforces the Neumann condition for the Poisson–Boltzmann equation, also implying an amplification of the magnitude of $\Psi |{}_{y=H}$.

Fig. 9.

Zeta potential as a function of pH for H = 1 nm and two salinities

Fig. 10.

Zeta potential as a function of pH for NaCl concentration 0.1 M and two thicknesses

Finally, in Figs. 11 and 12 we portrait the dependency of the swelling pressure with the pH for two values of salinities and particle separation. From Eq. (4.8) the swelling pressure (equal to ${\Pi }_d^0$ in the stratified arrangement) is mainly dictated by the magnitude of ${\Psi }_{*}$, which has a similar dependence of the $\zeta$-potential with salinity, pH, and $\mathrm{H}$. Thus, ${\Pi }_S^0$ exhibits high sensitivity and grows with the pH in the acid range (conversely, a weak dependence in the alkaline range) and is strongly influenced by the $\mathrm{NaCl}$ concentration for high pH. The computational simulations presented herein generalize the EDL results by incorporating the additional dependence of the disjoining pressure with pH.

Fig. 11.

Swelling pressure as a function of pH for H = 1 nm and two salinities

Fig. 12.

Swelling pressure as a function of pH for H = 4 nm and two salinities

7. Applications to a pH-Sensitive Free Swelling Experiment

Finally, the two-scale model is applied to numerically simulate the free swelling experiment of the cation exchange resin (Purolite C104E [48]). In the one-dimensional setting, the macroscopic constraint of the absence of overburden enforces equality between the total stress and the surrounding bulk pressure (${\sigma }_y^0=-p_b$). Consequently, the only counterstress to the disjoining pressure acting to mitigate swelling is Terzaghi's contact stress. Thus, setting ${\sigma }_y^0=-p_b$ in the system (Eq. (4.3)) and using the equality (Eq. ((4.10)) between ${\Pi }^0$ and ${\Pi }_S^0$ gives the constraint

$${\Pi }_S^0=E_s\frac{{\mathit{du}}^0}{\mathit{dx}}=-E_s\mathit{ln}\left(\frac{1-{\phi }_f}{1-{\phi }_{f0}}\right)$$ (7.1)

where ${\phi }_f=H/(H+\delta /2)$ in the stratified arrangement. The numerical solution of the free swelling problem includes the additional unknown H and thus requires an iterative procedure described below:

• Given the triplet $(C_{\mathrm{H}b},C_{\mathrm{Na}b},p_b)$ in the bulk fluid, choose the particle separation $H^k$ at the $k\mathrm{th}$-iteration.

• Given $H^k$, solve the discrete Poisson–Boltzmann problem supplemented by the nonlinear Neumann condition to compute the electric potential ${\Psi }_{*}^k$.

• Compute the swelling pressure ${\Pi }_S^{0k}$ at the $k\mathrm{th}$-iteration using Eqs. (4.8) and (4.9).

• Enforce local equilibrium by imposing the equality between the ${\Pi }_S^{0k}$ and the elastic restoring force (Eq. (7.1)).

• Given ${\phi }^k=H^k/(H^k+\delta /2)$, update the porosity ${\phi }^{k+1}$ (or particle distance $H^{k+1}$) using the mass balance of the solid phase (Eq. (7.1)).

• Restart the algorithm with the new $H^{k+1}$ and repeat the loop until convergence is reached for a given tolerance.

Our numerical results are expressed in terms of the swelling index I which, in the stratified arrangement, is given by

$$I:=\frac{m_f}{m_s}=\left(\frac{\mathrm{H}}{\delta /2}\right)\left(\frac{{\rho }_f}{{\rho }_s}\right)$$ (7.2)

where $(m_i,{\rho }_i)$ ($i=f,s$) denote the mass and density of fluid and solid, respectively. In Table 2, we present the values used for these parameters in the free-swelling experiment. It is worth noting that under the local incompressible assumption I is a linear function of H.

Table 2.

Parameters adopted in the free-swelling experiment

Water density	${\rho }_f$	$1000{\mathrm{\hspace{1em}}\mathrm{Kg}/\mathrm{m}}^3$
Solid density	${\rho }_s$	$1180{\mathrm{\hspace{1em}}\mathrm{Kg}/\mathrm{m}}^3$
Surface area	$A_s$	$900{\mathrm{\hspace{1em}}\mathrm{m}}^2/\mathrm{g}$
Young's modulus	$E_s$	$3.2\mathrm{\hspace{1em}}\mathrm{MPa}$
Particle thickness	$\delta$	$2\times {10}^{-9}\mathrm{m}$
Reference porosity	${\phi }_{f0}$	$0.45$

The plots in Fig. 13 show the swelling index I as a function of pH for three different salinities and compare them with the experimental results reported in Ref. [50]. In a similar fashion, we observe a high sensitivity of I with a pH in the acid range $2\le p\mathrm{H}\le 6$, which substantially decreases in the basic regime $p\mathrm{H}\ge 6$ where the swelling index reaches a plateau. Thus, one may observe that I inherits the similar behavior of the swelling pressure and surface charge, mainly dictated by two somewhat different patterns inherent to the acid and alkaline ranges.

Fig. 13.

Swelling index in the free swelling experiment as a function of pH for different salinities

8. Conclusion

In this work, we developed a two-scale equilibrium model for pH-sensitive expansive porous media. Within the framework of the homogenization procedure based on formal asymptotic expansions applied to upscale the pore-scale electrochemical model based on the elasticity problem for the macromolecules coupled with the Poisson–Boltzmann problem for the electrolyte solution supplemented by a nonlinear surface charge ruled by pH-dependent ion exchange reactions, an effective constitutive law for the swelling pressure was rigorously derived. The effective medium behavior constructed herein brought new physics to our knowledge on swelling systems since it incorporated the additional dependence of the expansion of the medium on the pH of the external bulk solution, which was inherited from the local behavior of the electrochemical stress.

The model was applied to numerically simulate a free-swelling experiment. Numerical results obtained by the finite element method showed good agreement with experimental data, illustrating a strong dependence of swelling on the pH in the acid region of low pH and a more pronounced sensitivity on salinity in the alkaline range of high pH.

Further work is in progress to extend the model to double porosity expansive media.