A Most Useful Device of Studying Electrode Processes: The Rotating Disk Electrode

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1 A Most Useful Device of Studying Electode Pocesses: The Rotating Disk Electode the theoetical basis Soma Vesztegom Laboatoy of Electochemisty & Electoanalytical Chemisty Eötvös Loánd Univesity of Budapest 10 th Octobe / 30

2 Oveview 1 Intoduction As a Reminde... Histoy 2 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile 3 4 Mixed Kinetic and Tanspot Contol Steady State) Tansients at the RDE Potential Step Linea Sweep 5 Constuction of the RRDE A Common Collection Expeiment 2 / 30

3 A Simple Polaization Cuve As a Reminde... Histoy Let us conside a simple electode pocess that involves a single electon tansfe between a educed species R and an oxidized species O, O + e R, and let us then assume that the net ate of this electode pocess is affected only by i.) the ate of the chage tansfe, ii.) the tanspot to o away fom the electode suface and iii.) nothing else. How should a polaization cuve of such a system, taken fom an RDE, look like? 3 / 30

4 As a Reminde... Histoy Ealy Investigations on Tanspot Contol Hemann Ludwig von Helmholtz Übe die Bewegungsstöme am polaisieten Platina. Vogelegt de Beline Akademie am 11. Mäz / 30

5 As a Reminde... Histoy The Fist Desciption of the RDE The famous Levich equation: I l = 0.620) z i F A D 2 3 i ν 1 6 c i, ω 1) Veniamin Gigo evich Levich The Theoy of Concentation Polaization, Acta Physicochim. URSS, 1942, vol. 17, pp / 30

6 Flux at an RDE Suface I. The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The flux at an RDE suface can geneally be given as j i = c i v + D }{{} i gad c i + µ }{{} i c i z i gad ϕ +..., 2) }{{} convection diffusion migation which equation can, when a suppoting electolyte is pesent in the system in a highe concentation, be simplified to j i = c i v D i gad c i. 3) 6 / 30

7 Flux at an RDE Suface II. The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile Based on Eqn. 3., it can be stated that since c i t dv = j i da = div j i dv, 4) and, thus, V A V c i t = div j i, 5) c i t = v gad c i + div D i gad c i ). 6) 7 / 30

8 Flux to the RDE Suface III. The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile If we assume that the flow is stationay thus, c i t D i is independent fom and c ), Eqn. 6 yields = 0) and that D i div gad c i = v, t) gad c i 7) Eqn. 7. is the vectoial fom of a system of secound-ode, patial diffeential equations, holding the functional coëfficient v, t). Detemining this latte function is a peequisite of any futhe investigations. 8 / 30

9 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Geneal Fom The Navie Stokes equations, in case of an incompessible fluid, can be witten as d v d t = gad p ϱ + ν div gad c + f ϱ, 8) which should be consideed togethe with the equation of continuity, div v = 0. 9) 9 / 30

10 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Coodinate System Used fo the Solution The poblem itself suggests the use of cylindical coodinates: Szőllőskislaki Kámán Tódo Übe laminäe und tubulente Reibung. Zeitschift fü Angewandte Mathematik und Mechanik, 1921, vol. 1., pp / 30

11 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom v t + v v = p + ν + vϕ [ 1 v ϕ + v vz z v ) + 1 v 2 ϕ = 2 v 2 ϕ v z 2 v 2 2 ] v ϕ 2 + ϱf ϕ v ϕ t + v v ϕ = 1 p ϕ + ν + vϕ v ϕ ϕ + v ϕ vz z [ 1 vϕ ) vvϕ = 2 vϕ 2 ϕ vϕ z ] v 2 ϕ vϕ 2 + ϱf ϕ v z t + v z v + vϕ = 1 [ p 1 ϱ z + ν v z ϕ + v z vz z = vz ) vz 2 ϕ 2 + ] 2 vz z 2 + ϱf z 1 ϱv) + 1 ϱv ϕ) ϕ + ϱvz) z = 0 11 / 30

12 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom v t + v v = p + ν + vϕ [ 1 v ϕ + v vz z v ) + 1 v 2 ϕ = 2 v 2 ϕ v z 2 v 2 2 ] v ϕ 2 + ϱf ϕ v ϕ t + v v ϕ = 1 p ϕ + ν + vϕ v ϕ ϕ + v ϕ vz z [ 1 vϕ ) vvϕ = 2 vϕ 2 ϕ vϕ z ] v 2 ϕ vϕ 2 + ϱf ϕ v z t + v z v + vϕ = 1 [ p 1 ϱ z + ν v z ϕ + v z vz z = vz ) vz 2 ϕ 2 + ] 2 vz z 2 + ϱf z 1 ϱv) + 1 ϱv ϕ) ϕ + ϱvz) z = 0 11 / 30

13 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom v t + v v = p + ν + vϕ [ 1 v ϕ + v vz z v ) + 1 v 2 ϕ = 2 v 2 ϕ v z 2 v 2 2 ] v ϕ 2 ϕ v ϕ t + v v ϕ = 1 p ϕ + ν + vϕ v ϕ ϕ + v ϕ vz z [ 1 vϕ ) vvϕ = 2 vϕ 2 ϕ vϕ z ] v 2 ϕ vϕ 2 v z t + v z v + vϕ = 1 [ p 1 ϱ z + ν v z ϕ + v z vz z = vz ) vz 2 ϕ 2 + ] 2 vz z 2 1 ϱv) + 1 ϱv ϕ) ϕ + ϱvz) z = 0 11 / 30

14 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom v t + v v = p + ν + vϕ [ 1 v ϕ + v vz z v ) + 1 v 2 ϕ = 2 v 2 ϕ v z 2 v 2 2 ] v ϕ 2 ϕ v ϕ t + v v ϕ = 1 p ϕ + ν + vϕ v ϕ ϕ + v ϕ vz z [ 1 vϕ ) vvϕ = 2 vϕ 2 ϕ vϕ z ] v 2 ϕ vϕ 2 v z t + v z v + vϕ = 1 [ p 1 ϱ z + ν v z ϕ + v z vz z = vz ) vz 2 ϕ 2 + ] 2 vz z 2 1 ϱv) + 1 ϱv ϕ) ϕ + ϱvz) z = 0 11 / 30

15 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom v t + v v + v v z z v2 ϕ = p [ 1 + ν v ϕ t + v v ϕ + v v ϕ z z + v [ v ϕ 1 = ν v z t + v v z + v v z z z = 1 ϱ p z + ν [ 1 v ) + 2 v z 2 v ] 2 v ) ϕ + 2 v ϕ z 2 v ] ϕ 2 v ) ] z + 2 v z z 2 1 ϱv ) + ϱv z) = 0 z 11 / 30

16 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom v t + v v + v v z z v2 ϕ = p [ 1 + ν v ϕ t + v v ϕ + v v ϕ z z + v [ v ϕ 1 = ν v z t + v v z + v v z z z = 1 ϱ p z + ν [ 1 v ) + 2 v z 2 v ] 2 v ) ϕ + 2 v ϕ z 2 v ] ϕ 2 v ) ] z + 2 v z z 2 1 ϱv ) + ϱv z) = 0 z 11 / 30

17 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom [ v t + v v + v v z z v2 ϕ 1 = ν v ϕ t + v v ϕ + v v ϕ z z + v [ v ϕ 1 = ν v z t + v v z + v v z z z = 1 ϱ p z + ν v ) + 2 v z 2 v ] 2 [ 1 v ) ϕ + 2 v ϕ z 2 v ] ϕ 2 v ) ] z + 2 v z z 2 1 ϱv ) + ϱv z) = 0 z 11 / 30

18 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom [ v t + v v + v v z z v2 ϕ 1 = ν v ϕ t + v v ϕ + v v ϕ z z + v [ v ϕ 1 = ν v z t + v v z + v v z z z = 1 ϱ p z + ν v ) + 2 v z 2 v ] 2 [ 1 v ) ϕ + 2 v ϕ z 2 v ] ϕ 2 v ) ] z + 2 v z z 2 1 ϱv ) + ϱv z) = 0 z 11 / 30

19 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: The Cylindical Fom [ v v + v v z z v2 ϕ 1 = ν v ϕ v + v v ϕ z z + v [ v ϕ 1 = ν v z v z z = 1 p ϱ z + ν v ) + 2 v z 2 v ] 2 v ) ϕ + 2 v ϕ z 2 v ] ϕ 2 [ 2 ] v z z 2 1 ϱv ) + ϱv z) = 0 z 11 / 30

20 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: Assumptions & Simplifications In ode to povide a solution of the Navie Stokes equations, we have made some assumptions, namely: 1 that we have neglected all natual foces acting on the system, since these ae all consideed small when compaed to the foces aising fom convection, thus, we have assumed that f = 0; 10) 2 that by axial symmety, the flow cannot depend explicitly on φ, thus, all the patials φ should disappea; 3 that since the fluid is incompessible and the lamina hoizontal, the pessue is a function of z only; 4 that in addition to being independent fom φ, v z is also independent fom ; 5 and, finally, that since ou pimay inteest is to povide a solution fo a non-tubulent, stationay state, d v = 0. 11) d t 12 / 30

21 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: Bounday Conditions fo RDE The bounday conditions used fo solving the pevious system of diffeential equations ae as follows: at z = 0 at z = v 0 0 v φ ω 0 v z 0 X Table: Bounday conditions expessed in cylindical coodinates. 13 / 30

22 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: Casting by Kámán Let us cast the components of v into a dimensionless fom, based on the following elations: v = ω F ξ), 12) v φ = ω G ξ), 13) v z = ω νh ξ), 14) p = ϱ ν ω P ξ), 15) whee ω ξ = z ν. 16) 14 / 30

23 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Navie Stokes Equations fo an Incompessible Fluid: Dimensionless Fom The peviously witten Navie Stokes equations ae shown below in a dimensionless fom. Using numeical methods, these can aleady be solved. Diffeential equations Bounday conditions 2F + d H d ξ = 0 F 0) = H 0) = P 0) = 0 F 2 + H d d F ξ G2 = d 2 F d ξ 2 G 0) = 1 2FG + H d d G ξ = d 2 G d ξ 2 F ) = H ) = 1 2 FH d F d ξ ) = d P d ξ Table: The poblem in dimensionless fom. 15 / 30

24 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Concentation Pofile Nea an RDE Suface Now that the velocity pofile is known, we focus again on the aleady discussed Eqn. 7. We should point out, that just like v, the concentation is also independent fom the angula and the adial coodinate. Eqn. 7. can thus be ewitten in ode to eflect this as d 2 c i D i d z 2 = v z z) d c i d z, 17) and it can be shown that the axial velocity can be well appoximated as v z z) = ων z 2. 18) 16 / 30

25 The Limiting Role of Tanspot Calculating the Velocity Pofile Calculating the Concentation Pofile The Concentation Pofile Nea an RDE Suface The diffeential equation, D i d 2 c i d z 2 = ων z 2 d c i d z, 19) has a solution that is { 1 c i z) = Γ 3, z3 ω 2 )} D c. 20) ν ω 17 / 30

26 Today s Conclusion Aiving to the Levich Equation The Levich equation can be acquied fom Eqn. 20. in the following way: ) d ci I l,i = ±z F A D i d z ±z F A D 2 3 i ν 1 6 ci, ω 21) Thank You fo the attention! 18 / 30

27 Today s Conclusion Aiving to the Levich Equation The Levich equation can be acquied fom Eqn. 20. in the following way: ) d ci I l,i = ±z F A D i d z ±z F A D 2 3 i ν 1 6 ci, ω 21) Thank You fo the attention! 18 / 30

28 Mixed Kinetic and Tanspot Contol Steady St Tansients at the RDE Potential Step Linea Sweep Mixed Kinetic and Tanspot Contol Steady State thus, I I l = c c 0 c, 22) ) I c 0 = c z F A D 3 2 ν 1. 23) 6 ω c Substituting this to the Edey-Gúz Volme-equation and eaanging yields [ I = F A k 0 exp α z F ] R T E E0 ) } {{ } k ct ) I c z F A D 3 2 ν 1. 24) 6 ω c Afte defining I k = n F A k ct c, we get to the Koutecký Levich-equation: 1 I = ) I k I k 19 / 30

29 Tansients at the RDE Mixed Kinetic and Tanspot Contol Steady St Tansients at the RDE Potential Step Linea Sweep In a geneal case, when studying tansient effects e.g., effects fo which c t is finite), solving Eqn. 26. is neccessay: c i t = D i div gad c i v gad c i. 26) Without going vey fa into the details, the solution fo the two following cases will be pesented: fo the case of a potential step expeiment; and fo the case of a linea voltammetic esponse. 20 / 30

30 Potential Step at an RDE Mixed Kinetic and Tanspot Contol Steady St Tansients at the RDE Potential Step Linea Sweep On an electode that is held at a potential whee no cuent flows, let us set anothe potential, so that the O + e R eaction can apidly go on. Thus, the new potential should be in the limiting cuent egion ). In ode to calculate the cuent as a function of time, solving Eqn. 26. is neccessay, with espect to the following bounday conditions c denotes the concentation of species O): c x, 0) = 0 27) lim c x, t) = c x 28) c 0, t) = 0 29) Remembe: in case of a stagnant not otating) electode, solution of the above poblem was caied out in a staightfowad way by applying Laplace tansfoms, yielding the Cottell-equation: I t) = n F A c D π t 30) 21 / 30

31 Potential Step at an RDE Mixed Kinetic and Tanspot Contol Steady St Tansients at the RDE Potential Step Linea Sweep Unlike the case of stagnant electodes, the solution fo an RDE can only be given in tems of infinite seies: I t) I l = ϑ 3 0, q), 31) whee ϑ 3 0, q) is the ) elliptic theta function at z = 0 having a so-called nome D 2 3 π q = exp 2. ϑ 3 z, q) is defined as the following sum: 1.61 ν 1 6 ω ϑ 3 z, q) = n= q n2 e 2nız, 32) thus, ) I t) D = exp n2 3 2 π 2 I l n= 1.61 ν 1 6 ω Watch animation. 33) 22 / 30

32 Linea Sweep at an RDE Mixed Kinetic and Tanspot Contol Steady St Tansients at the RDE Potential Step Linea Sweep It is known 1 that the dimensionless) cuent esponse function in linea sweep voltammety with Nenstian bounday conditions) can be calculated by an infinite-seies appoximation, with the use of the Lech tanscendent defined as Φ z, s, a) = m=0 z m a + m) s, 34) in the case ) F E z = exp R T 35) s = ) a = 1 + k 37) In this solution we have supposed that k = D 2 3 ν 1 6 ω R T F v 38) 1 J. Mocak, A. M. Bond, Use of Mathematica softwae fo theoetical analysis of linea sweep voltammogams, Jounal of Electoanalytical Chemisty 561 7), pp / 30

33 Constuction of the RRDE Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE 24 / 30

34 Constuction of the RRDE Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE 25 / 30

35 A Common Collection Expeiment Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE Let us conside a eaction, O + ne R, and let us ecod a steady-state) polaization cuve on the disk while keeping the ing while the ing is maintained at a sufficiently positive potential, E ing, that any R eaching the ing is apidly oxidized R O + ne ), so that the concentation of R at the ing suface is essentially zeo. We ae inteested in the magnitude of the ing cuent, I ing, unde these conditions; that is, we want to know how much of the disk-geneated R is collected at the ing. The appoach is again to solve the steady-state ing convective-diffusion equation, this time, fo species R: ) ) cr cr D R y = y 0.51 ω 2 3 ν c R y 2. 39) 26 / 30

36 Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE A Common Collection Expeiment Bounday Conditions Disk egion. Fo 0 < < 1, ) ) cr co D R = D O. 40) y y=0 y y=0 }{{} I D π 1 2nF Insulating gap egion. Fo 1 < 2, Ring egion. Fo 2 < 3, ) cr = 0. 41) y y=0 c R 0, t) = 0. 42) 27 / 30

37 Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE A Common Collection Expeiment Solution This poblem can be solved in tems of dimensionless vaiables using the Laplace tansfom method the solution shall not be discussed hee in details. It tuns out that the ing cuent is elated to the disk cuent by a quantity N, the collection efficiency, N = I R I D, 43) that depends only on 1, 2 and 3 and is independent of ω and all othe quantities. The collection efficiency can be calculated fom: N = 1 F ) { α +β 2 3 [1 F α)] 1 + α + β) F β whee α = 2 1 ) 3 1, β = 3 1 ) ) 3 and ϑ F ϑ) = 4π ln 1 + ϑ ) π actg Watch animation. [ ) ]} α 1 + α + β) 44) β [ 2 3 ϑ 1 3 ] ) 28 / 30

38 Tansients at the RRDE Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE Mathematical teatment of tansients occuing at an RRDE is a had job. Howeve, the poblem can be addessed using digital simulation methods. Basics of digital simulation will be discussed on-the-go, whith the example pesented hee. 29 / 30

39 Tansients at the RRDE Constuction of the RRDE A Common Collection Expeiment Tansients at the RRDE 30 / 30