12 The transport selectivity of different cations through cation exchange membranes (CEMs) could be 13 estimated with the partition coefficient (𝐾!") and the cation mobility ratio in the membrane (𝑢#"𝑢#!⁄), 14 which in turn can be related to corresponding membrane conductivity and dimensional swelling 15 degree data [Journal of Membrane Science, 2020, 597, 117645]. This method has been validated in 16 two hydrocarbon-based CEMs, and the obtained K+/Na+ selectivity equals to the one obtained with 17 conventional electrodialysis (ED) method. However, the K+/Na+ selectivity of perfluorosulfonic acid 18 (PFSA) membranes, and the bi-/monovalent cation (Mg2+/Na+) selectivity of all three types of CEMs 19 estimated with this ionic conductivity experimental approach deviate noticeably from corresponding 20 values obtained with ED. In this work, it is proved that this deviation is mostly due to the simplification 21 of cation activity coefficients in the membrane. Here, the cation activity coefficients in three types of 22 CEMs are calculated according to Manning`s counter-ion condensation model. In this model, the 23 Manning parameter (𝜉) characterizing the dimensionless linear charge density is determined by the 24 * Corresponding author at: School of Chemical Engineering, Sichuan University, 610065 Chengdu, China E-mail address: wangxl@scu.edu.cn (X.L. Wang) * Corresponding author at: Chemical Process Engineering AVT.CVT., RWTH Aachen University, Forckenbeckstraße 51, 52074 Aachen, Germany. E-mail address: Manuscripts.cvt@avt.rwth-aachen.de (M. Wessling) 2 average distance between two adjacent fixed sulfonate groups (𝑏) and the permittivity of hydrated 1 membranes (𝜀). In hydrocarbon-based CEMs, the average distance between fixed sulfonate groups 2 can be estimated by assuming homogeneous distribution of the fixed groups, while in PFSA 3 membranes three representative structure models are employed to estimate this average distance. 4 After accounting for the cation activity coefficients in the membrane, the cation transport selectivity 5 obtained with the ionic conductivity experimental approach agrees well with the selectivity obtained 6 with the ED method. This work shows the importance of cation activity coefficients in the membrane 7 phase in interpreting the membrane transport properties, and complements the proposed 8 conductivity approach to characterize the counter-ion selectivity of ion exchange membranes. 9 Keywords: cation exchange membrane, activity coefficient, selectivity, counter-ion condensation, 10 structure models of perfluorosulfonic acid membranes 11 12 3 1. Introduction 1 The transport characteristics of ions in ionomers and/or polyelectrolyte systems is interesting 2 and also important [1]. Such systems can be seen in the biological field like the delicate 3 sodium/potassium ion pump in cell membranes [2], and also can be witnessed in the artificial world 4 including the synthetic ion exchange membranes (IEMs) [3, 4]. Though the ability of IEMs to control 5 ion transport is not as perfect as their biological archetype in the cell membrane of living organisms, 6 IEMs can still achieve ion transport selectivity at two levels [5]: 1. facilitate the transport of 7 counter-ions while block the transport of co-ions (that bear the same sign of charge as the 8 membrane fixed charges) [6]; 2. achieve partial fractionation between counter-ions of different 9 valences [7, 8]. These two properties have already popularized IEMs in diverse applications for the 10 water – energy nexus [9-11], for example redox flow batteries for the renewable energy storage. 11 Therefore, further study of ion transport characteristics in such IEMs is definitely required to help 12 reach the realm of perfect ion transport selectivity in synthetic membranes. 13 Study of ion transport in IEMs relies on experimental methods to characterize the ion transport 14 selectivity [10, 12]. First of all, these methods shall be reliable, then ideally be convenient and swift. 15 For the counter-/co-ion permselectivity of IEMs, extensive research efforts in the past have 16 suggested the membrane potential method as a routine characterization method [5, 13]. For the 17 selectivity of IEMs towards different counter-ions, a subject not fully studied yet, both electrodialysis 18 (ED) [7, 8, 14] and the bi-ionic potential method [15] can be considered, with the former being the 19 benchmark technique. As summarized in a recent review [10], ED is time-consuming and requires 20 dedicated set-ups, and the bi-ionic potential method is not so popular though it is actually an easy 21 estimation method. Alternatively, in a previous contribution [16] we propose that the membrane 22 conductivity and dimensional swelling degree can predict the counter-ion selectivity of IEMs, by 23 assuming linear dependence of membrane conductivity on the equivalent counter-ion composition in 24 the membrane [17]. This new method will be referred as the ionic conductivity experimental approach 25 in following discussions. The key idea of this method is to decouple the transport selectivity of two 26 counter-ions into two parts: the partition selectivity between the membrane and contiguous aqueous 27 solutions (𝐾!"), and the mobility ratio of counter-ions in the membrane phase (𝑢"#𝑢!#'). The latter part 28 4 can be calculated from the ionic conductivity of membranes at corresponding counter-ion forms. 1 While the former part, the partition selectivity, can also be obtained from the ionic membrane 2 conductivity when the membrane is in equilibrium with a solution of two different types of 3 counter-ions with a known equivalence ratio. Though this new method has been validated in two 4 hydrocarbon-based cation exchange membranes (CEMs) for K+/Na+ transport, it is observed that the 5 K+/Na+ mobility ratio in one perfluorosulfonic acid (PFSA) membrane and the Mg2+/Na+ mobility ratio 6 in all the three types of CEMs are underestimated by this ionic conductivity experimental approach 7 [16]. This underestimation of ion mobility ratio is very likely due to the simplification of cation activity 8 in CEMs, which of course needs to be confirmed. 9 The ion activity, instead of ion concentration, is used to reflect the non-ideal behavior of ions in 10 electrolyte solutions. The deviation of ion activity from ion concentration is quantified through the ion 11 activity coefficient [18]. This is important for interpreting both the thermodynamic (ion partition) and 12 kinetic (ion transport) properties of IEMs [19]. Even though swollen IEMs can be approximated as 13 concentrated polyelectrolyte solutions [20], however, as compared to extensive study and theoretical 14 frameworks for the ion activity coefficient in electrolyte solutions, there is much less research focused 15 on ion activity coefficient in IEMs. Back in the 1950s, Mackie and Meares [21] applied the theory of 16 Katchalsky and Lifson originally developed for polyelectrolyte solutions [22, 23] in one type of CEM, 17 their theoretical results agreed well with experimental ion activity coefficients. However, with a similar 18 approach, the results of Lakshminarayanaiah led to an opposite conclusion on the same type of CEM 19 [24]. Shortly later, Gregor and his co-workers proposed a new model for the ion activity coefficient in 20 IEMs [20], however this model found limited penetration in the field possibly due to its complexity. A 21 significant advancement on this topic comes very recently. Kamcev and Freeman et al. adopted 22 Manning`s counter-ion condensation theory that was also originally for polyelectrolyte solutions in 23 the calculation of ion activity coefficients in IEMs [25-32]. Without adjustable parameters, Manning`s 24 counter-ion condensation theory could predict both the magnitude and variation trend of ion activity 25 coefficients as a function of the salt concentration in external solutions [26]. This approach has been 26 applied in several hydrocarbon-based CEMs and anion exchange membranes (AEMs) for 27 interpreting the ion partition [33], ion (especially co-ion) transport nicely [27, 28, 31, 34, 35]. The 28 validity of this approach in another considerably different, yet important class of CEMs – PFSA 29 5 membranes (for example Nafion) – remains to be checked [36]. There are already a few literature 1 reports that claim a large portion of protons (72 ~ 98%) in Nafion 117 equilibrating with pure water 2 are condensed [37], though the conclusion is at question due to simplification of cation activity 3 coefficient both in the membrane and in the solutions to be 1. A very recent work reports 4 experimental evidence revealing similar condensation of counter-ions in Nafion membranes [38], 5 which makes the application of Manning`s counter-ion condensation theory in PFSA membranes 6 quite interesting. Besides Manning`s counter-ion condensation theory, for IEMs of low water 7 contents (therefore small relative dielectric constants of hydrated membranes), Freger recently 8 proposes a unified model that aims to describe the ion association between mobile ions and fixed 9 charge groups within a charged membrane [39]. This seems an interesting step towards predictive 10 modeling of ion transport in charged membranes, yet some necessary parameters that shall be 11 determined by experiments are not available at the moment. 12 In this work, the objective is to apply the Manning`s counter-ion condensation theory for the 13 calculation of cation activity coefficients in not only hydrocarbon-based CEMs, but also in PFSA 14 membranes. Further, the inclusion of these cation activity coefficients in the proposed ionic 15 conductivity experimental approach for the characterization of counter-ion transport selectivity in 16 CEMs can testify whether the previously observed underestimation of cation mobility ratio in CEMs is 17 due to the simplification of the cation activity. On one hand, this work can complement the proposed 18 ionic conductivity experimental approach as a potentially reliable, swift and convenient approach for 19 the counter-ion transport selectivity characterization. On the other hand, it is attempted in such a first 20 trial to test the feasibility of applying the counter-ion condensation theory in PFSA membranes. It is 21 necessary to note that the Manning`s counter-ion condensation framework is based purely on 22 electrostatic interactions, and there are other types of interactions – hydration, ion association (as 23 called by Freger [39]) or non-electrostatic specific physical interaction (as termed by Weber et al. 24 [40]), and internal pressure in membranes like Nafion [41, 42]. At the present stage, with the 25 Manning`s theoretical framework as a predictive model, it is meaningful to explore the potential of 26 this model for the applications in PFSA membranes. 27 28 6 2. Background 1 2.1. The ionic conductivity experimental approach for the counter-ion selectivity 2 determination 3 The transport selectivity between counter-ions 𝑖 and 𝑗 measured in electrodialysis (ED), 𝑃!", is 4 defined as [5, 7, 8]: 5 𝑃!"=$!"$#"%&!'!&#'#⁄ (1) 6 where 𝑡"# is the transport number of counter-ion 𝑖 in the membrane phase, 𝐶" stands for the 7 molarity of counter-ion 𝑖 (mol m-3) at the membrane surface on the desalination side, and 𝑧" is the 8 valence of the counter-ion. As is generally known, the transport selectivity of counter-ions measured 9 in ED depends on the experimental conditions [10], which include but are not limited to the current 10 density, the equivalence ratio of counter-ions in the solutions [43], the ionic strength of solutions, the 11 mixing conditions (with or without flow) and the temperature. If one aims at characterizing the 12 counter-ion transport selectivity of the membrane itself, it is necessary to choose carefully the 13 experimental conditions to screen out the influence from other factors as much as possible. Though 14 there are different practices in literature, we employ such one: equal equivalence ratio of 15 counter-ions in the starting solutions (both the diluate and the concentrate), a current density well 16 below its limiting value, fast liquid flow through the ED cell, duration that is long enough to determine 17 the counter-ion concentration change in the solutions in ED (for several times of measurements at a 18 regular time interval), but not so long that the counter-ion concentration changes exceed 5% as 19 compared to their starting values (so 𝑐" in Equation 1 can be approximated with the starting values). 20 When the current density is well below the limiting one, it is rationale to approximate the counter-ion 21 concentration at the membrane surface with its concentration in the bulk solution [44]. The 22 counter-ion transport number in the membrane can be expressed with the ratio of ionic fluxes, which 23 can be described by the Nernst-Planck equation [45]. With these considerations, the contribution of 24 ionic diffusion and convection to the ionic fluxes can be neglected, then the counter-ion transport 25 selectivity can be simplified to the following form [16]: 26 7 𝑃!"=)!")!$%)#")#$% ∙+!"+#"=𝐾!"∙+!"+#" (2) 1 𝐾!" is the partition coefficient of counter-ions 𝑖 and 𝑗 between the solution and the membrane 2 phase, 𝑥"# is the equivalent ratio of counter-ion 𝑖 among all counter-ions in the membrane phase, 3 and 𝑥", its equivalent ratio in the bulk solution. 𝑢"# is the mobility of counter-ion 𝑖 in the membrane 4 phase (m2 s-1 V-1), and it is related closely to the membrane conductivity, 𝜅"# (S m-1), as [18]: 5 𝜅"#=𝐹𝑎"#𝑢"#=𝐹𝛾"#𝐶"#𝑢"# (3) 6 𝐹 is the Faraday constant (96485 C mol-1), 𝑎"# is the activity of counter-ion 𝑖 in the membrane, 𝛾"# 7 is the counter-ion activity coefficient in the membrane. In our previous work [16], the activity 8 coefficient was simplified as 1. For the concomitant presence of two different counter-ions, 𝑖 and 𝑗, 9 in the membrane phase, if we assume the membrane conductivity (𝜅",!#) is linearly dependent on the 10 counter-ion activity (a more stringent assumption than the use of concentration as reported in [16]), 11 then: 12 𝜅",!#=𝐹𝑎"#𝑢"# +𝐹𝑎!#𝑢!#= 𝐹𝛾"#𝐶"#𝑢"#+𝐹𝛾!#𝐶!#𝑢!# (4) 13 A parameter, the equivalent concentration of all counter-ions in the membrane, 𝐶# 14 (equivalence∙m-3), is defined to unify the notations for both mono/bivalent counter-ions. It is defined 15 as: 16 𝐶#=|𝑧"|𝐶"#+9𝑧!9𝐶!# (5) 17 𝑧" is the valence of counter-ion 𝑖. Rearrangement of Equation 5 links the counter-ions` equivalent 18 ratios as: 19 |&!|'!"'" + /&#/'#"'"= 𝑥"#+ 𝑥!#=1 (6) 20 The equivalent concentration of all counter-ions, 𝐶#, can also be calculated directly from the 21 membrane parameters including ion exchange capacity (𝐼𝐸𝐶, mmole∙g-1 dry polymer), the dry 22 polymer density (𝜌012, kg m-3), and the dimensional swelling degree (𝑆𝐷) by the following correlation 23 [10]: 24 8 𝐶#=𝐼𝐸𝐶∙𝜌012∙ 3%&'3()*=𝐼𝐸𝐶∙ 𝜌012∙ (44567)8 (7) 1 𝑉 is the volume of membranes (m3), and the subscript denotes the hydration state (dry or wet). In the 2 derivation of Equation 7, there is one another assumption that the membranes swell uniformly in 3 three directions. Though there is proof that PFSA membranes have different dimensional swelling 4 degrees in the normal and tangential directions to the membrane surface [46], we keep this 5 assumption here to simplify the calculation. The combination of Equations 3 - 7 gives (the deduction 6 details are given in the Supporting Information): 7 𝑥"#=9!,#"(4567!,#),<9#"(4567#), 9!"(4567!),< 9#"(4567#), (8) 8 This means that the equivalent ratio of counter-ion 𝑖 in the membrane, 𝑥"#, can be obtained from 9 the membrane conductivity and dimensional swelling degree data, when 𝛾"#𝑢"# is not dependent on 10 the membrane counter-ion composition. This result is exactly the same as when the counter-ions` 11 activity coefficients in the membrane are simplified to 1 (i.e., in Equations 3 and 4, 𝛾"#=𝛾!#=1) [16]. 12 One step further, the partition coefficient in Equation 2 can be expressed as: 13 𝐾!"=)#$)!$∙9!,#"(4567!,#),<9#"(4567#), 9!"(4567!),< 9!,#"(4567!,#), (9) 14 The deductions shown above for the partition coefficient with the simplification of counter-ion 15 activity coefficients (𝛾"#=𝛾!#=1) have been validated experimentally for K+/Na+ in cation exchange 16 membranes (CEMs), and for Mg2+/Na+ in two hydrocarbon-based CEMs, as reported in our previous 17 work [16]. However, the mobility ratios of K+/Na+ or Mg2+/Na+ obtained from the conductivity data are 18 underestimated, when the cation activity coefficients are simplified [16]. Therefore, based on 19 Equation 3, the counter-ion mobility ratio shall be expressed in the following way: 20 +!"+#"= 9!"(4567!),|&!|9#"(4567#),/&#/∙=#"=!" (10) 21 As shown by Equation 2, the transport selectivity of counter-ions 𝑖 and 𝑗 can be expressed as 22 the product of the partition coefficient (𝐾!") and the counter-ion mobility ratio in the membrane phase 23 (𝑢"#𝑢!#'). Because the ionic conductivity experimental approach can already ensure the 24 determination of the partition coefficient reasonably well (via Equation 9), so this study will then focus 25 9 on the counter-ion activity coefficients in the membrane phase, 𝛾"#, to increase the accuracy when 1 determining the counter-ion mobility ratio in the membrane phase via Equation 10. The calculation of 2 counter-ion activity coefficients in the membrane phase is detailed in the following section. 3 4 2.2. Calculation of the counter-ion activity coefficient 5 The counter-ion (gegen-ion) and co-ion activity coefficients in an IEM, and respectively, 6 can be calculated according to Manning`s model [25, 28, 47] as: 7 (11) 8 9 (12) 10 where is the molarity of fixed charges in a hydrated membrane (mol m-3), is the mobile salt 11 concentration in the membrane (mol m-3), and are the gegen-ion and co-ion valences, 12 respectively. and indicate the stoichiometric number of counter-ions and co-ions in one 13 mobile salt, respectively, and . is the so-called Manning parameter that denotes the 14 dimensionless fixed linear charge density on a polymer chain of the membrane [47]. 15 When the amount of imbibed co-ion into an IEM can be neglected (namely ), the 16 counter-ion activity coefficient in the membrane can then be simplified as: 17 gmgcmg2expmmmAAgg sgmgmm m mAg g s Ag gsCCzCzCzC Cz Cnxgn n nxæöéù+ç÷êúç÷=-êúç÷++êúç÷êúç÷ëûèø22expmcAgmcmmAg gszCzCz Cgnn xéùæöêúç÷ç÷êúèø=-êú+êúêúëûmACmsCgzczgncngcvnn=+x0msC» 10 (13) 1 Under such a condition, the counter-ion activity coefficient is only related to the counter-ion valence 2 and the Manning parameter, and the latter is defined as follows [47]: 3 (14) 4 where is the elementary charge (1.60 10-19 C). is the vacuum dielectric constant (8.85 × 5 10-12 F m-1), is the relative dielectric constant (permittivity) of the hydrated membrane. is the 6 Boltzman constant (1.38 × 10-23 J K-1), 𝑇 stands for the absolute temperatue (K). is the 7 average distance between two adjacent fixed charges in a hydrated membrane (m). is the 8 Bjerrum length (m) that characterizes the critical distance at which the thermal energy of a 9 counter-ion is equal to the attractive electrostatic energy between this counter-ion and an interacting 10 fixed charge group. When the distance between a counter-ion and a fixed charge group (eg. on a 11 polymer chain) is smaller than the Bjerrum length, the counter-ion is considered as “condensed” with 12 the fixed charge group. When the distance is larger than the critical value, the counter-ion is 13 regarded as “free”. In Manning`s model, the dimensionless linear charge density is in fact the 14 ratio of Bjerrum length to the average distance between two adjacent fixed charge groups (on a 15 polymer chain). 16 The model has an important idea of the critical linear charge density, , which discriminates 17 the situation whether or not a counter-ion will be condensed with these consecutive fixed charge 18 groups on a polymer chain. The critical linear charge density is correlated to the valence of the 19 counter-ion and corresponding fixed charge groups as [47]: 20 (15) 21 is the valence of the fixed charge groups on polymer chains. For fixed sulfonate groups (𝑧>=−1) 22 and a monovalent cation (eg. Na+), the critical linear charge density is 1; for the same fixed charge 23 11exp2mggzgxæö=-ç÷èø 24BoekTb blxpe e==e´oeekbBl x critx1critgpzzx=pz 11 groups and a bivalent cation (eg. Mg2+), the value is 0.5. Within this framework, counter-ions will be 1 condensed when the Manning parameter is larger than its critical value, and these condensed 2 counter-ions could then neutralize some fixed charges and increase the average distance between 3 two adjacent fixed charge groups, consequently the Manning parameter is reduced until its critical 4 value is reached. Though Manning`s theory was originally developed for polyelectrolyte systems, it 5 has also been proved valid in similar systems like highly hydrated IEMs, since these IEMs can be 6 treated as crosslinked ionomers with high fixed charge concentrations. Kamcev and Freeman et al. 7 have recently extended Manning`s theory to IEMs, and obtained excellent results in interpreting the 8 thermodynamic [25, 29] and transport properties [28, 31] of several types of hydrocarbon-based 9 CEMs. 10 In Equation 14, the permittivity of a hydrated membrane () can either be measured 11 experimentally [48-50], or alternatively be approximated by the following additive rule [25]: 12 (16) 13 𝛷" is the volume fraction of component 𝑖 in a hydrated membrane, is the permittivity of 14 component 𝑖. The subscripts 𝑤 and 𝑝 stand for water and polymer, respectively. The value of 15 is chosen at 78, the permittivity of bulk water at room temperature. In Equation 14, the average 16 distance between two adjacent fixed charge groups in an IEM () can be estimated geometrically, 17 based on known knowledge of the chemical structure and the membrane composition for 18 hydrocarbon-based IEMs [25]. However, for PFSA membranes with significant hydrophobic and 19 hydrophilic nanophase separation upon hydration, structure models are necessary for estimating the 20 parameter . 21 22 3. Characteristics of cation exchange membranes 23 The chemical schemes of three representative types of CEMs are shown in Figure 1. CMX 24 (Astom Corp., Japan) is made from sulfonated polystyrene crosslinked by divinylbenzene (Figure 1, 25 a), and further reinforced with polyvinyl chloride fabric. It is conventionally used as a standard CEM in 26 eww p pee e=F +Fiewebb 12 ED. SPEEK is short for sulfonated poly(ether ether ketone) (Figure 1, b), and it is representative of 1 homogeneous CEMs. PFSA membranes are another important class of CEMs, including Nafion from 2 DuPont and F9120 from Fumatech, and so on. The difference of these PFSA membranes is the 3 density of side chains (m, n values), and the length of side chains (x, y values in Figure 1, c) [46]. The 4 spanning length of a monomer residue (𝑏?) in CMX and SPEEK is also shown in the figure. 5 6 7 Figure 1. Chemical schemes of the three different types of cation exchange polymers that form the 8 corresponding cation exchange membranes (CEMs): a. CMX, b. SPEEK and c. perfluorosulfonic 9 acid (PFSA) polymers including F9120. All the polymers are shown in their proton forms. 10 Table 1 lists the basic physicochemical characteristics of CEMs, including the density of the dry 11 polymer membrane (𝜌!"#), the water contents and the dimensional swelling degree. The data of 12 water contents and swelling degree are based on 10 samples for each condition. The standard 13 deviations of water contents are all less than 3.5%. The numbers of water molecules per fixed 14 sulfonate groups (λ) in different CEMs are listed in the Supporting Information (Table S1). The 15 swelling degree is obtained by measuring the thickness of membranes before and after hydration. 16 There is such a concern that PFSA membranes are anisotropic in swelling [51], which is also 17 reflected by the inconsistency between F9120 swelling degree in the normal direction and water 18 contents if isotropic swelling was assumed (Table S2). In this work, the change of membrane 19 thickness is still chosen as the representative swelling degree because of the convenience in its 20 measurements. Delicate measurements in the future are necessary to clarify the swelling behaviour 21 of PFSA membranes. 22 13 Table 1. The characteristics of cation exchange membranes (CEMs) 1 Membranes 𝜌!"# [g cm-3] Water contents [g H2O · g-1 dry polymer, %] Swelling degree (𝑆𝐷) [%] K+ Na+ Mg2+ K+ Na+ Mg2+ CMX 1.34 18.2 22.0 21.9 14.1 ± 1.0 11.1 ± 2.1 12.0 ± 1.2 SPEEK 1.44 15.4 25.1 22.8 13.8 ± 3.3 19.4 ± 8.4 17.0 ± 10.0 F9120 1.77 7.6 16.9 17.4 19.1 ± 1.3 21.3 ± 1.1 17.2 ± 5.2 2 4. Structure models of perfluorosulfonic acid membranes 3 The average distance between two adjacent fixed sulfonate groups is an important parameter in 4 the Manning`s counter-ion condensation theory, as detailed in Section 2.2. In the case of hydrated 5 PFSA membranes, ion clusters have been observed [46, 52], which indicates that this average 6 distance can`t be estimated based on the assumption that fixed sulfonate groups are 7 homogeneously distributed. To this end, knowledge of the PFSA membrane microstructure is 8 necessary. However, the microstructure of PFSA membranes has long been a subject of hot debate, 9 one possible reason of which might be the adaptive nature of the PFSA membrane structure as a 10 response of environmental stimuli, especially the water contents [51]. 11 In this work, three important and influential structure models have been included for the 12 estimation, and their characteristic features are illustrated in Figure 2. The first one is the famous and 13 generally accepted cluster-network model proposed by Gierke et al. (Figure 2, a) [53]. In this model, 14 spherical inverted micelles (called clusters) distribute with a paracrystalline pattern through the 15 membrane, and these clusters are interconnected by narrow cylindrical channels with a diameter of 16 around 1 nm (which also depends on the hydration level) [52]. Upon further hydration, clusters grow 17 and coalesce. The second model introduced by Schmidt and Chen in 2008 is the first quantitative 18 model obtained by fitting the small angle X-ray scattering (SAXS) data of a hydrated Nafion [54], and 19 has attracted a lot of attention by far. This model proposes cylindrical water nanochannels arranged 20 14 in parallel, with cylinder lengths on the order of some 20 nm [54]. Swelling results in generally the 1 enlargement of the cylinder diameter (Figure 2, b). This model could explain the fast ion transport in 2 these PFSA membranes naturally. Despite its popularity, the parallel cylindrical water channel model 3 is also at question. Kreuer points out that the hydration level of the exact sample whose data have 4 been employed by Schimdt and Chen in their simulation is not correct, and also the simulation is 5 limited to only one sample [51]. With the same approach of direct numerical simulation of the 6 scattering data, Kreuer further developed his semi-quantitative structure model of PFSA membranes. 7 It is found that the microstructure of the water domains in PFSA should be locally flat, without any 8 curvature as proposed in the case of cluster-network and parallel cylindrical water channel models 9 [51]. Because the swelling is observed to be in one direction, so the locally planar water channels 10 increase their spacing distance as a result of swelling (Figure 2, c) [51]. Such planar structures might 11 be with a length on the order of 30 -100 nm, as reflected by the tortuosity of PFSA membranes 12 deduced from SAXS data [51]. 13 14 15 15 Figure 2. Three representative structure models of PFSA membranes and the structure evolution 1 upon swelling with water. (a) The classical cluster-network model proposed by Gierke et al. (modified 2 from Reference [52] with permission), (b) the newly introduced model of parallel cylindrical water 3 channels (simplified from Reference [54] with permission) and (c) the structure model with locally flat 4 water layer as suggested by Kreuer (modified from Reference [51] with permission). 5 6 5. Results and Discussions 7 5.1. The ion partitioning selectivity factor and the ion mobility ratio 8 Figure 3 shows the equilibrium partition coefficient of Mg2+ over Na+ (a), of K+ over Na+ (b) 9 between three types of CEMs and the surrounding aqueous solution with varying equivalent ratios of 10 two different cations. The definition of the partition coefficient is given in Equation 2. For the CMX 11 and SPEEK membranes, Mg2+ ions are preferentially distributed in the membrane phase for the 12 whole solution composition range investigated, and the partition coefficient 𝐾@A-BC.- decreases with 13 the increase of equivalent Mg2+ ratio in the solution. This is in accordance with the general principle 14 of ion exchange [55]. However, for the PFSA membrane F9120, the partition coefficient drops from 15 1.6 at an equivalent solution Mg2+ ratio of 0.1 to values below 1 when the equivalent solution Mg2+ 16 ratio is not less than 0.2. This reversal of the counter-ion partition selectivity in the membrane phase 17 is not frequently seen, and a reasonable explanation has not been achieved yet. As shown in Figure 18 3(b), there is no preferential partition of K+ between the CMX membrane and the solution, while K+ 19 ions are clearly enriched in the membrane phase of SPEEK and F9120. The partition coefficients 20 𝐾@A-D- in the F9120 membrane are quite large, especially at low equivalent solution K+ ratios. 21 Because ion exchange membranes prefer the counter-ion with smaller equivalent solvated volume 22 when different counter-ions have the same valence [55], therefore K+ is preferentially enriched over 23 Na+ in the membrane. The underlying reason for this preference of IEMs is the favorable elastic 24 relaxation of the polymeric matrix when the more hydrated counter-ions are replaced by less 25 hydrated ones [55]. The large partition coefficients 𝐾@A-D- at low equivalent solution K+ ratios for the 26 F9120 membrane as compared to the SPEEK membrane indicate that the polymeric matrix of the 27 F9120 membrane tends to reorganize relatively easier. This is in line with the observation that the 28 16 polytetrafluorene ether side chains in PFSA membranes (Figure 1, c) are easier to reorganize as 1 compared to the stiffer poly(ether ether ketone) hydrocarbon chains [56], which is also reflected by 2 their corresponding glass transition temperatures. 3 4 5 Figure 3. The equilibrium partitioning selectivity factor of (a) Mg2+ over Na+ (𝐾@A-BC.-), and (b) K+ over 6 Na+ (𝐾@A-D-) between three CEMs (CMX, SPEEK and F9120) and aqueous solutions with varying 7 equivalent cation ratios. The anion is sulfate, and its total molar concentration is 0.5 M. The dashed 8 lines in the figures indicate no preferential partitioning of cations between the membrane and the 9 solution phases. 10 11 The cation mobility ratio in the membrane phase (𝑢"#𝑢!#') can be calculated via Equation 2, 12 when the partition coefficient of cations (𝐾!") and the transport selectivity (𝑃!") are known. Due to 13 experimental difficulties in measuring the actual partition coefficient during electrodialysis (ED), the 14 equilibrium partition coefficient is generally used instead [7]. Table 2 lists the transport selectivity of 15 Mg2+/Na+ and K+/Na+ measured by ED. Mg2+ ions are preferentially transported as compared to Na+ 16 ions through all three types of CEMs, and so are K+ ions. In the ED process, when the current density 17 is well below its limiting value, the partition of counter-ions between the membrane and the solution 18 phase can be considered as close to equilibrium [44, 57]. In our ED experiments the current density 19 at 10 mA cm-2 is well below the limiting current densities of systems investigated [16], so the 20 17 experimentally measured equilibrium partition coefficients (as shown in Figure 3) can be used to 1 calculate the counter-ion mobility ratio in the membrane 𝑢"#𝑢!#'(𝐸𝐷), via Equation 2. As also shown 2 in Table 2, the Mg2+/Na+ and K+/Na+ mobility ratios determined with the ED selectivity data are larger 3 than 1, except for the Mg2+/Na+ pair in the SPEEK membrane and the K+/Na+ pair in the F9120 4 membrane. The unusual phenomenon of smaller K+ mobility as compared to Na+ in the membrane 5 phase has also been observed in other PFSA membranes like Nafion [58]. On the other hand, the 6 cation mobility ratios in the membrane can also be calculated from the membrane conductivity data 7 at pure counter-ion forms (Equation 10, without cation activity coefficients), the results of 𝑢"#𝑢!#'(𝜅) 8 are tabulated in Table 2. It is clear that only the K+/Na+ mobility ratios in the CMX and SPEEK 9 membranes are close to those calculated with the ED selectivity data. The Mg2+/Na+ mobility ratios 10 and the K+/Na+ mobility ratio in the F9120 membrane calculated from the membrane conductivity 11 data (without cation activity coefficients) are all smaller than corresponding values calculated from 12 the ED selectivity data. This discrepancy should be due to the simplification of cation activity 13 coefficients in the CEMs when the membrane conductivity data are employed for the calculation of 14 cation mobility ratios in the membrane. Therefore, the calculation of cation activity coefficients in 15 these CEMs becomes necessary. 16 17 Table 2. The measured cation transport selectivity and calculated cation mobility ratio for CEMs 18 Membranes 𝑃!"(𝐸𝐷) 𝑢"#𝑢!#'(𝐸𝐷) 𝑢"#𝑢!#'(𝜅) Mg2+/Na+ K+/Na+ Mg2+/Na+ K+/Na+ Mg2+/Na+ K+/Na+ CMX 1.34 1.43 1.07 1.51 0.17 1.62 SPEEK 1.09 1.31 0.83 1.11 0.15 1.19 F9120 1.32 1.30 1.48 0.89 0.37 0.34 19 20 5.2. The average distance between two adjacent fixed sulfonate groups 21 The average distance between two adjacent fixed sulfonate groups (Equation 13, b) in CEMs is 22 necessary for the calculation of cation activity coefficients in CEMs. For the hydrocarbon-based 23 CEMs (CMX and SPEEK) it is calculated based on the assumption that these ionic groups are 24 18 distributed randomly throughout the whole membrane, and the distribution can be considered as 1 homogeneous on a microscopic scale. Therefore, with known molar ratio of charged and 2 non-charged groups, the b values can easily be estimated. Table 3 lists the parameters necessary 3 for the calculation in the hydrocarbon-based CEMs. The CMX membranes are made from sulfonated 4 polystyrene cross-linked with divinyl benzene (Figure 1), and these cross-linked copolymers are 5 further reinforced with inert fabric. The weight fraction of styrene and divinyl benzene copolymers in 6 the composite membrane is reported to be 45 – 65% [59], and an average value of 55% is used here. 7 The homogeneous distribution of fixed charge groups in these hydrocarbon-based ionomers is a 8 rationale assumption, because it is already known that fixed chare groups in these ionomers are 9 more distributed and show no ion clustering [56]. This assumption has also been adopted by Kamcev 10 et al. in their calculations, and the calculated cation activity coefficients in the hydrocarbon-based 11 CEMs show remarkably good agreement with experimental values [25]. As shown in Table 3, though 12 the CMX and SPEEK membranes have similar IECs, the b value of SPEEK is larger than that of 13 CMX because the unit distance (𝑏E) in SPEEK is much larger (Figure 1). A recent study on the small 14 angle X-ray scattering profiles of SPEEK membranes reveals a scattering peak corresponding to the 15 average distance between two adjacent sulfonate groups on the polymer backbone [60]. For SPEEK 16 membranes with IEC in the range of 1.3 - 1.6 mmol∙g-1, the average distance is around 2.4 nm [60], 17 which accredits the value calculated in this study. 18 19 Table 3. The average distance between two adjacent fixed sulfonate groups () in CEMs 20 Membranes IEC [mmol·g-1 dry polymer] * [mmol·g-1] * [-] [nm] [nm] CMX 1.66 ± 0.06 1.62 0.98 0.25 [25] 0.50 SPEEK 1.62 ± 0.01 1.28 0.79 1.70 [61] 3.04 bnonncn o nnnobb 19 * and stand for the molar amount of charged and non-charged groups per 1 g of the whole 1 dry ion exchange polymer. is the unit distance between two adjacent fixed charge groups as 2 schemed in Figure 1. 3 4 Different from the situation in the hydrocarbon-based CEMs, estimation of the average distance 5 between two adjacent fixed sulfonate groups in PFSA membranes relies on the membrane structure 6 models derived from the scattering data [51, 52, 62]. Since ion clustering is observed in these 7 membranes, the assumption of homogeneous distribution of fixed charge groups throughout the 8 whole membrane does not hold any more. The cluster-network model initiated by Gierke et al. [52], 9 the parallel cylindrical water channel model [54] and the layered locally planar channel model 10 proposed by Kreuer et al. [51, 56, 62] are typical examples. In these models, the water volume 11 fraction of the hydrated membranes (𝛷F) is the most important parameter. As shown in Table 1, the 12 water content of the F9120 membrane in Na+ form is 16.9 wt.%, which is close to 17.8 wt.% of Nafion 13 in Na+ form with an equivalent weight of 1200 g∙mol-1 (Table 1 from Ref. [52]). For this type of Nafion 14 membrane at such a hydration level, Gierke et al. estimated a cluster diameter of 3.88 nm and 73 15 fixed charge sites within a single cluster [52]. By assuming homogeneous distribution of fixed charge 16 groups within a single spherical cluster (as sketched in Figure 2(a)), an average distance of 0.91 nm 17 between these fixed charge groups can be obtained. Though the equivalent weight of the F9120 18 membrane (950 g∙mol-1) used in this study is different from this type of Nafion described in literature, 19 we assume that the microstructure of the F9120 membrane resembles this type of Nafion, because 20 of their similar water contents at Na+ form. Following the same procedure of Gierke, and use the 21 equivalent weight of 950 g∙mol-1 for the calculation of fixed charge group numbers, the cluster 22 diameter and the number of fixed charge groups per cluster of the F9120 membrane in Na+ and K+ 23 forms are obtained, and results are presented in Table 4. The calculation details are provided in the 24 Supporting Information. There are no scattering data about Mg2+-form Nafion membranes [52], 25 consequently the Bragg spacing distance necessary for the estimation of the cluster spacing 26 distance in Nafion is not available. Alternatively, the Bragg spacing distance measured from 27 cnnonnob 20 Na+-form Nafion is adopted for the calculation. Because of similar water contents, the 𝑏 value of the 1 Mg2+-form F9120 membrane is almost identical to that of the Na+-form membrane. 2 3 Table 4. The average distance between two adjacent fixed sulfonate groups () in PFSA 4 Structure model Cation Cluster/cylinder diameter [nm] Fixed charges per cluster/cylinder [-] 𝑏 [nm] Cluster–network [52] K+ 3.22 93 0.67 Na+ 3.73 81 0.83 Mg2+ 3.75 81 0.84 Parallel cylindrical water channels [54] K+ 1.91 3.98 × 106 0.56 Na+ 2.76 3.98 × 106 0.69 Mg2+ 2.80 3.98 × 106 0.68 Layered locally planar channels [51] K+ 0.4* / 0.79 Na+ 1.0 / 0.79 Mg2+ 1.0 / 0.79 * This is the water layer thickness in the layered locally planar channels model. 5 6 The parallel cylindrical water channels model suggests that the water nanochannels in PFSA 7 membranes are parallel cylinders of a few nanometer diameter. This model is based on the direct 8 numerical simulation of small angle X-ray scattering data of a Nafion117 membrane with 20 vol.% of 9 water [54]. For such a membrane, the authors find the average diameter of cylinders to be 2.4 nm 10 [54]. The length of these cylindrical water nanochannels is around 20 nm, however whether these 11 nanochannels align along the normal direction to the membrane surface is not clear [54]. To simplify 12 b 21 the calculation, it is assumed that these nanochannels align almost in series across the whole 1 membrane thickness. So, the estimated value of the nanochannel length is the minimum (and also 2 equals to the hydrated membrane thickness), and consequently the estimated value of the average 3 distance between two adjacent fixed charge groups is also the minimum. With the amount of fixed 4 charge groups (IEC value), the number of fixed charge groups within such a water-filled cylindrical 5 nanochannel (Figure 2, b) penetrating the membrane thickness is calculated as much as 4.20 × 106, 6 by taking into account of 20 vol.% water in the membrane. Then the average distance between two 7 adjacent fixed charge groups is estimated to be 0.62 nm. This structure model is also transferred to 8 the F9120 membrane by comparing the water volume fractions, together with the adoption that 9 PFSA membranes swell/shrink in a fashion that only the diameter of cylindrical water nanochannels 10 increases/decreases (Figure 2, b). For the K+-form of the F9120 membrane with 7.6 wt.% of water 11 (12.7 vol.%, Table 5), the diameter of the water nanochannels in the F9120 membrane is 1.91 nm, 12 and the average distance between two adjacent fixed charge groups is 0.56 nm. In a similar way, the 13 average distances in Na+-form and Mg2+-form F9120 membranes are obtained, and these two values 14 are quite close (Table 4) because of similar water contents in these two membranes. 15 Even though the parallel cylindrical water nanochannels model of Nafion is the first quantitative 16 structure model and has become popular, it is still at criticism because only a few scattering patterns 17 were employed for the comparison of simulation and experimental results, and the water volume 18 fraction of the important sample was also faulty [51]. Kreuer et al. propose a different structure model 19 by simulations of the scattering patterns in a similar fashion, but with an array of membrane samples 20 with large variations of the water volume fraction and pre-treatment/relaxation history [51]. Their 21 results exclude the possibility of the parallel cylindrical water nanochannels model, and instead 22 suggest cautiously that the water domain is locally flat, and the water and the polymer domains are 23 most likely layered lamella [51]. According to this model, only the thickness of the water layers 24 changes with the alteration in the membrane water contents. Therefore, the calculated average 25 thickness of the water layers changes as the counter-ions of the F9120 membrane change (Table 4). 26 However, the average distance between two adjacent fixed charge groups pervading 27 homogeneously on the planar water-polymer interface (Figure 2, c) is not affected, and is constant at 28 around 0.79 nm. 29 22 These estimated values of 𝑏, based upon three different structure models of PFSA membranes, 1 are all at the same order of magnitude as the reported 0.47 nm obtained by molecular dynamics 2 simulations [40, 46, 63, 64]. This agreement substantiates our estimations. 3 4 5.3 The Manning parameter and cation activity coefficients 5 With the estimated average distance between two adjacent fixed sulfonate groups (𝑏) in CEMs, 6 it is then possible to calculate the cation activity coefficients in these CEMs. According to Equation 7 14, the Manning parameter (ξ) can be calculated when the b values and the permittivity of hydrated 8 membranes (𝜀) are known. The values of 𝜀 can be measured experimentally [48-50] or alternatively 9 can be approximated with the average of the membrane polymer and the water permittivity (Equation 10 16). Table 5 lists the approximated 𝜀 values, Bjerrum length (𝜆G) and ξ values for the three types of 11 CEMs. For the calculation of the water volume fraction in the hydrated membranes (ΦF), the water 12 density of 1 g∙cm-3 is chosen. The permittivity of water at room temperature is 78, and for simplicity, 13 the permittivity of the dry membrane polymer is chosen at 6 as in Reference [25]. One recent work 14 reports different experimental dielectric constants of crosslinked poly(glycidyl methacrylate) 15 polymers and Nafion 117 at similar water volume fraction, which suggests that polymer water content 16 alone may be insufficient for estimating the permittivity of hydrated polymers [50]. This means that 17 polymer chemistry is also influencing. However, in this work, all the cation exchange polymers have 18 the same functional groups (sulfonate groups, though slightly different pKa values) and the additivity 19 rule has been proved valid in similar CEMs [25], we keep the use of additivity rule at the present 20 stage. 21 22 23 24 25 26 27 23 Table 5. The calculated Bjerrum length (𝜆G) and Manning parameter (ξ) in CEMs 1 Membranes Cations ΦF [%] 𝜀 [-] 𝜆G [nm] ξ [-] CMX K+ 19.6 20.1 2.79 5.6 Na+ 22.8 22.4 2.50 5.0 Mg2+ 22.7 22.3 2.51 5.0 SPEEK K+ 18.2 19.1 2.93 0.86 Na+ 26.5 25.1 2.23 0.66 Mg2+ 24.7 23.8 2.35 0.69 F9120 K+ 12.7 15.1 3.70 5.52/6.61/4.68 Na+ 26.5 25.1 2.23 2.69/3.23/2.83 Mg2+ 27.2 25.6 2.19 2.61/3.22/2.77 2 As can be seen in Table 5, most of the 𝜆G values are in the range of 2.20 – 3.00 nm, because 3 similar 𝜀 values are found for these hydrated CEMs at room temperature. However, the ξ values 4 differ. For the CMX membrane, the Manning parameters are all larger than corresponding critical 5 values (1 for K+ and Na+, 0.5 for Mg2+), so the three types of cations in the CMX membrane are all 6 condensed. A recent work reports that the Manning parameter (ξ) for the CMX membrane is 1.0±0.1, 7 obtained by fitting the experimental co-ion (Cl-) concentrations in the membrane according to the 8 counter-ion condensation theory [33]. This difference is mainly due to smaller water contents in our 9 case (18.0 – 22.0%) as compared to more absorbed water in literature (around 40%), because more 10 concentrated external salt solutions of bivalent sulphate anions (therefore larger ionic strength) are 11 employed in this work. While for the SPEEK membrane, only bivalent Mg2+ ions are condensed 12 because ξ is larger than 0.5. The K+ and Na+ ions in the SPEEK membrane are not condensed, just 13 24 because the average distance between two adjacent fixed sulfonate groups (b values) is large (Table 1 3), and which is a result of the relatively low IEC values for these SPEEK membranes. For the F9120 2 membrane, b values estimated from 3 different structure models (Table 4) are used to calculated the 3 Manning parameter, and the results are grouped in Table 5. As shown, all three types of cations are 4 condensed in the F9210 membrane, irrespective of which structure model is employed. 5 When the counter-ions are not condensed, i.e. the Manning parameter ξ < 𝜉H1"$ , then the 6 counter-ion activity coefficient in the membrane phase is calculated with the following formula [47]: 7 ln𝛾C#=−4I𝜉'/"'/"5I'0" (17) 8 Since the free salt concentration in the membrane phase can be neglected (𝐶,#≈0) when bivalent 9 co-ion SO42- at a total molar concentration of 0.5 M is employed in this study, Equation 17 can then 10 be simplified as: 11 𝛾C#=exp (−4Iξ) (18) 12 For the case of the SPEEK membranes with K+ or Na+ ions, the cation activity coefficients should be 13 calculated with Equation 18, instead of Equation 13 [47]. The calculated results are listed in Table 6. 14 As can be seen, when the cations are condensed in the membrane, the resultant activity coefficients 15 are generally only about 10 – 25% of the ideal values (1), which is in general agreement with the 16 results published in literature [25, 47]. For the Mg2+ ions in the SPEEK membrane, the Manning 17 parameter is slightly larger than the critical value (0.5), so limited portion of counter-ions are 18 condensed and the Mg2+ activity coefficient is therefore larger than 0.5. While for the K+ and Na+ ions 19 in the SPEEK membrane, because these cations are not condensed at all in this CEM with a large 20 distance between two adjacent fixed sulfonate groups, their activity coefficients are as large as 0.651 21 and 0.719, respectively. In a word, with the aid of Manning`s counter-ion condensation theory, all the 22 cation activity coefficients in CEMs are obtained. 23 24 25 25 Table 6. The cation activity coefficients in CEMs calculated via Manning`s theory 1 Membranes Structure model 𝛾C# K+ Na+ Mg2+ CMX / 0.131 0.147 0.074 SPEEK / 0.651 0.719 0.533 F9120 Cluster-network 0.110 0.225 0.116 Parallel cylindrical channel 0.092 0.188 0.094 Layered locally planar channel 0.130 0.214 0.110 2 5.4 The revised ion mobility ratio and resultant ion transport selectivity 3 As shown in Section 5.1, the cation mobility ratios calculated from the membrane ionic 4 conductivity data are obviously smaller than corresponding values obtained from the ED data, except 5 for the cases of K+/Na+ transport in the CMX and SPEEK membranes (Table 2). This discrepancy 6 could be ascribed to the simplification of the cation activity coefficients in CEMs. Now these cation 7 activity coefficients have been obtained with the Manning`s counter-ion condensation theory, and are 8 listed in Table 6. Then the revised mobility ratios (𝑢!"(𝜅,𝛾)) with the consideration of the cation activity 9 coefficient can be calculated with Equation 10, and the results are shown in Table 7. For the F9120 10 membrane, only the results based on the cluster – network structure model are presented in Table 7. 11 The revised mobility ratios are generally in quantitative agreement with the mobility ratios calculated 12 from the ED data. As compared to the non-revised mobility ratios, the revised Mg2+/Na+ mobility 13 ratios of the CMX and SPEEK membranes have increased, but are still smaller than the 14 corresponding ratios from the ED data. At the present stage, we think this might possibly be due to 15 oversimplification in estimating the average distance between adjacent fixed sulfonate groups (𝑏) by 16 assuming homogeneous distribution of fixed charge groups in these hydrocarbon-based CEMs. In 17 the future, it is necessary to figure out how to include the influence of membrane water volume 18 26 fraction (𝜙F) on the average distance between fixed sulfonate groups (𝑏) in these 1 hydrocarbon-based CEMs. 2 3 Table 7. The revised cation mobility ratios considering the cation activity coefficients 4 Membranes 𝑢"#𝑢!#'(𝐸𝐷) 𝑢"#𝑢!#'(𝜅) 𝑢"#𝑢!#'(𝜅,𝛾) Mg2+/Na+ K+/Na+ Mg2+/Na+ K+/Na+ Mg2+/Na+ K+/Na+ CMX 1.07 1.51 0.17 1.62 0.34 1.82 SPEEK 0.83 1.11 0.15 1.19 0. 20 1.31 F9120 1.48 0.89 0.34 0.34 0.72 0.70 5 The ion transport selectivity values obtained by the conventional ED method and the ionic 6 conductivity experimental approach we proposed before are illustrated in Figure 4, and numerical 7 values are provided in the Supporting Information (Table S3). The benchmark values are those 8 obtained by ED. The cation transport selectivity values 𝑃@A-BC.- and 𝑃@A-D- for the three types of CEMs 9 are all larger than 1 (indicated by the dashed horizontal line in Figure 4). The 𝑃@A-BC.- calculated from 10 the membrane conductivity data (𝜅) without the consideration of cation activity coefficients are all 11 significantly smaller than their corresponding reference values (Figure 4, a). While for the 𝑃@A-D- 12 calculated in the same fashion, the selectivity values are quite close to the reference values for the 13 CMX and SPEEK membranes, however not for the F9120 membrane (Figure 4, b). With the 14 inclusion of the cation activity coefficients in the calculation, the 𝑃@A-BC.- for CEMs all increase to 15 values quite close to the