V. Electrostatics Lecture 27a: Diffuse charge at electrodes
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1 V. Electrstatcs Lecture 27a: Dffuse charge at electrdes Ntes by MIT tudent We have talked abut the electrc duble structures and crrespndng mdels descrbng the n and ptental dstrbutn n the duble layer. Nw we start t nvestgate the effect f duble layer n the reactn knetcs, especally hw the dffuse nc charges affect the Faradac halfcell reactn knetcs. 1. Frumkn Crrectn t Butler Vlmer Knetcs Frst we assume that Faradac reactn takes place at a reactn plane, whch s rughly the same as stern plane. The charge carrers need t hp ver ths stern layer barrer (wth a thckness f ne slvent mlecule) and pck up an electrn t fnsh the Faradc reactn. Ths regn s bascally a slvatn shell f the electrde surface. emember that the Butler Vlmer (B-V) knetcs s a mlecular scale equatn, thus, n rder t ntrduce the Frumkn crrectn, we apply the B-V equatn nly acrss ths cmpact layer and the reactn rate, where the reactn ccurs. Then the B-V equatn can be descrbed as: e e (1 ) ne ne [ ] prefactr k a a 1 verptental acrss stern layer : eq a ln( ) ne a A Cmbnng all these equatns we wll get: eq (1 ) ne ( ne ) ( ) [ ae ae ] A Nte that fr every charged speces, actvtes r cncentratns are all evaluated lcally at the stern plane, whch dffers frm the crrespndng values n the neutral bulk electrlyte as we used n BV equatn befre. Nw, n rder t relate t t the bulk cncentratns, we need a duble layer mdel. Fr example, f the dffuse part f the duble layer satsfes dlute slutn apprxmatn, then fr nc speces wth charge z e
2 Lecture 27a: Dffuse charge at Electrdes c ( x) where c c e ze ( x) c s bulk cncentratn s cncentratn at stern plane ( x) bulk D( x) e ze s the Bltzman factr fr charge acrss the dffuse layer. eq Ths makes, and thus and all depend n zeta ptental r surface charge. At surface. Ths was frst nted by Frumkn n the 193s, but Zeta ptental s x=, ( x) D( x) always related t charge, and r qs are usually ftted and assumed cnstants. Hwever, as we seen n the prevus lecture, must depend n the current and shuld be predcted by the mdel. 2. Cmplete mathematcal mdel based n tern bundary cndtn After we splt duble layer nt tern/cmpact layer and dffuse layer, ptental bundary cndtn at the nterface f these tw layers s always mssng n electrchemcal system mdelng. T cmplete the mathematcal mdel, we have t ntrduce a reasnable bundary cndtn. T an apprxmatn, snce n charge s lcated wthn the cmpact layer, the electrc feld wthn t can be treated as cnstant, s f we dente x = t be the tern plane, ur bundary cndtn can be wrtten n terms f ur effectve cmpact layer thckness as fllws based n a lnear extraplatn f the ptental acrss the cmpact layer. n n E e ( x) ( x) where n s the utward nrmal pntng frm slutn t the electrde h s the effectve thckness f the stern layer Ths makes reactn knetcs dependent n nrmal electrc feld E Apply Gauss s law, we wll fgure ut that t als depends n surface charge densty q s. n ( E slutn E metal ) q qs n Eslutn slutn metal s slutn Then we relate qs t : n E q s h q s slutn 2
3 Lecture 27a: Dffuse charge at Electrdes A. Helmhltz lmt emember the dmensnless grup s the rat f stern layer effectve thckness and D Debye screenng length. When ges t nfnty D, and we recver the usual BV knetcs usng bulk cncentratns wthut any Frumkn crrectn fr dffuse charge. But surface charge s nt ndependent. The mdel predcts surface charge self-cnsstently by q s h, q eq s eq h B. Guy-Chapman lmt When ges t zer, D, and the flux bundary cndtn lses ts explct vltage dependence. Cnsder the reactn belw: s M z ne Flux bundary cndtn s: (1 ) ne s n F s ae ae A k c k c (Chang-Jaffe BC) r ne ( ) ( ) where (1 ) ne k r e A ne kr re A In the dervatn, t seems that vltage n lnger matters fr the reactn knetcs, but actually t des. The current r reactn rate stll depends mplctly n the ttal duble layer vltage va the cncentratns c, c. Wth k, kr cnsts ths s Chang-Jaffe bundary cndtn (just standard chemcal knetcs appled at the stern plane) 3
4 Lecture 27a: Dffuse charge at Electrdes Example: Cnsder a cathde reactn M M e (M=L, H, etc) Assume dlute slutn and quas-equlbrum, c s the stern plane cncentratn, and bulk cncentratn. ed c c c e D c bulk I e e/ kc kr c e ( e s Frumkn crrectn) ea Ths just lks lke BV equatn wth =1 At equlbrum =, we get krc a eq ln( ) ln( ) Ths s the Nernst equatn. e k c e a eq I kc k c k c ( ) e ea e kc I k c ea(1 e ) I (1 e ) e e/ c s the Based n ths current expressn, we see that n the Guy-Chapman lmt, duble layer acts lke an deal semcnductr dde (lke p-n junctn behavr), and pass current n ne drectn nly. Ths can be further explaned frm mcrscpc pnt f vew n fgure 2 and 3. Fg. 1 In Guy-Chapman lmt duble layer nly allws current n ne drectn 4
5 Lecture 27a: Dffuse charge at Electrdes In the frward bas, vltage keeps ncreasng the actve speces (cunterns) at surface, and the dffuse layer s based t accumulate a lt f actve speces (here catns) at the reactn plane due t the negatve vltage relatve t the bulk. Therefre, the lcal resstance s lw and t allws a large current. Fg. 2 In dstrbutn n frward bas Hwever, n the reverse bas, the actve speces (catns) are depleted by the pstve ptental, thus lcal resstance becmes large and nly a small cnstant current can g thrugh, snce we assume a neutral reduced state, whse cncentratn s unaffected by the electrc feld. Fg. 3 In dstrbutn n negatve bas 5
6 Lecture 27a: Dffuse charge at Electrdes We can als wrte verptental as a functn f current fr the Guy-Chapman lmt f the Frumkn-Butler-Vlmer thery: I ln(1 ) e I Ths lks very smlar t the dde crcut element fr dffusn lmtatn (n cncentratn plarzatn), but nte that here I s actually a reactn lmted current. eferences 1. M. Z. Bazant, K. T. Chu and B. J. Bayly, Current-vltage relatns fr electrchemcal thn flms, IAM J Appl Math 65 (25) P. M. Besheuvel, M. van estbergen, and M. Z. Bazant, Impsed currents n Galvanc cells, Electrchemca Acta 54 (29)
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