Membrane Potential in Multi-Ionic Mixtures Y. Lanteri, † A. Szymczyk,* ,‡,§ and P. Fievet † Institut UTINAM, UMR CNRS 6213, UniVersite´ de Franche-Comte´, 16 route de Gray, Besanc¸on Cedex 25030, France, Sciences Chimiques de Rennes, UMR 6226 CNRS - UniVersite´ de Rennes 1 - ENSCR, 263 AVenue du Ge´ne´ral Leclerc, Baˆtiment 10 A, CS 74205, 35042 Rennes Cedex, France, and UniVersité Européenne de Bretagne, 12, AVenue Jean JanVier, 3500 Rennes, France ReceiVed: February 6, 2009; ReVised Manuscript ReceiVed: April 30, 2009 The membrane potential arising through charged porous membranes separating electrolyte mixtures at identical hydrostatic pressures but different concentrations was investigated theoretically by means of the steric, electric, and dielectric exclusion (SEDE) model. Transport phenomena were described through the Nernst-Planck formalism, while ion partitioning at the membrane/solution interfaces was accounted for by means of modified Donnan equations including steric and dielectric effects. The high concentration limit of the membrane potential depends on the mixture composition and the pore size as well. A specific feature of membrane potential in multi-ionic systems is the dependence at high concentration on the effective dielectric constant of the solution confined inside pores. Indeed, the effective dielectric constant inside pores does not affect the high concentration limit of the membrane potential in the case of single salt solutions. The low concentration limit of the membrane potential is independent of the mixture composition, the effective dielectric constant inside pores, and the pore radius, but it is ruled by counterions with the highest charge number. The membrane potential measured at high salt concentration with single salt solutions and electrolyte mixtures could be used to determine the pore size and the effective dielectric constant inside pores, respectively. This may constitute an alternative way for membrane characterization with the advantage of avoiding the need for additional rejection rate measurements. 1. Introduction Investigating ion transport induced by a concentration gradient has proved to be an efficient way to characterize the surface electrical properties of ion-exchange 1-7 and porous membranes like microfiltration, 8,9 ultrafiltration, 10-12 and nanofiltration 13-16 membranes. Theoretical studies on the membrane potential phenomenon have been performed using either the Teorell-Meyer-Sievers (TMS) model 17,18 or the space charge (SC) model. 19 Wester- mann-Clark and Anderson compared membrane potentials predicted by these two approaches in the case of 1-1 electro- lytes. 20 Their findings showed that the TMS model and the much more complex SC model led to similar results for sufficiently low surface charge densities. Shang et al. showed that TMS and SC models coincide with each other for slightly charged membranes, separating symmetric or asymmetric electrolytes, provided the pore radius is less than 5.0 nm. 21 Otherwise, the TMS model (which neglects radial variations of both electrical potential and ion concentrations) was shown to overestimate the membrane potential. Steric hindrance is known to play a role in the separation of solutes by nanofiltration membranes and is taken into account in modern transport models. 22,23 Cervera et al. 24 investigated the membrane potential phenomenon with binary electrolytes by including ion size effects in the SC model. They concluded that ion size effects lead to an increase in the membrane potential. The same conclusion was drawn by Lanteri et al. who investigated the membrane potential phenomenon by means of the steric, electric, and dielectric exclusion (SEDE) model, which is an improved version of the TMS model describing the exclusion mechanism of charged solutes as the result of ion size effects, the Donnan exclusion, and dielectric effects. 25 For charged membranes separating two solutions of the same single electrolyte, it was shown that both steric and dielectric effects increase the membrane potential at a given concentration, but the high concentration limit of the membrane potential is affected only by steric effects. All the above mentioned works were performed by consider- ing membranes separating single solutions of binary electrolytes. In the present work, we extend our previous analysis based on the SEDE model 25 (i.e., including both steric and dielectric effects) in the case of the membrane potential arising through charged membranes separating multi-ionic mixtures. Special attention is paid to the low and high concentration limits of the membrane potential in ternary mixtures made of a symmetric electrolyte (KCl) and an asymmetric one (CaCl 2 ) with a common anion. 2. Theoretical Background The electrical potential difference arising through a membrane that separates two solutions of the same electrolyte(s) at the same temperature and hydrostatic pressure but at different concentrations is termed membrane potential. 1 The current work deals with the theoretical analysis of the membrane potential phenomenon occurring when a charged membrane separates multi-ionic mixtures. Within the scope of the TMS model and related models like the SEDE model, the membrane potential (∆Ψ m ) can be split in two components, namely the difference between the Donnan potentials at each interface (∆Ψ D ∆x and ∆Ψ D 0 : superscripts ∆x * Corresponding author: anthony.szymczyk@univ-rennes1.fr. † Universite´ de Franche-Comte´. ‡ Universite´ de Rennes 1. § Universite´ Europe´enne de Bretagne. J. Phys. Chem. B 2009, 113, 9197–9204 9197 10.1021/jp901110c CCC: $40.75  2009 American Chemical Society Published on Web 06/11/2009