caa a22 4500 463162951 CHVBK 20180406164753.0 cr unu---uuuuu 170326e20070901xx s 000 0 eng 10.1007/s11242-006-9071-6 doi (NATIONALLICENCE)springer-10.1007/s11242-006-9071-6 Investigation of Donnan equilibrium in charged porous materials—a scale transition analysis [Elektronische Daten] [Peter Pivonka, David Smith, Bruce Gardiner] 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. Springer Science+Business Media, Inc., 2006 Donnan potential nationallicence Anion exclusion nationallicence Electrochemical potential nationallicence Poisson equation nationallicence Charged porous materials nationallicence Volume averaging nationallicence c i : Concentration of ions in pore solution, mol/m3 nationallicence c i 1 : Concentration of ions in pure solution (phase 1), mol/m3 nationallicence c i 2 : Concentration of ions in porous material (phase 2), mol/m3 nationallicence d : Characteristic length of heterogeneities, m nationallicence $${\ell}$$ : Characteristic length of RVE, m nationallicence D f : Electric displacement of fluid, C/m2 nationallicence E Don : Donnan potential, V nationallicence E Don,+, E Don,- : Donnan potential computed from cations and anions, V nationallicence F : Faraday constant, C/mol nationallicence L : Characteristic length of structure, m nationallicence n f : Unit normal vector nationallicence R : Universal gas constant, J/(K mol) nationallicence S sf : Area of solid-fluid interface within RVE, m2 nationallicence T : Absolute temperature, K nationallicence x : Macroscopic spatial coordinates, m nationallicence z : Microscopic spatial coordinates, m nationallicence z i : Valence of ion i nationallicence α : Index for α-phase (i.e., solid or liquid) nationallicence β : Index for β-material (i.e., solution or charged porous material) nationallicence ε : Permittivity of the medium, C2/(J m) nationallicence $$\varepsilon_{\rm eff\beta}$$ : Effective permittivity of β-material, C2/(J m) nationallicence ε f : Relative permittivity of fluid nationallicence ε w : Relative permittivity of water nationallicence ε0 : Permittivity of free space, C2/(J m) nationallicence μ i β : Electrochemical potential, J/mol nationallicence ρβ : Volume charge density in fluid phase of β-material, C/m3 nationallicence σβ : Surface charge density on solid-fluid interface of β-material, C/m2 nationallicence $$\phi_{\alpha}$$ : Volume fraction of α-phase nationallicence χ0, χα : Indicator functions nationallicence ψβ : Microscopic voltage, V nationallicence ω : Sign of intrinsic fixed charge concentration nationallicence Ω( 0 ), Ω( x ) : Domain of the RVE at x=0 and x nationallicence Ωα : Domain occupied by the α-phase of the RVE nationallicence ∂Ω sf : Solid-fluid interface within the RVE nationallicence $$\mathcal{V}, \mathcal{V}_{\alpha}$$ : Total volume and volume of α-phase of RVE, m3 nationallicence $${\overline{e_{\alpha}}^{\alpha}({\bf x},t)}$$ : Intrinsic phase average of e nationallicence $${\overline{e_{\alpha}}({\bf x},t)}$$ : Apparent phase average of e nationallicence $${\overline{c_{i\beta}}^{f}}$$ : Intrinsic actual concentration, mol/m3 nationallicence $${\overline{\hat{c}_{i\beta}}^{f}}$$ : Intrinsic effective concentration, mol/m3 nationallicence $${\overline{X_{\beta}}^{f}}$$ : Intrinsic fixed charge concentration, mol/m3 nationallicence $${\overline{\hat{X}_{\beta}}^{f}}$$ : Intrinsic effective fixed charge concentration, mol/m3 nationallicence $${\overline{\psi_{\beta}}^{f}}$$ : Intrinsic voltage, V nationallicence $${\Delta \overline{\psi}^{f}}$$ : Difference in intrinsic voltage, V nationallicence Pivonka Peter Department of Civil and Environmental Engineering, The University of Melbourne, 3010, Melbourne, VIC, Australia aut Smith David Department of Civil and Environmental Engineering, The University of Melbourne, 3010, Melbourne, VIC, Australia aut Gardiner Bruce Department of Civil and Environmental Engineering, The University of Melbourne, 3010, Melbourne, VIC, Australia aut Transport in Porous Media Kluwer Academic Publishers 69/2(2007-09-01), 215-237 0169-3913 69:2<215 2007 69 11242 https://doi.org/10.1007/s11242-006-9071-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11242-006-9071-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Pivonka Peter Department of Civil and Environmental Engineering, The University of Melbourne, 3010, Melbourne, VIC, Australia aut NATIONALLICENCE 700 1- Smith David Department of Civil and Environmental Engineering, The University of Melbourne, 3010, Melbourne, VIC, Australia aut NATIONALLICENCE 700 1- Gardiner Bruce Department of Civil and Environmental Engineering, The University of Melbourne, 3010, Melbourne, VIC, Australia aut NATIONALLICENCE 773 0- Transport in Porous Media Kluwer Academic Publishers 69/2(2007-09-01), 215-237 0169-3913 69:2<215 2007 69 11242 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer