1. Chapter 3
transport phenomena in electrolytic systems　and concentration overpotential

2. 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.

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

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

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

6. 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.

7. 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

8. When

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

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

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

12. +1 -1 
re -

13. 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 –

14. For nitrate ion:
At stable state
For silver cation:

15. 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.

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

17. 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 !

18. 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.

19. 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

20. 3.6.1 Potential step / jump t t rxn No rxn Instantaneously jumpV V1 V2
t = 0 t
No rxn
rxn I
t = 0 t
Instantaneously jump
Current response

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

22. To solve Fickian second law using Laplacian transform

23. 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

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

25. when

26. 0.0
1.0  x
cis/ci0

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

28. 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.

29. 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

30. I
t = 0 t

31. 3.6.2 Current step / jump I
t = 0 t
I = 0
I = I0 cs c0 x

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

33. At cis = 0, complete polarization
Transition time

34. For same reactant in another reaction:
For other reaction:

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

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

37. When t , 
t = 
 = re 
When 1

38. Applications For determination of bulk concentration
For determination of n

39. 3.6.3 Linear sweep voltammetry1 E 2 I E
Important parameters:
Initial potential, final potential
Sweep rate

40. 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.

41. For reaction
Boundary conditions

42. 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

43. 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.

44. For irreversible process at different scan rates

45. 3.6.4 cyclic voltammetry (CV) I
Triangle wave
Lower limit and upper limit

46. 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 .

47. p = pa - pc = 59 mV/n

49. 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

50. More than one electrochemical couple?

51. Irreversible process

52. 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

53. Double Layer effect

54. 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

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

56. At complete polarization:

57. 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)

58. 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.

59. 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

60. 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 V; V; V; 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

61. a(Pb2+) = 3.310-49 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

62. 3.8.3 The basic principle of polarography
Dropping mercury cathode N2 A + 
Hg anode
Cu2+
Tl+
E1/2
Imax
Polarographic wave

63. A modern computer-aided polarography

64. 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

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

66. 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

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

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

69. 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

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

71. 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

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

73. 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.

74. 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

75. 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:

76. 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.

77. 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.

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

81. 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)

82. 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.

84. Microdisk electrode
Microsylinder electrode

85. boron-doped diamond (BDD) :decreased fouling

87. 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

88. Nanoelectrode arrays:
Is it possible?

90. 4) Application of microelectrodes
Mapping

91. Intracellular analysis and in vivo analysis