Abstract
We propose a new theory describing how the macroscopic Donnan equilibrium potential can be derived from the microscale by a scale transition analysis. Knowledge of the location and magnitude of the charge density, together with the morphology of the pore space allows one to calculate the Donnan potential, characterizing ion exclusion in charged porous materials. Use of the electrochemical potential together with Gauss’ electrostatic theorem allows the computation of the ion and voltage distribution at the microscale. On the other hand, commonly used macroscopic counterparts of these equations allow the estimation of the Donnan potential and ion concentration on the macroscale. However, the classical macroscopic equations describing phase equilibrium do not account for the non-homogeneous distribution of ions and voltage at the microscale, leading to inconsistencies in determining the Donnan potential (at the macroscale). A new generalized macroscopic equilibrium equation is derived by means of volume averaging of the microscale electrochemical potential. These equations show that the macroscopic voltage is linked to so-called “effective ion concentrations”, which for ideal solutions are related to logarithmic volume averages of the ion concentration at the microscale. The effective ion concentrations must be linked to an effective fixed charge concentration by means of a generalized Poisson equation in order to deliver the correct Donnan potential. The theory is verified analytically and numerically for the case of two monovalent electrolytic solutions separated by a charged porous material. For the numerical analysis a hierarchical modeling approach is employed using a one-dimensional (1D)macroscale model and a two-dimensional (2D)microscale model. The influence of various parameters such as surface charge density and ion concentration on the Donnan potential are investigated.
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Abbreviations
- c i :
-
Concentration of ions in pore solution, mol/m3
- c i1 :
-
Concentration of ions in pure solution (phase 1), mol/m3
- c i2 :
-
Concentration of ions in porous material (phase 2), mol/m3
- d :
-
Characteristic length of heterogeneities, m
- \({\ell}\) :
-
Characteristic length of RVE, m
- D f :
-
Electric displacement of fluid, C/m2
- E Don :
-
Donnan potential, V
- EDon,+, EDon,-:
-
Donnan potential computed from cations and anions, V
- F :
-
Faraday constant, C/mol
- L :
-
Characteristic length of structure, m
- n f :
-
Unit normal vector
- R :
-
Universal gas constant, J/(K mol)
- S sf :
-
Area of solid-fluid interface within RVE, m2
- T :
-
Absolute temperature, K
- x :
-
Macroscopic spatial coordinates, m
- z :
-
Microscopic spatial coordinates, m
- z i :
-
Valence of ion i
- α:
-
Index for α-phase (i.e., solid or liquid)
- β:
-
Index for β-material (i.e., solution or charged porous material)
- ε:
-
Permittivity of the medium, C2/(J m)
- \(\varepsilon_{\rm eff\beta}\) :
-
Effective permittivity of β-material, C2/(J m)
- ε f :
-
Relative permittivity of fluid
- ε w :
-
Relative permittivity of water
- ε0 :
-
Permittivity of free space, C2/(J m)
- μiβ :
-
Electrochemical potential, J/mol
- ρβ :
-
Volume charge density in fluid phase of β-material, C/m3
- σβ :
-
Surface charge density on solid–fluid interface of β-material, C/m2
- \(\phi_{\alpha}\) :
-
Volume fraction of α-phase
- χ0, χα :
-
Indicator functions
- ψβ :
-
Microscopic voltage, V
- ω:
-
Sign of intrinsic fixed charge concentration
- Ω(0), Ω(x):
-
Domain of the RVE at x = 0 and x
- Ωα :
-
Domain occupied by the α-phase of the RVE
- ∂Ω sf :
-
Solid-fluid interface within the RVE
- \(\mathcal{V}, \mathcal{V}_{\alpha}\) :
-
Total volume and volume of α-phase of RVE, m3
- \({\overline{e_{\alpha}}^{\alpha}({\bf x},t)}\) :
-
Intrinsic phase average of e
- \({\overline{e_{\alpha}}({\bf x},t)}\) :
-
Apparent phase average of e
- \({\overline{c_{i\beta}}^{f}}\) :
-
Intrinsic actual concentration, mol/m3
- \({\overline{\hat{c}_{i\beta}}^{f}}\) :
-
Intrinsic effective concentration, mol/m3
- \({\overline{X_{\beta}}^{f}}\) :
-
Intrinsic fixed charge concentration, mol/m3
- \({\overline{\hat{X}_{\beta}}^{f}}\) :
-
Intrinsic effective fixed charge concentration, mol/m3
- \({\overline{\psi_{\beta}}^{f}}\) :
-
Intrinsic voltage, V
- \({\Delta \overline{\psi}^{f}}\) :
-
Difference in intrinsic voltage, V
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Pivonka, P., Smith, D. & Gardiner, B. Investigation of Donnan equilibrium in charged porous materials—a scale transition analysis. Transp Porous Med 69, 215–237 (2007). https://doi.org/10.1007/s11242-006-9071-6
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DOI: https://doi.org/10.1007/s11242-006-9071-6