Saline retention and permeability of nanofiltration membranes versus resistance and capacitance as obtained from impedance spectroscopy under a concentration gradient

Ci a ion: Pé ez, M.-Á.; Gallego, S.; Palacio, L.; He nández, A.; P ádanos, P.; Ca mona, F.J. Saline Re en ion and Pe meabili y o Nano il a ion Memb anes Ve sus Resis ance and Capaci ance as Ob ained om Impedance Spec oscopy unde a Concen a ion G adien . Memb anes 2023,13, 608. h ps://doi.o g/ 10.3390/memb anes13060608 Academic Edi o s: Jie Shen, Zongyao Zhou and Sheng Zhou Recei ed: 23 May 2023 Re ised: 13 June 2023 Accep ed: 16 June 2023 Published: 18 June 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). memb anes A icle Saline Re en ion and Pe meabili y o Nano il a ion Memb anes Ve sus Resis ance and Capaci ance as Ob ained om Impedance Spec oscopy unde a Concen a ion G adien Miguel-Ángel Pé ez , Sil ia Gallego, Lau a Palacio , An onio He nández, Ped o P ádanos and F ancisco Ja ie Ca mona * G upo de Supe icies y Ma e iales Po osos, Depa amen o de Física Aplicada, Facul ad de Ciencias, Uni e sidad de Valladolid, Paseo de Belén 7, 47011 Valladolid, Spain; [email p o ec ed] (M.-Á.P.); [email p o ec ed] (S.G.); [email p o ec ed] (L.P.); [email p o ec ed] (A.H.); [email p o ec ed] (P.P.) *Co espondence: coja ie [email p o ec ed] Abs ac : Impedance spec oscopy has been widely used o he s udy o he elec ical p ope ies o memb anes o hei cha ac e iza ion. The mos common use o his echnique is he measu e o he conduc i i y o di e en elec oly e solu ions o s udy he beha io and mo emen o elec ically cha ged pa icles inside he po es o memb anes. The objec i e o his in es iga ion was o obse e i he e is a ela ion p esen be ween he e en ion ha a nano il a ion memb ane possesses o ce ain elec oly ic solu ions (NaCl, KCl, MgCl 2 , CaCl 2 , and Na 2 SO 4 ) and he pa ame e s ha a e ob ained h ough IS measu emen s o he ac i e laye o he memb ane. To achie e ou objec i e, di e en cha ac e iza ion echniques we e pe o med o ob ain he pe meabili y, e en ion, and ze a po en ial alues o a Desal-HL nano il a ion memb ane. Impedance spec oscopy measu emen s we e pe o med when a g adien concen a ion was p esen be ween bo h sides o he memb ane o s udy he a ia ion ha he elec ical pa ame e s had wi h he ime e olu ion. Keywo ds: nano il a ion; e en ion; pe meabili y; ze a po en ial; impedance spec oscopy 1. In oduc ion Nano il a ion (NF) has been p o en o be a good al e na i e o e e se osmosis (RO) o poo b ackish wa e s ha ha e di alen ions o solu es wi h low molecula weigh (~300 Da), due o he low p essu es equi ed [ 1 , 2 ] and he high pe mea e luxes ob ained. When NF is applied o ion e en ion, i s pe o mance o ha pu pose depends s ongly on he cha ge o he memb ane [ 3 – 5 ] and hei in e ac ion wi h he cha ges o he ions. The e ec s o he cha ges o he memb ane su ace appea mainly in he case o NF memb anes h ough wo p ocesses: he ioniza ion o unc ional g oups [ 6 – 8 ] and he abso p ion o cha ged molecules [7,9]. The many applica ions o nano il a ion make i highly desi able o ully cha ac e ize NF memb anes so ha he e en ion phenomena can be unde s ood and a be e il a ion p ocess can be achie ed. When cha ac e izing a memb ane, he mos s udied p ope ies a e hose ela ed o he p ope ies o e en ion ha he memb ane p esen s o he passing o ce ain molecules. Some o he mos measu ed pa ame e s ha gi e an accoun o he anspo o molecules h ough a NF memb ane a e e en ion, pe meabili y, and ze a po en ial. The la e is qui e ele an as i has been p o en o be s ongly co ela ed wi h he alue o e en ion [ 10 , 11 ]. The measu emen o hese pa ame e s has been commonly accomplished by di e en au ho s and exposed in mul iple a icles [12–14] o di e en ypes o memb anes. Elec ochemical impedance spec oscopy (EIS), o simply impedance spec oscopy (IS), is a ela i ely mode n cha ac e iza ion echnique o many ma e ials and has been used in many scien i ic and echnological a eas [ 15 , 16 ]. In he pa icula case o memb anes Memb anes 2023,13, 608. h ps://doi.o g/10.3390/memb anes13060608 h ps://www.mdpi.com/jou nal/memb anes Memb anes 2023,13, 608 2 o 16 o po ous ma e ials, EIS [ 17 , 18 ] has been used by di e en au ho s o he cha ac e iza ion o he beha io o some elec oly e solu ions inside he po es o NF memb anes, which can gi e a g ea amoun o in o ma ion abou he elec ical p ope ies o he sys em and he mo emen o sal s inside he po es [ 14 , 19 – 21 ]. In 2001, Fie e e al. [ 22 ] used EIS o s udy, bo h expe imen ally and heo e ically, he dependance ha exis s be ween he conduc i i y o di e en sal s inside he po es o an ul a il a ion (UF) memb ane and he alue o ze a po en ial ha he memb ane p esen ed when i was imme sed in o such solu ions. In ha s udy, hey showed how he ela ion be ween conduc i i y and ze a po en ial could de ine a possible echnique o ze a po en ial measu emen which could be applied e en in he case o NF memb anes. This la es applica ion o EIS would suppose o emo e he disad an ages ha p esen s he lux g adien me hod wi h small po es, as i can commonly lead o se ious p oblems a low concen a ions. He e, we will s udy he dependence and possible co ela ion be ween he elec i- cal pa ame e s ha a e no mally used in EIS o cha ac e ize a NF memb ane when he memb ane is imme sed in di e en elec oly e solu ions (NaCl, KCl, MgCl 2 , CaCl 2 , and Na 2 SO 4 ). This s udy will lead us o co ela e he EIS da a wi h impo an pa ame e s o he desc ip ion o memb ane pe o mance such as pe meabili y, ze a po en ial, o e en ion. EIS measu emen s we e pe o med in he p esence o a concen a ion g adien be ween bo h sides o he memb ane, o cing a lux o ions h ough un il equilib ium was eached. This expe imen al se up was p e iously compa ed wi h he usual EIS expe imen al se up, whe e a solu ion wi h a ixed concen a ion is pumped o bo h sides o he memb ane un il equilib ium is eached. The g adien echnique allows a de ailed analysis o he cha ac e is ics o he ac i e laye and hei e olu ion wi h ime, he e o e app oaching he eal nano il a ion p ocedu e. EIS measu emen s we e aken o he memb ane + solu ion sys em and ollowed wi h ime so as o obse e he ime e olu ion o he elec ical p ope - ies o he sys em, especially hose ela ed o he memb ane’s ac i e laye , and o see i he e exis s a ela ionship be ween he calcula ed elec ical pa ame e s and he e en ion alues measu ed h ough con en ional echniques. This ela ionship could imply ha EIS could be used in he s udy o memb anes o a deep unde s anding o he beha io o elec oly ic solu ions inside he po es. 2. Ma e ials and Me hods In his wo k, expe imen al measu emen s o impedance spec oscopy, saline pe me- abili y, and angen ial s eaming po en ial we e ca ied ou o he cha ac e iza ion o a nano il a ion memb ane using i e di e en saline solu ions. 2.1. Memb ane The memb ane used in his wo k was he DESAL-HL, which is a plane nano il a ion memb ane manu ac u ed by GE-Osmonics (Minne onka, MN, USA). I is a polyamide memb ane wi h a molecula weigh cu o (MWCO) be ween 150 and 300 g/mol [ 23 ] and ha can be used in media wi h a pH anging be ween 3 and 9 and a a maximum empe a u e o 50 ◦ C, acco ding o he manu ac u e . I has a p e iously e alua ed mean po e adius o 0.47 nm [19]. 2.2. Elec oly ic Solu ions The sal s used in his expe imen we e NaCl and KCl (elec oly e ype 1:1), MgCl 2 and CaCl 2 ( ype 1:2), and Na 2 SO 4 ( ype 2:1). The sal s MgCl 2 , CaCl 2 , and Na 2 SO 4 we e supplied by Pan eac (ITW, Glen iew, IL, USA), KCl by VMR Chemicals (Radno , PA, USA), and NaCl by Me ck (Da ms ad , Hesse, Ge many). Saline aqueous solu ions we e p epa ed wi h ion exchange and e e se-osmosis- ea ed demine alized wa e using a miliQ (Millipo e, Subsidia y o Me ck KGaA, Bille ica, MA, USA). Memb anes 2023,13, 608 3 o 16 2.3. Elec ochemical Impedance Spec oscopy EIS measu emen s we e pe o med a oom empe a u e using a Sola on 1260 impedance/gain-phase analyze (AMETEK, Be wyn, Pennsyl ania) in an in-ci cui impedance measu emen scheme. Each o he spec al measu emen s was conduc ed wi h an AC signal o 50 mV o ampli ude and a ange o dec easing equencies be ween 0.1 Hz and 1 MHz. Ten poin s pe decade we e aken so Nyquis plo s o he impedance beha io had enough esolu ion o dis inguish he beha io a bo h laye s o he memb ane. Samples o ou sys em unde es , a Desal-HL memb ane, we e placed be ween wo ma ch- ing me hac yla e semicells wi h an a ea o 10.18 cm 2 open o he lux o he elec oly e solu ion; his same expe imen al se up was p e iously desc ibed by Mon al illo e al. [ 24 ]. In ou case, wo Cl − -selec i e elec odes we e placed close o he memb ane o pe o m EIS measu emen s. To educe he signal e o , a ypical ou -wi e impedance measu emen was designed o elimina e he inhe en impedance o he lead wi e. Two ecipien s, placed a bo h sides o he memb ane, we e illed wi h 0.56 L o deion- ized wa e in which we added a highly concen a ed elec oly e solu ion, un il he desi ed concen a ions we e eached. The concen a ions we e designed as high concen a ion ( chigh ) on he side o he same name, and low concen a ion ( clow ) o he concen a ion on he o he side. These we e in ended o ake ini ial alues o 10 −3 mol/L and 10 −5 mol/L, espec i ely, in he case o mono alen sal s and 5 × 10 −4 mol/L and 5 × 10 −6 mol/L o he di alen sal s. These concen a ion anges co espond o he elec odes’ ange o wo king, and hey a e also he app op ia e ange o he s udy o ze a po en ial. The saline solu ions o he i e di e en sal s p epa ed in his way con ained he same cha ge concen a ion in all cases (NaCl, KCl, CaCl 2 , MgCl 2 , CaCl 2 , and Na 2 SO 4 ). The i s ou sal s we e chosen o ha e Cl − as a common anion in o de o analyze he ca ion e ec : wo wi h one posi i e cha ge and wo wi h wo posi i e cha ges. In each pai o sal s wi h he same elemen al cha ge, he size and di usi i y we e di e en . The Na 2 SO 4 sal was chosen o he sake o analyzing he e ec o he anion when i is mo e oluminous and mo e nega i e. The eal concen a ions, sligh ly di e en om he desi ed ones, we e measu ed expe imen ally as indica ed in Sec ion 2.4. The memb ane was p e iously subme ged o se e al days in he low-concen a ion solu ion o p econdi ion i . In o de o educe he concen a ion pola iza ion, i is e y impo an o keep bo h sides p ope ly s i ed; his was ensu ed by eci cula ing he solu ion wi h a pump wi h a angen ial lux o 0.6 L/min on each side o he memb ane. A scheme o he expe imen al de ice is shown in Figu e 1. Memb anes 2023, 13, x FOR PEER REVIEW 4 o 16 Figu e 1. Impedance spec oscopy expe imen al se up o he measu emen o elec ical pa ame e s o ou NF memb ane o he di e en elec oly e solu ions. The measu emen s o he impedance spec um we e au oma ically eco ded wi h a compu e . The adjus men o he expe imen al esul s o he co esponding equi alen elec ical ci cui o ob ain he elec ical pa ame e s o his ci cui was ca ied ou using he comme cial Z iew so wa e (AMETEK, Be wyn, PA, USA). This i p ocedu e used he ‘Calc-Modulus’ da a weigh ype; each da a poin ’s weigh was no malized by i s magni- ude. 2.4. Re en ion and Pe meabili y Sal e en ion and pe meabili y we e measu ed simul aneously and using he same cell as o he impedance spec oscopy measu emen s. When pe o ming he e en ion and pe meabili y measu emen s, he concen a ion was measu ed using a HI 5522 con- duc i i y me e (Hanna Ins umen s, Padua, I aly). De ails o he expe imen al de ice can be ound in a p e ious wo k [14]. The p ocess was de eloped o e 30 h, du ing which, in ou ep esen a i e momen s, he EIS measu emen s we e made. This was a ime poin s = 0 h, = 6 h, = 23 h, and = 29 h. The i s wo o hem we will call “ini ial imes”, and he las wo we will call “ inal imes”, o he e en ion and pe meabili y p ocess. 2.5. Tangen ial S eaming Po en ial and Ze a Po en ial Wi h he aim o e alua ing he ze a po en ial, angen ial s eaming po en ial meas- u emen s we e ca ied ou [25–27]. The expe imen al se up used o hese measu emen s consis ed o a memb ane holde whe e he ac i e aces o wo shee s o he memb ane we e placed ace o ace, lea ing a single channel h ough which he solu ion was ci cu- la ed. This holde did no allow pe mea ion h ough he memb ane. The expe imen al p ocedu e consis ed o applying di e en p essu e g adien s and measu ing he p oduced elec ical po en ial. The solu ion lowed angen ially o he mem- b ane su aces hanks o he p essu e g adien gene a ed wi h a pump. The elec ical po- en ial was measu ed by using a high-impedance (g ea e han 1010 Ω) Hewle -Packa d ol me e (HP3456A) wi h an accu acy o 1 µV. The sys em used o he expe imen al ob- en ion was he same se up used by Sil a e al. [28,29] and shown in Figu e 2. The meas- u emen s we e made a he na u al pH o he sal solu ion in deionized wa e . Figu e 1. Impedance spec oscopy expe imen al se up o he measu emen o elec ical pa ame e s o ou NF memb ane o he di e en elec oly e solu ions. Memb anes 2023,13, 608 4 o 16 The measu emen s o he impedance spec um we e au oma ically eco ded wi h a compu e . The adjus men o he expe imen al esul s o he co esponding equi a- len elec ical ci cui o ob ain he elec ical pa ame e s o his ci cui was ca ied ou using he comme cial Z iew so wa e (AMETEK, Be wyn, PA, USA). This i p ocedu e used he ‘Calc-Modulus’ da a weigh ype; each da a poin ’s weigh was no malized by i s magni ude. 2.4. Re en ion and Pe meabili y Sal e en ion and pe meabili y we e measu ed simul aneously and using he same cell as o he impedance spec oscopy measu emen s. When pe o ming he e en ion and pe meabili y measu emen s, he concen a ion was measu ed using a HI 5522 conduc i i y me e (Hanna Ins umen s, Padua, I aly). De ails o he expe imen al de ice can be ound in a p e ious wo k [14]. The p ocess was de eloped o e 30 h, du ing which, in ou ep esen a i e momen s, he EIS measu emen s we e made. This was a ime poin s = 0 h, = 6 h, = 23 h, and = 29 h. The i s wo o hem we will call “ini ial imes”, and he las wo we will call “ inal imes”, o he e en ion and pe meabili y p ocess. 2.5. Tangen ial S eaming Po en ial and Ze a Po en ial Wi h he aim o e alua ing he ze a po en ial, angen ial s eaming po en ial mea- su emen s we e ca ied ou [ 25 – 27 ]. The expe imen al se up used o hese measu emen s consis ed o a memb ane holde whe e he ac i e aces o wo shee s o he memb ane we e placed ace o ace, lea ing a single channel h ough which he solu ion was ci cula ed. This holde did no allow pe mea ion h ough he memb ane. The expe imen al p ocedu e consis ed o applying di e en p essu e g adien s and measu ing he p oduced elec ical po en ial. The solu ion lowed angen ially o he memb ane su aces hanks o he p essu e g adien gene a ed wi h a pump. The elec ical po en ial was measu ed by using a high-impedance (g ea e han 10 10 Ω ) Hewle -Packa d ol me e (HP3456A) wi h an accu acy o 1 µ V. The sys em used o he expe imen al ob en ion was he same se up used by Sil a e al. [ 28 , 29 ] and shown in Figu e 2. The measu emen s we e made a he na u al pH o he sal solu ion in deionized wa e . Memb anes 2023, 13, x FOR PEER REVIEW 5 o 16 Figu e 2. Tangen ial s eaming po en ial expe imen al se up o he expe imen al measu emen o ze a po en ial alues o he di e en elec oly e solu ions. 3. Resul s and Discussion 3.1. Re en ion and Pe meabili y Saline pe meabili y (P) is a cha ac e is ic pa ame e ela ing he lux (j) o a saline solu ion h ough a memb ane, when a concen a ion di e ence (𝑐high −𝑐low) exis s be- ween bo h aces o he memb ane. Acco ding o Fick’s law, 𝑗= 𝑃(𝑐high −𝑐low)=𝑉low 𝐴 d𝑐low(𝑡) d𝑡 =−𝑉high 𝐴 d𝑐high(𝑡) d𝑡 (1) In he expe imen al a angemen used he e, he olumes o he high-concen a ion side and he low-concen a ion sides a e equal, 𝑉low =𝑉high ≝V, and he exchange o sal occu s h ough he memb ane a ea open o he lux (A). The e o e he a ia ions o high and low concen a ions will be he same al hough o opposi e sign d𝑐low(𝑡)/d𝑡= −d𝑐high(𝑡)/d𝑡, being i s sum cons an a all imes, 𝑐high(𝑡)+𝑐low(𝑡)=𝑐high(0)+𝑐low(0). Then, bo h he concen a ions e ol e o he equilib ium alue, so ha o a long enough ime ( →∞), i could be conside ed ha he concen a ions o bo h solu ions will equal he alue o he ini ial a e age, 𝑐high(∞)=𝑐low(∞)=(𝑐high(0)+𝑐low(0))/2≝𝑐(∞). Wi h all his in mind, he concen a ion on he low-concen a ion side will e ol e o e ime ac- co ding o Equa ion (2), as used p e iously by Diaz e al. [14]: 𝑓(𝑐low(𝑡))≝ln[𝑐low(𝑡)−𝑐low(∞) 𝑐low(0)−𝑐low(∞)]=−(2𝐴𝑃 𝑉)𝑡 (2) The linea i s o 𝑓(𝑐low(𝑡)) agains ime allow us o know he pe meabili ies o each solu ion (Figu e 3). Fo all he elec oly es s udied in his wo k, he co esponding i showed a s ong linea ela ionship, indica ing ha he equa ion used o calculus is ap- plicable. The pe meabili y alues a e shown in Table 1 o he elec oly e solu ions used in his wo k. Figu e 3. Linea i s o he ime e olu ion o 𝑓(𝑐low(𝑡)) de ined in Equa ion (2). Figu e 2. Tangen ial s eaming po en ial expe imen al se up o he expe imen al measu emen o ze a po en ial alues o he di e en elec oly e solu ions. 3. Resul s and Discussion 3.1. Re en ion and Pe meabili y Saline pe meabili y (P) is a cha ac e is ic pa ame e ela ing he lux (j) o a saline solu ion h ough a memb ane, when a concen a ion di e ence ( chigh −clow ) exis s be ween bo h aces o he memb ane. Acco ding o Fick’s law, j=Pchigh −clow=Vlow A dclow( ) d =−Vhigh A dchigh( ) d (1) In he expe imen al a angemen used he e, he olumes o he high-concen a ion side and he low-concen a ion sides a e equal, Vlow =Vhigh de = V, and he exchange o sal occu s h ough he memb ane a ea open o he lux (A). The e o e he a ia ions o Memb anes 2023,13, 608 5 o 16 high and low concen a ions will be he same al hough o opposi e sign dclow( )/d = −dchigh( )/d ,being i s sum cons an a all imes, chigh( )+clow( )=chigh(0)+clow(0) . Then, bo h he concen a ions e ol e o he equilib ium alue, so ha o a long enough ime ( →∞ ), i could be conside ed ha he concen a ions o bo h solu ions will equal he alue o he ini ial a e age, chigh(∞)=clow(∞)=chigh0+clow0/ 2 de =c∞ . Wi h all his in mind, he concen a ion on he low-concen a ion side will e ol e o e ime acco ding o Equa ion (2), as used p e iously by Diaz e al. [14]: (clow( )) de =lnclow( )−clow(∞) clow(0)−clow(∞)=−2AP V (2) The linea i s o (clow( )) agains ime allow us o know he pe meabili ies o each solu ion (Figu e 3). Fo all he elec oly es s udied in his wo k, he co esponding i showed a s ong linea ela ionship, indica ing ha he equa ion used o calculus is applicable. The pe meabili y alues a e shown in Table 1 o he elec oly e solu ions used in his wo k. Memb anes 2023, 13, x FOR PEER REVIEW 5 o 16 Figu e 2. Tangen ial s eaming po en ial expe imen al se up o he expe imen al measu emen o ze a po en ial alues o he di e en elec oly e solu ions. 3. Resul s and Discussion 3.1. Re en ion and Pe meabili y Saline pe meabili y (P) is a cha ac e is ic pa ame e ela ing he lux (j) o a saline solu ion h ough a memb ane, when a concen a ion di e ence (𝑐high −𝑐low) exis s be- ween bo h aces o he memb ane. Acco ding o Fick’s law, 𝑗= 𝑃(𝑐high −𝑐low)=𝑉low 𝐴 d𝑐low(𝑡) d𝑡 =−𝑉high 𝐴 d𝑐high(𝑡) d𝑡 (1) In he expe imen al a angemen used he e, he olumes o he high-concen a ion side and he low-concen a ion sides a e equal, 𝑉low =𝑉high ≝V, and he exchange o sal occu s h ough he memb ane a ea open o he lux (A). The e o e he a ia ions o high and low concen a ions will be he same al hough o opposi e sign d𝑐low(𝑡)/d𝑡= −d𝑐high(𝑡)/d𝑡, being i s sum cons an a all imes, 𝑐high(𝑡)+𝑐low(𝑡)=𝑐high(0)+𝑐low(0). Then, bo h he concen a ions e ol e o he equilib ium alue, so ha o a long enough ime ( →∞), i could be conside ed ha he concen a ions o bo h solu ions will equal he alue o he ini ial a e age, 𝑐high(∞)=𝑐low(∞)=(𝑐high(0)+𝑐low(0))/2≝𝑐(∞). Wi h all his in mind, he concen a ion on he low-concen a ion side will e ol e o e ime ac- co ding o Equa ion (2), as used p e iously by Diaz e al. [14]: 𝑓(𝑐low(𝑡))≝ln[𝑐low(𝑡)−𝑐low(∞) 𝑐low(0)−𝑐low(∞)]=−(2𝐴𝑃 𝑉)𝑡 (2) The linea i s o 𝑓(𝑐low(𝑡)) agains ime allow us o know he pe meabili ies o each solu ion (Figu e 3). Fo all he elec oly es s udied in his wo k, he co esponding i showed a s ong linea ela ionship, indica ing ha he equa ion used o calculus is ap- plicable. The pe meabili y alues a e shown in Table 1 o he elec oly e solu ions used in his wo k. Figu e 3. Linea i s o he ime e olu ion o 𝑓(𝑐low(𝑡)) de ined in Equa ion (2). Figu e 3. Linea i s o he ime e olu ion o (clow( )) de ined in Equa ion (2). Table 1. Pe meabili y ob ained h ough a linea i acco ding o Equa ion (2) and he cha ac e is ic ime ( c). Elec oly e Solu ion Pe meabili y (10−7m s−1) c (h) NaCl 4.21 ±0.02 178 KCl 2.45 ±0.15 315 MgCl21.94 ±0.01 385 CaCl24.32 ±0.01 175 Na2SO40.49 ±0.14 1378 The obse ed e en ion coe icien p esen ed by he memb ane, R obs , is calcula ed in his case as: Robs =1−clow chigh (3) Acco ding o he expe imen al p ocedu e desc ibed abo e, he concen a ions on each side o he memb ane should e ol e acco ding o he ollowing exp essions, ob ained ia he in eg a ion o Equa ion (2) and aking in o accoun he limi s o in eg a ion indica ed. chigh( )=chigh(0)−0.5chigh(0)−clow(0)1−exp−2AP V  clow( )=clow(0)+0.5chigh(0)−clow(0)1−exp−2AP V  (4) Memb anes 2023,13, 608 6 o 16 Addi ionally, indeed, his is how hey beha e expe imen ally. In his way, e en- ion would e ol e acco ding o he ela ionship ob ained by subs i u ing Equa ion (4) in Equa ion (3): Robs =chigh(0)−clow(0)exp−2AP V  c(∞)+0.5chigh(0)−clow(0)exp−2AP V (5) These ime dependences o concen a ions and e en ion wi h ime-dec easing ex- ponen ials allow us o speak o a cha ac e is ic ime in he p ocess, c , de ined by he loga i hmic dec emen . The alues o he cha ac e is ic ime a e shown in Table 1. All o hem a e alues g ea e han hund eds o hou s. The equilib ium in he sys em can be es ima ed o each se e al imes his amoun . The measu emen s ca ied ou in his wo k ex ended un il 30 h, in ei he case—a si ua ion s ill a om equilib ium. The e o e, i is impo an ha e e ing o ou measu emen s as “ini ial imes” and “ inal imes” has no hing o do wi h ac ual equilib ium. The e en ion alues we e de e mined o he same ins an s o ime as he pe meabili y measu emen s, in wo “ini ial” and “ inal imes” pe iods, h oughou he i s 30 h o sys em e olu ion. 3.2. Elec ochemical Impedance Spec oscopy As al eady men ioned, EIS expe imen s o elec ical cha ac e iza ion o solu ions inside he po es o a memb ane a e usually pe o med by making a ixed concen a ion solu ion pass h ough he il a ion memb ane un il equilib ium is eached [ 14 , 24 ], which is enough o gi e he elec ical in o ma ion o he sys em. In ou case, as we wan ed o s udy he dependence be ween he obse ed e en ion and he elec ical p ope ies o he memb ane + solu ion, pe o ming EIS measu emen s in equilib ium by using a ixed concen a ion, migh no gi e he equi ed in o ma ion. Since, in his wo k, we wan o ela e he esul s di ec ly ob ained om he EIS measu emen s o he e en ion and pe mea ion p ope ies o he memb ane, he si ua ion o be s udied mus be a nonequilib ium one; hence, he e mus be a concen a ion g adien be ween bo h sides o he memb ane. Gi en ha he impedance measu emen s equi e a ew minu es o ob ain a comple e spec um o su icien ly ep esen a i e equencies, his s udy could no be ca ied ou on memb anes wi h a e y sho cha ac e is ic ime o he o de o seconds, as was p e i- ously shown [ 14 ]. In his case, he NF memb anes gi e a slow enough e olu ion owa ds equilib ium wi h a su icien ly long cha ac e is ic ime (Table 1). Se e al measu emen s o he impedance spec a o he sys em we e conduc ed a di e en ins an s. As men ioned, only he esul s a 0, 6, 23, and 29 h we e used o ep esen a ion and analysis in ou g aphs o allow an easy isualiza ion. Figu e 4shows a ypical example, o NaCl ( he es o he sal s p esen ed simila beha io s), o he empo al e olu ion o he impedance spec a. The igu e shows he ime e olu ion o he Nyquis diag am (imagina y impedance as a unc ion o i s eal pa ) o a high s a ing concen a ion o 8.3 × 10 −3 mol/L and a low concen a ion o 1.8 × 10 −5 mol/L. EIS cu es o h ee ins an s a e shown. The high concen a ion ba ely changed om 8.3 o 7.8 × 10 −3 mol/L. In he same pe iod, on he o he side o he memb ane, he low concen a ion changed om 1.8×10−5mol/L o 7.8×10−5mol/L. Memb anes 2023,13, 608 7 o 16 Memb anes 2023, 13, x FOR PEER REVIEW 7 o 16 Since, in his wo k, we wan o ela e he esul s di ec ly ob ained om he EIS meas- u emen s o he e en ion and pe mea ion p ope ies o he memb ane, he si ua ion o be s udied mus be a nonequilib ium one; hence, he e mus be a concen a ion g adien be- ween bo h sides o he memb ane. Gi en ha he impedance measu emen s equi e a ew minu es o ob ain a comple e spec um o su icien ly ep esen a i e equencies, his s udy could no be ca ied ou on memb anes wi h a e y sho cha ac e is ic ime o he o de o seconds, as was p e iously shown [14]. In his case, he NF memb anes gi e a slow enough e olu ion owa ds equi- lib ium wi h a su icien ly long cha ac e is ic ime (Table 1). Se e al measu emen s o he impedance spec a o he sys em we e conduc ed a di - e en ins an s. As men ioned, only he esul s a 0, 6, 23, and 29 h we e used o ep esen- a ion and analysis in ou g aphs o allow an easy isualiza ion. Figu e 4 shows a ypical example, o NaCl ( he es o he sal s p esen ed simila beha io s), o he empo al e olu ion o he impedance spec a. The igu e shows he ime e olu ion o he Nyquis diag am (imagina y impedance as a unc ion o i s eal pa ) o a high s a ing concen a ion o 8.3·10−3 mol/L and a low concen a ion o 1.8·10−5 mol/L. EIS cu es o h ee ins an s a e shown. The high concen a ion ba ely changed om 8.3 o 7.8·10−3 mol/L. In he same pe iod, on he o he side o he memb ane, he low concen- a ion changed om 1.8·10−5 mol/L o 7.8·10−5 mol/L. Figu e 4. Nyquis diag ams o HL memb ane and NaCl solu ion o di e en imes (0, 6 and 29 h). The impedances displayed by he sys em o each equency can be g ouped in o h ee lobes. A e y high equencies, he low eal impedance co esponds o he i s lobe. I ela es o he beha io o he solu ion inside he suppo laye o he memb ane and ou side i . All hese in e ac ions, indis inguishable in he Nyquis plo , come oge he in a CPE1/R1 lobe in Figu e 5. Fo e y low equencies, an incomple e lobe appea s; o he highes eal impedances, ha co esponds o he elaxa ion o he pola iza ion laye in con ac wi h he elec odes in esponse o he applied ield [24] (W/R3 in Figu e 5). Figu e 4. Nyquis diag ams o HL memb ane and NaCl solu ion o di e en imes (0, 6 and 29 h). The impedances displayed by he sys em o each equency can be g ouped in o h ee lobes. A e y high equencies, he low eal impedance co esponds o he i s lobe. I ela es o he beha io o he solu ion inside he suppo laye o he memb ane and ou side i . All hese in e ac ions, indis inguishable in he Nyquis plo , come oge he in a CPE1/R1 lobe in Figu e 5. Fo e y low equencies, an incomple e lobe appea s; o he highes eal impedances, ha co esponds o he elaxa ion o he pola iza ion laye in con ac wi h he elec odes in esponse o he applied ield [24] (W/R3 in Figu e 5). Memb anes 2023, 13, x FOR PEER REVIEW 8 o 16 Figu e 5. Equi alen elec ical ci cui o a sys em o an elec oly e solu ion inside he po es o a nano il a ion memb ane. The cen al lobe, co esponding o in e media e equencies, will be he ocus o in- e es o ou s udy. This is because i is he lobe co esponding o he mo e es ic i e pa o he memb ane, wi h he po es o he ac i e laye o he memb ane. The solu ion inside hese po es clea ly beha es di e en ly han he solu ion ou side hem. The in e p e a ion o hese expe imen al EIS esul s needs o assume an elec ical equi alen ci cui , which is he common way o hese measu emen echniques [17,18]. I is p esumed ha he elemen s o a ela i ely simple elec ical ci cui can explain he da a ob ained by a ending o hei physical meaning in ela ion o he sys em unde s udy. We chose he simples model ha could desc ibe he sys em. Mo e complex equi alen ci cui s could lead o a possible loss o he physical meaning o he pa ame e s ha desc ibe each elemen . Fo his case, he whole sys em o med by a NF memb ane subme ged in an elec o- ly e solu ion and measu ed using selec i e elec odes can be associa ed wi h an elec ical ci cui , as shown in Figu e 5. The model consis s o assuming ha he suppo and he ac i e laye a e in a se ies associa ion, each o med o a esis i e elemen in pa allel wi h a cons an -phase elemen , and he solu ion in con ac wi h he elec odes beha es as a esis i e elemen in pa allel wi h a Wa bu g elemen . O he au ho s [19] ha e explained in ull de ail he physical meaning o assuming such models, which we will use o he analysis o ou da a. In ou case unde s udy, he da a co esponding o he e ec o he Wa bu g elemen was igno ed since hey do no p o ide physical in o ma ion abou he NF memb ane. In he same way, he pa o he ci cui co esponding o ee dissolu ion was no de e mined ei he . The expe imen al da a in e al co esponding o he cen al lobe was i ed o a single se in pa allel. Since he lobe p esen s a shape easonably close o a semici cle (see Figu e 4), i was accep ed as desc ibed by a pa allel a angemen o a esis o (Ral) and a pu e capaci y (Cal), co esponding o he ac i e laye . This elec ical ci cui , deno ed he e as Ral/Cal, is shown in ed in Figu e 5 inside he g ay dashed ec angle. Unde he assump ion o he men ioned model wi h ou simpli ica ions, we can eas- ily ob ain he modelized impedance equa ion o each laye o he sys em memb ane + solu ion by applying he laws o elec ical ci cui s o he union o elemen s men ioned, which leads o a heo e ical impedance o he suppo and ac i e laye equal o Equa ion (6). Figu e 5. Equi alen elec ical ci cui o a sys em o an elec oly e solu ion inside he po es o a nano il a ion memb ane. The cen al lobe, co esponding o in e media e equencies, will be he ocus o in e es o ou s udy. This is because i is he lobe co esponding o he mo e es ic i e pa o he memb ane, wi h he po es o he ac i e laye o he memb ane. The solu ion inside hese po es clea ly beha es di e en ly han he solu ion ou side hem. The in e p e a ion o hese expe imen al EIS esul s needs o assume an elec ical equi alen ci cui , which is he common way o hese measu emen echniques [ 17 , 18 ]. I is p esumed ha he elemen s o a ela i ely simple elec ical ci cui can explain he da a ob ained by a ending o hei physical meaning in ela ion o he sys em unde s udy. We chose he simples model ha could desc ibe he sys em. Mo e complex equi alen ci cui s could lead o a possible loss o he physical meaning o he pa ame e s ha desc ibe each elemen . Memb anes 2023,13, 608 8 o 16 Fo his case, he whole sys em o med by a NF memb ane subme ged in an elec oly e solu ion and measu ed using selec i e elec odes can be associa ed wi h an elec ical ci cui , as shown in Figu e 5. The model consis s o assuming ha he suppo and he ac i e laye a e in a se ies associa ion, each o med o a esis i e elemen in pa allel wi h a cons an -phase elemen , and he solu ion in con ac wi h he elec odes beha es as a esis i e elemen in pa allel wi h a Wa bu g elemen . O he au ho s [ 19 ] ha e explained in ull de ail he physical meaning o assuming such models, which we will use o he analysis o ou da a. In ou case unde s udy, he da a co esponding o he e ec o he Wa bu g elemen was igno ed since hey do no p o ide physical in o ma ion abou he NF memb ane. In he same way, he pa o he ci cui co esponding o ee dissolu ion was no de e mined ei he . The expe imen al da a in e al co esponding o he cen al lobe was i ed o a single se in pa allel. Since he lobe p esen s a shape easonably close o a semici cle (see Figu e 4), i was accep ed as desc ibed by a pa allel a angemen o a esis o (R al ) and a pu e capaci y (C al ), co esponding o he ac i e laye . This elec ical ci cui , deno ed he e as R al /C al , is shown in ed in Figu e 5inside he g ay dashed ec angle. Unde he assump ion o he men ioned model wi h ou simpli ica ions, we can easily ob ain he modelized impedance equa ion o each laye o he sys em memb ane + solu ion by applying he laws o elec ical ci cui s o he union o elemen s men ioned, which leads o a heo e ical impedance o he suppo and ac i e laye equal o Equa ion (6). Zal =Ral[1−iRalCalω] 1+Ral2Cal2ω2(6) The expe imen al da a o he cen al lobe can be i ed o Equa ion (6), which co e- sponds o he ac i e laye esponse. In ou case, we could obse e ha a good i ing can be ob ained o all he expe imen al da a and he p oposed model. The men ioned i o expe imen al alues o he complex impedance o Equa ion (6) was pe o med o all he elec oly e solu ions a di e en imes. As can be seen in Figu e 6, he e was a ime gap wi hou measu emen s. Since he beha io o he sys- em was su icien ly slow and smoo h, he measu emen s we e ocused on he “ini ial” and “ inal” momen s o he s udied p ocess, wi h he beha io o each one being easily in e pola ed du ing he pe iod o nonexis ence o measu emen s. The a ia ion in he elec ical esis ance and capaci ance ob ained o he di e en solu ions inside he ac i e laye o he memb ane is shown in Figu e 6 o all he sal solu ions used in his expe imen . Memb anes 2023, 13, x FOR PEER REVIEW 9 o 16 𝑍 = 𝑅󰇟1−𝑖𝑅𝐶𝜔󰇠 1+𝑅𝐶𝜔 (6) The expe imen al da a o he cen al lobe can be i ed o Equa ion (6), which co e- sponds o he ac i e laye esponse. In ou case, we could obse e ha a good i ing can be ob ained o all he expe imen al da a and he p oposed model. The men ioned i o expe imen al alues o he complex impedance o Equa ion (6) was pe o med o all he elec oly e solu ions a diffe en imes. As can be seen in Figu e 6, he e was a ime gap wi hou measu emen s. Since he beha io o he sys em was su - icien ly slow and smoo h, he measu emen s we e ocused on he “ini ial” and “ inal” momen s o he s udied p ocess, wi h he beha io o each one being easily in e pola ed du ing he pe iod o nonexis ence o measu emen s. The a ia ion in he elec ical e- sis ance and capaci ance ob ained o he diffe en solu ions inside he ac i e laye o he memb ane is shown in Figu e 6 o all he sal solu ions used in his expe imen . Figu e 6. Time e olu ion o esis ance (a) and capaci ance (b) o he ac i e laye o he s udied NF memb ane. Resis ance dec eased wi h ime in all cases, whe eas he capaci y inc eased, as shown in Figu e 6. The same expe imen al beha iou was obse ed in p e ious wo ks [14]. Ana- lyzing he ime e olu ion, acco ding o ou di ision in o “ini ial imes” (≤6 h) and “ inal imes” (≥23 h), a simila e olu ion in ime was obse ed o all he s udied saline solu ions. I we s udy he beha io o he ac i e laye esis ance, R al , he MgCl 2 solu ion beha ed somewha diffe en ly om he es . A “ini ial imes”, i s esis ance alues began by being conside ably lowe han hose shown by he es o he sal s. Fo “ inal imes”, he solu ions es ablished in simila alues o all he sal s, excep o Na 2 SO 4 . The s ill-nega i e slope a e 30 h o R al (Na 2 SO 4 ), as well as i s e y long cha - ac e is ic ime (see Table 1), would indica e ha he sys em was s ill e y a om eaching equilib ium. In any case, his beha io is expec ed o his sal since, as we will see la e , bo h om he elec ical and s e ic poin o iew, equilib ium is un a o able o he en y o his sal in o he po es. The alue o C al g ew om an almos equal alue o all solu ions o a alue o “ inal imes” ha showed he ollowing beha io : C al (CaCl 2 ) > C al (NaCl) > C al (KCl) > C al (Na 2 SO 4 ). Only he MgCl 2 solu ion p esen ed a sligh ly diffe en beha io o he alue o C al : i s ly, because he alue shown a he ini ial ime was somewha highe han ha o he o he elec oly ic solu ions, and secondly, because i s empo al e olu ion had a smoo he end han o he es o he solu ions. This gen le end inally caused he alue o pa ame e C al o he “ inal imes” o end up below ha o NaCl and CaCl 2 . Fo MgCl 2 , i is con enien o no e ha Mg 2+ is he smalles ion (see Table 2), which would allow a g ea e mobili y inside he po es and, hus, a low ini ial esis ance and a ela i ely high ini ial capaci y. In bo h cases, NaCl and CaCl 2 ga e mo e ou s andingly low esis ances and high capaci ies o he inal s ages. Figu e 6. Time e olu ion o esis ance ( a ) and capaci ance ( b ) o he ac i e laye o he s udied NF memb ane. Resis ance dec eased wi h ime in all cases, whe eas he capaci y inc eased, as shown in Figu e 6. The same expe imen al beha iou was obse ed in p e ious wo ks [ 14 ]. Analyz- ing he ime e olu ion, acco ding o ou di ision in o “ini ial imes” ( ≤ 6 h) and “ inal imes” ( ≥ 23 h), a simila e olu ion in ime was obse ed o all he s udied saline solu ions. Memb anes 2023,13, 608 9 o 16 I we s udy he beha io o he ac i e laye esis ance, R al , he MgCl 2 solu ion beha ed somewha di e en ly om he es . A “ini ial imes”, i s esis ance alues began by being conside ably lowe han hose shown by he es o he sal s. Fo “ inal imes”, he solu ions es ablished in simila alues o all he sal s, excep o Na 2 SO 4 . The s ill-nega i e slope a e 30 h o R al (Na 2 SO 4 ), as well as i s e y long cha ac e is ic ime (see Table 1), would indica e ha he sys em was s ill e y a om eaching equilib ium. In any case, his beha io is expec ed o his sal since, as we will see la e , bo h om he elec ical and s e ic poin o iew, equilib ium is un a o able o he en y o his sal in o he po es. The alue o C al g ew om an almos equal alue o all solu ions o a alue o “ inal imes” ha showed he ollowing beha io : C al (CaCl 2 ) > C al (NaCl) > C al (KCl) > C al (Na 2 SO 4 ). Only he MgCl 2 solu ion p esen ed a sligh ly di e en beha io o he alue o C al : i s ly, because he alue shown a he ini ial ime was somewha highe han ha o he o he elec oly ic solu ions, and secondly, because i s empo al e olu ion had a smoo he end han o he es o he solu ions. This gen le end inally caused he alue o pa ame e Cal o he “ inal imes” o end up below ha o NaCl and CaCl2. Fo MgCl 2 , i is con enien o no e ha Mg 2+ is he smalles ion (see Table 2), which would allow a g ea e mobili y inside he po es and, hus, a low ini ial esis ance and a ela i ely high ini ial capaci y. In bo h cases, NaCl and CaCl 2 ga e mo e ou s andingly low esis ances and high capaci ies o he inal s ages. Table 2. Ionic adius, hyd a ed adius, and in ini e dilu ion di usi i y, om he li e a u e, o he ions s udied in his wo k. Ion Ionic Radius (nm) [30] Hyd a ed Radius, h. (nm) [31] Ionic Di usi i y (10−5cm2s−1) [32] Na+0.098 ±0.003 0.2356 ±0.0060 1.334 K+0.134 ±0.004 0.2798 ±0.0081 1.957 Mg2+ 0.072 ±0.002 0.2090 ±0.0041 0.706 Ca2+ 0.103 ±0.003 0.2422 ±0.0052 0.792 SO42−0.240 ±0.005 0.3815 ±0.0071 1.065 Cl—0.183 ±0.003 0.3187 ±0.0067 2.032 3.3. Ze a Po en ial The ze a po en ial o ou NF memb ane o each one o he elec oly e solu ions was calcula ed using he me hod o angen ial s eaming po en ial. In his echnique, he g adien p essu e ( ∆ P) be ween bo h ex emes o he single channel d i es o a g adien o elec ic po en ial ( ∆ U); bo h magni udes a e measu ed. Assuming ha he e is a nonze o conduc i i y only in he sec ion o he channel whe e he elec oly e solu ion is lowing, he s eaming po en ial coe icien , ( ∆ U/ ∆ P) I=0 , is ela ed o ze a po en ial ( ζ ) wi h he Helmhol z–Smoluchowski equa ion [33–35]: ζ=ηκ ε ε0∆U ∆PI=0 (7) whe e η is he dynamic iscosi y, κ is he solu ion conduc i i y, and ε ε0 he pe mi i i y o he solu ion. Unde he assump ion o g ea ly dilu ed solu ions, he dynamic iscosi y and he elec ical pe meabili y o he solu ion can be assumed o be equal o he alues o pu e wa e [36]. Fo each elec oly e solu ion, he s eaming po en ial was measu ed, using solu e concen a ions close o he alue o he eed used o EIS measu emen s. Da a co esponding o he s eaming and he ze a po en ials a e shown in Table 3. Memb anes 2023,13, 608 16 o 16 27. Möckel, D.; S aude, E.; Dal-Cin, M.; Da co ich, K.; Gui e , M. Tangen ial low s eaming po en ial measu emen s: Hyd odynamic cell cha ac e iza ion and ze a po en ials o ca boxyla ed polysul one memb anes. J. Memb . Sci. 1998,145, 211–222. [C ossRe ] 28. Del Valle Sil a, V. Theo e ical Founda ions and Modelling in Nano il a ion Memb ane Sys ems, p. 1. 2010. A ailable online: h ps://dialne .uni ioja.es/se le / esis?codigo=295607&in o= esumen&idioma=SPA (accessed on 28 Janua y 2023). 29. Sil a, V.; Ma ín, A.; Ma ínez, F.; Mal ei o, J.; P ádanos, P.; Palacio, L.; He nández, A. Elec ical cha ac e iza ion o NF memb anes. A modi ied model wi h cha ge a ia ion along he po es. Chem. Eng. Sci. 2011,66, 2898–2911. [C ossRe ] 30. Nys öm, M.; Pihlajamäki, A.; Ehsani, N. Cha ac e iza ion o ul a il a ion memb anes by simul aneous s eaming po en ial and lux measu emen s. J. Memb . Sci. 1994,87, 245–256. [C ossRe ] 31. Nys öm, M.; Linds öm, M.; Ma hiasson, E. S eaming po en ial as a ool in he cha ac e iza ion o ul a il a ion memb anes. Colloids Su . 1989,36, 297–312. [C ossRe ] 32. Masliyah, J.H.; Bha acha jee, S. Elec okine ic and Colloid T anspo Phenomena; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2006. 33. The Ze a Po en ial o Solid Su ace Analysis–A P ac ical Guide o S eaming Po en ial Measu emen | Web-Books in he Aus ia-Fo um. A ailable online: h ps://aus ia- o um.o g/web-books/en/ze a00en2014iicm (accessed on 1 Feb ua y 2023). 34. S a zak, M.E. The Physical Chemis y o Memb anes; Academic P ess: Camb idge, MA, USA, 1984; p. 334. 35. Ma cus, Y. Ionic adii in aqueous solu ions. J. Solu . Chem. 1983,12, 271–275. [C ossRe ] 36. Ma cus, Y. Ionic adii in aqueous solu ions. Chem. Re . 1988,88, 1475–1498. [C ossRe ] 37. Ci e i, A.; Pe ico, A. Ionic In e ac ions in Na u al and Syn he ic Mac omolecules, 1s ed.; Wiley: Hoboken, NJ, USA, 2012. Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). 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