and Rotating Ring disk electrode

Bar-Ilan University
Theoretical concepts and
application of a Rotating disk
electrode (RDE) and Rotating
Ring disk electrode
Concentrated summer course in electrochemistry,
September, 2014
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
Migration
+
+
-
+
+
- +
- +
+
-
+
Diffusion
-
Convection
-
-
Nernst-Plank equation
𝒛𝒋 𝑭
𝑱𝒋 = −𝑫𝒋 𝜵𝑪𝒋 −
𝑫𝒋 𝑪𝒋 𝜵𝝋 + 𝑪𝒋 𝒗
𝑹𝑻
𝝏𝑪𝒋 (𝒙) 𝒛𝒊 𝑭
𝝏𝝋(𝒙)
𝑱𝒋 (𝒙) = −𝑫𝒋
−
𝑫𝒊 𝑪𝒊
𝝏𝒙
𝑹𝑻
𝝏𝒙
𝑰𝒋
𝑱𝒋 (𝒙) =
𝒛𝒋 𝑭𝒋 𝑨
𝝏𝑪𝒋 (𝒙)
𝑰𝒋 = 𝒏𝑭𝑨𝑫𝒋
𝝏𝒙
Steady State system
𝑶 + 𝒏𝒆 ↔ 𝑹
Stagnant
𝑪𝑶
*
𝑪𝑶
Bulk solution
(stirred)
Convective diffusion equation
𝒛𝒊 𝑭
𝑱𝒊 = −𝑫𝒊 𝜵𝑪𝒊 −
𝑫𝒊 𝑪𝒊 𝜵𝝋 + 𝑪𝒊 𝒗
𝑹𝑻
𝝏𝑪𝒊
= −𝜵𝑱𝒊
𝝏𝒕
𝝏𝑪𝒊
= 𝑫𝒊 𝜵𝟐 𝑪𝒊 − 𝒗𝜵𝑪𝒊
𝝏𝒕
𝑰𝒍 = 𝒏𝑭𝑨𝒎𝟎 𝑪𝟎∗
𝑫𝟎
𝒎𝟎 =
𝜹𝟎
Hydrodynamic electrode
Rotating disc electrode (RDE)
Wall-jet electrode (WJE)
The tubular and channel electrode
RDE- specialized hydrodynamic electrode used in the study of the
kinetics and mechanism of electrode reaction
• 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
Working
electrode
Insulator
(Teflon)
Convective diffusion equation: RDE
𝝏𝑪𝒊
= 𝑫𝒊 𝜵𝟐 𝑪𝒊 − 𝒗𝜵𝑪𝒊
𝝏𝒕
𝝏𝑪𝒊
=𝟎
𝝏𝒕
Convective diffusion equation expressed in cylindrical polar-co-ordinates (r, z, φ)
𝒗𝒓
𝒗𝝋 𝝏𝑪𝟎
𝝏𝑪𝟎
𝝏𝑪𝟎
𝝏𝟐 𝑪𝟎 𝝏𝟐 𝑪𝟎 𝟏 𝝏𝑪𝟎 𝟏 𝝏𝟐 𝑪𝟎
+
+ 𝒗𝒚
= 𝑫𝟎
+
+
+
𝝏𝒓
𝒓 𝝏𝝋
𝝏𝒚
𝝏𝒚𝟐
𝝏𝒓𝟐
𝒓 𝝏𝒓 𝒓𝟐 𝝏𝝋𝟐
=0
=0
=0
=0
=0
𝒗𝒚
𝝏𝑪𝟎
𝝏𝟐 𝑪𝟎
= 𝑫𝟎
𝝏𝒚
𝝏𝒚𝟐
Convective diffusion equation: RDE
𝜕 2 𝐶𝑂
𝑦 2 𝜕𝐶𝑂
=−
2
𝜕𝑦
𝐵 𝜕𝑦
𝐵 = 𝐷𝑂 𝜔 3/2 𝑣 −1/2 /0.51
𝑣𝑦 = 0.51𝜔 3/2 𝑣 −1/2 𝑦 2
𝐴 = 𝜋𝑟 2
Allen J. Bard and Larry R. Faulkner, ELECTROCHEMICAL METHODS, 2nd edition, 2001
Levich equation Diffusion limited Current
𝝏𝑪𝟎
𝑪*𝟎 =
𝝏𝒚
𝟎. 𝟖𝟗𝟑𝟒
𝟑𝑫𝟎
𝟏/𝟑
−𝟑/𝟐
𝟏/𝟐
𝝎
𝜸
𝒚=𝟎
𝒊 = 𝒏𝑭𝑨𝑫𝟎
𝝏𝑪𝟎 (𝒙)
𝝏𝒚
𝟎. 𝟓𝟏
𝒚=𝟎
𝒊𝑳 = 𝟎. 𝟔𝟐𝒏𝑭𝑨𝑫𝟐/𝟑 𝒗−𝟏/𝟔 𝑪𝟎* 𝝎𝟏/𝟐
Levich
𝟏 𝟏
𝟏
𝟏
𝟏
= +
= +
Koutecky-Levich
𝟐/𝟑
−𝟏/𝟔
𝟏/𝟐
𝒊 𝒊𝑲 𝒊𝒍,𝒄 𝒊𝑲 𝟎. 𝟔𝟐𝒏𝑭𝑨𝑫 𝒗
𝑪𝟎 𝝎
Levich study (For a simple electrochemical system where the
rate of the half reaction is governed only by mass transport to
the electrode surface)
𝑹 ↔ 𝑶 + 𝒏𝒆
i LA  0.620nFADR2 / 3 1 / 6 C R 1 / 2
The limiting current increases linearly with the square root of the
rotation rate and the line intercepts the vertical axis at zero.
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
1 / i  1 / ik  (1 / 0.620nFAD 2 / 3 1/ 6C ) 1/ 2
Determination of the activation controlled current density
1 / ik
𝑖𝑘 = 𝐹𝐴𝑘𝑓 (𝐸)𝐶0*
Application of RDE
i-E curves as a function of rotation rate
k3Fe(CN)6 10mM
&
il  0.62nFD 2 / 31/ 2 1/ 6C0
K4Fe(CN)6 20mM
il  1/ 2C0
Fe 2  Fe 3  1e
Fast electron transfer
Fe 3  1e  Fe 2
i-E curves as a function of K4Fe(CN)6 concentration.
Rotation rate: 2000 rpm
il  0.62nFD 2 / 31/ 2 1/ 6C0
Fe 2  Fe 3  1e
Fe 3  1e  Fe 2
Increase in RDE rotation speed and electroactive species concentration cause an
increase in the limiting current density.
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
IL vs. ω0.5
i  0.62nFD 2 / 3ω1/2 1/ 6C0
l
20.66 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒 2+ 2/3 × (1.1 × 10−6 )1/6 × 20
𝐷𝐹𝑒 2+ = 2.5 × 10−9 𝑚2 𝑠 −1
10.44 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒 3+ 2/3 × (1.1 × 10−6 )1/6 × 10
𝐷𝐹𝑒 3+ = 2.6 × 10−9 𝑚2 𝑠 −1
IL vs. C
il  0.62nFD 2 / 31/ 2 1/ 6C0
15.59 = 0.62 × 1 × 96500 × 𝐷𝐹𝑒 2+ 2/3 × (1.1 × 10−6 )1/6 × 14.47
𝐷𝐹𝑒 2+ = 2.7 × 10−9 𝑚2 𝑠 −1
RRDE
Disk
Teflon
Ring
RRDE
𝑶 + 𝒏𝒆 ↔ 𝑹
Disk- Ed, id
𝑹 ↔ 𝑶 + 𝒏𝒆
Ring-Er, ir
Steady-state ring convective-diffusion equation
𝒗𝒓
𝝏𝑪𝒐
𝝏𝑪𝑹
𝑫𝑹 𝟏 𝝏𝟐 𝑪𝑹
− 𝒗𝒚
=
𝝏𝒓
𝝏𝒚
𝑩′ 𝒚 𝝏𝒚𝟐
𝑩′ = 𝟎. 𝟓𝟏𝝎𝟑/𝟐 𝜸−𝟏/𝟐
lim 𝐶𝑅 = 0
𝑦→∞
R initially absent in solution
Bulk concentration of O is CO*
𝒊𝑹 = 𝒏𝑭𝑫𝑹 𝟐𝝅
𝒓𝟑
𝒓𝟐
𝝏𝑪𝑹
𝒓𝒅𝒓
𝝏𝒚
𝑵𝒆𝒎𝒑𝒊𝒓𝒊𝒄𝒂𝒍 = −𝒊𝑳,𝒓𝒊𝒏𝒈 /𝒊𝑳,𝒅𝒊𝒔𝒌
RRDE experiment
K3Fe(CN)6 10mM in 1M KNO3
3-
4-
𝐹𝑒(𝐶𝑁)6 + 𝒆 → 𝐹𝑒(𝐶𝑁)6 Reduction of ferricyanide to ferrocyanide at disk
4-
3-
𝐹𝑒(𝐶𝑁)6 → 𝐹𝑒(𝐶𝑁)6 + 𝒆 Oxidation of ferrocyanide to ferricyanide at ring
𝑵𝒆𝒎𝒑𝒊𝒓𝒊𝒄𝒂𝒍 = −𝒊𝑳,𝒓𝒊𝒏𝒈 /𝒊𝑳,𝒅𝒊𝒔𝒌

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