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.

ln ln ln ' lns s sO O O O

ir re re res s sR R R R

a f c cRT RT RT RT

nF a nF f nF c nF c y y y

0

0ln

sO R

re sR O

c cRT

nF c c

0 00

0 0ln ' lnO O

re re reR R

a cRT RT

nF a nF c y

' lnsO

ir re sR

cRT

nF c y

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

product R is dissoluble.

0 si i

iE

c cI nFD

0i

d iE

cI nFD

( )s d OO

O

I Ic

nFD

0sR

RR

cI nFD

0 ' ln lnO R dre

R O

D I IRT RT

nF D nF I

s RR

R

Ic

nFD

01/ 2 ' ln O R

reR O

DRT

nF D

0 ' ln lnO R dre

R O

D I IRT RT

nF D nF I

When 1

2 dI I

1/ 2 ln dI IRT

nF I

Half-wave potential

Special for system can be

used for qualitative

analysis.

1/2

I

Id/2

Id

The shape of the polarization curve

lg dI I

I

+1 -10

1/2

2.303RTtg =

nF

The linear relationship

If the oxidative form and the reductive form has similar structure so that DO DR, O R, and fO fR, then

0 01/ 2 're re

Therefore 1/2 can be used for qualitative analysis

This is usually the true case for organic system.

01/ 2 ' ln O R

reR O

DRT

nF D

2) When both O and R exist and are dissoluble.

At equilibrium condition, the flux of O and R should be just the same.

0 0s sO O R R

O RO R

c c c cI nFD nFD

Limiting current density 0

,

0

,

Od c O

O

Rd a R

R

cI nFD

cI nFD

00

O RO R

xx

c cD D

x x

,( )d c OsO

O

I Ic

nFD

,( )d a Rs

RR

I Ic

nFD

,0

,

' ln ln d cO Rre

R O d a

I IDRT RT

nF D nF I I

When , ,

1

2 d c d aI I I

,1/ 2

,

ln d c

d a

I IRT

nF I I

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:

0 si i

OO

c cI nFD

0O

d OO

cI nFD

0 1si i

d

Ic c

I

0 ' ln 1red

RT I

nF I

ln 1red

RT I

nF I

1s sR R Ra f c

0 ' ln sre O

RTc

nF

When I<<Id,

d

RT I

nF I

Linearity

Logarithm ln 1red

RT I

nF I

I

Id

lg dI I

I

+1 -10

re

-

2.303RTtg =

nF

ln 1red

RT I

nF I

ln 1red

RT I

nF I

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 –

,mJ c u E ,mJ c u E

For nitrate ion:

For silver cation:

i i

RTD u

zF

,2 2 d

dcI FD I

dx

At stable state

,d

dcJ D

dx

,d

dcJ D

dx

, , 0m d

dcJ J J D c E

dx

, ,m d

dcJ J J D c E

dx

,2 2 d

dcI FD I

dx

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

, ,i i d i m

dc IJ J J D c E

dx nF

'' ' ' 0M

M M M

dcD c u E

dx

0AA A A

dcD c u E

dx

MM M M

dc ID c u E

dx nF

cM + cM’ = cA, DM DM’ DA, uM uM’ uA,

'

11

2M M

MM

c dcI nFD

c dx

If cM’ > 50 cMM

M

dcI nFD

dx

The effect of electric migration can be neglected. Therefore,

when electrochemical measurement is conducted, the

concentration of reactant is usually low, such as 110-3

moldm-3, while the concentration of supporting electrolyte

0.1 moldm-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.

Instantaneous current

0,

( ) ii

x t

cI t nFD

x

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

V

V1

V2

t = 0 t

No rxn

rxn

Instantaneously jump

I

0

t = 0 t

Current response

cs

0x

c0

Time increase

ms

1s

10s

Variation of surface concentration

Instantaneous current

0

ii

xi

cnFI D

v x

Boundary conditions:

0( ,0)i ic x c

( , )i ic x t c

0( , )i ic t c

(0, ) 0si ic t c

To solve Fickian second law using Laplacian transform

0( , ) ( )2

s si i i i

i

xc x t c c c erf

D t

(0, ) 0si ic t c

0( , )2

i i

i

xc x t c erf

D t

About error function

2

0

2( ) yerf e dy

definition

Important properties

0, ( ) 0

2, ( ) 1

erf

erf

0.0

0.5

1.0

1 2 3

2

2

( )erf

( )erfc

0

( ) 2derf

d

Just as the same as concentration gradient

Conjugation function:

erfc = 1- erf

0

( ) 2derf

d

2( )erf

Integration gives 0( , )i ic x t c erf

2 i

x

D t

00 0

0

( , ) ( ) 2 1

2i i

x i i

i i

c x t cderf dc c

x d dx D t D t

0

0

ii

i

x

cD t

c

x

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

2 i

x

D t

0

( ) 2derf

d

2( )erf

2, ( ) 1

2erf

2 2 i

x

D t

ix D t

2, 42

i

i

xx D t

D t when

( ) 1erf

0( , )i ic x t c

0.0

1.0

4 iDtiDt

x

c is /ci0

0

0

ii

i

x

cD t

c

x

0 0s si i i i

i i

i

c c c cI nFD nFD

D t

Cottrell equation

discussion 1. When t 0, I

2. Id c0 concentration measurement

3. I t-1/2 diffusion control, meauring of Di

4. t , I 0

00i i

d i i

i

c DI nFD nFc

tD t

Time required for attaining stable state

Due to natural convection, the maximum thickness of diffusion layer can only attain 10-2 cm

iDt

Stable-state diffusion will achieved in several seconds.

limitations

1. Potential step requires time

2. Difficulty in measuring of current

3. Rs (resistance of solution) slow response.

4. Separation of charge current

V

V1

V2

t = 0 t

V

V1

V2

t = 0 t

I

0

t = 0 t

cs

0x

c0

3.6.2 Current step / jump

I

0

t = 0 t

I = 0

I = I00

ii

xi

cnFI D

v x

0 =constI I

0

const.i

x

c

x

0

0

i i

x i

c I v

x nFD

Boundary conditions

0( ,0)i ic x c

0( , )i ic t c

Initial condition

20 0( , ) 2 exp

42i

i ii i ii

v I x x t xc x t c erfc

nF D D DtD t

0 02(0, ) ii i

i

v I tc t c

nF D

At electrode surface

Surface concentration decreases linearly with t1/2.

(0, ) 0ic t At cis = 0, complete polarization

0 02(0, ) ii i

i

v I tc t c

nF D

01/ 2

02i i

i

nFc Dt

v I

2 20 2

2 20

( )4

ii

i

n F Dc

v I

Transition time

0 02(0, ) ii i

i

v I tc t c

nF D

1

20(0, ) 1i i

i

tc t c

1

20(0, ) 1i i

i

tc t c

For same reactant in another reaction:1

20(0, ) 1i i

i

tc t c

For other reaction:

11/ 22

0 0(0, ) j ij j i

i j i

v D tc t c c

v D

If electrochemical equilibrium remained

0 ln (0, )t re i

RTc t

nF

1

20(0, ) 1i i

i

tc t c

1 1

2 20

1

20

const. lnt

tRT

nF

Ox e Redn

If Red is indissoluble and aRed =1, then

0 Ox

Red

(0, )ln

(0, )t re

c tRT

nF c t

1 1

2 20

1

2

const. lnt

tRT

nFt

Ox e Redn

If Red is dissoluble, then1 1

2 20 0 Ox

Red Red Ox0 Red

(0, )Dt

c t c cD

Take c0Red = 0, DOx = DRed

1 1

2 20

1

2

const. lnt

tRT

nFt

When t , 1 1

2 20

1

2

t

t

0

1

4t When

1 1

2 20

1

2

t

t

1

t =

= re

0

1

4t

0const.t re

2 20 2

2 20

( )4

ii

i

n F Dc

v I

Applications

For determination of bulk concentration

1 1

2 20

1

2

lnt

t

t

For determination of n

1 1

2 20

1

2

const. lnt

tRT

nFt

3.6.3 Linear sweep voltammetry

t

1

E

1( )t t

2

Important parameters:

Initial potential, final potential

Sweep rate

I

1

2

2

P

PI

2

PI

E

PE

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.

Ox

Red

lnaRT

nF a y

Ox Ox

Red Red

1

(0, )( ) exp ( ) ln

(0, )

exp '

re

re

c t fnF RTf t t

c t RT nF f

nFt

RT

y

y

For reaction Ox e Redn

Boundary conditions 0

Ox Oxc ( ,0)x c Redc ( ,0) 0x

0Ox Oxc ( ,0) c Redc ( ,0) 0

P 1 1

2 2 2

28.24= =-1.109 mV

n

RT

nF

P– –

5 3/ 2 1/2 1/ 2 0Ox Ox2.69 10PI n D c

Criteria for the reversible wave

Randles-Sevcik equation

1/2 is located just about midway between p and P/2

A convenient diagnosis for a Nernstian wave is

P

2

56.5mV

n P At 25 oC

For different scan rates

I

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

5 3/ 2 1/2 1/ 2 0Ox Ox2.69 10PI n D c

v 1/2

Ip

I

For irreversible process at different scan rates

3.6.4 cyclic voltammetry (CV)

I

Triangle wave

Lower limit and upper limit

I cPI

aPI

Typical CV diagram for reversible single electrode

Potential separation

59Δ mVc a

P P n

Both P, c and P,a is independent on scan rate

cPaP

=1I

I

The reversed potential should be 35/n mV past P, c .

p = pa - pc = 59 mV/n

, ,'2

p c p a

y

Cyclic Voltammetry of Potassium Ferrocyanide

Ferrocycanide ( Fe(CN)64- )

and ferricyanide ( Fe(CN)6

3- ) 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?More than one electrochemical couple?

Irreversible processIrreversible process

3.6.5 effect Cdl and Ru

ch dl

dq dq dI C

dt d dt

Charge current flow

5 3/ 2 1/2 1/ 2 0Ox Ox2.69 10PI n D c

8 1/ 2ch

3 / 2 0Ox

(2.4 10 )

P

I

I n c

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 effectDouble Layer effect

3.7 Radial diffusion through a spherical shell

Microhemisphere Electrode (radius a)

I

transport zone thick c.f. a

rsemiinfinite

Inlaid Disk Electrode (radius r)

I often treated as a hemisphereof radius r/

Fick’s second law at spherical ordinate

At r + dr:

rr0

( , )( , )

c r tJ r t D

r

( , )( , ) ( , )

J r tJ r dr t J r dr t dr

r

( , ) ( , )c r t c r tD dr

r r r

At complete polarization:

0 00 0( , ) ( )2

sO O O O

O

r r rc r t c c c erfc

r D t

0 0 0( , ) 12

O O

O

r r rc r t c erfc

r D t

0

00

0

1 1( )O

r r O

cc

r r D t

0 0

1 1( ) O

d O O Or r O

cj t nFD nFD c

r r D t

Analysis:

1)The thickness of the diffusion layer is much less than the curvature radius

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)

0

O Od

nFD cj

r

0 0

1 1( ) O

d O O Or r O

cj t nFD nFD c

r r D t

0OD t r

Radial diffusion gives very high rates of mass transport to t

he electrode surface with a mass transport coefficient of the

order of D/r0. Therefore, even at rotation rates of 104 rpm, c

onvective transport to a rotating macroelectrode is smaller t

han diffusion to a 1 m microdisk.

0

1 1( )d O O

O

j t nFD cr D t

Radial diffusion vs. planar diffusion

2 / 3 1/ 6 1/ 2 00.62d i ij nFD c

Jaroslav Heyrovský1959 Noble Prize

Czechoslovakia

1890/12/20 ~ 1967/03/27

Polarography

3.8.1 Progress of the sensitivity of pol

arography

1920: 10-2 mol·dm-3

1935: 10-3 ~ 10-5 mol·dm-3

1957: 10-8 ~ 10-9 mol·dm-3

1957: 210-10 mol·dm-3

At present: 1010~1012 moldm-3

3.8 dropping mercury electrode and polarography

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.

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

0.337

⊖ Cu2+/Cu

-0.126

⊖ Pb2+/Pb

0.799

⊖ Ag+/Ag

0.000

⊖ H+/H2

3.8.2 Order of liberation

a(Ag+) = 1.510-8

0.799 V

a(Cu2+) = 2.210-16

0.337 V

a(Pb2+) = 3.310-49

-0.126 V -1.56 V

The liberation order and the residual concentration of the ions upon negative shift of potential of cathode

Polarograph

Polarographic wave

Dropping mercury cathode

N2

A

+

Hg anode

Cu2+

Tl+

E1/2Imax

3.8.3 The basic principle of polarography

A modern computer-aided polarography

Dropping mercury electrode

4

8Hgr P

ml

Critical size of mercury drop

2H O

Hg

2

1drop

rt

mg

Optimum parameters:

r=25-40 m, l = 5-15 cm, h=30-80 cm, m=1-2 mgs-1, tdrop=3-6 s

2 23 2 3 336 0.850s V m t

Variation of surface area with time

2 1

3 30.567ds

m tdt

0

( ) ii

i x

cni t FsD

v x

Instantaneous current through dropping mercury electrode

Time-dependence of current for polarography

t

I 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

2 1 103 6 2( ) 0.48 ( )si iI t nFm t D c c

2 1 103 6 2

2 1 103 6 2

7( ) 0.48 ( )

3

0.732 ( )

si i

i

si i

i

nI t Fm t D c c

v

nFm t D c c

v

Considering the counteracting effect of the drop growing to

diffusion layer

The mean diffusion current2 1 1

03 6 20.627 ( )sdrop i ii

nI Fm t D c c

v

The mean limiting diffusion current2 1 1

03 6 20.627d drop ii

nI Fm t D c

v

Ilkovic equation

Cu2+

Tl+

E1/2Imax

1/ 2 ln di iRT

nF i

2 1 103 6 20.627d drop i

i

nI Fm t D c

v

Quantitative and qualitative analysis

Inlaid disk

Mantle in z=0 plane (extend beyond diffusion layer boundary)

Flux into mantle = 0

r0

z axis r = 0r axis z = 0

3.9 Microdisk electrode

2 2

2 2

( , , ) ( , , ) ( , , ) ( , , )1O O O Oc r z t c r z t c r z t c r z tD

t r r r z

0

2 20 0

2i i

z

c c

z r r

0

00

00

2

4

ri

d iz

i i

cI nFD rdr

z

nFD c r

004 s

i i iI nFD r c c

Boundary conditions:

cO(r, z, 0) = cO*

*lim ( , , )O Orc r z t c

*lim ( , , )O Ozc r z t c

0

( , , )0 ( )O

z 0

c r z tr > r

z

2 2

2 2

( , , ) ( , , ) ( , , ) ( , , )1O O O Oc r z t c r z t c r z t c r z tD

t r r r z

0 0 00

20 0 0

4 4

4

i i i i i id

nFD c r nFD c nFD cJ

r r r 0

4E

r

If the radius of the microdisc is 1 micron, the effective th

ickness of diffusion layer is 0.79 micron.

Inlaid Disk Electrode (radius r)

I often treated as a hemisphereof radius r/

0

2 20 0

2i i

z

c c

z r r

Current depends on the distance from the central point

Then the solution for current is

f() is a function which contains both constant and t to various powers.

For nonsteady state

0 1/ 2 1/ 20( ) 4 0.2146exp( 0.7823/ )

4 4d i iI t nFD c r

0

0

4( )O OnFAD c

I fr

20

4 OD t

r

1/ 2( ) 0.7854 0.8862f

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.

004d i iI nFD c r

But at long times, f() goes to 1.

1/ 2 0 2 1/ 20( )d i i

nFI D c r t

Equilibrium polarization curve can be obtained at high scan rate (10~50 mVs-1 ), which can be usually obtained at 1 mVs-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 compa

rable 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:

0

1 1( )d O O

O

I t nFD cr D t

0

( ) O Od

nFD cI t

r

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.

attributes of microelectrodes

a) small currents

b) steady state responses

c) short response time

The immunity of microelectrodes to ohmic drop (IR drop) phen

omena allows one to perform experiments in previously inacces

silbe samples such as nonpolar solvents, supercritical fluids, an

d solids or even wet gas.

The ability of microelectrodes to respond rapidly to changes in t

he applied potential makes microelectrodes useful in dynamic st

udies of short timescale (a low microsecond or even a nanoseco

nd timescale) homogeneous and heterogeneous electron transfer

processes.

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

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 gl

ass, epoxy resin, PTFE.

Microlithographic techniques

Immobilizing large numbers of metal wires within a no

nconductive 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

(a) 3×3 electrode array, (b) array of MWCNT bundles on on

e of the electrode pads,(c) and (d) array of MWCNTs at U

V-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