Electrical Double Layers: Effects of Asymmetry in Electrolyte Valence on Steric Effects, Dielectric Decrement, and Ion Ion Correlations

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1 Cite Thi: Langmuir 18, 4, pub.ac.org/langmuir Electrical ouble Layer: Effect of Aymmetry in Electrolyte Valence on Steric Effect, ielectric ecrement, an IonIon Correlation Ankur Gupta an Howar A. Stone* epartment of Mechanical an Aeropace Engineering, Princeton Univerity, Princeton, New Jerey 8544, Unite State ownloae via PRINCETON UNIV on October 9, 18 at :4:55 (UTC. See for option on how to legitimately hare publihe article. ABSTRACT: We tuy the effect of aymmetry in electrolyte valence (i.e., non z:z electrolyte on mean fiel theory of the electrical ouble layer. Specifically, we tuy the effect of valence aymmetry on finite ion-ize effect, the ielectric ecrement, an ionion correlation. For a moel configuration of an electrolyte near a charge urface in equilibrium, we preent comprehenive analytical an numerical reult for the potential itribution, electroe charge enity, capacitance, an imenionle alt uptake. We emphaize that the aymmetry in electrolyte valence ignificantly influence the iffue-charge relation, an prior reult reporte in the literature are reaily extene to non z:z electrolyte. We evelop caling relation an invoke phyical argument to examine the importance of aymmetry in electrolyte valence on the aforementione effect. We conclue by proviing implication of our fining on iffue-charge ynamic an other electrokinetic phenomena. INTROUCTION har-phere moel with a finite iameter of ion. A vat literature iffue charge refer to the itribution of ion in an electrolyte i available in thi area, an we only icu a few report in etail. For a more in-epth review of the literature on finite ion-ize olution ajacent to a charge oli urface. The charge profile i effect, incluing more ophiticate moel for pherical ion, critical to a variety of application; in colloi cience an we refer the reaer to ref, 1, an 46. For example, microfluiic application, iffue charge i important in electrophorei, 1 electroomoi, 15 an iffuiophorei, 6,7 Borukhov et al. while in 7 erive an expreion for the concentration of ion with a finite ion ize for a valence ymmetric (or z:z, a well energy torage evice, the charge layer form the bai of electrochemical capacitor 8,9 a valence aymmetric (or non z:z electrolyte near a charge an more recently emioli flow ubtrate. Kilic et al. 1,8 an Kornyhev erive a moifie capacitor. 1 The propertie of iffue charge are typically etermine by the wiely ue GouyChapman (GC theory, which ecribe the epenence of urface charge enity q an capacitance on the potential rop ψ acro the iffue-charge region The claical GC reult are bae on a mean-fiel approximation an provie analytical expreion for q(ψ an ( ψ. ue to their relatively imple nature, GC reult continue to be wiely ue, even though it i well-recognize that they uffer from everal limitation. 17 To improve the preiction of the GC reult, everal moification have been uggete in the literature while retaining the mean-fiel framework of the claical GC reult. In thi article, we focu on three uch moification for valence aymmetric (or non z:z electrolyte: finite ion-ize effect, ielectric ecrement, an ion ion correlation, an we inicate below earlier work in each area. We firt icu finite ion-ize effect, alo commonly known a teric effect. The claical GC reult are obtaine by olving the PoionBoltzmann equation where ion are treate a point charge. Therefore, the claical GC reult preict an unphyical outcome that ion concentration can increae inefinitely with increae in ψ. Thi limitation wa recognize alreay 9 year ago by Stern, 17 Bikerman 18 an Freie, 19 an thee author accounte for the finite ion-ize effect by auming a imple PoionBoltzmann equation, an provie explicit expreion for q(ψ an ( ψ for z:z electrolyte while auming that the cation an anion iameter are equal. In thee article, the author alo icue the implication of incluing teric effect on the ynamic of iffue charge. More recently, Han et al. 9 erive the reult for q(ψ an ( ψ for z:z electrolyte but allowe the cation an anion iameter to be unequal, thu extening the reult of Kilic et al. 1,8 an Kornyhev. We emphaize that equality of cation an anion valence i a common aumption. In thi article, we relax thi aumption an we further exten the above-mentione reult to non z:z electrolyte. We how that aymmetry in cation an anion valence ignificantly influence the behavior of q(ψ an ( ψ. Secon, we focu on the ielectric ecrement effect, which refer to the ecreae in the ielectric contant ue to a reuction in the orientational polarizability of the hyrate ion with increae in electrolyte concentration. A ecreae in the ielectric contant lower the ability to tore charge in the ouble layer. The ielectric ecrement ha been recognize in everal Receive: June 18, 18 Revie: Augut 6, 18 Publihe: Augut 8, American Chemical Society OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

2 report. 18,,6,4 Here, we focu on the recent reult of Nakayama an Anelman, which ecribe the interplay between finite ion-ize effect (with equal cation an anion iameter an the ielectric ecrement for z:z electrolyte. In particular, we erive general reult for the ielectric ecrement for non z:z electrolyte while alo allowing for unequal cation an anion iameter. We then focu on the effect of valence aymmetry an how that it trongly impact the ouble layer propertie. The ionion correlation effect, alo known a the overcreening effect, relate to the interaction between nearby ion. Thi effect can be accounte for in the mean-fiel framework by efining a creening length an incluing an aitional fourthorer term in the moifie PoionBoltzmann equation, a recently erive by Bazant, Storey, an Kornyhev,,5 who icue the competition between teric effect an ionion correlation. The author howe that ionion correlation give rie to ocillation in charge enity profile, epecially for large creening length.,5 We note that the article by Bazant an Storey 5 conier 1:1 an 1: electrolyte with equal cation an anion iameter. In thi article, we exten thee reult for non z:z electrolyte with ifferent cation an anion iameter an invetigate the effect of electrolyte valence on ionion correlation. Before proceeeing further, we acknowlege that we are certainly not the firt to invetigate non z:z electrolyte. The effect of electrolyte valence ha been invetigate for the claical PoionBoltzmann equation 11,641 (e.g., Gouy, 11 Levine an Jone, 7 an Grahame 6 analyze the cenario of z /z or z /z 1/. On the other han, Lyklema 8 an Levie 4 ecribe ( ψ for a general combination of z an z, though the reult are preente in an awkwar imenional form. However, the equilibrium relationhip for non z:z electrolyte in the moifie PoionBoltzmann ecription are not reaily available. Therefore, the aim of thi article i to invetigate the impact of aymmetry in electrolyte valence on the moifie PoionBoltzmann equation. We fin that incluion of aymmetry in electrolyte valence i critical a it affect all of the effect mentione above. We preent phyical argument to broaly highlight the importance of aymmetry in electrolyte valence. Since the magnitue of valence ictate the force experience by the ion, when cation an anion are of ifferent valence, the magnitue of force experience by the cation an anion are unequal, which create an aymmetry in ouble layer propertie, or q(ψ q(ψ an ( ψ ( ψ. Thi apparent breaking of ymmetry can have ignificant implication. For intance, everal experimental ata et publihe in the upercapacitor literature utilize valence aymmetric electrolyte uch a Na SO 4 an CaCl. 4,4 However, the moeling approache in thi area are till typically retricte to z:z electrolyte, 44 an thu tuie have not exploite valence aymmetry a a way to tune the energy an power enity of upercapacitor. Similarly, valence aymmetry can be important for capacitive eionization,,45 a proce where toxic ion migrate move from the bulk to the ouble layer. Here, capacitance woul influence the quantity of toxic ion eplete a well a the time require for the epletion. Since ( ψ ( ψ for valence aymmetric ion, the irection of the potential rop will impact the efficacy of the proce. We alo emphaize that our analyi i general ince we conier a combination of valence aymmetry with other effect (e.g., finite ion ize, ielectric ecrement, an ionion correlation. To highlight the relative importance of imultaneou effect, we now preent a phyical argument for the cenario where the combine effect of valence aymmetry an finite ion ize are relevant. Let u aume that in an electrolyte, anion have a higher valence than the cation. At the ame time, the anion have a ignificantly maller ion ize than the cation. When uch an electrolyte come in contact with a poitively charge urface, the anion migrate towar the charge urface an cation move away from the urface. The higher valence of the anion lea to a rapi increae in the anion concentration with increae in ψ until there i no longer pace to accommoate more anion. Therefore, a higher valence implie that the ouble layer aturate with anion at a maller value of ψ. In contrat, the maller anion ize implie that each ion occupie a maller volume, an thu aturation of the ouble layer with anion occur at larger value of ψ. Therefore, valence aymmetry can compete or cooperate with other aitional effect an can provie flexibility in eign of procee when the aitional effect are ignificant. In thi article, we tuy the influence of aymmetry in electrolyte valence on finite ion-ize effect (with unequal cation an anion iameter, ielectric ecrement, an ionion correlation. For each of thee effect, we firt erive the iffue-charge relation for a general valence electrolyte an provie analytical an numerical reult for the potential itribution, q(ψ, ( ψ an imenionle alt uptake α(ψ. Uner appropriate aumption, we recover previouly reporte reult for q(ψ an ( ψ, thu highlighting the generality of the propoe relation. Since ( ψ ha multiple local extrema, we provie caling relation to better explain the epenence of the extrema on ifferent parameter. Latly, we icu the implication of our reult on iffue-charge ynamic an imenionle parameter that govern the electrokinetic phenomena. We conclue by proviing limitation of the moel an irection for future reearch. PROBLEM SETUP We conier an electrolyte in equilibrium with a charge urface (Figure 1. ue to electrotatic attraction, oppoitely charge Figure 1. An electrolyte with cation valence z an anion valence z i near a charge urface. The Stern layer thickne i enote λ S, an the ebye length i enote λ. The potential at the electroe i taken a ψ ψ, the potential at the bounary between the Stern layer an iffue layer i ψ ψ, an the potential in bulk i zero OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

3 ion (alo referre a counterion migrate towar the charge urface, an compete with thermal or entropic effect to create a region of iffue charge. The typical thickne of the region of exce charge, or ouble layer, i given by the ebye length λ 15,16,8,46 λ εε kt B e z c, (1 i i i where ε i the electrical permittivity of vacuum, ε i the ielectric contant of the olution without the electrolyte, k B i the Boltzmann contant, T i temperature, z i i the valence of the i th ion type, c,i i the bulk concentration of the i th ion type, an e i the charge on an electron. The um i over all ionic pecie preent in the olution. The region cloe to the electroe where ion are aorbe at the urface i known a the Stern layer (Figure 1. The (molecular thickne of the Stern layer i enote a λ S. To be pecific, we conier an electrolyte with one cation type an one anion type. The cation an anion valence are enote z an z, repectively. Since the electrolyte in bulk i neutral, the ion concentration in bulk are c z c an c z c. For intance, K SO 4 i enote by z 1,z,c c, an c c. Therefore, for a z :z electrolyte, accoring to equation 1, λ i given a εε kt B λ zz ( z z ec ( FINITE ION-SIZE EFFECTS In thi ection, we conier finite ion-ize effect an pecifically focu on the effect of valence aymmetry on ouble layer propertie. For implicity, we aume that the relative permittivity ε i inepenent of c ±,orε(c ± ε. However, we icu the effect of a change in ielectric contant ε(c ± eparately in the next ection. Similarly, we alo exclue the ion ion correlation in thi ection but icu their effect in the ubequent ection. erivation. Potential itribution an Charge Accumulate. To inclue the finite ion-ize effect, we aume a harphere moel where ion are pherical an are ecribe with an effective iameter. epening on the interaction between ifferent ion type, the effective iameter may or may not be the ame a the ion iameter. We refer the reaer to ref, 1, an 8 for a more etaile icuion on effective iameter an the length cale of interaction. Here, we enote the effective 1 iameter of a cation a a. Typically, a (1 1 nm, an thu the concentration of cation cannot excee 1 4 (1 1 m. 1 We alo allow for aymmetry in a the effective iameter of anion an cation an efine the effective iameter of an anion a a. Therefore, the concentration 1 of anion cannot excee. a The free energy of the ytem per unit volume F i efine a F U TS ( where U i the internal energy per unit volume, S i the entropy per unit volume, an F(ψ,c ±. U i efine a,1, εε ψ U zec ψ zec ψ x (4 where ψ(x i the potential at a location x relative to a reference potential at x, orψ(. The firt term in equation 4 repreent the energy tore in the electric fiel, an the remaining two term account for the potential energy of the ion. To evaluate S, we ue Boltzmann formula of mixing, S k B ln ω, where ω i the number of microtate. To etimate ω, we firt etimate the number of way to arrange the larger ion an then multiply with the number of way to arrange the maller ion. Here, for convenience, we aume that a a. We etimate entropy a S 1 ac c ln( a c ln(1 ac kb a ac (1 ac ac c ln 1 ac a ac ac ln 1 1 ac An equivalent expreion of entropy for a a can be etimate by witching the poitive an negative ubcript. With F(c ± given by equation 5, we evaluate the chemical potential μ ± a μ μ F c (5 ac a ac ac ze ψ kt ln ln 1 B 1 ac a 1 ac (6a F c ac ze ψ kt B ln 1 ac ac (6b c, zc eψ. kt B We efine imenionle concentration a n c n, an the imenionle electric potential a Ψ zc At equilibrium, the chemical potential i contant for all x, μ ± (x μ ± (, an we obtain n n exp( z Ψ g( Ψ (7a exp( zψ f( Ψ g( Ψ (7b g( Ψ f( Ψ z a c[exp( z Ψ f( Ψ] zacf ( Ψ[exp( zψ 1] (7c f ( Ψ 1 zac Ψ (exp( z 1 1 zac a / a1 (7 Phyically, g(ψ account for the reuction in concentration ue to finite ion ize, an f(ψ account for the change in concentration ue to the contrat in ion ize [note f(ψ1 for a a ]. For a ± in equation 7, we recover the Boltzmann itribution. For a a an z z, we recover the tanar reult in ref 1, 7, an 8. Furthermore, for a a an z z, we recover the known reult in ref 9. To olve for c ± an ψ,we couple equation 7 with Gau law, 1197 OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

4 Table 1. Summary of Q( Ψ an ( Ψ Relation That Account for Ion Valence an Finite Ion Size a conition iffue charge relation ref z z z a Q ± z z a ± zψ inh z zψ coh Q gn( Ψ z exp( zψ Ψ z exp( z 1 zz z z exp( z Ψ exp( z Ψ ( z z Q z z z 1 a a a Q gn( Ψ ln 1 4za c inh zac inh zψ Ψ z 1 4za inh z Q z z z a a ( exp( zψ exp( zψ f( Ψ n, n g( Ψ g( Ψ zψ 11 an 1 8 an 4, 1, an 8 9 f ( Ψ 1 zac Ψ (exp( z 1 1 za c a / a 1 z z a a g( Ψ f( Ψ za c (exp( zψ f( Ψ za c f( Ψ(exp( zψ 1 1 Q gn( Ψ ln g( Ψ zac n ( Ψ n ( Ψ zq exp( z Ψ n, n g( Ψ exp( zψ f( Ψ g( Ψ thi work f ( Ψ 1 zac Ψ (exp( z 1 1 zac a / a 1 a For implicity, we aume Λ S. g( Ψ f( Ψ z a c (exp( z Ψ f( Ψ z a c f( Ψ(exp( z Ψ 1 Q gn( Ψ ln g( Ψ zz( z z ac n ( Ψ n ( Ψ ( z z Q ψ εε ezc ( zc x (8 which i to be olve with bounary conition ψ S x x ψ ψ( λ an ψ(, where ψ( ψ i the potential rop acro the iffue layer (Figure 1. We note that the bounary conition at the electroe aume a thin Stern layer. We nonimenionalize with Ψ eψ, X kt B λ Λ λ S λ S Ψ n X z to obtain n z with two bounary conition x, an ( OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

5 Ψ Ψ Λ Ψ S X (1a X Ψ( (1b Equation 7 an 9 are governe by ix imenionle parameter: z, z, a c, a /a, Λ an Ψ (the potential meaure on the oli bounary. To olve for Ψ an n ±,we aume z, z, a c, a /a, Λ S, an Ψ are pecifie. Typically, 1 1 a± c (1 1. For a more etaile icuion on the phyical interpretation of a ± c an the range of poible value of a ± c, we refer the reaer to ref 1. We multiply both ie of equation 9 by Ψ an integrate once (uing equation 1btofin X Ψ gn( Ψ ln g( Ψ X zz ( z z ac (11 where gn(ψ i the ign function. For a c, a /a 1, an z Ψ z Ψ z z, equation 11 become inh, an another X integration yiel the well-known relation z Ψ z Ψ tanh tanh exp( X, where Ψ i relate to Ψ through the bounary conition in equation 1a. For a, a /a 1, an z z, we numerically integrate equation 11 with the Stern layer bounary conition (equation 1a to obtain Ψ(X, an the reult are icue later. Next, we evaluate the urface charge enity on the electroe ψ a q εε. Similarly, we can alo calculate the x x capacitance (i.e., the charge tore in the electroe per unit q total potential rop, or ψ. It i convenient to q nonimenionalize Q an λ, uch that zz ( z z eλ c εε Q Ψ Q an X X Ψ. Thu, from equation 11, wefin the imenionle urface charge enity Q gn( Ψ ln g( Ψ zz ( z z ac (1 Capacitance. To calculate Ψ, we write the Stern layer bounary conition in equation 1a a Ψ Ψ Λ S Q. ifferentiating thi relation with repect to Ψ, we get Ψ Q Q Ψ Λ. Thu, Ψ,or Ψ 1 o that 1 S Ψ Q Ψ 1 Λ S Q Ψ ( z z ln g( Ψ 1 Λ zzac n ( Ψ n ( Ψ 1 (1 S (14 Equation 1 an 14 emontrate the well-known reult that we can characterize the ytem a an electrical circuit with a capacitor repreenting the Stern layer an a capacitor repreenting the iffue layer in erie. For a, a /a 1, an z z z, equation 1 an 14 take the form Q zψ inh z zψ ech Λ 1 S (15a (15b which are the claical GC relation. 11,1 For a an a /a 1 but ifferent ion valence, equation 1 an 14 become Q gn( Ψ z exp( zψ Ψ z exp( z 1 zz z z (16a 1 z z exp( z Ψ exp( z Ψ z exp( zψ Ψ z exp( z 1 ΛS zz z z (16b which are conitent with the reult reporte in ref 8. However, the reult analogou to equation 16 reporte in ref 8 are preente in an awkwar imenional form. Taking the limit of z z z in equation 16, it i eay to recover the GC relation in equation 15. Latly, for z z z an a a a, equation 1 an 14 are evaluate a 1 Q gn( Ψ ln 1 4za c inh zac 1 zψ z Ψ 1 4za c inh inh zψ 1 zψ ln 1 4 za c inh Λ za c S (17a (17b which agree with the relation preente in ref 1 an 8. We ummarize the valiity of the aforementione iffue charge relation for Q(Ψ an C(Ψ intable 1. To the bet of our knowlege, equation 1 an 14 are the mot general charge an capacitance relation reporte in the literature accounting for ion valence an finite ion ize. Salt Uptake. Both Q an are meaure of the net charge inie the ouble layer. However, the formation of a ouble layer alo eplete alt from the bulk. A note in ref an 45, the amount of alt uptake irectly ictate the ynamic of the ouble layer formation proce, a explaine later. We efine the length cale of the bulk a L an etimate a imenionle meaure of the exce alt uptake a L ( c c ( z z c x α ( z z c L (18 When α 1, the olution to ouble layer charging for an electrolyte between two parallel plate can be reliably approximate a an electrical circuit. However, if α (1, thi approximation i no longer reliable. Therefore, etimation of α i important for time-epenent problem. To etimate α, we utilize equation 7 an 11 to obtain λ α L acz ( z z Ψ z ( n 1 z ( n 1 Ψ ln g( Ψ (19 where n, n, an g(ψ are evaluate from equation 7. In the limit a, a /a 1, an z z z, we recover the well OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

6 ocumente reult 4 of α λ z inh Ψ. We numerically L 4 integrate equation 19 to evaluate the epenence of α on ifferent parameter, an the reult are icue later. Potential itribution. We now icu the numerical olution to equation 11 with the Stern layer bounary conition (equation 1a. We firt report the effect of change in a c on Ψ(X with Ψ 5,z 1,z,a /a 1, an Λ S ; ee Figure a. Phyically, for a larger value of a c (i.e., larger teric effect Q i maller. Since Q Ψ, a larger a X c implie a more X graual ecreae in Ψ with X. Figure. Effect of aymmetry in electrolyte valence on finite ion-ize effect for Q(Ψ, a given by equation 1 an. The oli line repreent reult from equation 1, an the otte line enote reult from equation. (a ifferent a c with z 1,z, an a /a 1. (b ifferent z an z with a c. an a /a 1. (c ifferent a /a with z 1,z, an a c.. Figure. Effect of aymmetry in electrolyte valence on finite ion-ize effect for Ψ(X, a given by the numerical olution of equation 11 with bounary conition (equation 1a for Ψ 5. (a ifferent a c with z 1,z,a /a 1, an Λ S. (b ifferent z an z with a c., a /a 1, an Λ S. (c ifferent a /a with z 1,z,a c., an Λ S. ( ifferent Λ S with z 1,z,a c., an a /a 1. The effect of change in z an z on Ψ(X with Ψ 5,a c., a /a 1, an Λ S i provie in Figure b. The tren how that the fatet ecay in Ψ occur for z z 1, wherea the ecay i lowet for z,z 1. Phyically, for a c., the ion concentration i high even in the bulk, an thu, finite ion-ize effect are important. We aume that for Ψ 5, c 1/a, z c ( z c (, an it can be etimate that Ψ Ψ X z ( z z a c ; ee equation 9. Thi approximation explain the tren we oberve in Figure b. Similarly, for z 1, z,a c., an Λ S,wefin that a maller a increae Ψ the magnitue of, an thu the change in Ψ i more rapi for X a maller a, a oberve in Figure c. The effect of Λ S on Ψ(X enter through the bounary conition (equation 1a. We preent the reult of change in Λ S on Ψ(X with Ψ 5,a c., z 1,z, an a /a 1 in Figure. A thicker Stern layer, or a larger Λ S, implie a larger potential rop acro the Stern layer. Therefore, we ee that Ψ Ψ( ecreae for an increae in Λ S. Further, a maller Ψ inicate a lower Q (ee below, an thu for a larger Λ S, the rate of ecay of Ψ i maller. Charge Accumulate. We preent the epenence of accumulate charge Q on ifferent parameter accoring to equation 1. Figure a how the epenence of Q with Ψ for ifferent value of a c with z 1,z, an a /a 1. A larger a c implie that teric effect are tronger, an thu Q i maller. Increaing Ψ increae the concentration of ion, an thu Q increae. For a large Ψ, teric effect tart to become more important, an the increae in Q i maller ince the rate of change in ion concentration i lower. Next, we conier valence aymmetry for a c. an a /a 1. Since finite ion-ize effect are ignificant here, for Ψ >,we aume c ( 1/a an z c ( z c ( to obtain Ψ Ψ Q X x z ( z z ac ( We fin goo agreement between compute value from equation 1 an approximate value from equation, epecially for large Ψ, ince concentration approximation are more accurate for large Ψ. Q i highet for z z 1 followe by z 1,z, an z,z 1 (ee Figure b. Similarly, for a c., z 1, an z, a maller a lea to a larger Q, a preicte by equation. Thi obervation i corroborate in Figure c. Latly, Λ S oe not influence the variation of Q veru Ψ. However, for the ame value of Ψ, value of Ψ will be maller for a larger Λ S (equation 1a, ee Figure. Capacitance. Capacitance i a meaure of the amount of Q charge tore per unit total potential rop (i.e.,. We icu Ψ the epenence of on ifferent parameter bae on equation 14. Figure 4a plot the variation of with Ψ for ifferent value of a c with z 1,z,a /a 1, an Λ S a contant. epening on the value of a c, capacitance exhibit ifferent behavior. For ilute ion concentration in the bulk (i.e., a c (1, (Ψ iplay a camel hape with one local minimum an two local maxima. Phyically, thi occur becaue for mall value of Ψ, counterion concentration increae with increae in Ψ. For large value of Ψ, the counterion concentration aturate aroun Ψ Ψ,max, beyon which the capacitance ecreae. Figure 4a how that the curve are aymmetric when the cation an anion valence are not equal, or ( Ψ ( Ψ for z z. We note that the location of the minimum Ψ Ψ,min for z 1 an z, unlike valence ymmetric electrolyte. Similarly, the location of two maxima are not equal an oppoite for unequal cation an anion valence. For large bulk ion concentration (i.e., a c (1 1, ( Ψ curve how a bell hape with no local minimum an one local maximum. Since the ion concentration i high even in bulk, the local minimum iappear an only one local maximum remain. We fin that ince cation an anion valence are unequal, Ψ,max. Though the camel hape an bell hape curve have been OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

7 an increae in Λ S reult in the effective capacitance to be ictate by the Stern layer capacitance. Salt Uptake. In thi ection, we ecribe the variation of the imenionle alt uptake α(ψ a given by equation 19. A note previouly, α ictate the ynamic of ouble layer charging. Figure 5a how the epenence of α for ifferent a c Figure 4. Effect of aymmetry in electrolyte valence on finite ion-ize effect for ( Ψ, a given by equation 14 an 1. The oli line repreent reult from equation 14, an the otte line enote reult from equation 1. (a ifferent a c with z 1,z,a /a 1, an Λ S. (b ifferent z an z with a c., a /a 1, an Λ S. (c ifferent a /a with z 1,z,a c., an Λ S. ( ifferent Λ S with z 1,z,a c., an a /a 1. reporte previouly, here we emphaize that the hape an the Ψ,max an Ψ,min are ignificantly influence by z an z. In the ubequent ubection, we ue a caling analyi to etail a more quantitative etimate of extrema an their epenence on z an z. Next, we preent the reult for a c., a /a 1, an Λ S but with ifferent cation an anion valence in Figure 4b. We fin that the poition of a local maximum in the bell hape capacitance i alo ictate by the valence an Ψ,min > when z > z an Ψ,min < when z > z. We note that an approximation for i poible by auming c ( 1/a, z c ( z c ( for Ψ < an c ( 1/a, z c ( z c ( for Ψ >. We can etimate by ifferentiating equation to obtain 1/ ( Ψ z ( z z a c for Ψ < 1/ ( Ψ z ( z z a c for Ψ > (1a (1b Equation 1 i more accurate for large a ± c an Ψ ince the aumption for ion concentration are more reaily atifie. Therefore, equation 1 preict a ecreae in with an increae in Ψ an oe not preict the extrema near Ψ. However, it correctly capture the tren an relative poition of reporte in Figure 4b for Ψ (1 for ifferent combination of z an z. We note that equation 1 ugget that alo epen on a / a. Typical reult are preente in Figure 4c with a c., z 1, z, an Λ S for ifferent a /a. Equation 1 explain the collape of curve for Ψ < an the increae in for higher a /a for Ψ >. Latly, we icu the effect of Λ S. For ifferent value of Λ S, Figure 4 preent the variation of with Ψ for ifferent with a c., z 1,z, an a /a 1.Wefin that ecreae with an increae in Λ S an become almot inepenent of Ψ for larger value of Λ S. Thi change in behavior occur ince the Stern layer capacitor an the iffue layer capacitor are in erie; Figure 5. Effect of aymmetry in electrolyte valence on finite ion-ize effect for α(ψ, a given by equation 19. (a ifferent a c with z 1, z, an a /a 1. (b ifferent z an z with a c. an a /a 1. (c ifferent a /a with z 1,z, an a c.. with z 1,z, an a /a 1. A expecte, an increae in a c ecreae alt uptake ince the ion concentration aturate ue to finite ion-ize effect. In Figure 5b, we preent the epenence of valence on alt uptake for a c an a /a 1.Wefin that αλ /L i larget for z 1,z 1 followe by z,z 1, an z 1,z. Thi tren occur ince for Ψ >, anion concentration aturate inie the ouble layer. Thi aturation occur for maller value of Ψ for z 1,z, an thu, lower alt i eplete from the bulk when compare to z,z 1, an z z 1. Further, though thee two cae might achieve anion concentration aturation at imilar value of Ψ, a larger number of anion in the bulk for z,z 1 lea to a lower level of alt epletion. Figure 5c ummarize the effect of a /a for a c., z 1, an z.wefin that alt epletion i maximum for maller value of a ince the aturation concentration of anion i larger, an thu more alt can be eplete. A quantitative preiction of α(ψ, imilar to equation an 1 for Q an, i challenging ince we nee to fin an approximate ecription for ψ/x for all x (an not jut at x a in equation an 1; ee equation 18. Scaling Anlayi. A unique feature of the erive iffuecharge relation i the preence of extrema in the epenence of with Ψ an their epenence on the value of z an z.we now preent phyical argument to preict the location of local extrema. Local maxima occur when the ion concentration inie the iffue layer i on the orer of 1/a ±. For Ψ >, negative ion will be attracte an the conition for a local maximum implie c (1/ a. On the other han, for Ψ <, the conition for a local maximum become c (1/ a. Thu, auming the c ± follow the Boltzmann itribution, we etimate or,max z c exp( z Ψ (1/ a for Ψ,max z c exp( z Ψ (1/ a for Ψ 1 Ψ ( z ln( z a c forψ,max 1 Ψ ( z ln( z a c forψ,max (a (b (a (b Equation emontrate that Ψ,max i trongly influence by z an z. We oberve a goo quantitative agreement between OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

8 the preiction of equation an the compute value (obtaine from equation 14, a illutrate in Figure 6 (panel Figure 6. Scaling analyi of finite ion-ize effect. Comparion of compute Ψ,max with eq for (a Ψ < an (b Ψ >. (c Comparion of compute Ψ,min with equation 4. The re point are cae with the camel hape (i.e., two local maxima an one local minima, an the blue point are cae with the bell hape (i.e., one local maximum; ee Figure 4. For computation, 1 6 a c., 1 a / a 8, 1 z, an 1 z. a an b. The caling relation accurately capture the behavior, epecially for the camel-hape capacitance curve (i.e., with two local maxima an one local minima. However, the caling relation i not a accurate for the bell hape capacitance curve (i.e., only one local maximum an no local minimum, ince the aumption that concentration follow a Boltzmann itribution i le accurate. Neverthele, equation correctly capture the epenence of Ψ,max on z, z, a c, an a /a. We now preent a phyical argument to preict Ψ,min, which i only oberve in the camel hape capacitance curve. We know that the magnitue of charge per unit volume carrie by the cation an anion i proportional to z c an z c, repectively. For intance, in bulk, by efinition, z c z c, an the charge balance. However, the magnitue of the rate of change in charge with Ψ of cation an anion i proportionate to z c an z c (auming the Boltzmann itribution, ee equation 7. We argue zc that Ψ,min can be etimate when (1. Auming the zc Boltzmann itribution, thi conition yiel ln( z / z Ψ,min z z (4 Therefore, phyically, Ψ,min i the potential at which the rate of change of both poitive an negative charge with Ψ are equal. ue to ymmetry, thi occur at Ψ for z z 1.We how that preiction of equation 4 are in goo agreement with compute reult (obtaine from equation 14 infigure 6c. Phyical Significance. We now icu the phyical ignificance an implication of valence aymmetry of finite ion-ize effect. For thi icuion, we briefly retore imenion for charge q(ψ an capacitance ( ψ. By converting equation an 1 to imenional form, we obtain q ( εεz a ψ for ψ < 1/ 1/ q ( εεz a ψ for ψ > 11/ (.5 εεza ψ for ψ < 11/ (5a (5b (5c (.5 εεza ψ for ψ > (5 Equation 5 clearly how that q(ψ q(ψ an ( ψ ( ψ only when z a z a. Phyically peaking, z /a an z /a are, repectively, a meaure of the maximum poitive charge enity an negative charge enity that can be tore inie the ouble layer. Therefore, the conition z a z a implie that the ouble layer formation i ymmetric only when the magnitue of maximum charge enitie accumulate inie the ouble layer are the ame irrepective of the ign of the potential rop. Moreover, the iniviual factor z a an z a combine the relative importance of ion valence an finite ion ize an ugget that a higher valence an a lower ionic iameter increae the amount of charge tore an the capacitance. Thi reult i conitent with phyical intuition ince increaing the valence increae the charge tore per ion, an a maller ion ize allow for a larger number of ion per unit volume to accumulate in the ouble layer. Moreover, the quare root epenence highlight a quantitative feature that i important for practical application. A an example, we conier the cae of CaCl, which i a poible electrolyte caniate for upercapacitor, 4 among many application. For thi alt, z,z 1,a.11 nm, an ( 1/ za.71 a.17 nm. Here, the factor za, which inicate that for the ame magnitue potential rop acro the charge urface, when the urface i negatively charge, the ouble layer will accumulate almot thrice a much net charge. Thi break of ymmetry i ignificant for upercapacitor application where the amount of charge tore ictate the energy enity. We note that the aymmetry in the ouble layer propertie that arie when z z i the novel apect of thi work an create opportunitie for future reearch. IELECTRIC ECREMENT EFFECT In thi ection, we relax the aumption of a contant ielectric contant (i.e., ε(c ± ε. A previouly icue, change in the ielectric contant with ion concentration can reuce the ability to tore charge inie a ouble layer, an thu can have a major impact on iffue layer propertie. Here, we aume that the ecreae in the ielectric contant i linear with ion concentration, or ε( c± ε γ c γ c (6 where γ ± are contant. Though equation 6 i not obtaine from a theoretical erivation, experiment have hown that thi epenence work reaonably well for ion concentration up to a few molar.,4,47 Typical value of γ ± range from 7 6 γ (1 1 ± m (ee reference. erivation. To erive charge an capacitance relationhip for a variable ielectric contant, equation 5 remain ientical except ε i replace by ε(c ±, a given by equation 6. ue to the epenence of the ielectric contant on c ±, the chemical potential μ ± alo have a epenence on ε, or ε ε ψ μ F ac a zeψ kt B ln c c x 1 ac a ac ac ln 1 1 ac (7a ε ε ψ μ F ac zeψ kt B ln c c x 1 ac ac (7b Though it i poible to fin an explicit relationhip for ψ concentration by equating μ ± (xμ ± (, the preence of x in equation 7 make the expreion inconvenient. Therefore, we OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

9 exploit the relationhip evelope in ref 4 for omotic preure π(x, i.e. εε ψ π( x z ec ψ z ec ψ TS c μ c μ x Uing equation 5, 6, an 7 in equation 8 yiel (8 equation ha been erive in the limit a ± an a a, the reult are reaily extene to the general cae. Alo, we fin that in the limit γ c an γ /γ 1 (the abence of the ielectric ecrement, equation give reult conitent with equation 16. Potential itribution. We preent the variation of Ψ(X for Ψ 5inFigure 7. Firt, we icu the effect of the change in ε ψ π( x ε γ c γ c ( x 1 kt ln(1 1 1 ac ac ln(1 ac B a a a (9 Furthermore, utilizing the equilibrium requirement π(x π( an μ ± (x μ ± (, we obtain three equation to relate c ± (x, ψ(x, an. Thu, uing thee three equation at x, ψ x ψ x we can evaluate c ± (ψ an (ψ, an by extenion evaluate q(ψ an ( ψ. To implify our calculation, we aume that teric effect an Stern layer effect are negligible. However, a clear from the above erivation, no uch retriction i neceary an the reult can alo be evaluate for the general cenario. We note that ince we have ue omotic preure to generate an aitional relationhip, we have not utilize Gau law with variable ielectric contant. We firt evaluate π(x π( an μ ± (x μ ± ( in the abence of teric effect (i.e., in the limit a c an a /a 1. Conitent with the earlier icuion, we efine imenionle variable a n, n zc obtain c c, zc X x λ, an Ψ eψ, an kt B Ψ ε ( z ( n 1 z ( n 1 X ( z z zz( ε γ czn γ czn (a z ( n 1 z ( n 1 γ c Ψ z ln( n ε γ cz γ n czn (b z ( n 1 z ( n 1 γ c Ψ z ln( n ε γ cz γ n czn (c Equation i governe by the imenionle parameter z ±, γ c, γ /γ, an ε. We olve equation numerically for pecifie value of Ψ( Ψ, z ±, γ c, γ /γ, an ε 8 (typical of water an obtain the function n ± (Ψ an Ψ ( Ψ X. Once we have obtaine Ψ ( Ψ X, it i traightforwar to obtain Ψ(X through numerical integration. Next, we evaluate Q ( ε γ cz n γ czn Ψ ε X X. Moreover, ince we aume Λ S, then Ψ Ψ (ee equation 1a, an Ψ, which i evaluate through numerical ifferentiation. We alo numerically calculate the imenionle alt uptake a α λ L Ψ z ( n 1 z ( n1 Ψ ( z z X Q Ψ. We note that though Figure 7. Effect of aymmetry in electrolyte valence on ielectric ecrement for Ψ(X, a given by the olution of equation for Ψ 5. (a Effect of γ c with z 1,z, an γ /γ 1. (b Effect of z an z with γ c an γ /γ 1. (c Effect of γ /γ with γ c with z 1 an z.ε 8 i aume for all calculation. γ c with z 1,z, an γ /γ 1. The change in Ψ(X i le rapi with an increae in γ c a hown in Figure 7a. In the ielectric ecrement effect, much like the finite ion-ize effect, the concentration of the counterion aturate beyon ome Ψ. Here, the aturation occur becaue a lower ielectric contant implie that the charge torage capacity of the olution i reuce, an thu the concentration of the counterion aturate. Since it i ifficult to pare the epenence of ifferent parameter from equation, we preent a implifie moel to unertan the effect of the ielectric ecrement. Since Ψ >, we aume that z c ( z c ( an c. Thee aumption phyically imply that the x x majority of the repelle ion have been eplete an that the ielectric ecrement lea to a aturation of the counterion at the urface, an thu the graient of the counterion vanihe. Thee aumption allow u to implify Gau law at x a ψ εε( c± ezc ( zc x x x ψ εε zec x x x x (1 Next, we invoke the chemical potential equality μ (x μ ( with a ± inequation 6 an 7 to get εγ ψ c( x ze ψ ktln B x zc ( We ifferentiate equation with repect to x an utilize equation 1 to evaluate c ( an ψ a x x ε c ( γ (a ε( ( ε / c ψ x x gn( ψ ze ψ ktln εγ ε B γ zc ( (b (c OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

10 Nonimenionalizing equation c, we arrive at ( 1 ε ε Ψ Ψ ln z γ z c gn( Ψ X X z ( z z γ c (4 We emphaize that equation 4 i an approximation an c aume that z c ( z c ( an. Moreover, the x x argument inie the quare root nee to be poitive, or 1 ε Ψ >, an thu the relation i only applicable for ln ( z γ z c large Ψ. However, there are ueful inight to be gaine from equation 4. The equation ugget that increaing γ c lea to a more graual ecay in Ψ, conitent with the numerical obervation in Figure 7a. The variation of Ψ with z an z i preente in Figure 7b for Ψ γ c an γ /γ 1. From equation 4, we learn that i X X larget for z 1,z 1, followe by z 1,z, an z,z 1. Thi tren i conitent with the reult hown in Figure 7b. However, equation 4 i only vali at X an the variation in Ψ for other value of X are not capture in equation 4. Nonethele, the reult in Figure 7b an equation 4 clearly how that Ψ(X epen on the cation an anion valence. Next, we icu the effect of γ /γ with γ c,z 1, an z.we oberve in Figure 7c that an increae in γ /γ lea to a more rapi ecay in Ψ with X, although the ifference i relatively minor. Thi obervation i conitent with the preiction of equation 4. Charge Accumulate. We now icu the epenence of the charge accumulate Q on Ψ. Since the ielectric ecrement aturate the counterion concentration, an increae in γ c reuce Q; ee Figure 8a where the tren are preente for z 1,z, an γ /γ 1. By utilizing equation, we preict Q gn( Ψ ( 1 ε ε Ψ ln z γ z c 1 ε for Ψ > ln z ( z z γ c z γ z c (5 Figure 8. Effect of aymmetry in electrolyte valence on ielectric ecrement for Q(Ψ, a given by the olution of equation an 5. The oli line repreent reult from equation, an the otte line enote reult from equation 5. (a Effect of γ c with z 1,z, an γ /γ 1. (b Effect of z an z with γ c an γ /γ 1. (c Effect of γ /γ with γ c with z 1 an z 1.ε 8 i aume for all calculation. Equation 5 how that increaing γ reuce Q, conitent with the tren oberve in Figure 8a. The epenence of Q on z an z i preente in Figure 8b. We fin qualitative agreement between the compute value from equation with the preiction of equation 5. Figure 8c how the variation of Q on γ /γ for γ c,z 1, an z. The reult ugget a larger Q for a maller γ, qualitatively conitent with the preiction of equation 5. The iagreement between the olution from equation an 5 occur ince the approximation of z c ( z c ( an ε c ( i more accurate for Ψ γ (1. Nonethele, equation 5 provie a convenient analytical expreion to infer the epenence of ifferent parameter on Q. Capacitance. We now focu on the epenence of capacitance on Ψ. Figure 9a how the epenence of Figure 9. Effect of aymmetry in electrolyte valence on ielectric ecrement for ( Ψ, a given by the olution of equation an 6. The oli line repreent reult from equation, an the otte line enote reult from equation 6. (a Effect of γ c with z 1,z, an γ /γ 1. (b Effect of z an z with γ c an γ /γ 1. (c Effect of γ /γ with γ c with z 1 an z 1.ε 8 i aume for all calculation. on γ c for z 1,z. an γ /γ 1. Firt, we note that for γ c (i.e., no ielectric ecrement, ha only one local minimum at Ψ Ψ,min <. Thi repone ha been ecribe in etail in the previou ection; ee equation 4. We oberve that increae in γ c lea to a ecreae in. In aition, with finite ielectric ecrement, we tart oberving a maximum in for Ψ Ψ,max, leaing to the camel hape curve. For very large γ c, we fin that Ψ,min iappear an only one of the maxima Ψ,max remain, leaing to a bell hape curve, imilar to the finite ion-ize effect (ee Figure 4a. However, increae in γ c alo influence the at Ψ unlike the increae in a c for finite ion-ize effect; ee Figure 4a. To unertan the capacitance repone more quantitatively, we buil on our implifie moel for ielectric ecrement. Here, we aume that for Ψ >,z c ( z c ( an ε c ( γ (ee equation, an thu by extenion for Ψ <,z c ( z c ( an c ε (. On the bai of thee aumption, we γ Q previouly erive Q(Ψ (ee equation 5 an Ψ i thu calculate a ε ε γ 1 Ψ 1 8 z ( z z c ln γ for z z c 1 ε Ψ > ln, z γ z c 1 ε ε 1 8 z ( z z γ c Ψ ln for z γ z c 1 ε Ψ > ln z γ z c (6 We invetigate the effect of z an z on for γ c an γ /γ 1inFigure 9b. We fin that change in z an z create aymmetry in the capacitance curve. We are able to qualitatively capture the aymmetry an relative poition for ifferent combination of z an z in equation 6. However, equation 6 oe not preict a maximum an ugget that i a trictly 1198 OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

11 ecreaing function with Ψ. Thi icrepancy between the compute reult from equation an approximate reult from equation 6 arie ue ince the aumption of cation an anion concentration are le accurate for Ψ (1. InFigure 9c, we ecribe the epenence of with Ψ for ifferent γ /γ, γ c, z 1, an z.wefin that ecreaing γ increae for Ψ >, wherea keeping γ contant collape curve for Ψ <, conitent with the preiction of equation 6. Salt Uptake. An overview of the effect of ielectric ecrement on imenionle alt uptake i provie in Figure 1. Firt, we focu on the effect of γ c for z 1,z, an γ /γ ε Ψ 1,max z ln Ψ γ for, z c ε 1 Ψ,max z ln for Ψ z γ c (8 We hypotheize that to preict a local minimum follow the ame argument a before; ee equation 4. We ummarize our reult from the caling analyi in Figure 11. We fin that both equation 8 an 4 are in goo agreement with the compute value. Figure 1. Effect of aymmetry in electrolyte valence on ielectric ecrement for α(ψ, a given by the olution of equation. (a Effect of γ c with z 1,z, an γ /γ 1. (b Effect of z an z with γ c an γ /γ 1. (c Effect of γ /γ with γ c,z 1, an z 1.ε 8 i aume for all calculation. 1. We oberve in Figure 1a that an increae in γ c ecreae αl/λ. Thi repone i expecte ince an increae in ielectric ecrement lea to a larger aturation of ion concentration, an thu le alt i aborbe in the iffue layer. Figure 1b preent the variation of αl/λ on z an z. On the bai of our analyi for finite ion-ize effect, here alo, we expect αl/λ to be lowet for z 1,z ince aturation woul occur at the mallet value of Ψ. However, we fin a ifferent tren in Figure 1b. Though αl/λ for z 1,z oe become lowet for large Ψ, it i maximum for mall Ψ. Furthermore, αl/λ for z, z 1 i higher than z 1,z 1, in contrat to finite ion-ize effect. Thee ifference arie ue to tren in Ψ ; ee equation a an Figure 7b. X The effect of γ /γ on αl/λ i ummarize in Figure 7c for γ c,z 1, an z 1. The imenionle alt uptake αl/λ increae for ecreae in γ ince for Ψ >, the aturation concentration of anion i larger; ee equation. Therefore, a larger amount of alt can be taken up by the ouble layer. Scaling Analyi. We now evelop caling relation to better analyze the effect of the ielectric ecrement. To etimate the location of extrema in veru Ψ an their epenence on the value of z an z, we invoke phyical argument. For Ψ >, negative ion will be attracte to the urface an the conition for ε a local maximum implie c. In contrat, for Ψ <, ( γ ε ( the conition for a local maximum become c. Thu, γ auming the c ± follow the Boltzmann itribution, we get ε zc exp( z Ψ,max for Ψ, γ ε zc exp( zψ,max for Ψ, γ (7 or Figure 11. Scaling analyi of ielectric ecrement effect. Comparion of compute Ψ,max with equation 8 for (a Ψ < an (b Ψ >. (c Comparion of compute Ψ,min with equation 4. The re point are cae with the camel hape (i.e., two local maxima an one local minima, an the blue point are cae with the bell hape i.e., one local minimum; ee Figure 9. For computation,.5 γ c, 1 γ /γ 4, 1 z, an 1 z. ε 8 i aume for all calculation. In thi ection, we evaluate the influence of electrolyte valence on the ielectric ecrement effect. Though we analyze the cae of a linear ielectric ecrement, it i traightforwar to exten thi analyi to the cae of nonlinear ielectric ecrement. We refer the reaer to ref for more etail. Phyical Significance. We now preent phyical argument to explain the ignificance of electrolyte valence on the ielectric ecrement, which relate to the ecreae in ielectric contant ue to reuction in orientational polarizability. Simply put, a ecreae in ielectric contant relate to reuction in the ability of the electrolyte to accumulate charge. The ielectric contant ecreae with increae in ion concentration, an in thi article, we aume a linear ecrement; ee equation 6. When the electrolyte come in contact with a charge urface, ue to electrotatic attraction, the concentration of counterion increae cloer to the urface. Conequently, the ielectric contant, an the ability to tore charge, ecreae cloer to the urface. For a large potential rop acro the ouble layer, thee two effect are comparable an reult in aturation of the counterion near the urface; ee equation. It might appear that thi effect i very imilar to thoe from finite ion ize where the concentration of counterion alo aturate, an thu equivalent expreion can be erive by replacing the maximum 1 ion concentration with ε 1 an with ε. However, upon a γ a γ comparion of equation an 5, we note two ifference, (i a reuction by a factor of an (ii an apparent ecreae in Ψ. The reuction by a factor of occur becaue the ielectric contant i reuce by a factor of at the urface; ee equation. On the other han, the apparent ecreae in Ψ occur becaue when the ielectric ecrement i inclue, the energy tore in electric fiel alo varie with ion concentration. Therefore, to keep the ouble layer in equilibrium, the electric fiel energy negate the potential energy, leaing to a maller effective potential rop acro the ouble layer OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,

12 The effect of aymmetry in electrolyte valence i nontrivial, a hown in Figure 8b an Figure 9b. We oberve that the effect of electric fiel energy ominate for mall potential rop uch that Q for z 1,z 1 i lower than Q for z 1,z, unlike Figure b. Furthermore, incluion of valence aymmetry lea to aymmetry in capacitance value, ee Figure 9b, imilar to the cae with finite ion ize. Therefore, regarle of which effect (the finite ion ize or the ielectric ecrement ominate, the incluion of valence aymmetry lea to aymmetry in capacitance value. A note before, the aymmetry in capacitance i valuable ince capacitance influence the charge torage capacity an time cale of ouble layer formation. Therefore, for application uch a capacitive eionization,,45 it will be crucial to account for valence aymmetry ince it ignificantly impact the proce variable. In ummary, ielectric ecrement lea to the counterion aturation, the ecreae in ielectric contant at the urface, an the ecreae in the effective potential rop acro the ouble layer ue to variation in electric fiel energy with ion concentration. The effect of valence i nontrivial, epecially for mall Ψ, when the effect of variation in electric fiel energy i ominant. Furthermore, an aymmetry in electrolyte valence reult in aymmetric ouble layer propertie. Next, we analyze the effect on electrolyte valence of ion-ion correlation on ouble layer propertie. IONION CORRELATIONS In thi ection, we conier the combine effect of ionion correlation an the teric effect. For implicity, we o not conier the effect of the ielectric ecrement an Stern layer in thi ection. For ionion correlation, we buil on the work of Bazant, Storey, an Kornyhev,,5 an we refer the reaer to thee reference for the erivation of the moifie Gau law, which i given a ψ ψ εε lc 4 zec zec 4 x x (9 where l c i the correlation length that quantifie the effect of ion ion correlation. Equation 9 i to be olve with bounary ψ conition ψ( ψ,, an ψ(. We x x nonimenionalize equation 9 with n c, zc 4 c 4 lc c λ L to arrive at Ψ Ψ n n L X X z z X x λ, n c, zc (4 where n ± are given by equation 7. We numerically integrate equation 4 to fin Ψ(X an n ± (X. From the numerical integration, we evaluate the imenionle charge an alt uptake a n n Q X z z (41a of equation 4 are only poible for Ψ 1, an we refer the reaer to ref 5 for more etail. Potential itribution. We firt the icu the effect of L c l c λ 1 on the potential Ψ(X. For a c., a /a 1,z 1, an z,wefin that for mall value of L c (i.e., L c.5, the variation in Ψ(X are not ignificant. However, for larger value of L c, we tart to ee ocillation in Ψ(X, a previouly ecribe by Bazant an co-worker.,5 Similar to finite ion-ize an ielectric ecrement effect, the effect of aymmetry in electrolyte valence i ignificant for ion ion correlation effect. Figure 1b how the effect of change in z Figure 1. Effect of aymmetry in electrolyte valence on ionion correlation. Variation of Ψ with X a given by the olution of equation 4 for Ψ 5. (a Effect of L c with a c., a /a 1,z 1, an z. (b Effect of z an z with L c 1,a c., an a /a 1. an z for L c 1,a c., an a /a 1.Wefin that combination of cation an anion valence alo influence the Ψ(X an the egree of ocillation. The parameter a c an a /a can alo be varie. However, thee effect have alreay been icue in etail in the previou ection, an we expect the qualitative tren to remain the ame even with the incluion of ionion correlation. Charge Accumulate. The ocillation in Ψ(X ue to ion ion correlation impact the charge accumulate inie the ouble layer. The effect of L c on Q for a c., a /a 1,z 1, an z i provie in Figure 1a. We expect ocillation in Ψ(Xto Figure 1. Effect of aymmetry in electrolyte valence on ionion correlation for Q(Ψ, a given by the olution of equation 4 an 41, plotte here in oli line. We alo repreent the approximate olution a given by equation for L c with otte line. (a Effect of L c with a c., a /a 1,z 1, an z. (b Effect of z an z with L c 1, a c., an a /a 1. αl zn zn 1X λ z z (41b Latly, we can alo numerically evaluate the imenionle Q capacitance a Ψ. We alo note that analytical olution 1198 be larger for larger value of L c. Therefore, Q ecreae with increae in the value of L c. However, we emphaize that the hape of the curve are imilar to the cenario without ionion correlation an for L c.5. Equation can be ue a a firtorer approximation of Q, a hown in Figure 1a. OI: 1.11/ac.langmuir.8b64 Langmuir 18, 4,