G f. 1 Electric double layer effect for the redox species in solution. Electric double layer effect (EDL) for the redox species in solution

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1 M M Elecric doule layer effec EDL for he redox species in soluion Masahiro Yamamoo April 8, : 0 am In his repor I will reformulae he elecric doule layer effec of ulk redox reacion. 1 Elecric doule layer effec for he redox species in soluion I f e l e c r o d e n e M r e a c i o n p l a n e O x z R d z n s 0 O x z + n e z F s n F z F n F M = s = 0 G f G f = zfφ s nfφ M z nfφ s = nfφ M φ s = nfe = nfφ M φ = nfe φ + φ s = nfe φ G G R d z n z n F s z n F Figure 1: Lef: Redox reacions are occurred a he reacion plane z = z. Righ: Energy diagram along he reacion pah. In his figure he reducion sae is akes as he energy reference. We assume he redox reacion Ox z + ne Rd z n akes place a he reacion plane a z = z. When he elecric doule layer effec is no considered surface charge densiy is low and he reacion plane is no so close o he elecrode, he poenial φ a he reacion plane can e negligile, and he poenial is equal o he poenial φ s a ulk soluion φ = φ s = 0 1 When he exernal poenial E = φ M φ s is applied o he elecrode he difference 0 of he elecrochemical poenial of he reacans and he producs ecome 0 = zfφ s nfφ M z nfφ s = nfφ M φ s = nfe 1

2 Here F is he Faraday consan, φ M is he poenial a he elecrode, and E is he elecrode poenial. In Buler-Volmer heory he difference 0 conriue he decrease of he acivaion arrier for he reacion of Ox z + ne Rd z n y α 0 from he arrier G f when E = 0, and 0 conriue he increase of he reacion arrier for oxidaion y 1 α 0. When he elecrode inerface is srongly elecrified, he poenial a he reacion plane is no negligile and he reacion rae will change. In he following formulaion, we simply pu φ ino φ s φ in he reacion rae heory y Buler-Volmer. Then 0 should e replaced y in he following way, 1.1 kineics = zfφ nfφ M z nfφ = nfφ M φ = nfe φ 3 The acivaion arriers for he forward reducion G f and ackwardoxidaion reacions G are given y G f = G f α = G f + nfαe φ 4 G = G G f + + G f = G G f + + G f α = G + 1 α = G nf1 αe φ 5 Here G f and G is he acivaion arrier for he forwardreducion and ackwardoxidaion reacion when = 0. Figure : α parameer The curren for he forward reducion reacion I f is I f = nfa kt h G f c Oxz 6 Here A is he area of he elecrode, k is Bolzman consan, h is Plank consan, R is gas consan, T is emperaure, c Ox z is he concenraion of Ox a he reacion plane z = z. From he equilirium condiion we have we will show laer c Ox z = zfφ 7 Here is he concenraion of Ox in he ulk phase. We have I f = nfa kt h G f = nfak 0 f zfφ k 0 f kt h G f c Oxz = nfa kt h G f αnfe The curren for ackward reacion oxidaion is given y I = nfa kt h G c Rdz = nfa kt h G nfαe φ c 0Ox zfφ 8 1 αnfe φ z nfφ 9

3 = nfak 0 zfφ k 0 kt h G 1 αnfe Here c 0 is he concenraion of Rd in he ulk phase. We used Rd 10 1 αnfe φ z nfφ = 1 αnfe nfφ + αnfφ znfφ + nfφ = 1 αnfe + zfφ 1. From equilirium condiion Ox z z + nez = 0 : elecrode Rd z n z µ Ox z = µ 0 Ox + ln a Oxz + zfφ 14 µ Ox z = + = µ 0 Ox + ln a0 Ox + zf φ s }{{} =0 15 µ Ox z = µ Ox z = +, a Ox z = a 0 Ox zfφ 16 c Ox z = zfφ, in he case of γ Ox = 1 17 µ e = µ 0 e nfφ M 18 µ Rd z = µ 0 Rd + ln a Rdz + z nfφ 19 µ Rd z = + = µ 0 Rd + ln a0 Rd + z nf φ s }{{} =0 0 µ Rd z = µ Rd z = +, a Rd z = a 0 Rd z nfφ 1 c Rd z = z nfφ, in he case of γ Rd = 1 µ 0 Ox + µ0 e + ln a Ox z + zfφ nfφ M = µ 0 Rd + ln a Rdz + z nfφ 3 The oal curren is zero in equilirium and for he poenial E = E eq k 0 f zfφ αnfe0 k 0 f zfφ φ M φ s φ φ s = µ0 Rd µ0 Ox µ0 e nf αnfe eq E eq φ = E 0 + nf ln a Oxz a Rd z + nf ln a Oxz a Rd z 4 5 = E 0 + nf ln a0 Ox a 0 z n φ + z n n φ 6 Rd E eq = E 0 + nf ln a0 Ox a 0 Rd E eq = E 0 + nf ln c0 Ox 7 8 I f + I = 0 9 = k 0 zfφ 1 α α = k 0 zfφ k 0 f αnfe 0 = k 0 1 αnfe 0 1 αnfeeq 1 αnfe α α k

4 1.3 Elecric doule layer EDL effec Then we have I f = nfak 0 f zfφ = nfa k 0 zfφ } {{ } = k app αnfe αnfe E0 I = nfak 0 zfφ 1 αnfe = nfa k 0 zfφ 1 αnfe E 0 } {{ } = k app The oal curren of he sysem can e wrien in he same way as Buler-Volmer-like equaion I = I f + I = nfak app [ αnfe ] E0 1 αnfe E c 0 0 Ox In he aove equaion, please noe ha he ulk concenraions are used. If we wrie he apparen rae consan y The EDL effec for he redox reacion rae ecomes k app = k 0 zfφad In he original Buler-Volmer equaion he surface concenraions are used I = I f + I = nfak 0 [ αnfe E0 c Ox z, k 0 k 0 f αnfe 0 1 αnfe = k αnfe E 0 ] c Rd z, In his sense he concenraion profile of redox species should e reconsidered. The doule layer region diffuse layer is much shor-ranged compared o he diffusion layer as shown elow. If he elecric doule layer EDL effec is on, he redox concenraion is grealy enhanced or suppressed a he diffuse layer in such a way c Ox z = zfφz /. 4

5 Figure 3: some examples of CV and he concenraion profile Figure 4: some examples of CV and he concenraion profile 5