Advanced Electroanalytical Chemistry HUBERT H. GIRAULT

Sion, August 20151

3 cell configurations

• Thin layer cell - Microfluidics - Sensors

• Unstirred cell

• Cell with steady state diffusion

2

Electrochemical cell

CellUnstirred

Stirred Steady-state

No mass transfer Thin layer -adsorption

Mass transfer

Thermodynamics Nernst equation

Kinetics Butler-Volmer equation

Thermodynamics Nernst equation

Kinetics Butler-Volmer equationThermodynamics Nernst equation

Kinetics Butler-Volmer equation

3

3-electrode cell

Potentiostat compares the measured and the desired potential difference, and drives the feedback loop.

V

Sourcede courant

WE

RE

CE

Potentiostat

Voltmeter measures the potential difference between the working & the reference electrode

Potentiostat drives a current between the working and the counter electrode

4

Microelectrode Instrumentation

When using a microelectrode, the current is small enough not to destroy the reference electrode.

In this case, a voltage source and a ammeter are sufficient.

5

Electrode reactions in a thin layer cell

E = E o

cO = cR

R O

Elec

trod

e

Equilibrium in the standard state

R

O

E < E o

cO < cREl

ectr

ode

Cathodicreduction

O

R

E > E o

cO > cR

Elec

trod

e

Anodicoxidation

6

Thin layer voltammetry

1.2

1.0

0.8

0.6

0.4

0.2

0.0

c O e

t cR

-0.4 -0.2 0.0 0.2 0.4

E - Eo / V

T = 298 K

cR cO

At equilibrium, the Nernst equation is obeyed through the finite volume on the electrode

ctot = cR + cO

cR

cO

= exp −nF

RT(E − E o /

)⎡

⎣⎢⎤

⎦⎥

cR = ctot

exp −nF

RT(E − E o /

)⎡

⎣⎢⎤

⎦⎥

1+ exp −nF

RT(E − E o /

)⎡

⎣⎢⎤

⎦⎥

7

Linear sweep voltammetry

E = Ei + νtE

t

I =dqdt

= − nFV dcR(t)dt

⎛⎝⎜

⎞⎠⎟

= − nFVν dcR(t)dE

⎛⎝⎜

⎞⎠⎟

Current response

I =n2F2ν V ctot

RT

exp −nF

RT(E − E o /

)⎡

⎣⎢⎤

⎦⎥

1+ exp −nF

RT(E − E o /

)⎡

⎣⎢⎤

⎦⎥⎡

⎣⎢

⎤

⎦⎥

2

8

Thin layer

-1.0

-0.5

0.0

0.5

1.0

( I ·

V–1 )

/

A·d

m–3

-0.2 -0.1 0.0 0.1 0.2

(E – Eo') / V

Ip =n2F2 ν V ctot

4RT

Peak current

Charge passed

Q(E) =nF V ctot

1+ exp −nF

RTE − E o /( )⎡

⎣⎢⎤

⎦⎥

Maximum charge Qmax = nFVctot

9

Electrode Kinetics Short summary

10

Ox

Red

Electrodereaction

e-

Electrode Solution

Diffusion

Diffusion ConvectionMigration

Migration

11

Mass transfer vs kinetics

Electrode kinetics

Anodic current Ia = FA ka cR (0)

Cathodic current Ic = − FA kc cO(0)

The electrochemical rate constants are given in m·s–1

e– Ia

e– Ic

12

Standard reaction

Standard case: cO(∞) = cR (∞) = c

At equilibrium : kacR (0) = kccO(0)

Reaction coordinate

OR

Gib

bs e

nerg

y

Oxidation

�

ΔGacto

Eeq = EO/Ro + RTFln cO

cR⎛⎝⎜

⎞⎠⎟

�

k o = δkTh

⎛ ⎝

⎞ ⎠ e

−ΔGacto /RT

Transition state theory

13

Activation barrier - Standard case

Reaction coordinate

Gib

bs e

nerg

y

!µRo !µO

o + !µe_

ΔGacto

!µe_ = µe_ − FφM

Oxidation

Eapp > Eo Anodic oxidationIa > Ic

Increase fM by F(E–Eo)

F(E − E o )

ΔGa ΔGc

ΔGa = ΔGacto −αF(E − E o ) ΔGc = ΔGacto + (1−α )F(E − E o )

14

Butler-Volmer for the standard case

The electrode kinetics depends on the applied voltage

Current = anodic current + cathodic current

I = nFA k o c eαnF (E−E o )/RT − e−(1−α ) nF (E−E o )/RT⎡⎣

⎤⎦

-1000

-500

0

500

1000

j /

A·m

–2

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

η / V

jo = 100 ( A·m–2 )Anodic current

Cathodic current

(E – E o ) / V

15

�

Io = n FA k o cR(∞)[ ]1−α cO(∞)[ ]αExchange current density

I = Io eα nF η / RT − e − (1 − α ) nF η / RT[ ]

-1000

-500

0

500

1000

j /

A·m

–2

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

η / V

jo = 100 ( A·m–2 )Anodic current

Cathodic current

16

�

η = E – EeqOvervoltage

Butler-Volmer for the general case

Charge transfer resistance

I = IonFη / RT

-1500

-1000

-500

0

500

1000

1500

j /

A·m

–2

-0.4 -0.2 0.0 0.2 0.4

η / V

jo = 1000, 100, 10, 1 (A·m-2)

17

Tafel plot

6

4

2

0

log

| j / A

·m–2

|

-0.4 -0.2 0.0 0.2 0.4

η / V

jo = 1000, 100, 10, 1 (A·m–2 )

Julius Tafel Born 2 June 1862 Courrendin, CH

18

Corrosion of iron

19

Tafel slopes for H2 evolution

H3O+ + S + e− kV

kV−

⎯ →⎯← ⎯⎯ H2O + S-HVolmer:

S-H + H3O+ + e− kH

kH−

⎯ →⎯← ⎯⎯ H2 + S + H2OHeyrovsky:

2 S-H kH

kH−

⎯ →⎯← ⎯⎯ H2 + 2 S Tafel :

Volmer = rds slope = 1/120mV

Tafel = rds slope = 1/30mV at low overpotentials

Heyrovsky = rds slope = 1/40mV at low overpotentials slope = 1/120 mV at high overpotentials

20

Slow reaction - Thin layerCurrent

8

6

4

2

0

(I ·

A–

1 )

/ µ

A·m

–2

0.60.40.20.0-0.2

(E – Eo') / V

δ = V/A = 10–5m, α = 0.5, n = 1mV·s–1

Reversible and ko = 10–8 m·s–1, ko = 10–10 m·s–1, ko = 10–12 m·s–1

Ia = nFA k

octot exp α nF(E − E o /

) / RT − Kcmα−1eα nF(E−E o / )/RT⎡

⎣⎢⎤

⎦⎥

21

Kcm =

RTA k o

nFνV=

RTk o

nFνδ

Thin layer

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

(Ep

a –

Eo )

/ V

&

( Ep

c –

Eo )

/ V

-4 -3 -2 -1 0 1 2 3

log(! / (V·S–1)) !

Trumpet plot

Irreversible behaviour

Reversible behaviour

22

Amperometry in solution

Steady-state voltammetry for reversible reactions

23

Ox

Red

Electrodereaction

e-

Electrode Solution

Diffusion

Diffusion ConvectionMigration

Migration

24

Mass transfer vs kinetics

Diffusion layer

Elec

trode

Close to any solid surface there is a convection free zone called the diffusion layer where concentration gradients can occur.

Bulk is homogenised by the convection

25

Electrode reactions in solution

R O

Elec

trod

e E < E o

cO < cR

R

O

Cathodicreduction

E > E o

cO > cRO

R

Anodicoxidation

E = E o cO = cR

R O

Equilibrium in the standard state

The Nernst equation fixes the surface concentrations

26

Diffusion controlled reaction

Ox + ne– Red

�

E = E o / +RTnF ln

cO(0)cR(0)

⎛

⎝ ⎜

⎞

⎠ ⎟

The Nernst equation always applies to the interfacial concentrations

27

Diffusion controlled reaction

�

E = E o / +RTnF ln

cO(0)cR(0)

⎛

⎝ ⎜

⎞

⎠ ⎟

bulk concentrations

Surface concentrations imposed by the Nernst

equation

concentration gradient in solution

x

Oxidation

cR

cO

The diffusion flux is given by Fick’s law

j = − Dgradc = − D∂c∂x

⎛⎝⎜

⎞⎠⎟

28

Diffusion controlled current

Ia = n FA DR∂cR(x)∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟ x=0

Anodic oxidation

Equality of interfacial fluxes

Ia = n FA DR∂cR(x)∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟ x=0

= − n FA DO∂cO (x)∂x

⎛ ⎝ ⎜ ⎞

⎠ ⎟ x=0

29

Nernst layer

Ia = nFA DR cR(∞) − cR(0)( ) /δR = − nFA DO cO (∞) − cO (0)( ) /δO

Concentration

0.5

xδ

�

E = E o /

�

cR(∞, t)

�

cO(∞, t)

Concentration

0.5

0.91

0.09

x

�

cR(∞,t)

�

cO (∞,t)

�

cO (x, t)

�

cR(x, t)

δ

�

E = E o / + 0.06V

30

Diffusion timescale

δ / μm τ / s

1 10–3

10 0.1

100 10

τ =δ 2

2D

Statistical approach to diffusion Einstein’s equation

31

Limiting diffusion current

δ x

Concentration

Ida =nFA DRcR(∞)

δR= nFA mRcR (∞)

32

Current-Potential relationship

Ida = nFA DR cR(∞) /δR

Idc = − nFA DO cO (∞) /δO

cR (0) = cR(∞) −δRI

nFADR=

δRnFADR

Ida − I( )

cO(0) = cO(∞) +δOI

nFADO=

δOnFADO

I − Idc( )

Anodic limiting current

Cathodic limiting current

Interfacial concentrations:

33

Current-Potential relationship

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

I / n

FAD

c Rδ–1

1.00.80.60.40.2

E / V

E1/2 = 0.6 V

Half-wave potential

�

I =Ida + Idc2

cO(∞)= 2 cR(∞) δ = δO = δR D = DO =DR

E = E

O/R

o /+

RT

nFln

DRδ

O

DOδ

R

⎡

⎣⎢⎢

⎤

⎦⎥⎥+

RT

nFln

Idc

− I

I − Ida

⎡

⎣⎢⎢

⎤

⎦⎥⎥

E

1/2= E

O/R

o /+

RT

nFln

DRδ

O

DOδ

R

⎡

⎣⎢⎢

⎤

⎦⎥⎥

34

Simple oxidation

1.0

0.8

0.6

0.4

0.2

0.0

I / I d

a

1.00.80.60.40.2

E / V

E1/2 = 0.6 V

E = E1/2

+RT

nFln

I

Ida− I

⎡

⎣⎢

⎤

⎦⎥

35

Slopy results

1.2

1.0

0.8

0.6

0.4

0.2

0.0

I / I d

a

1.00.80.60.40.2

E / V

E1/2 = 0.6 V

I = Ida 1+ B ⋅ E − E1/2( )( )exp

nF E−E1/2( )/RT

1+ expnF E−E1/2( )/RT

⎡

⎣⎢⎢

⎤

⎦⎥⎥

36

Hydrodynamic control

• Rotating disc electrode

• Wall-jet electrode

• Band electrode in a channel flow

• Polarography

37

Rotating disc electrode

disc ring

δ = 1.61D1/ 3 ν1/6 ω−1/2

38

Rotating Electrode II

14

12

10

8

6

4

2

0

I / µ

A

0.50.40.30.20.10.0

E / V

Oxidation of ferrocenemethanol 0.25mM in 50mM NaCl on a rotating gold electrode (Diameter = 3mm). Angular velocity : 200, 400, 800, 1400, 2000, 2600 & 3000 rpm. Potential scan rate =10 mV·s–1 outward and return. At high rotation speeds, instabilities of current are noticeable

14121086420

I / µ A

86420

f 1/2 / Hz 1/2

E1/2

= Eo /+RT

nFln

DR

2/3

DO

2/3

⎡

⎣⎢

⎤

⎦⎥

39

Band electrode in a microchannel

c the bulk concentration, D the diffusion coefficient,l and L the width and length of the microband,2h and d the height and width of the channelFV the volumic flow rate

2h

l

L

l

d

I = 0.6454 nFcbL

D2l2FV

h2d

⎛

⎝⎜

⎞

⎠⎟

1/3

d2h

40

Electrochemical flow meter

100x10 -9

80

60

40

20

0

I / A

0.300.200.100.00

E vsAg|AgCl / V

100x10-9

80

60

40

I / A

1208040

Flow /µL�h-1

0.5mM ferrocenemethanol + 0.1M KCl

41

Geometric control

• Microelectrode with spherical diffusion fields

• Membrane covered electrode

42

Microhemispheres

�

Id = 2π nFD c rhs

c(rhs,t) = c(∞, t) − J2πDrhs

Diffusion in spherical coordinates

Limiting diffusion current

�

δ = rhs

�

Jm = 2πr 2Ddcdr

c[ ]r=rhs

r=∞=

−J

2π Dr

⎡

⎣⎢⎤

⎦⎥r=rhs

r=∞

43

Microelectrodes

Microdisc Microhemisphere

�

Id = 2π nFD c rhsId = 4 nFD c rd

Example : c = 1mM, D =10–9 m2·s–1, F =105, rd=10μm, I = 4 nA

44

Microhemisphere II

Diffusive flux for a reversible reaction

�

J = 2πDrhsc(∞,t)

Diffusion layer thickness

δ = rhs

Diffusion limiting current

Id = 2π nFD c rhs

45

Microdisc electode

1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

I /

nA

0.50.40.30.20.10.0

E / mV

Diffusion layer thickness

�

δ =πrd4

Diffusion limiting current Id = 4 nFD c rd

46

Recessed microdisc

L

�

Id = nFADcL / L = nFAD c − cL( ) /δ

�

cL

�

Id =nFADcL +δ

Continuity of the current

47

Microelectrode array

www.multichannelsystems.com48

Microelectrode array

2L 2r

When using a microelectrode array, the measurement time scale should be small enough not to have an overlap of the diffusion layers

49

Membrane covered electrode

Membrane

SolutionEl

ectro

de

I = nFAcDm /δm

50

Conclusion

A single equation Id =nFADcδ

Control the diffusion layer thickness

The diffusion coefficient is temperature dependent

Dynamic range : From micromolar to molar

51