10. Tools for Reactions at Surfaces
Single Crystal Surfaces
Pourbaix Diagrams
Galvanostatic Measurements
(aka constant current potentiometry)
Electrochemical Impedance Spectroscopy (EIS)
Current Voltage Curves
In order to understand the reactions of surfaces
One must consider the reactivity of various crystal
Faces on the surface
For example the energy holding a single
Atom in the solid is greater within the
Bulk of the solid than at the surface
In a similar fashion the surface roughness
Will play a role in the reactivity of the surface
Paunovic and Schlesinger, Fundamentals of Electrochemical Deposition, 2nd ed
One also has to consider the energetics of the surface that arise
from the edges of the crystal
Widepedia
http://stm2.nrl.navy.mil/~lwhitman/images/miller.gif
http://www.cem.msu.edu/~cem924sg/LowIndexPlanes.html
10. Tools for Reactions at Surfaces
Single Crystal Surfaces
Pourbaix Diagrams
Galvanostatic Measurements
(aka constant current potentiometry)
Electrochemical Impedance Spectroscopy (EIS)
Current Voltage Curves
Some Surface Reactions of Interest
Fuel Cells
Electroless Plating
Pourbaix Diagrams
Consider a metal, M, at a concentration of 10-6 M
M 2  2e 
 Ms
E Mo z  / M
Which also undergoes the reactions

M 2   H2 O 
MO

2
H

2
2
MO  H2 O MO


 2H

K1
K2
Marcel Pourbaix
1904-1998
Since this is a corrosion reaction
2
Ms 
 2e
 M
E   E Mo z  / M
 E Mo z  / M
6
0.059  10  
0.354
o
o



log


E



E
 0177
.

M z / M
M z / M
2
2
 1 
Plot this as a horizontal line independent of pH
M 2+
 E Mo z  / M  0177
.
metal
pH
Arrange the second reaction in terms of pH
MO  H 


 2
K1
s
M 
2


pK1  p MO,s  p H
pH 
1
2
pK1  3
pK1  2 p H    p10  6
2 p H    pK1    6


 2
 p M 2  
pK1    6
p H  
2

M
Plot this line at
At the solved for
pH as a a vertical
Originating from
The previous
horizontal line
MO,s
2+
 E Mo z  / M  0177
.
metal
pH2
pH
We have to consider that the metal can oxidize directly to the oxide
2
Ms 
M
 2e


 E Mo z  / M

M 2   H2 O 
 MOs  2 H
 2 F  E Mo z  / M
 RT 2.303 log K1 
K1

M s  H2 O 
 MOs  2 H  2e
pH 
1
2



E rxo   E M0 2  / M 
pK1  3


E   E M0 2  / M 


0.059
log K1 
2
0.059
0.059
2
log K1  
log H  
2
2
E   E M0 2  / M  0.0295 log K1   0.0591pH
MO,s
M 2+
Plot this line as originating
At the intersection of the
First two lines
 E Mo z  / M  0177
.
metal
pH2
Slope=0.0591
pH
The conversion of MO to MO2-2 has a similar equation as we derived for
M2+ to MOs
Calculate it!
pHa 
1
2
A plot like this can
Be used to help
Figure out what types
Of compounds you
Should encounter
When applying a potential
scan
pK1  3
MO,s
M 2+
 E Mo z  / M  0177
.
metal
pHa
MO22


E   E M0 2  / M  0.0295 log K1   0.0591pH
pHb
pH
In some fields (environmental chemistry)
Pourbiux plots are
Known as Eh/pH plots
10. Tools for Reactions at Surfaces
Single Crystal Surfaces
Pourbaix Diagrams
Galvanostatic Measurements
(aka constant current potentiometry)
Electrochemical Impedance Spectroscopy (EIS)
Current Voltage Curves
Some Surface Reactions of Interest
Fuel Cells
Electroless Plating
Luigi Galvani
1737-1798
Tools: Galvanostatic measurement
Analytical and physical electrochemists would call this “constant current
chronopotentiometry”
If allowed to proceed will maintain a constant
potential until all of the material is reacted
and then it will shift to a new potential, at
time tau

Eo (formal potential) occurs at ¼ time
required to consume the material under
a controlled current measurement.
Chronopotentiometry allows you to monitor major change in potential as one
Species is consumed and another is formed over time (rather like a titration curve)
Consider a solution containing different compounds
E
o
Re d3 / Ox3
o
E Re
d2 / Ox2
E so ln  E
o
do min ant species
RT  Re d 1

ln
 Ox1
nF
Here the dominant species is the one most easily
reduced, Red1
Re d1  O2  Ox1
o
E Re
d1 / Ox1
EOo2 / H2O


d Re d1  Re d , Ox  0
1
1
 E solution
dt
Re d   0
d  Re d   
time
1
2

Re d 2 , Ox2  0
dt
 E solution
time
Reversible Reaction, Er
nFADO 
i
* 
2
CO
1/ 2
For an analytical application
1/ 2
 855
. nAin cm2 D
1/ 2
o
mA  s
mM
1/ 2
The shape of the of the curve is described
E  E o ( E / 4 )
RT   1/ 2  t 1/ 2 

ln
nF  t 1/ 2 
Problem: Show that when

t
4
E  E o ( E / 4 )
The concentration can
Be measured from the
Transition time by the
Sand Equation
H. J. S. Sand, Phil. Mag, 1, 45, 1901
Reversible Reaction, Er
nFADO 
i
* 
2
CO
1/ 2
For an analytical application
1/ 2
 855
. nAin cm2 D
The shape of the of the curve is described
E  E o ( E / 4 )
mA  s
mM
The concentration can
Be measured from the
Transition time by the
Sand Equation
H. J. S. Sand, Phil. Mag, 1, 45, 1901
RT   1/ 2  t 1/ 2 

ln
nF  t 1/ 2 
 1/ 2    1/ 2 
 2 1/ 2  1/ 2
     
 2  2
4
  ln 
ln 
1/ 2
1/ 2
 



  


2
4


t
1/ 2
o
1/ 2

4
E  E o ( E / 4 )


 2 1/ 2   1/ 2 
  ln 
  ln1  0
1/ 2





Irreversible reaction, Ei
 2 k o  RT
RT
1/ 2
1/ 2

E  Eo 
ln 

ln


t


1/ 2
F  D   F
o



 F E  E o
i  nFAk o CO, x  0,t exp

RT
Corresponding potential
log iapp  cons tan t 
Applied current
F  E  E o 
RT
Can get alpha
Measure Ei as a function of the input current
Repeat for multiple experiments in which the input current
is varied both in magnitude and sign
Plot and obtain the Tafel slope
Results of a “galvanostatic”
deposition for copper.
This linearized portion of the
experiment gives the Tafel
Slope!
For this experiment the Tafel
slope was 0.125 V vs log I
and 0.05 V vs log I
Calculate alpha and indicate whether
or not bond breaking
Is likely in the mechanism
F 

cathodic Tafel slope   

 2.303RT 
F 

anodic Tafel slope  1    

 2.303RT 
cathodic Tafel slope
anodic Tafel slope
Results of a “galvanostatic”
deposition for copper.

1 
 cathodic Tafel slope 

 1     
 anodic Tafel slope 
 cathodic Tafel slope 


 anodic Tafel slope 


 cathodic Tafel slope  
1 


 anodic Tafel slope  

This linearized portion of the
experiment gives the Tafel
Slope!
For this experiment the Tafel
slope was 0.125 V vs log I
and 0.05 V vs log I

The low value of
Alpha is indicative
Of a bond breaking
And/or concerted
mechanism
 0.05 
 .125 



 0.28
.
05




1 


.  
 0125
Problem: (Bard and Faulkner) An analyst determines a mixture of lead and cadmium
at a mercury pool cathode by chrono potentiometry. In the cell used in the
determination, a 1.00 mM solution of Pb2+ reduced at a current of 273 mA yielded τ25.9s and Eτ/4 = -0.38 V vs SCE. A 0.69 mM solution of Cd2+ reduced with a current of
136 mA gave τ= 42.0 s and Eτ/4 = -0.56 V vs SCE. An unknown mixture of Pb2+ and
Cd2+ reduced at a current of 56.5 mA produced a double wave, with τ1 – 7.08 s and τ27.00 s. Calculate the concentrations of Pb2+ and Cd2+ in the mixture.
10. Tools for Reactions at Surfaces
Single Crystal Surfaces
Pourbaix Diagrams
Galvanostatic Measurements
(aka constant current potentiometry)
Electrochemical Impedance Spectroscopy (EIS)
Current Voltage Curves
Some Surface Reactions of Interest
Fuel Cells
Electroless Plating
Electrochemical Impedance Spectroscopy (EIS) is based on a small amplitude AC
perturbation at the electrode surface

f Hz 
e  E sint 
i  I sin
2
Current can lag: be out of phase with the applied potential
90o out of phase
t  
Capacitive reactance With applied potential

For example, at a capacitor
q  Ce
and
de
iC
dt
E


i
sin t  
Xc 
2
1
Xc 
Cd
15000
10000
5000
Amplitude
d  E sint  
iC
 CE cost 
dt
0
0
0.2
0.4
0.6
-5000
From Ohm’s Law
-10000
E
i
R
-15000
And from
Time


cost   sin t  

2
0.8
1
1.2
e  E sint 
For a capacitor and resistor in series
i  I sint   

f Hz 
2

R

R Total  R  X c
Xc
RTotal
Since these are 90o out of phase

R Total 
R  X 
2
2
c
 1 
2
R 

 C 
2

 Z
This total resistance is called the impedance, Z
The phase angle is for the resulting total resistance is:
R
Xc


Z
R
Xc


Z
Can be expressed
In terms of real and
Imaginary terms
Calculate the impedance of a an RC circuit for which
The capacitance is 10F and the resistance is 10 ohms
at frequencies of 1, 10, 100 Hz. What are the phase
angles?
Randles Equivalent circuit cell description of an electrode system
Z faradaic
R  solution resis tan ce
RCT  R P
Warburg
impedance
Is a kind of
resistance to mass
transfer
RCT  ZW
Electron transfer kinetics
RCT
RT

nFio
Cd

ZW 

Surface charging
where
 1
1
1 
 * 1/ 2  * 1/ 2 
 
2
CR DR 
nF A 2  CO Do
For 1 electron process
The resistance to mass transfer is related to the diffusion coefficients
It drops out at high frequencies
The impedance has a real part and an imaginary part:
Z Re
1
RCT  

 R 
2
2
1 
2 2
Cd    1   Cd  RCT  





2
Z Im 
1 
1

 Cd  RCT  
Cd    1
 





1 
2 2
Cd    1   Cd  RCT  





2
2
Reduce the two equations for low frequency limit, and eliminate the frequency term to
Show:
ZIm  Z Re  R  RCT  2 2 Cd
A plot of the real vs imaginary impedance should be a line with a slope of 1
ZIm  Z Re  R  RCT  2 2 Cd
In the high frequency limit

 Z Re

2
RCT 
 RCT 
2
 R 
  Z Im  

 2 
2 
2
This is the equation of circular plot
What do you
Observe?
12
Rsoln=10
RCT =20
Rsoln=5
RCT =20
10
Zimaginary
8
6
4
2
0
0
5
10
15
Zreal
20
25
30
Application of AC
Sin wave across
This circuit leads to
“Nyquist Plot”
R  RCT  2 2 Cd
For 1 e process
Bard and Faulkner
Harry Nyquist
1889-1976
An alternative name is an “Argand” diagram
z  x  iy  z e i
Jean-Robert Argand, 1768-1822
Wikepedia
What are Batteries, Fuel Cells, and Supercapacitors, Chem Rev, 2004, 104, 4245, Martin Winter and
Ralph J. Brodd
R is related to the exchange
current
C is typically of the order of
200 uF/cm ~ 10 times larger
than double layer capacitance
R (“ohmic”) is due to
electrolyte
current collectors
terminals
contact between active
particle & conductive
elements
resistive film on the surface
f max
1

RC
Warburg impedance arises as
conc of active species
changes
What are Batteries, Fuel Cells, and Supercapacitors, Chem Rev, 2004, 104, 4245, Martin Winter and
Ralph J. Brodd
A Current voltage curve (discharge curve) “can be used to determine the cell
Capacity, the effect of the discharge rate” and can be used to illustrate
The impact of the various forms of “resistance”: ET, diffusion, contact or ohmic
What are Batteries, Fuel Cells, and Supercapacitors, Chem Rev, 2004, 104, 4245, Martin Winter and
Ralph J. Brodd
An example of a electron transfer controlled reaction (left)
And a diffusion controlled reaction (right)
Bard and Faulkner
For the image at the left what is
Rct? Rsoln? The exchange current, io?
For the image on the right, what is the
Slope?
J. Ross Macdonald, Impedance spectroscopy: Emphasizing Solid Materials and Systems,
This is a system of zirconia-ytttria compounds.
R1 is the charge transfer resistance at
the electrode
C1 is the double layer capacitance of
the electrode
R2 is the intergranular resistance
corresponding to resistance of
conduction across two different
grains due to impurities
C2 is the capacity across the
intergranular region
R3 is resistance to conduction within the
grains
The point is that you can “see” multiple
Processes occurring
J. Ross Macdonald, Impedance spectroscopy: Emphasizing Solid Materials and Systems,
This is a system of zirconia-ytttria compounds.
From the data they can “image”
The surface
R1 is the charge transfer resistance at
the electrode
C1 is the double layer capacitance of
the electrode
R2 is the intergranular resistance
corresponding to resistance of
conduction across two different
grains due to impurities
Semiconducting
particles
Electrode
C2 is the capacity across the
intergranular region
R3 is resistance to conduction within the
grains
The point is that you can “see” multiple
Processes occurring
J. Ross Macdonald, Impedance spectroscopy: Emphasizing Solid Materials and Systems,
1 et process
Shows that after 164 hours that two
Different processes coexist that carry
Charge to the surface (as seen by
The near double loop)
Time
2nd et process developing
Cu-Ni alloy corrosion in sea water
Influence of fluoride concentration and pH on corrosion behavior of titanium
in artificial saliva; Alain Robin Æ Jeferson Paulino Meirelis, J. Appl.
Electrochem. 2007 37:511-517
Example: Dental corrosion
Study that begins with
Pourbaix diagram
Example calculation here
Predict what the corrosion
Pathway will be for…..
Capacitive semicircles, size decreases
With increasing FStraight line is diffusion through the oxide
Solution
resistance
Increasing rate of corrosion as Fincreases
Oxide
layer
14. Tools for Reactions at Surfaces
Single Crystal Surfaces
Pourbaix Diagrams
Galvanostatic Measurements
(aka constant current potentiometry)
Electrochemical Impedance Spectroscopy (EIS)
Current Voltage Curves
Some terminology for batteries
Capacity: = is the charge stored in the battery
QD  I d   Amp  hours  Ah
  some time of disch arg e
Power = energy per unit time
J
Watts 
s
because
J
C
V 
and I  Amps 
C
s
 J   C J
P  VI        Watts
 C  s 
s
Discharge Curve
If one discharges at a constant current this
Is simply the Galvanostatic chronopotentiometry just discussed
E  E o ( E / 4 )
RT   1/ 2  t 1/ 2 

ln
nF  t 1/ 2 
An analytical chemist thinks this way
QD  I d   Amp  hours  Ah
http://www.mpoweruk.com/performance.htm
One can set Qd constant and then
Calculate the voltage at different
Discharge currents
The formal potential of the compound controlling the voltage (via the
Nernst eq) should be that observed at ¼ discharge
Example calculation here
Discharge curves calculated for different discharge currents for a battery with a net
Capacity of 20A-hr, and Eo of 2
2.15
Capacity = 20 A-hr
2.1
Voltage
2.05
0.25 Amp
2
1.95
0.5 Amp
1.9
1.5 Amp
1.0 Amp
1.85
1.8
0
5
10
15
20
Time
25
30
35
40
(Hours)
Note that at smaller discharge currents the time for the voltage drop is extended