Physical & Interfacial Electrochemistry 2013. Lecture 8 Hydrodynamic Voltammetry Hydrodynamic voltammetry • Hydrodynamic voltammetry • Examples of hydrodynamic deals with voltammetric electrodes include the rotating measurements conducted under disc electrode (RDE), the conditions where there is rotating ring disc electrode, the controlled hydrodynamic flow of wall jet electrode, the tubular solution near the electrode. electrode and the channel • If convection is applied to the electrode. solution in the vicinity of the electrode then the rate of mass • The transport of reactant and transport is increased which product can in principle be means that faster reactions may calculated quantitatively under be studied. such circumstances by solving the mass transport equations • If the fluid flow is well defined and the relevant hydrodynamic (laminar rather than turbulent), equations either using the rate of mass transport can approximate analytical methods be controlled quantitatively by (for particularly simple varying the solution flow rate. geometries) or, more often, via This is much more convenient numerical simulation (via finite than varying the size of the element modelling). electrode. Also steady state conditions are achieved at hydrodynamic electrodes. Transport and kinetics Transport via Molecular diffusion in electrochemical systems. only Diffusion layer (stagnant) Simple diffusion layer Double layer model neglects region convection effects and also simplifies c analysis of Fick Interfacial A A k diffusion equations. ET kET D B Bulk solution Electrode c0 (well stirred) D c c Material 0 Hydrodynamic f flux layer Transport and kinetics Diffusion layer approximation used. at electrodes. Current density 1.0 i J f ET & MT nFA nF D 0.5 Net flux D = i / i k MT D dc D 0.0 f D kET c0 c c0 kD c c0 -6 -4 -2 0 2 4 6 dx 0 = F(E-E0)/RT ET c c0 kET kET kDc 1 1 1 1 f D kD kET f kET c kDc f kET c0 Normalised 0 kET k exp potential Typically, for an aqueous Electrode Distance scales solution D = 10-9 m2s-1 in electrochemistry and = 10-6 m2s-1 and so ET / H ≈ 0.16. kinetics 1/3 0.1 nm Inner Helmholtz Plane D 1.61 Outer Helmholtz Plane H 1 nm RDE 10 nm Diffuse Double Layer 0.1 m Aqueous solution 1 m 10 m Diffusion Layer 0.1 mm 1 mm Hydrodynamic Boundary Layer H distance Convective diffusion equation. • In most cases a numerical approach has to be • Material flux near an adopted. electrode surface computed • Commercial software by solving the convective packages utilising finite diffusion equation (this element and finite neglects electromigration difference methods are effects). available to solve the • To do this the velocity convective diffusion and profile in the fluid must be Navier Stokes equations. evaluated using the Navier These include FEMLAB and Stokes equation of FLUENT. hydrodynamics, which is non Convection linear and therefore difficult Diffusion to solve analytically. c • The convective diffusion D2c v c equation admits analytical t solutions for only a few problems in electrochemistry such as the rotating disc 2c 1 c 2c c c electrode. D v v r 2 r r z2 r r z z RDE CV eqn. c c 2c v r v z D 2 r z z vCrzr 2 vCzz Normal diffusion C 0.513/2 1/2 2 c 2 c c D 2 Cz Crz z z r Normal Radial convection convection Either of these equations may be solved analytically for a Number of well defined geometries such as RDE, RRDE, ChE TE etc. http://physchem.ox.ac.uk/~rgc/john/Thesis/1/1.html Tubular electrode (TE) Rotating disc electrode (RDE) Channel electrode (CE) Wall Jet electrode (WJE) The Rotating Disc Electrode . • RDE has well defined hydrodynamic flow to electrode surface . • Fluid velocity profile near Insulating mantle disc well defined . • Transport of reactant to Angular surface obeys steady state velocity Convective diffusion equation which may be rigorously solved . • Rate of material transport depends in a well defined Disc manner on the rotation speed D of the electrode . k D 1/3 1/ 6 1/ 2 0.643D = kinematic viscosity -1 = rotation speed (Hz) kD= diffusive rate constant (cms ) D = substrate diffusion coefficient = Nernst diffusion layer thickness Rotating disc electrode (RDE) The reactant is conveyed to the electrode The RDE is constructed from a disk of electrode material surface by a combination of two types of (e.g. gold, glassy transport : convection and diffusion. carbon or platinum) imbedded in a rod of insulating material Vortex flow in the bulk solution continuously (e.g. Teflon). brings fresh reactant to the outer edge of the stagnant diffusion layer. The electrode is attached to a motor and rotated at a The reactant diffuses across stagnant certain frequency. layer. The thinner the stagnant layer, the The movement of rotation leads to a very well defined faster the reactant can diffuse across it solution flow pattern. and reach the electrode surface. The rotating device acts as a pump, pulling the solution Faster electrode rotation makes the upward and then stagnant layer thinner. Higher rotation rates permit the reactant to diffuse to the throwing it outward. electrode faster, resulting in a higher current being measured at the electrode. RDE z = 0 Configuration r + z Cylinderical polar co-ordinate system Rotating-ring-disk electrode (RRDE). RRDE is a variant of the rotating-disk electrode which includes a second electrode -a concentric ring electrode – that is placed outside the disk and used to analyze the species generated on the disk. The ring is electrically insulated from the disk so that their potentials can be controlled independently. Abbreviated as RRDE RRDE is a convenient way to measure post- electron transfer reactions of products. The relationship between disk current and ring current depends on rate of movement of product from the disk and is quantified in terms of the Collection Efficiency which can Be computed analytically and depends only on the geometry of the RRDE. RDE Configuration RRDE Configuration Rotating electrode assembly. The whole disc acts as a pump, sucking solution towards it, spinning the solution around and then flinging it out sideways. r rF Radial rG Angular Normal z H H() G() F() z z H 23/21/222 Velocity profiles near RDE vaz 0.51 zCz Surface. 3/2 1/2 varr 0.51 rzCrz Convective diffusion equation : Rotating disc electrode (RDE) Neglect radial Fluid velocity profile (approximate) Diffusion & Steady state time independent Convection. 2 equation. c 2 c D Cz 3/2 1/2 radial z 2 z C 0.51 Convective diffusion equation expressed in cylinderical polar co-ordinates (r,z,) Rotation frequency (Hz) normal rcc = 2f angular c r 00 r Angular velocity (rad/s) zcc Normalised distance c zfDkc0 ET 0 1/3 1/3 z zC D x r cos y rsin z z z0 r 00 2 z cc 3 cc 0 exp d 2 2 1 y 0 r x y tan 0 3 x Reactant concentration profile near RDE C = convective constant c c c c c 0 0 1/3 1/3 1/6 1/2 1/3 z z 0 KC D 1.288 1. 2 5 D c c 0 1. 61 1/6 D1/3 1/2 K = 31/3 (4/3) = 1.288. Nernst diffusion layer c cc 0 Approximation. z z0 Diffusion layer thickness for RDE (Levich) KC1/3 D 1/31.61 D 1/3 1/6 1/2 a c i 2nFD dr Levich equation L 0 z z 0 Diffusion limited Current. 0.62nFa 2 D2/3 1/6c1/2 Transport and kinetics at electrodes. A A k kET D Linear Plot: B B 11/2 fvs i kkcET D fkc ET 0 Kinetic info (k ,k0) got nFA kET k D ET from intercept c 11 Diffusion coefficient got from slope. fk ET k D D kD can be computed k D accurately provided 0 k k exp ET 1/3 1/ 6 1/ 2 Convective-diffusion 0.643D equation can be solved. Specific result for RDE, RRDE geometry. D 2/3 1/6 1/2 2/3 1/6 1/2 kDD 0.62 1.55 DW ZD Hz Rad s-1 Levich equation. Veniamin Grigorievich (Benjamin) Levich. a c i 2nFD dr Levich equation L 0 z z 0 1/ 2 0.62nFa 2 D2/3 1/6c1/2 iL SL Levich slope 2 2/3 ∞ -1/6 SL = 0.62nFa D c iL w1/2 Data Analysis using the RDE. Low • Apply constant potential to electrode. 1/i • Measure limiting current i at various rotation speeds . High • Plot 1/i versus -1/2 . • Kinetic information obtained from S intercept corresponding to infinite KL rotation speed . • Transport information obtained I from slope . KL -1/2 nFAc 1/2 SIKL KL iL Koutecky-Levich Plot . 1 IKL kET Diffusion coefficient calculated 1 from slope, ET rate constant SBKL calculated from intercept. B Levich constant 1.55D2/3 1/ 6 Tube Electrode (TE) The TE system consists of an annular electrode that is inserted into the length of a cylinderical tube down which the solution under study flows. The volume flow rate Vf is Since the flow is laminar chosen to ensure laminar The diffusion layer thickness flow of solution in the tube. Increases with the Because of friction with the Distance x downstream tube walls, the velocity of From the upstream edge of the liquid v in the tube varies in Tube electrode and ~ x1/3. a parabolic manner with the The mass transport controlled distance r from the tube flux varies as x-1/3 across the centre. electrode surface and so the TE Velocity at tube centre is not uniformly accessible.