Weierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN

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1 Weertraß-Inttut für Angewandte Analy und Stochatk Lebnz-Inttut m Forchungverbund Berln e. V. Preprnt ISSN A new perpectve on the electron tranfer: Recoverng the Butler-Volmer equaton n non-equlbrum thermodynamc Wolfgang Dreyer, Clemen Guhlke, Rüdger Müller ubmtted: December 22, 215 Weertra-Inttute Mohrentr Berln Germany E-Mal: Wolfgang.Dreyer@wa-berln.de Clemen.Guhlke@wa-berln.de Ruedger.Mueller@wa-berln.de No. 224 Berln Mathematc Subject Clafcaton. 35Q6, 35Q79, 8A17. Key word and phrae. electrolyte, double-layer, Butler-Volmer, thermodynamc.

2 Edted by Weertraß-Inttut für Angewandte Analy und Stochatk WIAS Lebnz-Inttut m Forchungverbund Berln e. V. Mohrentraße Berln Germany Fax: E-Mal: preprnt@wa-berln.de World Wde Web:

3 Abtract Undertandng and correct mathematcal decrpton of electron tranfer reacton a central queton n electrochemtry. Typcally the electron tranfer reacton are decrbed by the Butler-Volmer equaton whch ha t orgn n knetc theore. The Butler-Volmer equaton relate nterfacal reacton rate to bulk quantte lke the electrotatc potental and electrolyte concentraton. Snce n the clacal form, the valdty of the Butler-Volmer equaton lmted to ome mple electrochemcal ytem, many attempt have been made to generalze the Butler-Volmer equaton. Baed on non-equlbrum thermodynamc we have recently derved a reduced model for the electrode-electrolyte nterface. Th reduced model nclude urface reacton and adorpton but doe not reolve the charge layer at the nterface. Intead t locally electroneutral and contently ncorporate all feature of the double layer nto a et of nterface condton. In the context of th reduced model we are able to derve a general Butler-Volmer equaton. We dcu the applcaton of the new Butler-Volmer equaton to dfferent cenaro lke electron tranfer reacton at metal electrode, the ntercalaton proce n lthum-ron-phophate electrode and adorpton procee. We llutrate the theory by an example of electroplatng. 1 Introducton Energy converon n battere, fuel-cell or redox-flow-cell requre electrochemcal urface reacton to take place at the contact between electrode and electrolyte. In electroly or electroplatng, on the other hand, electrcal energy ued to drve a chemcal reacton. In all thee applcaton, t oberved that the urface reacton rate R, or equvalently the electrc current denty j e, related to a potental dfference at the nterface, the urface overpotental η S. The mot mple relaton of th knd the emprcal Tafel-equaton [Taf5] η S = a + b logj e, 1 where the coeffcent b called the Tafel-lope and a a further phenomenologcal coeffcent. A more general relaton that account for multaneou anodc oxdaton reacton and cathodc reducton reacton on the ame electrode urface the Butler-Volmer equaton [BF1, BRGA2, NTA4], j e = A exp αa e η S C exp α C e η S. 2 Heren A/C are called anodc and cathodc exchange current, repectvely, whch can be general functon of the temperature T and the concentraton of the dfferent chemcal pece. The tranfer coeffcent α A and α C are condered a phenomenologcal coeffcent. The Boltzmann 1

4 contant denoted by k and e the elementary charge. If the exchange current are contant, the Butler-Volmer equaton predct contant Tafel-lope at larger overpotental. Such a behavor can n fact be oberved, mot pronounced n the cae of the hydrogen reducton reacton whch how contant Tafel lope over a range of everal decade of the current j e [Vet61]. In other reacton a tronger dependence of the Tafel lope on potental and temperature can be oberved. Thee devaton from the lnear Tafel behavor are explaned wthn the Marcu-Huh theore [Mar56, Mar65, Mar93, Hu58, Hu99]. We do not conder thee theore here n detal but only refer to the comment n the concludng dcuon of our reult n Sect. 7. The Butler-Volmer equaton condered to be the central equaton n phenomenologcal electrode knetc [BRGA2, p. 153]. All the more t urprng to fnd n the lterature dfference n the way the potental are defned, cf. [NTA4] v. [BvSB9] or [LZ11], and how the exchange current depend on the concentraton, cf. [NTA4] v. [LZ13]. Orgnally, the dervaton of the Butler-Volmer equaton baed on knetc theory [But24, EGV3]. Recently ome approache have been made to gve the Butler-Volmer equaton a thermodynamc jutfcaton. A thermodynamc defnton of an overpotental a a urface exce quantty wa ntroduced n [KRB96] and n [KR3] an attempt wa taken to gve a jutfcaton of the Butler-Volmer equaton wthn a meocopc thermodynamc theory. Recently a Butler-Volmer equaton for oxdaton reacton n fuel cell wa derved wthn the GENERIC framework [BKÖ14], and n the context of phae eparatng electrode materal, a Butler-Volmer equaton n non-equlbrum thermodynamc wa derved [Baz13]. In th paper we derve a general Butler-Volmer of the form 2 whch not retrcted to a pecfc applcaton cenaro. The dervaton baed on a thermodynamc content model for electrochemcal ytem n non-equlbrum and follow from ratonal argument. By th procedure we want to clarfy characterzaton of the dfferent quantte n the Butler-Volmer equaton and the tructure of functonal dependence between them. For the dervaton of 2 t mportant to bear n mnd: 1 The defnton of η S not evdent from modelng baed on non-equlbrum thermodynamc where the et of thermodynamc varable contan the electrc feld E = ϕ but not a potental ϕ telf or ome potental dfference. 2 The nterfacal Maxwell equaton requre that the electrc potental ϕ contnuou at an nterface. Therefore no natural potental dfference ext, whch can be ued to defne an overpotental. 3 The generalty of A/C retrcted by the 2nd law of thermodynamc. In conequence, the forward and backward reacton can not be modeled ndependently. 4 In order to have a general form of the Butler-Volmer equaton that doe not depend on a pecfc expermental etup, the functon have to be expreed n term of the chemcal potental ntead of concentraton or partcle dente. Uually Butler-Volmer equaton eem to mplctly aume that materal are modeled a deal mxture of gae. The underlyng model for electrochemcal ytem n thermodynamc non-equlbrum wa formulated [DGM15]. We call th the complete model nce the dffue charge layer are patally 2

5 reolved. The conttutve law for the nterfacal reacton rate are not of Butler-Volmer type. Under the aumpton of approprate calng relaton, the double layer can aymptotcally be condered n equlbrum. Th allow the dervaton of a locally electroneutral reduced bulk model uch that all feature of the double layer are contently ncorporated nto a et of jump condton at the nterface, ee [DGM15]. Wthn th reduced model, we are able to defne an overpotental and to recover relaton of Butler-Volmer type for the nterfacal reacton rate. Our approach to generalze the Butler-Volmer equaton contrary to thoe uggeted prevouly. Intead of tartng from tandard Butler-Volmer equaton and then ung a-pror aumpton about a pecfc electrochemcal ytem wth t partcular double-layer tructure to derve new Butler-Volmer equaton, we tart from a general thermodynamcally content model and can derve a general Butler-Volmer equaton whch under approprate aumpton reduce to the clacal varant. Outlne. After the ntroducton of the notaton n the followng ecton, we recaptulate n Secton 3 the reduced bulk model of [DGM15]. In Secton 4, we derve a general Butler-Volmer equaton and then dcu n Secton 5 t applcaton to dfferent relevant cenaro. For llutraton of the Butler-Volmer equaton and the nteracton wth the bulk tranport, we conder the example of copper depoton and doluton n Secton 6. Fnally, n Secton 7, we dcu the range of valdty of the underlyng aumpton, compare our reult wth the lterature and dcu the role of the double layer tructure. 2 Decrpton of reactng mxture Here we ntroduce the notaton for the decrpton of reactng mxture n ubdoman eparated by planar electrochemcal nterface. For mplcty we conder a planar tuaton where two onedmenonal regon Ω ± R are eparated by an nterface I = Ω + Ω. For quantte defned n the bulk doman there wll often be correpondng quantte on the nterface I, ndcated by a ubcrpt. 2.1 Conttuent and chemcal reacton In each of the two doman Ω ± and on the nterface I we conder a mxture of everal conttuent. The total number of conttuent n the ubdoman Ω ± denoted by N + 1 and the et of conttuent M = {A, A 1,, A N }, uually ndexed by α {, 1,, N}. In general we have dfferent conttuent n Ω + and Ω, but for the mplcty of notaton th fact wll only be ndcated f neceary. We aume that each conttuent of Ω ± alo preent on I, but n addton there may be conttuent that are excluvely preent on I. Accordngly, the number of conttuent on I N S N and the et of conttuent M S = {A, A 1,, A NS }. A conttuent A α ha the atomc ma m α and may be carrer of charge z α e, where z α the charge number and e the elementary charge. We may have chemcal reacton among the conttuent. There are M bulk reacton and n addton there may be M S urface reacton of 3

6 the general form a A + + a N A N a A + + a NS A NS R f ba + + b N A N for {1,, M}, 3a R b R f R b b A + + b NS A NS for {1,, M S }. 3b The contant a α, b α are potve nteger and γ α = b α a α denote the tochometrc coeffcent of the reacton. The reacton from left to rght called forward reacton wth reacton rate Rf >. The reacton n the revere drecton wth rate R b > the backward reacton. The net reacton rate defned a R = Rf R b. Snce charge and ma have to be conerved by every ngle reacton n the bulk and on the nterface, we have N z α γα = α= N m α γα = α= and and N S α= N S α= z α γα =, m α γα =. 4a 4b 2.2 Bac quantte In each pont x Ω ± and at any tme t >, the tate of the mxture characterzed by the number dente n α α=,1,,n, the barycentrc velocty v, the temperature T and the electrc potental ϕ. On the nterface I the mxture characterzed at any t by the number dente of the nterfacal conttuent, n α α=,1,,ns, the velocty w of the nterface, the nterfacal temperature T and electrc potental ϕ. Multplcaton of the number dente n α by the ma m α gve the partal ma dente: ρ α = m α n α and ρ α = m α n α. 5 For the mxture, the ma denty defned by ρ = N α= The free charge dente are defned a n F = ρ α and ρ = N z α e n α and n F = α= N α= ρ α. 6 N z α e nα. 7 α= 4

7 .8.6 complete model reducded model ϕ [V].4.2 λ L metal electrolyte x [nm] Fgure 1: Electrc potental ϕ at an nterface n equlbrum. The complete model reolve the boundary layer and the electrc potental contnuou dahed lne. The lmt λ yeld a reduced model, where the electrc potental can be dcontnuou at the nterface old lne. 2.3 Jump at nterface We ntroduce the boundary value and the jump of a generc functon ut, x n Ω ± at the nterface I a u ± I = lm u and [u ] = u + x Ω ± I u I. 8 I In cae that the functon u not defned n ether Ω + or n Ω, we et the correpondng value n 8 to zero. The normal ν to the nterface I alway pont from Ω to Ω +. In th one-dmenonal ettng, we have ν = ±1. 3 Reduced bulk model for the thn double layer lmt In a prevou work [DGM15] we derved a thermodynamc content model, whch decrbe the electrochemcal nterface between two arbtrary mxture. Th model we call the complete model, becaue t patally reolve the charge layer n the vcnty of the nterface, the double layer. The charactertc length cale for the charge layer λl ref = ε e 2 n ref, 9 where k the Boltzmann contant, n ref denote a charactertc value for the number denty and L ref a charactertc length of the ytem. For example, L ref can be the dtance between two electrode and n ref can be related to the anon and caton denty n an electrolyte. Then, the length λl ref repreent the well known Debye length. For a oluton of.1mol per lter, λl ref m. Snce often λ 1, we appled n [DGM15] the method of formal aymptotc analy to derve from the complete model a reduced bulk model for the lmt λ. Th reduced model characterzed by mplfed bulk equaton and new urface equaton for the thn double layer nterface. The Fgure 1 how the the electrotatc potental at a metal-electrolyte nterface gven by the complete model and t approxmaton by the reduced model. 5

8 Model aumpton. The reduced model derved from the complete model under the followng aumpton, cf. [DGM15]: 1 The parameter λ atfe the condton λ 1. 2 The ytem under conderaton can be treated a a one dmenonal problem. 3 The electrc feld qua-tatc, the magnetc feld can be gnored. 4 Qua-tatc momentum balance, vcoty n the bulk doman neglgble. 5 Iothermal cae, the bulk temperature T and urface temperature T are contant and atfy T = T. 3.1 Bulk and nterface equaton The reduced model rele on unveral balance equaton whch are ndependent of the pecfc materal. In addton we need conttutve equaton for the materal at hand. In the othermal cae the unveral equaton are the balance equaton of ma and momentum. In addton there a local electroneutralty condton that a conequence of Maxwell equaton n the aymptotc lmt λ. Bulk equaton. In the bulk Ω ±, the electrc potental ϕ, the velocty v and the number dente n α for α =, 1,, N are determned by t m α n α + x m α n α v + J α = M γαm α R for α =,..., N, 1a =1 x σ =, N z α e n α =. α= 1b 1c The quantte J α and σ are the ma flux of the conttuent A α and the Cauchy tre tenor, repectvely. Note that there are only N ndependent dffuon fluxe and the flux J determned by the de condton N α= J α =. Interface equaton. dente n α for α =, 1,, N S we have For the nterfacal peed w, the electrc potental ϕ and the number M S [m α n α v wν + J α ν ] = γ α m α R, α =,, N S, 11a =1 q + + n F + q =. [σ ] =, 11b 11c 6

9 The lat equaton the electroneutralty condton of the double layer I, whch a conequence of the Maxwell equaton n the aymptotc lmt λ, cf. [DGM15]. In general the determnaton of q ± requre the oluton of an addtonal ytem of equaton that patally reolve the layer tructure. Therefore the nterface equaton 11a-11c are not a cloed equaton ytem contanng excluvely the varable of the reduced ytem that were ntroduced above. But n many relevant cae, e.g. for the nterface at a metal electrode, an explct oluton of the nterface electroneutralty condton 11c not neceary. Snce t can be decoupled from the ret of the equaton ytem, and then erve only to determne one remanng urface number denty wthn a pot proceng tep. 3.2 Conttutve equaton The unveral equaton 1a 11b need to be upplemented by conttutve equaton for the ma fluxe J α, the reacton rate R, R and the Cauchy tre tenor σ. The conttutve equaton are retrcted by the prncple of materal objectvty and the 2 nd law of thermodynamc. Free energy and chemcal potental. In order to cover a wde cla of dfferent materal we ue free energy functon of the general form ρψ = ρ ˆψT, ρ,..., ρ N, ρψ = ρ ˆψ T, ρ,..., ρ N. 12 S Note that due to the aymptotc lmt, the free energy functon are ndependent of the electrc feld, cf. [DGM15]. The chemcal potental of the bulk and urface materal are defned by µ α = ρψ ρ α and µ α = ρψ. 13 ρ α Conttutve equaton n bulk. For the dffuon fluxe, the preure and the reacton rate we chooe the followng relaton that guarantee the contency wth the 2 nd law of thermodynamc partcularly the non negatvene of the entropy producton N µβ µ J α = M αβ x + e zβ z x ϕ, α = 1,, N, 14a T T m β m β=1 σ = p wth p = ρ ˆψ R = R [ exp β A α= N m α n α µ α, α= N γαm α µ α exp 1 β A N ] γαm α µ α. α= 14b 14c Here R and A denote potve phenomenologcal coeffcent and M αβ defne the moblty matrx that mut be potve defnte. Although not requred by the 2 nd law of thermodynamc, we aume < β < 1. The quantty p the preure whch gven by the Gbb-Duhem equaton 14b 2. All conttutve equaton are related to each other nce they hare the dependency on the chemcal potental and thu on the free energy ρψ. 7

10 Conttutve equaton for the nterface. A n the volume, the conttutve relaton are all related to the urface free energy ρψ and can not be modeled ndependently of each other. We chooe the followng thermodynamc content conttutve equaton for the reacton rate and the ma fluxe on I [ A R = R exp β N S γ αm α µ α α= A exp 1 β N S ] γ αm α µ α, 15a α= ρv w ± ± = L µ + z e I m ϕ ± µ I + z e m ϕ, 15b ρα v wν + J α ν ± I = M ± α exp β α ± mα B ± α D α ± exp β ± α 1 mα B ± α D α ±. 15c where R, A, L ±, M α and B ± α denote potve phenomenologcal coeffcent. The coeffcent β and β α ± are uually called ymmetry factor. Although not requred by the 2nd law of thermodynamc, we aume < β, β α ± < 1. By D α ± we denote the drvng force for adorpton, D ± α = µα + zαe m α ϕ ± I µ + z e m ϕ ± I µ α + zαe m α ϕ µ + z βe m ϕ. 16 We hghlght, that n [DGM15] lnear relaton were choen for all conttutve law except for the reacton n the volume and on the urface, where an Arrhenu-type exponental form wa choen. Here we alo chooe the exponental relaton for the adorpton n 15c. We note that near to equlbrum,.e. D ± α 1, 15c can be lnearzed reultng n the conttutve equaton NS ρα v w + J α ± = I β=1 M ± αβ [ µβ + z βe m β ϕ ± µ I + z e m ϕ ± I µ β + z βe m β ϕ µ + z βe m ϕ ], 17 ± whch concde wth the repectve equaton n [DGM15, eq. 3], f the matrx M αβ choen dagonal. 3.3 Remark Electrotatc potental. When dervng a trctly thermodynamcally content model, the proper thermodynamc varable the electrc feld E = ϕ, not the electrotatc potental ϕ telf [MR59, dm84, Mül85]. The explct dependence of the conttutve relaton 15b and 15c on ϕ only jutfed a a conequence of the formal aymptotc approach n [DGM15]. Moreover, n the reduced model above, ϕ not requred to be contnuou acro the nterface and hence we have to ntroduce on I the varable ϕ that n general not an one-ded lmt of ϕ n the bulk. 8

11 Electrochemcal potental. It remarkable that the conttutve relaton 14a, 15b and 15c depend explctly on the electrochemcal potental µ e α = µ α + zαe m α ϕ and µ α = µ e α + zαe m α ϕ. 18 If we ue the electroneutralty condton 1c and the charge conervaton for chemcal reacton 4b we can alo expre the conttutve equaton of the Cauchy tre tenor σ and the reacton rate R and R n term of electrochemcal potental. Such that all conttutve equaton 14 and 15 can be expreed n term of electrochemcal potental ntead of chemcal potental. Electrc current. In the reduced model, the electrc current gven by the mple relaton j e = N α= z α e m α J α. 19 Moreover, we can derve from 1 and 11 the followng tatonary balance equaton for the electrc current j e n the bulk and on the nterface x j e = and [j e ν ] =. 2 The electrc current contant n each ubvolume Ω ± and contnuou at the nterface I. Hence j e even a global contant n the whole electrochemcal ytem. Electrc current reacton rate. In the reduced model there a mple relaton between the electrc current j e and the reacton rate R. We ntroduce the ubet J + M of all pece that are preent n Ω +. Then multplyng n Ω + the urface balance 11a for the pece A α J + by z α e /m α and ubequent ummaton together wth 1c yeld M S γα z α e R = =1 α J + α J + zα e n α v wν + zαe m α J α ν + I 1c,19,2 = j e ν I. 21 If there only one urface reacton,.e. M S = 1, we get the drect proportonalty j e ν I = γ α J + α z α e R n Ω ±, 22 that allow the alternatve formulaton of the Butler-Volmer equaton 2 a a logarthmc relaton between electrc current and the overpotental. 4 Dervaton of the Butler-Volmer equaton The above reduced model form the thermodynamc content ba for a dervaton of the Butler-Volmer equaton. The dervaton not retrcted to a ngle urface reacton. Therefore 9

12 we derve for each urface reacton a correpondng Butler-Volmer equaton. The am of the, dervaton to dentfy exchange rate R f/b, coeffcent α f/b and the overpotental η S n term of bulk quantte: number dente n α and electrc potental ϕ. In contrat, the conttutve relaton 15a for the reacton rate depend only on the nterfacal quantte: nterfacal number dente nα and urface electrc potental ϕ. The neceary relaton between bulk and nterfacal quantte etablhed by the conttutve relaton 15b and 15c for the ma fluxe. Decompoton of the et of bulk conttuent. For the dervaton of the Butler-Volmer equaton, t neceary to decompoe the et of all bulk conttuent nto two djont et M = M + M, M + M =, uch that all conttuent A α M + are preent n Ω + and all conttuent A α M are preent n Ω. In general, t poble that a conttuent preent n both bulk doman Ω + and Ω. Then, the decompoton of the et M nto the ubet M ± not unquely defned. Aumpton of fat adorpton. The conttutve equaton 15b and 15c for the ma fluxe are the boundary condton that decrbe the adorbton of the conttuent at the nterface I. Here we are only ntereted n the lmt cae of fat adorpton,.e. L ± and M ± α. Then, we obtan the contnuty of the electrochemcal potental at the nterface µα + zαe m α ϕ ± I = µ α + zαe m α ϕ for α M. 23 Equlbrum, Nernt equaton. At frt we conder the equlbrum,.e. R =, to how the bac tep for the dervaton of the Butler-Volmer equaton wthn the reduced model. We denote all quantte that depend on the equlbrum tate by an overbar. For every urface reacton, we deduce from R = and 15a the law of ma acton, vz. = αm α µ α. 24 γ α M S Accordng to 4a, each reacton conerve electrc charge. Therefore = αm α µ α + zαe m α ϕ. 25 α M S γ Next we ue the fat adorpton lmt 23 to replace the nterfacal quantte by bulk quantte, = γα mα µ α + z α e ϕ ± + γ I α mα µ α + z α e ϕ, 26 α M α M S \M where n the frt um the bulk value taken from Ω + f α M + and ele taken from Ω. A an abbrevaton we defne the coeffcent Γ + = αz α, Γ = αz α and Γ = γ αz α 27 γ α M + γ α M α M S \M 1

13 The conervaton of charge 4a mple Γ + + Γ + Γ =. Now we obtan for the th reacton the alternatve repreentaton of the ma acton law = γαm α µ α ± + γ I αm α µ α + Γ +e [ ϕ ] + Γ e ϕ ϕ I. 28 α M α M S \M We oberve that by ung the conttutve equaton 23, we are able to expre the urface law of ma acton 24 n term of bulk quantte. By th we automatcally ntroduce an electrc potental dfference [ ϕ ] acro the nterface. In the cae M S \ M =, and hence Γ = and Γ +, 28 reduce to the well known Nernt-equaton Γ +e [ ϕ ] = α M γ α m α µ α ± I. 29 Generalzed Butler-Volmer equaton. Next we conder an arbtrary tme dependent tate. In order to replace the urface chemcal potental n 15a by the bulk chemcal potental, we apply the ame tep lke n the dervaton of the Nernt equaton n the above veron 28. Therefore we frt rewrte the drvng force of the chemcal reacton 15a a αm α µ α = γαm α µ α ± + γ I αm α µ α + Γ +e [ϕ ] + Γ e ϕ ϕ I α M γ α M S α M S \M Becaue we want to ntroduce the overpotental η S a the devaton from an equlbrum potental, we ubtract the law of ma acton 28 from 3 to get γαm α µ α = γαm α µα µ α ± + γ I αm α µ α µ α 31 α M S α M + Γ +e [ϕ ϕ ] + Γ e ϕ α M S \M ϕ I ϕ ϕ For each urface reacton we defne t repectve overpotental by [ϕ ϕ ] + Γ ηs Γ + ϕ ϕ I := ϕ ϕ I, f Γ + ϕ ϕ I ϕ ϕ I, f Γ+ = Inertng the dentty 31 nto 15a yeld net reacton rate R n the form of a Butler-Volmer equaton, vz. R = R, f wth the tranfer coeffcent a β A Γ αf +, f Γ +, = β A Γ, f Γ + =, exp α f e η S R, b I 3 32 exp + α b e η S β A Γ and αb +, f Γ +, = 1 β A Γ, f Γ + =, 34 11

14 and the exchange rate R, f R, b = R exp = R exp β 1 β [ A γ αm α µα µ α ± + I A α M α M S \M [ γ αm α µα µ α ± + I α M α M S \M γαm α µ α µ γαm α µ α µ ] α α, 35a ]. 35b Contency of conttutve equaton. Due to the knetc approach, Butler-Volmer equaton n the tandard lterature uually contan an explct dependency on the number dente n α, cf. [But24, EGV3, NTA4, BRGA2, BF1]. In contrat, the general Butler-Volmer equaton above ha only an ndrect dependency on n α becaue of the chemcal potental. Such a formulaton n term of the chemcal potental can alo been found n Butler-Volmer equaton whch have been propoed recently for pecfc applcaton lke L-on battere or fuel cell, cf. [LZ13, Baz13, BKÖ14]. We want to emphaze the mportance of the chemcal potental for a content modelng: In the formulaton of the conttutve equaton 14 and 15, we followed the approach of non-equlbrum thermodynamc that mpoe compatblty contrant. Th ha the advantage, that the modelng conderably mplfed, becaue once the free energy denty pecfed, all conttutve equaton are determned unquely and n a compatble way, up to the choce of phenomenologcal coeffcent. Compared to 15a, urface chemcal potental µ α have been replaced by bulk chemcal potental µ α n By th, we ntroduced a dependency of the conttutve law for the urface reacton on the bulk free energy ρψ. Hence the urface reacton rate are alo related to the conttutve equaton n the bulk 14 and can not be modeled ndependently of each other. Wthn our framework, a Butler-Volmer equaton formulated n term of n α would mply a pecfc form of the free energy dente and thu mpoe compatblty contrant for the other conttutve equaton n the bulk and the urface. Smplfcaton n the abence of excluve urface pece. Although mot of the urface quantte have been replaced by volume quantte, the general Butler-Volmer equaton tll contan urface quantte. However, n many electrochemcal ytem the conttuent on the urface are alo preent n at leat one of the bulk phae,.e. M S \ M =. Th lead to a repreentaton of the Butler-Volmer equaton that ndependent from the urface quantte and we get a ngle overpotental for all urface reacton a well a mplfed reacton coeffcent whch excluvely depend on bulk quantte, η S = [ϕ ϕ ], 36 A = R exp β γ αm α µα µ α ±, 37 I α M A = R exp 1 β γ αm α µα µ α ±. 38 I R, f R, b α M 12

15 The potental dfference η S then decrbe the devaton of the actual potental dfference ϕ + I ϕ I from the equlbrum voltage ϕ + I ϕ I of the bulk phae that n accordance wth uual defnton n electrochemtry [BRGA2, NTA4]. 5 Adapton to dfferent electrochemcal ytem In th ecton, we tudy two cenaro for the applcaton of Butler-Volmer equaton: Frt, we conder a prototypcal metal-electrolyte nterface lke t can be found n many electrochemcal applcaton. Next we turn to the electron tranfer reacton n modern lthum-on-battere. Here we are ntereted n the decrpton of the ntercalaton materal FePO 4 that undergoe a phae tranton durng the ntercalaton proce. Moreover, we dcu the orgn of Butler-Volmer type relaton n the cae of lthum depoton from an electrolyte to a metallc lthum. In partcular we demontrate that Butler-Volmer type equaton do not alway orgnate from an electron tranfer reacton, but t can reult from an ntercalaton proce or a low adorpton proce ntead. 5.1 General metal electrode We conder a metal-electrolyte nterface wth an electron tranfer reacton of the form R f O + n e R. 39 Rb Here O and R denote the oxdzed and the reduced pece, repectvely, and n the number of tranfered electron. The backward reacton R b n 39 the oxdaton and the forward reacton R f the reducton. The tochometrc coeffcent related to 39 are γ O = 1, γ e = n and γ R = We decrbe the metal a a mxture of free electron e and metal on M. The electrolyte a mxture contng of a olvent S a well a anon and caton. The oxdzed pece O alway a potvely charged caton of the electrolyte. If the reduced pece doe not concde wth M, then we aume R to be a further caton pece of the electrolyte. Becaue there are no excluve urface pece nvolved n the reacton 39, we can aume that all pece whch are preent on the metal-electrolyte-nterface, are alo preent n Ω + or Ω,.e. M S \ M =, and hence Γ =. Followng the conventon [NTA4, pp. 7] we want the anodc oxdaton reacton to be domnant for potve overpotental η S. From 33 and 34 we ee that Γ + ha to be potve and thu we chooe Ω + to repreent the metal electrode. We denote the electrc potental n the metal by ϕ M = ϕ + I and the electrc potental on the electrolyte de by ϕ E = ϕ I. The overpotental then gven by η S = ϕ M ϕ E ϕ M ϕ E

16 Specfc bulk materal model for metal and electrolyte. To get a fully explct Butler-Volmer equaton we have to pecfy free energy dente for the metal and the electrolyte. Here we wll only ntroduce the chemcal potental and refer for the correpondng free energy dente to [DGL14a, DGL14b, DGM13]. The elatc model for the metal appled n [DGL14b] mple that the number denty of the metallc on n M contant. Then, the electro-neutralty condton 1c requre that the electron denty n e alo contant. Thu n an othermal proce, alo the chemcal potental of the metal on and electron n the bulk are contant n pace and tme, and we et µ M = µ M and µ e = µ e. 42 The electrolyte aumed to be an deal mxture. Then the chemcal potental of the electrolyte pece are gven by [DGM15, DGL14a] µ α = g α T, p + m α ln y α for α = A, C, S. 43 Here y α = n α / β n β denote the mole fracton of conttuent A α and g α the Gbb free energy of the correpondng pure ubtance. In general the preure dependence of g α of outmot mportance n the complete model, ee [DGM13, DGL14a], whch contan more nformaton on the thermodynamc modelng of electrolyte. Here, n the reduced model, the preure contant due to the momentum balance equaton 1b and the mple conttutve equaton 14b. Cae I: The metal on are not nvolved n reacton,.e. M R. We obtan the coeffcent Γ + = n and α f = βan and α b = 1 βan. 44 In equlbrum,.e. R =, the Nernt equaton 28 take the form ϕ M ϕ E = ne ln ȳo ȳ R + 1 ne m O g O m R g R + n m e µ e. 45 Snce the chemcal potental of the electron are contant, the equlbrum potental ϕ M ϕ E vare logarthmcally wth mole fracton ȳ O and ȳ R of the on when preure p and temperature T are kept fxed. In Butler-Volmer equaton 33 for the reacton 46 the contrbuton of the electron vanh due to 42. Wth γ O = 1, γ R = 1 and µ O and µ R accordng to 43, the Butler-Volmer equaton read β βan e ȳr y 1β 1 βan e O A R = R exp ȳo ȳ O y R η y R A S R exp η S ȳ R y O 46 wth the overpotental η S gven by 41. When applyng 22, we get a Butler-Volmer equaton a a relaton between overpotental and electrc current,.e. j e ν I = jc exp β A n e η S 1 βan e ja exp η S, 47 14

17 wth the repectve anodc and cathodc exchange current j C = ne R ȳr ȳ O y O y R β A ȳo and ja y R = ne R ȳ R y O 1β A. 48 When R >, then the domnant reacton the reducton, whch depend on the avalablty of uffcently many partcle of pece O. Keepng R > fxed, then we conclude from 46 that y O requre η S. Analogouly, aumng R < a precrbed contant rate, then the oxdaton reacton domnant and y R mple η S +. Further, for y O 1 we ee that already a mall overpotental, η S /e, uffcent to reach a rate R <. Therefore the Butler-Volmer equaton 46 mple the expected behavor that the oxdaton favorable for a low concentraton of the oxdzed pece. Cae II: The meal on are the reduced pece,.e. M = R. We obtan the coeffcent Γ + = z R + n = z O and α f = βaz O and α b = 1 βaz O. 49 In equlbrum,.e. R =, the Nernt equaton 28 take the form ϕ R ϕ E = z O e ln ȳ O + 1 z O e m O g O m R µ R + n m e µ e. 5 Snce the chemcal potental of the metal on and electron are contant, there reman only a logarthmc dependence of the equlbrum potental ϕ M ϕ E on ȳ O. The Butler-Volmer equaton 33 for the reacton 46 mplfe conderably becaue the contrbuton of the chemcal potental of e and R vanh due to 42. Wth γ O = 1 and µ O accordng to 43, the Butler-Volmer equaton read R = R yo ȳ O βa exp β Az O e ȳo η S R wth the overpotental η S gven by 41. From 22, we get j e ν I = jc exp β y O Az O e η S ja exp 1βA exp 1 β 1 β wth the repectve anodc and cathodc exchange current 1βA ȳo ja = z O e R and jc = z O e R y O Az O e η S 51 Az O e η S, 52 yo ȳ O βa. 53 We hghlght two dfference to the prevou cae and 46. Frt, n 51 there no mechanm for η S to blow up f an oxdaton at a fxed rate R < precrbed. Second, the tranfer coeffcent n front of η S dffer between 46 and 51 f z R and hence z O n. 15

18 5.2 Lthum ron phophate electrode The electrode materal L y FePO 4 LFP a cheap and afe electrode materal for lthumon battere [PNG97]. A LFP electrode cont of a metallc carrer fol, carbon-coated LFP nano-partcle, bnder and further addtve mprovng the electrc and onc conductvty and the mechancal properte of the battery. We decrbe the LFP nano partcle a a bnary mxture of crytallne FePO 4 - and L-atom. The lthum atom can move freely through the FePO 4 crytal lattce [HVdVMC4, Baz13, DJG + 1]. Snce FePO 4 ha a low electrc conductvty, we can neglect the electron tranport nde the nano partcle. To etablh electrc conductvty of the LFP, the LFP partcle are coated wth a thn carbon layer [MDCK + 7]. Therefore we have free electron e on the urface, n addton to the electrolyte and electrode pece, a well a the non-reactng carbon. The electrolyte cont of an organc olvent S, lthum on L + and the aocated anon A. At the ron phophate-electrolyte nterface, we conder the electron tranfer reacton The tochometrc coeffcent are L + + e L. 54 γ L + = 1, γ e = 1 and γ L = We let Ω + repreent the doman of the ron phophate electrode. Accordngly the electrolyte occupe the doman Ω. Then we get the coeffcent Γ + =, Γ = 1, and Γ = For the phenomenologcal coeffcent α f,b we obtan α f = βa and α b = 1 βa. 57 Snce we model the LFP-electrode a mxture of electrcally neutral pece, there no pace charge layer nde the electrode. In conequence the electrc potental patally contant and atfe ϕ + I = ϕ. Denotng the electrc potental n the electrolyte by ϕ E = ϕ I and n the ron phophate by ϕ LFP = ϕ + I, the overpotental n 32 can be expreed a η S = ϕ LFP ϕ E ϕ LFP ϕ E. 58 Specfc materal model. A before, the electrolyte modeled a a mple mxture wth chemcal potental gven by the conttutve equaton 43. The chemcal potental of the urface electron aumed to be contant,.e. µ e = µ e. Lthum-on battere wth LFP-electrode are characterzed by a phae tranton between lthum-poor and lthum-rch phae nde the LFP-electrode [PNG97, WSvA + 12, LML + 15]. There are everal approache to model for LFP electrode takng the phae tranton nto account [ZB14, BCB11, HVdVMC4, DGH11, DJG + 1]. A common feature of all thee model a non-monotone chemcal potental of lthum 16

19 n LFP. The mplet conttutve equaton for the chemcal potental µ L, that account for heat of oluton and the entropy of mxng, gven by µ L = L m L 1 2y L + m L lnyl ln1 y L. 59 Here y L = n L /n FePO4 the mole fracton of lthum and L the heat of oluton. Mechancal contrbuton due to the volume change of LFP durng the lthum ntercalaton are not condered here. Butler-Volmer equaton for LFP. The Butler-Volmer equaton for the reacton 54 at the ron phophate-electrolyte nterface gven by βae 1 βa e R = Rf exp η S Rb exp η S 6 wth the coeffcent R f = R R b = R yl + ȳ L + ȳl + y L + 1 y L ȳ L βa exp + 2L yl ȳ L, 61a 1 ȳ L y L 1 ȳ L y L exp 1 y L ȳ L 2L yl ȳ L 1βA. 61b Compared to the Butler-Volmer equaton 46 for the metal-electrolyte nterface, the tructure of the tranfer coeffcent more complex becaue of the dependency on the lthum mole fracton y L. For ntercalaton electrode, the overpotental hould blow up f the electrode completely flled or empty, cf. [LZ13]. Th requrement atfed by the Butler-Volmer equaton 6: Durng the ntercalaton of lthum n the electrode, the reducton the domnatng reacton and thu the reacton rate n 6 ha to be potve. Aume an ntercalaton proce wth a fxed potve rate R >. For y L 1, the tranfer coeffcent yeld and Rf and R b +. Then, the Butler-Volmer equaton 6 mple η S. Analogouly, for dentercalaton proce wth a contant negatve reacton rate R <, the overpotental ha to ncreae η S + for y L. 5.3 Lthum electrode The metallc lthum electrode n contact wth an lthum conducton electrolyte erve a an example where the Butler-Volmer equaton doe not orgnate from an electron tranfer reacton. Intead, the Butler-Volmer equaton here can reult from two dfferent procee: urface reacton wthout charge tranfer and adorpton. Let the doman Ω + repreent the metallc lthum electrode and Ω repreent the electrolyte doman. The electrolyte cont of lthum on L +, the aocated anon A and a olvent S. A already decrbed n Secton 5.1, the metal electrode cont of potve metal on L + M and free electron e. Snce L + and L + M have the ame charge number, an electrc current can not orgnate from an electron tranfer reacton 17

20 between thee two pece. However meaurement how, that alo th proce generate an exponental relatonhp of Butler-Volmer type between the overpotental of the lthum-electrolyte nterface and electrc current [MBS87]. Therefore th behavor ha to orgnate form one of the three mechanm: 1 Adorpton of the lthum on from the electrolytc oluton to the metal-electrolyte nterface. 2 Intercalaton of lthum from the electrolyte phae to the metal phae,.e. L + M L + at I Adorpton of lthum on and electron from the metal to the metal-electrolyte nterface. In general the thrd mechanm aumed fat compared to the other, and therefore can not the orgn of the non-lnear relatonhp between overpotental and current. The fat adorpton modeled by L + + n 15b and M e n 15c. We obtan on the lthum metal de the equaton µ L + M + e m L + M ϕ + I = µ L+ M + e m L + M ϕ and µ e e m e ϕ + I = µ e e m e ϕ. 63 We denote the potental n the lthum metal wth ϕ M = ϕ + I and n the electrolyte wth ϕ E = ϕ I. Accordngly, we defne the overpotental a η S = ϕ M ϕ E ϕ M ϕ M. Now we how that both cae, a low reacton and a low adorpton, reult n a Butler-Volmer equaton of the ame type. We ue the ame conttutve relaton for the metal and the electrolyte a n Secton 5.1. Cae 1: Slow reacton and fat adorpton. Let the reacton 62 be the lmtng proce. We can aume that the fat adorpton aumpton 23 atfed,.e. µ α + zαe m α ϕ + I = µ α + zαe m α ϕ α = L +, A, S, 64 and the reult of Secton 4 are applcable. In partcular we can ue the reult of Secton 5.1, nce the dervaton hold alo n the cae where no electron are nvolved n the reacton 39. We have j e ν I = jc exp wth anodc and cathodc exchange current β Az L + e η S ja exp 1 β Az L + e η S, 65 j A = z L +e R ȳl + y L + 1βA and j C = z L +e R yl + ȳ L + βa

21 Cae 2: Fat reacton and low adorpton. The reacton 62 fat compared to the adorpton. We can aume that the knetc parameter for the reacton rate meet R n 15a. Then the law of ma acton read µ = µ. 67 L+ L+ M To mplfy the argumentaton n the followng, we aume that the adorpton of the olvent S= A to the metal-electrolyte nterface fat,.e. L + n 15b, and we have µ S I = µ S. 68 Further, we aume that the anon cannot adorb at the metal urface,.e. M A = n 15c, and we obtan from the conttutve relaton 15c ρa v wν + J A ν =. I 69 Thee aumpton yeld that the electrc current j e at the nterface gven by the lthum flux j e ν I = e m L + ρl +v wν + J L +ν I. 7 Th relaton follow from the mple relaton 19 for the electrc current, the local electroneutralty condton 1c and the equaton 69. The lthum flux n 7 gven by the conttutve relaton for the adorpton, equaton 15c, ρl +v wν + J L +ν I =M L + exp β m L + L B + L D + L exp β + Here, the drvng force defned a µl D L = z L + e ϕ m L + I L + 1 m L + B L + D L µ L+ + z L + e m L + ϕ. 72 Ung the relaton 63 1 we can wrte the drvng force a µl D L = z L + e ϕ µl m L + I + + z L + e ϕ M m L + I A n the dervaton of the Butler-Volmer µ equaton, we ntroduce an equlbrum tate n whch the drvng force vanhe, leadng to L + z + L + e ϕ = µ m L + I L + + z L + e ϕ +. The drvng M m L + I force 73 can be wrtten a D L = µ + L + µ L + µ I L + µ L + + z L + e η I m S, 74 L + where we have replaced the electrc potental by the overpotental η S = ϕ ϕ + I ϕ ϕ I. Wth the chemcal potental 43 for the lthum on of the electrolyte and the contant chemcal potental for lthum metal accordng to 42, we get for the current β j e ν I = jc L B exp + L z + L +e 1 β η S ja L B exp + L z + L +e η S 19 75

22 wth the exchange coeffcent j A = M L + z L +e m L + 1β ȳl + L + B y L + L + and j C = M L + z L +e m L + β yl + L + B L Thee relaton are dentcal to the Butler-Volmer equaton 65 reultng from the urface reacton. Th remarkable, becaue of the thermodynamc orgn of the non-lnear relaton, whch are derved n one cae from an adorpton proce and n the other cae from an urface reacton. Therefore we can not conclude from a Tafel-plot whether the adorpton or the urface reacton the lmtng urface proce. ȳ L + 6 Example Electroplatng The electroplatng of metal erve a a mple example of an electrochemcal proce wth a urface reacton that can be decrbed by a teady tate. We conder an aqueou copper ulfate oluton between two parallel copper plate, a hown n Fgure 2. The doman of the electrolyte Ω and we let Ω + denote both electrode. We model copper electrode a a bnary mxture of free electron e and cuprou on Cu +. The electrolyte cont of water H 2 O a the olvent S, the anon SO 2 4 and the caton Cu 2+, n the followng alo denoted by A and C, repectvely. j e Ω + e A ν + Cu Cu 2+ I A Ω SO 2 4 H 2O Fgure 2: Expermental etup for electroplatng: aqueou copper ulfate oluton bounded by two copper electrode. The urface normal ν alway pont to the electrode. The role of the electrode a anode or cathode depend on the drecton of the mpoed current j e ν. When a current j e appled to the electrode, copper oxdzed at the anode A and cuprc on are dolved from the the anode nterface I A nto the electrolyte. On the other de, at the nterface I C, the cuprc on are reduced and ncorporated nto cathode C. If the plate are uffcently large the proce one dmenonal. The overall doluton/depoton proce can be plt nto everal tep: I C Cu + adorpton/deorpton of Cu + between electrode bulk and urface, ν e C + Ω j e electron tranfer reacton Cu 2+ + e Cu adorpton/deorpton of Cu 2+ between urface and caton of the electrolyte. The adorpton n the frt and thrd tep condered nherently fat compared to 77 uch that aumpton 23 vald. 2

23 Bulk tranport. The momentum balance 1b mple that the preure n Ω + and Ω contant p = p. Snce we do not conder olvated on here, we may aume that all partcle are of the ame ze. Then the conttutve model for an ncompreble mple mxture [DGM15, DGL14a] together wth global electroneutralty accordng to 1c lead to the relaton n A + n C + n S = n ref, 78a z A n A + z C n C =, 78b where n ref contant total number denty of partcle. The pece A, C and S of the electrolyte atfy the tatonary veron of the ma balance equaton 1a. In the abence of bulk reacton we have to olve n Ω x manav + JA =, 79a x m C n C v + J C =, x ρv =, 79b 79c where we replaced the partal balance of S by the total ma balance. In addton there are two more contrant. Frt, we have to pecfy the total amount of copper ulphate dolved nto the water by precrbng the average number denty n C of caton n the electrolyte. Second, an abolute reference value for ϕ ha to be defned, e.g. ϕ = at I C. The boundary condton 11a for the pece A, C and S at the nterface are m α γ α R A/C = m α n α v w A/C ν + J α ν I A /I C, 8 where R A/C and w A/C denote the reacton rate and the nterfacal peed at I A and I C, repectvely. For the reacton 77 we have γ C = 1 and γ A = γ S =. We multply 8 wth z α e /m α for each α. Then ummaton and the electroneutralty 78b yeld z C e RA/C = zαe m α J α ν I A/C, 81 α {A,C,S} Wth 21 we conclude that z C e R A/C = j e ν and get the boundary condton m α γ α j e = z C e m α n α v w A/C + J α I A/C for α {A, C, S}. 82 Summng the boundary condton 82 for α = A, C, S how m C γ C j e = z C e ρ v w A/C I A/C. 83 From 79c we conclude w A = w C. Thu we can chooe a coordnate ytem uch that w A = w C =. The explct form of the boundary condton 82 m A n A v + J A =, z C e m C n C v + J C = m C γ C j e, z C e ρ v = m C γ C j e. 84a 84b 84c 21

24 n C x [ mol /l] j e [ A/m 2 ] x [ cm ] ϕx [ V ] j e [ A/m 2 ] x [ cm ] Fgure 3: Soluton of 84 over pace for dfferent appled current and a alt concentraton of.5 mol/l. Left: We oberve almot lnear concentraton profle n C wth a lope proportonal to j e. Rght: electrotatc potental ϕ n the electrolyte doman Ω. The boundary condton can be extended to hold n the whole electrolyte doman Ω by applyng 79. We emphaze the drect proportonalty between the electrc current j e and the barycentrc velocty v n the electrolyte n 84c. For th reaon a long a electrc current flow, the barycentrc velocty can not be neglected, a t uually done n the lterature. To olve the ytem 84 numercally, we chooe a dagonal moblty matrx wth M αα = B α T m 2 αn α leadng to the dffuve fluxe J α = B α m α x n α mα n α e m n x n + n α z α xϕ, α = A, C. 85 Note that there no preure dependence n 85 becaue x p =. Smulaton for Ω of length L = 1cm and ung materal parameter accordng to Table 1 yeld oluton n C and ϕ a plotted n Fgure 3. We oberve a nearly lnear patal dtrbuton of the caton wth a lope that proportonal to the mpoed current j e. Wth ncreang current, we oberve a nonlnear behavor of the electrotatc potental ϕ near the cathode, where the caton concentraton get low. A: SO 2 4 C: Cu 2+ aq S: H 2 O m A = u m C = u m S = u B A = /kg B C = /kg Table 1: Materal parameter ued for the calculaton, cf. [Ld5]. Butler-Volmer equaton. A Butler-Volmer equaton for the reacton 77 can be derved n the context of cae II of Sect The choce of Ω + above and 77 mply Γ + = z C = 2. Expermental meaurement n [BM59] how that α A 1.5 and α C.5. We thu chooe A = 1 and β = 1/4 to get from 51 at I A and I C 1/4 nc R = R exp 1 n C 2 e η S 3/4 nc 3 e R exp n C 2 η S

25 5 η [ mv ] anode catode logj e j e [ A / m 2 ] Fgure 4: Tafel plot from the reacton 77 and the computed electrolyte concentraton at I A and I C accordng to 84 wth n C =.5mol/l. Dahed lne for fxed caton concentraton n C = n C at I A and I C how the lnear Tafel lope n the large overpotental regme. Ung the electrolyte concentraton at the nterface from the prevou computaton, we can determne from 86 the overpotental at the anode and cathode, ee Fgure 4. In the Tafel plot we oberve a nearly lnear lope over one decade of the mpoed current dente. When j e get larger, the Tafel plot devate from the lnear behavor. Th effect due to the dependency of the exchange current on the concentraton of the pece. The oberved devaton much more pronounced at the cathode, where n C for larger mpoed current dente. Polarographc curve. From the computed overpotental we can not drectly nfer the potental dfference [ϕ ] over the double layer, nce by defnton η S alo depend on the choen equlbrum value. But, nce n the expermental etup condered here the electrode cont of the ame materal, the equlbrum potental [ ϕ ] the ame at I A and I C. We denote the electrotatc potental n the bulk of the electrode A and C by ϕ A and ϕ C, repectvely. Then the voltage over the complete electrochemcal cell ϕ A ϕ C = η A S + ϕ I A ϕ I C η C S, 87 where η A/C S denote the overpotental at I A/C, repectvely. At a certan mpoed current where n C approache at I C, the overpotental at the cathode, and thu alo ϕ A ϕ C, ha to blow up, a dcued n Sect Motvated by the dffuon lmted current n [BBC5], we defne j e d := 4z Ce B C n C L. 88 For a dluted electrolyte, we oberve that the blow up of the cell voltage occur cloe to j e = jd e, ee Fgure 5. Increang the alt concentraton,.e. ncreang n C, the characterzaton of the lmtng current by jd e get le and le harp, a the blow up of the cell voltage occur at hgher mpoed current. Th dcrepancy ha to attrbuted to the mpler bulk model n [BBC5], where v = aumed n Ω and the on olvent nteracton neglected. 23

26 n C [ mol /l] ϕ A - ϕ C [V] j e [A / m 2 ] n C [ mol /l] ϕ A - ϕ C [V] j e / j e d Fgure 5: Lmtng current caung a blow up of the cell voltage. Left: polarographc curve for dfferent alt concentraton. we oberve a blow up of the cell voltage for dfferent appled current j e. Rght: polarographc curve where the current recaled by jd e defned n Dcuon 7.1 Valdty of the Butler-Volmer equaton The Butler-Volmer equaton not a unveral natural law but t valdty lmted to certan applcaton cenaro. Our general Butler-Volmer equaton 33 can only be vald a long a the aumpton of the underlyng reduced bulk model hold. Frt of all, the reduced bulk model requre that the Debye length mall. That mean, we chooe a charactertc length cale L ref for the electrochemcal ytem under conderaton, uch that the overall ze of the ytem comparable to L ref and the curvature of urface le than 1/L ref. Then the Debye length, whch control the wdth of boundary layer, ha to be maller than L ref by ome order of magntude. In conequence, the Butler-Volmer equaton 33 can not be appled n the context of nano-ytem. Second, the dervaton of the reduced bulk model baed on qua-equlbrum of the boundary layer. For th, t neceary that relaxaton tme of the layer are mall compared to the macrocopc expermental tmecale. In [DGM15], a macrocopc tme cale of t ref = 1 wa ued. Th certanly rule out the applcaton of 33 n procee where exctaton by hort pule or medum to hgh frequence are appled. Moreover n the reduced model, we aumed othermal ytem. If the urface reacton are trongly endothermc or exothermc, the energy balance can not be neglected and the procedure of the aymptotc analy of [DGM15] ha to be appled to the larger ytem of equaton and mght pobly lead to dfferent relaton. Fnally, we remark that n ome cae t mght be more approprate to aume that the number dente of the urface pece are comparable to thoe n the volume. Th would necetate the ncluon of a materal model for the urface pece and lead to dfferent couplng condton of the urface pece to the bulk equaton. The Marcu-Huh theory often ued n tuaton where the clacal Butler-Volmer equaton fal. In partcular, the Marcu-Huh theory able to decrbe curved Tafel lope whch are nterpreted a a dependency of the tranfer coeffcent on the overpotental. Curved Tafel lope have been reported e.g. n [ST75], but uually th requre non-teady tate technque. 24

27 Expermental condton to oberve curved Tafel lope under teady-tate condton were decrbed n [Fel1] and expermental reult have been reported for a redox couple that ha a pathologcally mall rate contant [FV11]. We note that becaue the Butler-Volmer equaton 33 formulated n term of chemcal potental ntead of number dente, t eem poble that applyng a materal model dfferent from the mple mxture would allow for curved Tafel behavor. The ame alo hold true for temperature dependence of Tafel lope. In a ere of comparatve tude [LWH + 11, HWLP + 11, HLRC12], expermental data of quare wave voltammetry and cyclc voltammetry wa ftted to Marcu-Huh and Butler-Volmer theory. The reult ndcate ome quanttatve weakne of the Marcu-Huh theory. We remark that cyclc voltammetry or quare-wave voltammetry requre the ncluon of a tme cale nto the model equaton that at leat one order of magntude lower than the t ref = 1 condered here. Thu, n our pont of vew, a mulaton of thee procee hould be baed on the the complete model of [DGM15]. 7.2 Comparon wth the lterature General metal electrode. The textbook lterature provde Nernt- and the Butler-Volmer equaton for the electron tranfer reacton 39 at a metal electrode whch we can compare wth the repectve equaton n Sect For the Nernt equaton, we fnd [NTA4, 8.2]: ϕ M ϕ E = kc ȳ O ln. 89 ne k A ȳ R Here k A/C are the rate contant related to the anodc and cathodc reacton, repectvely. The ame tructure of the Nernt equaton can be found e.g. [BRGA2, 7.4] and [BF1, 3.2.2], only the rate contant are replaced by a reference potental. All thee equaton how the ame logarthmc dependency on the equlbrum concentraton ȳ O and ȳ R a 45. For the Butler-Volmer equaton, the overpotental η S defned n [NTA4, 8.21] dentcal to 41 and n our notaton the author tate the Butler-Volmer equaton: [NTA4, 8.24]: j e ν I = j exp βn e 1 βn η S j e exp η S. 9 The ame tructure of the equaton can be found n [BRGA2, 7.23]. We oberve that the exchange current are the ame for the anodc and the cathodc current,.e. j A = j C = j, what not compatble wth our approach to guarantee the contency wth the 2 nd law of thermodynamc. Moreover j e then alway mple η S. In contrat, the overpotental defned n Secton 4 depend on the choen equlbrum tate wth t repectve electrolyte concentraton at the nterface. A devaton from the concentraton of the reference tate then mple a non-vanhng overpotental even for j e =. The dependency on the concentraton pecfed a [NTA4, 8.23]: j = ne k β A k1β C y β R y1β O. 91 A a conequence, for a low amount of the oxdzed pece y O 1, the Butler-Volmer equaton 9 mple that a hgh overpotental η S requred n order to mantan an oxdaton 25