Chapter 3 transport phenomena in electrolytic systems　and concentration overpotential

3.4 Polarization curve under diffusion control If diffusion is r.d.s and the electrode is under electrochemical equilibrium, therefore, Nernst equation is applicable.

1) When there is no R (c0R = 0) in the bulk, and the product R is dissoluble.

When Half-wave potential Special for system can be used for qualitative analysis.

The linear relationship  1/2 I Id/2 Id The linear relationship +1 -1  1/2  The shape of the polarization curve

If the oxidative form and the reductive form has similar structure so that DO DR, O  R, and fO  fR, then Therefore 1/2 can be used for qualitative analysis This is usually the true case for organic system.

2) When both O and R exist and are dissoluble. At equilibrium condition, the flux of O and R should be just the same. Limiting current density

When

 I Id,c Id,a The typical polarization curve for electrochemical reaction under diffusion control with both O and R dissoluble.

3) when product R is indissoluble and form new phase (such as gas or solid) For pure phase:

Logarithm When I<<Id, I Id Linearity 

+1 -1  re -

3.7 Effect of electric migration on stationary diffusion c+, u+, D+, c–, u –, D –, represents the concentration, ionic mobility and diffusion coefficient of Ag+ and NO3 –

For nitrate ion: At stable state For silver cation:

This suggests that the limiting current doubles due to electric migration and independent of ionic mobility. For oxidation of anion on anode and reduction of cation on cathode, the current density will double due to electric migration, while for oxidation of cation on anode and reduction of anion on cathode, electric migration will decrease the overall current density.

Use of supporting electrolyte For reactant MA with supporting electrolyte M’A cM + cM’ = cA, DM  DM’  DA, uM  uM’  uA,

If cM’ > 50 cM The effect of electric migration can be neglected. Therefore, when electrochemical measurement is conducted, the concentration of reactant is usually low, such as 110-3 moldm-3, while the concentration of supporting electrolyte 0.1 moldm-3 A basic requirement !

3.6 Nonsteady state diffusion process For many electrochemical measurement, such as LSV, CV, AC, etc. stable state will not achieved during measuring. Diffusion characteristics of some transition EC techniques Voltammetry: a voltage or a series of voltages are applied to the W.E with the corresponding current that flows monitored.

Semi-infinite diffusion Instantaneous current Assumption and boundary conditions: 1) Di does not depend on c 2) initial condition: ci(x=0) = ci0 3) finial condition: ci(x=) = ci0 To solve Fickian second law Semi-infinite diffusion

3.6.1 Potential step / jump t t rxn No rxn Instantaneously jump V V1 V2 t = 0 t No rxn rxn I t = 0 t Instantaneously jump Current response

x Variation of surface concentration Boundary conditions: cs x c0 Time increase ms 1s 10s Variation of surface concentration Boundary conditions: Instantaneous current

To solve Fickian second law using Laplacian transform

definition About error function Important properties 0.0 0.5 1.0 1 2 3 About error function definition Important properties Just as the same as concentration gradient Conjugation function: erfc = 1- erf

Integration gives This suggests that the thickness of diffusion layer increases with t.

when

0.0 1.0  x cis/ci0

Cottrell equation discussion When t 0, I Id  c0 concentration measurement I  t-1/2 diffusion control, meauring of Di t , I 0

Time required for attaining stable state Due to natural convection, the maximum thickness of diffusion layer can only attain 10-2 cm Stable-state diffusion will achieved in several seconds.

t limitations Potential step requires time Difficulty in measuring of current Rs (resistance of solution) slow response. Separation of charge current V V1 V2 t = 0 t

I t = 0 t

3.6.2 Current step / jump I t = 0 t I = 0 I = I0 cs c0 x

Boundary conditions Initial condition At electrode surface Surface concentration decreases linearly with t1/2.

At cis = 0, complete polarization Transition time

For same reactant in another reaction: For other reaction:

If electrochemical equilibrium remained If Red is indissoluble and aRed =1, then

If Red is dissoluble, then Take c0Red = 0, DOx = DRed

When t ,  t =   = re  When 1

Applications For determination of bulk concentration For determination of n

3.6.3 Linear sweep voltammetry 1 E 2 I E Important parameters: Initial potential, final potential Sweep rate

I E 1 2 3 4 5 6 c x 1 2 3 4 5 6 Diffusion limit – by Cottrell equation No stable current achieved. Dropping of the current: diffusion layer grows, the flux of reactant to the electrode is not fast enough. The rate of electron transfer is fast in comparison to scan rate, equilibrium sustains.

For reaction Boundary conditions

Randles-Sevcik equation 1/2 is located just about midway between p and P/2 A convenient diagnosis for a Nernstian wave is At 25 oC Criteria for the reversible wave

For different scan rates  I v 1/2 Ip For reversible case, P keeps constant at different scan rates, while IP increase with v1/2 for rapid electron transfer kinetics.

For irreversible process at different scan rates 

3.6.4 cyclic voltammetry (CV)  I Triangle wave Lower limit and upper limit

Typical CV diagram for reversible single electrode  I Potential separation Both  P, c and P,a is independent on scan rate The reversed potential should be 35/n mV past P, c .

p = pa - pc = 59 mV/n

Cyclic Voltammetry of Potassium Ferrocyanide Ferrocycanide ( Fe(CN)64- ) and ferricyanide ( Fe(CN)63- ) are a classic redox couple. The cyclic voltammograms show a reversible reaction. CV in 3  10-3 mol dm-3 K4[Fe(CN)6] & 1 mol dm-3 KCl at various scan rates, geometric area of working electrode = 20 mm2

More than one electrochemical couple?

Irreversible process

3.6.5 effect Cdl and Ru Charge current flow Assuming that Cdl = 20 F cm-2 and Dox = 10-5 cm2 s-1 High  and low C0Ox will distort LSV and CV wave

Double Layer effect

3.7 Radial diffusion through a spherical shell Microhemisphere Electrode (radius a) I transport zone thick c.f. a r semiinfinite Inlaid Disk Electrode (radius r) I often treated as a hemisphere of radius 2r/p

Fick’s second law at spherical ordinate At r + dr:

At complete polarization:

Analysis: The thickness of the diffusion layer is much less than the curvature radius 1) 2) When r0 decreases or for prolonged reaction time When t  , r0 < hundreds of microns 3) When r0 is very small (of several microns or even of nanometer scales)

Radial diffusion vs. planar diffusion Radial diffusion gives very high rates of mass transport to the electrode surface with a mass transport coefficient of the order of D/r0. Therefore, even at rotation rates of 104 rpm, convective transport to a rotating macroelectrode is smaller than diffusion to a 1 m microdisk.

3.8 dropping mercury electrode and polarography 3.8.1 Progress of the sensitivity of polarography 1920: 10-2 mol·dm-3 1935: 10-3 ~ 10-5 mol·dm-3 1957: 10-8 ~ 10-9 mol·dm-3 1957: 210-10 mol·dm-3 At present: 1010~1012 moldm-3 Jaroslav Heyrovský 1959 Noble Prize Czechoslovakia 1890/12/20 ~ 1967/03/27 Polarography

3.8.2 Order of liberation 0.799 ⊖ Ag+/Ag 0.000 ⊖ H+/H2 For liberation of metal, the overpotential is usually very low, and the reversible potential can be used in stead of irreversible potential. For evolution of gas, the overpotential is relatively large, therefore, the overpotential should be taken into consideration. 0.337 ⊖ Cu2+/Cu Ag+, Cu2+, H+, and Pb2+ will liberates at 0.799 V; 0.337 V; 0.000 V; -0.126 V, respectively without consideration of overpotential; Overpotential of hydrogen liberation on Cu is 0.6 V, on Pb is 1.56 V -0.126 ⊖ Pb2+/Pb

a(Pb2+) = 3.310-49 -0.126 V -1.56 V a(Cu2+) = 2.210-16 0.337 V a(Ag+) = 1.510-8 0.799 V The liberation order and the residual concentration of the ions upon negative shift of potential of cathode

3.8.3 The basic principle of polarography Dropping mercury cathode N2 A +  Hg anode Cu2+ Tl+ E1/2 Imax Polarographic wave

A modern computer-aided polarography

Dropping mercury electrode Critical size of mercury drop Optimum parameters: r=25-40 m, l = 5-15 cm, h=30-80 cm, m=1-2 mgs-1, tdrop=3-6 s

Variation of surface area with time Instantaneous current through dropping mercury electrode

Time-dependence of current for polarography 1. Instantaneous current 2. Variation of current detected by instrument with low time resolution. 3. Averaged current. If  << r, and do not take area variation into consideration

Considering the counteracting effect of the drop growing to diffusion layer Ilkovic equation The mean diffusion current The mean limiting diffusion current

Quantitative and qualitative analysis Cu2+ Tl+ E1/2 Imax

3.9 Microdisk electrode z axis r = 0 r axis z = 0 Flux into mantle = 0 Inlaid disk Mantle in z=0 plane (extend beyond diffusion layer boundary) Flux into mantle = 0 r0 z axis r = 0 r axis z = 0

Boundary conditions: cO(r, z, 0) = cO*

Current depends on the distance from the central point If the radius of the microdisc is 1 micron, the effective thickness of diffusion layer is 0.79 micron. Inlaid Disk Electrode (radius r) I often treated as a hemisphere of radius 2r/p Current depends on the distance from the central point

For nonsteady state Then the solution for current is f() is a function which contains both constant and t to various powers.

At short times, there is a solution that looks like that from the infinite planar electrode described before, i.e. current is proportional to 1/t1/2. But at long times, f() goes to 1. Equilibrium polarization curve can be obtained at high scan rate (10~50 mVs-1 ), which can be usually obtained at 1 mVs-1.

The time required for the steady state current : Using D = 1 10-5 cm2 s-1, for a 5 mm radius electrode, the experimental timescale must be longer than 80 seconds. Reducing the electrode radius by a factor of a thousand to 5 m, a steady state response can be observed for 80 s. Mass transport rates to a microedisk electrode are comparable to those of a conventional millimetric electrode that is being rotated at several thousand rpm

3.10 Ultra-microelectrodes (UME) The differences in the electrochemical responses observed at macroscopic and microscopic electrodes arise because of the relative importance of the time dependent and time-independent terms at conventional electrochemical timescales. At relative long time, the current attains a time-independent steady state value given by:

attributes of microelectrodes a) small currents b) steady state responses c) short response time The immunity of microelectrodes to ohmic drop phenomena allows one to perform experiments in previously inaccessilbe samples such as nonpolar solvents, supercritical fluids, and solids or even wet gas. The ability of microelectrodes to respond rapidly to changes in the applied potential makes microelectrodes useful in dynamic studies of short timescale (a low microsecond or even a nanosecond timescale) homogeneous and heterogeneous electron transfer processes.

The immunity of microelectrodes to ohmic drop (IR drop) phenomena allows one to perform experiments in previously inaccessilbe samples such as nonpolar solvents, supercritical fluids, and solids or even wet gas. The ability of microelectrodes to respond rapidly to changes in the applied potential makes microelectrodes useful in dynamic studies of short timescale (a low microsecond or even a nanosecond timescale) homogeneous and heterogeneous electron transfer processes.

r = 0.01 V s-1 r = 1000 V s-1

Cyclic voltammagrams for reduction of anthracene (2 Cyclic voltammagrams for reduction of anthracene (2.22 m) in acetonitrile with 0.6 M TEAP at a gold microdisk electrode (r0 = 6.5 m ): scan rates in V s-1. (a) 1000; (b) 2000; (c) 5000; (d) 10000; (e) 20000; (f) 50000; (g) 100000.

Fabrication of microelectrodes Microelectrodes are commonly fabricated by sealing a fine wire or foil into a nonconducting electrode body such as glass, epoxy resin, PTFE. Microlithographic techniques Immobilizing large numbers of metal wires within a nonconductive support Electrodeposition of mercury and platinum within the pores of a polymer membrane.

Microdisk electrode Microsylinder electrode

boron-doped diamond (BDD) :decreased fouling

MWCNT electrode array: SEM images of array of MWCNT bundles on one of the electrode pads, and (d) array of MWCNTs at UV-lithography and e-beam patterned Ni spots, (e) and (f) the surface of polished MWCNT array electrodes grown on 2m and 200 nm spots

Nanoelectrode arrays: Is it possible?

4) Application of microelectrodes Mapping

Intracellular analysis and in vivo analysis