Theoretical concepts and application of a Rotating disk electrode (RDE) and Rotating

Ring disk electrode

Concentrated summer course in electrochemistry, September, 2014

Bar-Ilan University

Outline

• Introduction & Determination of general equations for hydrodynamic systems

• RDE equation and application

• RRDE equation and application

Not steady state system (not stirred)

*

𝑶+ 𝒏𝒆 ↔ 𝑹

* o

Mass Transport Mechanisms

Diffusion

-

-

-

- -

- -

-

-

+

+

+

+

+

+

+

+

Migration

Convection

𝑱𝒋(𝒙) = −𝑫𝒋𝝏𝑪𝒋(𝒙)

𝝏𝒙−𝒛𝒊𝑭

𝑹𝑻𝑫𝒊𝑪𝒊

𝝏𝝋(𝒙)

𝝏𝒙

𝑰𝒋 = 𝒏𝑭𝑨𝑫𝒋𝝏𝑪𝒋(𝒙)

𝝏𝒙

𝑱𝒋 = −𝑫𝒋𝜵𝑪𝒋 −𝒛𝒋𝑭

𝑹𝑻𝑫𝒋𝑪𝒋𝜵𝝋 + 𝑪𝒋𝒗

Nernst-Plank equation

𝑱𝒋(𝒙) =𝑰𝒋

𝒛𝒋𝑭𝒋𝑨

Steady State system

𝑪𝑶 Bulk solution

(stirred)

Stagnant 𝑪𝑶 *

𝑶 + 𝒏𝒆 ↔ 𝑹

Convective diffusion equation

𝑱𝒊 = −𝑫𝒊𝜵𝑪𝒊 −𝒛𝒊𝑭

𝑹𝑻𝑫𝒊𝑪𝒊𝜵𝝋 + 𝑪𝒊𝒗

𝝏𝑪𝒊𝝏𝒕= −𝜵𝑱𝒊

𝝏𝑪𝒊𝝏𝒕= 𝑫𝒊𝜵

𝟐𝑪𝒊 − 𝒗𝜵𝑪𝒊

𝑰𝒍 = 𝒏𝑭𝑨𝒎𝟎𝑪𝟎∗

𝒎𝟎 =𝑫𝟎𝜹𝟎

Hydrodynamic electrode

Rotating disc electrode (RDE) Wall-jet electrode (WJE)

The tubular and channel electrode

• Well-defined hydrodynamics flow to electrode

surface.

• The mathematical equations are available for the calculation of different parameters.

• Rate of material transport depends in a well defined manner on the rotation speed of the electrode.

• Systems with RDE are relatively simple for fabrication and operation.

RDE- specialized hydrodynamic electrode used in the study of the

kinetics and mechanism of electrode reaction

Working electrode

Insulator (Teflon)

RDE

Convective diffusion equation: RDE

𝒗𝒚𝝏𝑪𝟎𝝏𝒚

= 𝑫𝟎𝝏𝟐𝑪𝟎𝝏𝒚𝟐

𝝏𝑪𝒊𝝏𝒕= 𝟎

𝒗𝒓𝝏𝑪𝟎𝝏𝒓

+𝒗𝝋

𝒓

𝝏𝑪𝟎𝝏𝝋

+ 𝒗𝒚𝝏𝑪𝟎𝝏𝒚

= 𝑫𝟎𝝏𝟐𝑪𝟎𝝏𝒚𝟐

+𝝏𝟐𝑪𝟎𝝏𝒓𝟐

+𝟏

𝒓

𝝏𝑪𝟎𝝏𝒓

+𝟏

𝒓𝟐𝝏𝟐𝑪𝟎𝝏𝝋𝟐

=0

=0

=0

=0

=0

𝝏𝑪𝒊𝝏𝒕= 𝑫𝒊𝜵

𝟐𝑪𝒊 − 𝒗𝜵𝑪𝒊

Convective diffusion equation expressed in cylindrical polar-co-ordinates (r, z, φ)

Convective diffusion equation: RDE

𝜕2𝐶𝑂𝜕𝑦2

= −𝑦2

𝐵

𝜕𝐶𝑂𝜕𝑦

𝑣𝑦 = 0.51𝜔3/2𝑣−1/2𝑦2

𝐴 = 𝜋𝑟2

𝐵 = 𝐷𝑂 𝜔3/2𝑣−1/2/0.51

Allen J. Bard and Larry R. Faulkner, ELECTROCHEMICAL METHODS, 2nd edition, 2001

𝒊 = 𝒏𝑭𝑨𝑫𝟎𝝏𝑪𝟎(𝒙)

𝝏𝒚𝒚=𝟎

Levich equation Diffusion limited Current

𝑪𝟎 =𝝏𝑪𝟎𝝏𝒚

𝒚=𝟎

𝟎. 𝟖𝟗𝟑𝟒𝟑𝑫𝟎𝝎

−𝟑/𝟐𝜸𝟏/𝟐

𝟎. 𝟓𝟏

𝟏/𝟑

*

𝒊𝑳 = 𝟎. 𝟔𝟐𝒏𝑭𝑨𝑫𝟐/𝟑𝒗−𝟏/𝟔𝑪𝟎𝝎

𝟏/𝟐 *

𝟏

𝒊=𝟏

𝒊𝑲+𝟏

𝒊𝒍,𝒄=𝟏

𝒊𝑲+

𝟏

𝟎. 𝟔𝟐𝒏𝑭𝑨𝑫𝟐/𝟑𝒗−𝟏/𝟔𝑪𝟎𝝎𝟏/𝟐

Levich

Koutecky-Levich

Levich study (For a simple electrochemical system where the

rate of the half reaction is governed only by mass transport to the electrode surface)

The limiting current increases linearly with the square root of the rotation rate and the line intercepts the vertical axis at zero.

2/1

R

6/13/2

RLACnFAD620.0i

𝑹 ↔ 𝑶+ 𝒏𝒆

Measuring Limiting Currents

Koutecky-Levich Analysis - the rate of a half reaction

occurring at an electrode surface is limited by a combination of mass transport and sluggish kinetics

2/16/13/2 )620.0/1(/1/1 CnFADii k

ki/1

Determination of the activation controlled current density

𝑖𝑘 = 𝐹𝐴𝑘𝑓(𝐸)𝐶0 *

Application of RDE

0

6/12/13/262.0 CnFDil 0

2/1 Cil

i-E curves as a function of rotation rate

23 1 FeeFe

eFeFe 132

Fast electron transfer

k3Fe(CN)6 10mM & K4Fe(CN)6 20mM

i-E curves as a function of K4Fe(CN)6 concentration. Rotation rate: 2000 rpm

eFeFe 132

Increase in RDE rotation speed and electroactive species concentration cause an increase in the limiting current density.

23 1 FeeFe

0

6/12/13/262.0 CnFDil

Plots iL vs. ω0.5 , working solution: K3Fe(CN)6 (10mM)+ K4Fe(CN)6 (20mM)in Na2SO4(0.1M)

Plots iL vs. C. working solution: K3Fe(CN)6 (10mM)+ K4Fe(CN)6 (20-60mM)in Na2SO4(0.1M).

Rotation speed 2000 rpm.

Calculation of diffusion coefficient

0

6/13/262.0 CnFD 1/2ωliIL vs. ω0.5

0

6/12/13/262.0 CnFDil IL vs. C

20.66 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒2+2/3 × (1.1 × 10−6)1/6× 20

𝐷𝐹𝑒2+ = 2.5 × 10−9𝑚2𝑠−1

10.44 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒3+2/3 × (1.1 × 10−6)1/6× 10

𝐷𝐹𝑒3+ = 2.6 × 10−9𝑚2𝑠−1

15.59 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒2+2/3 × (1.1 × 10−6)1/6× 14.47

𝐷𝐹𝑒2+ = 2.7 × 10−9𝑚2𝑠−1

RRDE

Teflon Ring

Disk

RRDE

𝑶 + 𝒏𝒆 ↔ 𝑹 Disk- Ed, id

𝑹 ↔ 𝑶+ 𝒏𝒆 Ring-Er, ir

𝒗𝒓𝝏𝑪𝒐𝝏𝒓

− 𝒗𝒚𝝏𝑪𝑹𝝏𝒚

=𝑫𝑹𝑩′

𝟏

𝒚

𝝏𝟐𝑪𝑹𝝏𝒚𝟐

Steady-state ring convective-diffusion equation

𝒊𝑹 = 𝒏𝑭𝑫𝑹𝟐𝝅 𝝏𝑪𝑹𝝏𝒚

𝒓𝟑

𝒓𝟐

𝒓𝒅𝒓

𝑵𝒆𝒎𝒑𝒊𝒓𝒊𝒄𝒂𝒍 = −𝒊𝑳,𝒓𝒊𝒏𝒈/𝒊𝑳,𝒅𝒊𝒔𝒌

𝑩′ = 𝟎. 𝟓𝟏𝝎𝟑/𝟐𝜸−𝟏/𝟐

lim𝑦→∞

𝐶𝑅 = 0 R initially absent in solution

Bulk concentration of O is CO*

𝐹𝑒(𝐶𝑁)6 + 𝒆 → 𝐹𝑒(𝐶𝑁)6 3- 4-

Reduction of ferricyanide to ferrocyanide at disk

𝐹𝑒(𝐶𝑁)6 → 𝐹𝑒(𝐶𝑁)6 + 𝒆 4- 3-

Oxidation of ferrocyanide to ferricyanide at ring

K3Fe(CN)6 10mM in 1M KNO3

RRDE experiment

𝑵𝒆𝒎𝒑𝒊𝒓𝒊𝒄𝒂𝒍 = −𝒊𝑳,𝒓𝒊𝒏𝒈/𝒊𝑳,𝒅𝒊𝒔𝒌

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