1. LINEAR SWEEP VOLTAMMETRY AND CYCLIC VOLTAMMETRY
(LSV & CV)
Good morning Ladies and Gentleman. My name is Katalin Neuróhr, I will talk about the Co and Pb electrochemical codeposition.
Katalin Neuróhr
Research Institute for Solid State Physics and Optics (RISSPO),
Hungarian Academy of Sciences

2. Introduction LSV CV voltammetric methode
the current at the working electrode is measured while the potential between the working electrode and the reference electrode is swept linearly in time
ends when it reaches a set potential CV
a type of potentiodynamic electrochemical measurement
the working electrode potential is ramped linearly versus time like linear sweep voltammetry
when cyclic voltammetry reaches a set potential, the potential of the working electrode ramp is inverted
the current at the working electrode is plotted versus the applied voltage to give the cyclic voltammogram trace
The complete electrochemical behaviour of a system can be obtained through a series of steps to different potentials with recording of the current-time curves. It is not easy to recognise the presence of different species from the recorded i-t curves alone. More information can be gained in a single experiment by sweeping the potential with time and recording the i-E curve directly.
Linear sweep voltammetry is a voltammetric method where the current at a working electrode is measured while the potential between the working electrode and a reference electrode is swept linearly in time. Oxidation or reduction of species is registered as a peak or trough in the current signal at the potential at which the species begins to be oxidized or reduced.
Cyclic voltammetry or CV is a type of potentiodynamic electrochemical measurement. In a cyclic voltammetry experiment the working electrode potential is ramped linearly versus time like linear sweep voltammetry. Cyclic voltammetry takes the experiment a step further than linear sweep voltammetry which ends when it reaches a set potential. When cyclic voltammetry reaches a set potential, the working electrode's potential ramp is inverted. This inversion can happen multiple times during a single experiment. The current at the working electrode is plotted versus the applied voltage to give the cyclic voltammogram trace. 2 2

3. LSV – Nernstian (Reversible) System
Fundamentals
Consider the reaction We assume here the Erdey-Grúz–Volmer equation
and (1)
The summary of the kinetics parameters:
where l is a dimensionless parameter,
if l > 10, the voltammogram is reversible
if l < 0.1, the voltammogram is irreversible
if 0.1 < l > 10, the voltammogram is quasireversible 3 3 3

4. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
The equations governing this case are (2)
(3)
(4)
The initial condition, (3), merely express the homogeneity of the solution before the experiment starts at t = 0, and the semi-infinite condition, (4), is an assertion that regions distant from the electrode are unperturbed by the experiment. The third condition, (3), expresses the condition at the electrode surface after the potential transition, and it embodies the particular experiment we have at hand. 4 4 4

5. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
and the fluxe balance is
(5)
It is convenient to rewrite (1)
(6)
Consider reaction, linear diffusion, only species O, with the electrode held initially at Ei potential
(7)
the potential is swept linearly at v
If the rate of electron transfer is rapid at the electrode surface
(8) 5 5 5

6. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
after Laplace transformation of (2), the application of conditions (3) and (4) yields
(9)
(10) 6 6 6

7. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
The time dependence is significant because the Laplace transformation of (8) cannot be obtained. The problem was first considered by Randles and Sevcik; the tratement and notation here follow the later work of Nicholson and Shain.The boundary condition
(11)
Where and and Laplace transformation of the diffusion equations and application of the initial and semi-infinite conditions leads to [see (9)]:
(12) 7 7 7

8. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
After the transform of the current is given by
(13)
Laplace transformation (14)
(15)
An initial conditions (t = 0) and two boundary conditions in x. Typically one takes for the initial state:
(16)
and one uses the semi-infinite limit:
(17) 8 8

9. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
Generally:
(18)
For variable t, we obtain
(19)
(20)
(21) 9 9 9

10. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
The semi-infinite limit can be transformed to
(22)
hence, B’(s) must be zero for the conditions at hand. Therefore,
(23)
and
(24)
Final evolution of and C(x,t) depends on the boundary condition. 10 10 10

11. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
Combining (13) with (12) and inverting, we obtain
(25)
By letting
(26)
(25) can be written
(27) 11 11 11

12. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
An expression for CR(0,t) can be obtained (assuming R is initially absent):
(28)
The derivation of (27) and (28) employed only the linear diffusion equations, initial conditions, semi-infinite conditions, and the flux balance. No assumption related to electrode kinetics or technique was made; hence (27) and (28) are general. From these equations and the boundary condition for LSV (11), we obtain
(29)
(30)
where, as before, 12 12

13. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
The solution of this last integral equation would be the function i (t), embodying the desired current/time curve, or since potential is linearly related to time, the current/potential equation. A closed-form solution of (30) cannot be obtained, and a numerical method must be employed.
Before solving (30) numerically, it is convenient (a) to change from i(t) to i(E), since that is the way in which the data are usually considered, and (b) to put the equation in a dimensionless form so that a single numerical solution will give results that will be useful under any experimental conditions. This is accomplished by using the following substitution:
(31) 13 13 13

14. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
Let With , so that , , at and at , we obtain
(32)
so that (30) can be written
(33)
Of finally, dividing by , we obtain
(34)
where
(35) 14 14 14

15. LSV – Nernstian (Reversible) System
Solution of the Boundary Value Problem
Note that (34) is the desired equation in terms of the dimensionless variables , , , and Thus at any value of , which is a function of E, can be obtained by solution of (34) and, form it, the current can be obtained by rearrangement of (35):
(36)
At any given point, is a pure number, so
that (36) gives the functional relationship
between the current at any point on the LSV
curve and the variables. Specifically,i is
proportional to and v1/2.The solution of (34)
has been carried out numerically
(Nicholson and Shain), by a series solution
(Sevcik, Reinmuth), analitically in items of
an integral that must be evaluted numerically.
Fig.1. Linear potential sweep voltammogram in terms of dimensionless current function. Values on the potential axis are for 13oC 15 15 15

16. LSV – Nernstian (Reversible) System
Peak Current and Potential
The function , and hence the current, reaches a maximum where From (36) the peak current, ip, is
(37)
At 25 oC for in cm2, D0* in cm2/s, CO* mol/cm3, and v in V/s, ip is
(38)
The peak potential, Ep is
mV at 25 oC (39)
Difficult to determine
The half-peak potential, Ep/2, which is
mV at 25 oC (40) 16 16

17. LSV – Nernstian (Reversible) System
Peak Current and Potential
Note that E1/2 is located just about midway between Ep and Ep/2, and that a convenient diagnostic for a nernstian wave is
(41)
Thuse for a reversible wave, Ep is independent of scan rate and ip is proportional to v1/2. The latter property indicates diffusion control and is analogous to the variation of id in chronoamperometry. A convenient in LSV is , which deends on n3/2 and DO1/2. This constant can be used to estimate n for an electrode reaction, if a value of DO can be estimated. 17 17 17

18. LSV – Totally Irreversible Systems
The Boundary Value Problem
For a totally irreversible one-step, one-electron reaction ( ) the nerstian boundary condition, (8), is replaced by
(42)
Where
(43)
Introducing E(t) from (7) into (43) yields
(44)
where b = afv and
(45) 18 18 18

19. LSV – Totally Irreversible Systems
The Boundary Value Problem
The solution follows in an analogous manner to that described in Totally Reverse Systems and again requires a numerical solution of an integral equation. The current is given by
(46)
(47)
where is a function tabulated from a table. i at any point on the wave varies with vl/2 and D0*.
For spherical electrodes, values of the spherical correction factor, employed in the equation
(48) 19 19 19

20. LSV – Totally Irreversible Systems
Peak Current and Potential
The function goes throug a maximum at Introduction of this value into (36) yields te following for the peak current:
(49)
Where the units are as for (49). From a table, the peak potential is given by
mV at 25 oC (50) or
(51)
(52) 20 20 20

21. LSV – Quasireversible Systems
Matsuda and Ayabe coined the term quasireverible for raections that show electron transfer kinetic limitations where the reverse reaction has to be considered, and they provided the fisrt treatment of such systems. For the one-step, one-electron case, the corresponding boundary condition is
(53)
The shape of the peak and the various peak parameters were shown to be functions of a and a parameter L, defined as
(54) (55)
The current is given by (56)
or for 21 21 21

22. LSV – Quasireversible Systems
when L > 10, the behavior approachesthat of a reversible system. The values of ip, Eр, and Ep/2 depend on L and a. The peak current is given by
(57)
where ip(rev) is the reversible ip value (37). Note that for a quasireversible reaction, ip is not proportional to v1/2.
The peak potential is
(58)
For the half-peak potential, we have (59) 22 22 22

23. CV (Experimental basis)
The basic scheme involves application of a potential sweep to the working electrode. The various parameters of interest are shown in Fig. 2.
In linear sweep voltammetry the potential scan is done in only one, stopping at a chosen value, Ef for example at t = t1 in Fig. 2. The scan direction can be positive or negative and, in principle, the sweep rate can have any value.
In cyclic voltammetry, on reaching t = t1 the sweep direction is inverted
as shown in Fig. 2 and swept until Emin, then inverted and swept to Emax, etc.
Fig.2. Variation of applied potential with time in cyclic voltammetry, showing the initial potential, Ei9 the final potential, f, maximum, Emax, and minimum, Emin, potentials. The sweep rate \dE/dt\ = v. For linear sweep voltammetry consider only one segment. The fact that the initial sweep is positive is purely illustrative. 23 23 23

24. CV (Experimental basis)
The important parameters are
• the initial potential, Ei
• the initial sweep direction
• the sweep rate, v
• the maximum potential, Emax
• the minimum potential, Emin
• the final potential, Ef
It is not common, but can sometimes be convenient, to change the values
of Emax and Emin between successive cycles.
The total current is
(60) 24 24 24

25. CV (at planar electrode)
For simple electron transfer , with only O initially present in solution. The initial sweep direction is therefore negative. The observed faradaic current depends on the kinetics and transport by diffusion of the electroactive species. It is thus necessary to solve the equations
(61) and (62)
The boundary conditions are
(63 a)
(63 b)
(63 c)
(63 d) 25 25 25

26. CV (at planar electrode) – Reversible Systems
The final boundary condition for a reversible system is the Nernst equation
(64)
Solution of the diffusion equations leads to a result in the Laplace domain that cannot be inverted analytically, numerical inversion being necessary. The final result, after inversion, can be expressed in the form (65)
where and (66, 67)
Thus the current is dependent on the square root of the sweep rate. 26 26 26

27. CV (at planar electrode) – Reversible Systems
Indicate some quantitative parameters in the curve, which can be deduced from data in table. First, the current function, , passes through a maximum value of at a reduction peak potential Ep,c of (68)
(69)
Secondly, the peak current in amperes is
(70) 27 27 27

28. CV (at planar electrode) – Reversible Systems
With A measured in cm2, Do in cm2s-1, [O]oo in mol cm-3 and v in Vs-1, substituting T = 164 К in (65) and (66)—an equation first obtained by Randles and Sevcik. Thirdly, the difference in potential between the potential at half height of the peak, Ep/2,c (I = IP,c/2), and Epc is given by (71)
If the scan direction is inverted after passing the peak for a reduction reaction, then a cyclic voltammogram, as shown schematically in Fig. 3, is obtained. It has been shown that, if the inversion potential, El, is at least 35/n mV after Epc, then
(72)
In which x = 0 for El «Epc and is 3 mV for |Epc - El | = 80/n mV. In this case (73) 28 28 28

29. CV (at planar electrode) – Reversible Systems
The shape of the anodic curve is always the same, independent of El, but the value of El alters the position of the anodic curve in relation to the current axis. For this reason Ip,a should be measured from a baseline that is a continuation of the cathodic curve, as shown in Fig. 3.
We can summarize all the information in a diagnostic for linear sweep
and cyclic voltammograms of reversible reactions:
• Ip  v1/2
• Ep independent of v
•Ep - Ep/2= 56.6/n mV
and for cyclic voltammetry alone
•Ep,a – Ep,c = 57.0/n mV
•Ip,a/Ip,c = 1
Fig.3. Cyclic voltammogram for a reversible system 29 29 29

30. CV (at planar electrode) – Reversible Systems
As is clear from (65), Ip  T-1/2: so, if experiments are conducted at temperatures other than 298 K, the correction in Ip is easy to do. Sometimes, and this is one of the disadvantages of conventional analysis of cyclic voltammograms, it is not possible to measure the baseline with sufficient precision in order to measure Ipa. However, it is a good approximation to apply the following expression in terms of the peak current measured from the current axis (Ipa)0 and the current at the inversion potential (Il)0 (see Fig. 1) (74) 30 30 30

31. CV (at planar electrode) – Reversible Systems
The capacitive contribution to the total current as given in (60) should also be taken into account. Writing If = Ipc we have, from (60) and (75)
Substituting typical values (Cd = 20 mFcm-2, Do=10-5cm2s-1 and [О]∞ = 10-7 molcm-3 (10-4 M), and n = 1) we obtain
(76)
This ratio is 0.1 for v = 0.18Vs; if [О]∞ is an order of magnitude higher, i.e M, then the ratio is only This shows the advantage of using concentrations as high as possible, millimolar concentrations representing the upper limit. 31 31 31

32. CV (at planar electrode) – Irreversible Systems
In the case of an irreversible reaction of the type , linear sweep and cyclic voltammetry lead to the same voltammetric profile, since no inverse peak appears on inversing the scan direction. To solve (61) and (62), a fifth boundary condition to add to boundary conditions (63) is (77)
for a reduction, where
(78)
and
(79)
n' being the number of electrons transferred in the rate-determining step. 32 32 32

33. CV (at planar electrode) – Irreversible Systems
As for the reversible case, the mathematical solution in the Laplace domain cannot be inverted analytically. Numerical inversion leads to (80)
and the values of are tabulated, having a maximum of for E = Ep, see a Table. The voltammetric curve is shown in
Fig. 4.
The peak current in amperes is (81)
Fig.4. Linear sweep voltammogram for an irreversible system. In cyclic voltammetry, on inverting the sweep direction,
One obtains only the continuation of current decay 33 33 33

34. CV (at planar electrode) – Irreversible Systems
with the units the same as in (70). The peak potential is given by (82)
an alternative expression for Ip is obtained from combining (81) and (82), leading to
(83)
From data such as those in a Table it can be deduced that
mV and that mV.
With respect to reversible systems the waves are shifted to more
negative potentials (reduction), Ep depending on sweep rate. The peaks
are broader and lower. 34 34 34

35. CV (at planar electrode) – Quasi-reversible Systems
For quasi-reversible systems the kinetics of the oxidation and reduction reactions have to be considered simultaneously. The mathematical solution is, therefore, more complex, but there are numerical theoretical solutions. As a general conclusion, the extent of irreversibility increases with increase in sweep rate, while at the same time there is a decrease in the peak current relative to the reversible case and an increasing separation between anodic and cathodic peaks, shown in Fig.5.
Fig.5. The effect of increasing irreversibility on the shape of cyclic voltammograms. 35 35 35

36. CV (at planar electrode) – Quasi-reversible Systems
Peak shape and associated parameters are conveniently expressed by a parameter, L, which is a quantitative measure of reversibility, being effectively the ratio kinetics/transport, (84)
When DR = DO = D
(85)
showing that small L corresponds to large v (i.e. large s).
The following ranges were suggested for the different types of system at stationary planar electrodes: 36 36

37. CV (at planar electrode) – Quasi-reversible Systems
The transition between these zones is shown schematically in Fig.6. In the case of cyclic voltammograms and for 0.3 < a < 0.7, Ep is almost exclusively dependent on L and hardly varies with a. This can be useful for the calculation of k0 using a table, or by interpolation from a working curve drawn using these data.
Fig.6. Transition from a reversible to an irreversible system on increasing sweep rate. 37 37

38. CV – Adsorbed species
If the reagent or product of an electrode reaction is adsorbed strongly or weakly on the electrode, the form of the voltammetric wave is modified. There are two types of situation:
• the rate of reaction of adsorbed species is much greater than of
species in solution
• it is necessary to consider the reactions of both adsorbed species and of those in solution.
For a reversible reaction in which only the adsorbed species О and R contribute to the total current, the current-potential curve for О initially adsorbed and for fast electrode kinetics is given by
(86)
where Go,i is the surface concentration of adsorbed O, before the experiment begins, on an electrode of area А, s = (nF/RT)v, bo and bR express the adsorption energy of О and R respectively, and (87) 38 38 38

39. CV – Adsorbed species
The peak current for reduction, Ip,c, is obtained when (bo/bR) = 1, that is
(88)
having the same magnitude for an oxidation. The peak potential is then
(89)
The value of Ep is the same for oxidation and for reduction.
If the adsorption isotherm is of Langmuir type and (bO/bR) = 1, then the voltammetric profile is described by the function  (1 + )-2. From this it can be calculated that the peak width at half height is 90.6/n mV.
This is all shown schematically in Fig.7. 39 39
Fig.7. Cyclic voltammogram for a reversible system of species adsorbed on the electrode. 39

40. CV – Spherical electrodes
Considering first potential sweep for a reversible system one obtains
(90)
For a reduction, in which r0 is the electrode radius and f(st) a current
function different from (st). The peak current is
(91)
where ro is in cm and the other units are as in (70) for a planar electrode. Since the spherical correction does not depend on scan rate we can consider spherical electrodes as if they were planar, plus a spherical correction. 40 40 40

41. CV – Spherical electrodes
The current function for the spherical correction, f(st), is shown in a table; as can be seen it follows a sigmoidal I-Е profile. For irreversible systems the corresponding expressions are (92)
(93)
where r0 is in cm and the other units are as in (81) for a planar
electrode, the spherical correction being, once more, independent of scan
rate. Values of the current function for the spherical correction, f(bt)
are given in a table. As for reversible systems, the spherical correction
by itself corresponds to a sigmoidal I-Е profile, though of lower slope. 41 41 41

42. Thank you for your attention!42 42