Alternating current voltammetry of electrode reactions with constant surface activity: Application to electrolysis of molten electrolytes

Journal Pre-proof Alternating Current Voltammetry of Electrode Reactions with Constant Surface Activity: Application to Electrolysis of Molten Electrolytes

Andrew H. Caldwell, Antoine Allanore PII:

S1572-6657(19)30977-4

DOI:

https://doi.org/10.1016/j.jelechem.2019.113709

Reference:

JEAC 113709

To appear in:

Journal of Electroanalytical Chemistry

Received date:

17 October 2019

Revised date:

1 December 2019

Accepted date:

1 December 2019

Please cite this article as: A.H. Caldwell and A. Allanore, Alternating Current Voltammetry of Electrode Reactions with Constant Surface Activity: Application to Electrolysis of Molten Electrolytes, Journal of Electroanalytical Chemistry(2019), https://doi.org/10.1016/j.jelechem.2019.113709

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© 2019 Published by Elsevier.

Journal Pre-proof

Alternating Current Voltammetry of Electrode Reactions with Constant Surface Activity: Application to Electrolysis of Molten Electrolytes Andrew H. Caldwella , Antoine Allanorea,∗ of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139

of

a Department

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Abstract

Analytic solutions are derived for the alternating current voltammetry (ACV)

-p

harmonic waveforms of electrodeposition and gas evolution reactions where the surface activity of the reduced species R is constant. Reversible and quasi-

re

reversible charge transfer kinetics are considered, as well as the effects of ohmic drop and double layer charging. It is shown that ohmic drop produces a char-

lP

acteristic distortion of the waveform envelopes, resulting in the emergence of waveform extrema in the higher harmonics that are distinct from those of the conventional ACV theory of soluble redox couples. A quantitative connection

na

is made between the peak potential of the second harmonic and the electrode reaction parameters. The analytic solutions agree well with measurements of

ur

the fundamental, second, and third harmonic waveforms of Pb electrodeposition on liquid Pb and of Cl 2 evolution on graphite in molten PbCl 2 -NaCl-KCl

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at 700 ◦ C.

Keywords: AC Voltammetry, Electrodeposition, Gas evolution, Electrolysis, Ohmic drop

1. Introduction Electrolytic decomposition refers to an important class of electrochemical reactions in which species in solution are oxidized or reduced to higher energy constituent phases of the electrolyte or one of its components. These components are typically metal-nonmetal compounds that have been solubilized in a ∗ Corresponding

author Email address: [email protected] (Antoine Allanore)

Preprint submitted to Elsevier

December 10, 2019

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liquid supporting electrolyte. The products of electrolysis are therefore commonly a metal at the negative electrode (cathode) and a gas of an oxidized nonmetal anion at the positive electrode (anode), both of which are generally of low solubility in the electrolyte. Electrolysis reactions involving the formation of pure metals and gases were among the first electrochemical phenomena to be studied and were integral in the development of electrochemistry as a scientific discipline [1]. Today, in terms of tonnage of reaction product, electrolytic metal

of

extraction is the second most important electrochemical process, behind the chlor-alkali process [2]. Metals such as Al, Cu, Mg, Na, and Li are commercially

ro

made by electrolysis [3]. Research in this field has sought to extend the range of metals that may be industrially produced in order to improve the sustain-

-p

ability [4, 5, 6], selectivity [7], and cost [8] of the extraction process. These investigations have focused on the use of molten electrolytes, e.g., halide salts,

re

oxides, sulfides, in which a high solubility of the feedstock is necessary to be cost-competitive with traditional processing technologies. A properly designed

lP

molten electrolyte allows for electrodeposition of the target metal or alloy in the absence of side reactions while satisfying a host of other constraints on, for example, vapor pressure, liquidus temperature, viscosity, etc. [9]. The challenge

na

of identifying such an electrolyte is particularly acute for high temperature systems, for which comparatively little thermodynamic and transport data exist.

ur

The design of these electrolytes must be “rational,” i.e., predicated on a quantitative connection between composition and properties. The composition

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space for candidate electrolytes is too large to be tested without the aid of both component activity data and knowledge of the electrode reaction kinetics. This sort of information is necessary for predicting a number of critical features of the process, such as the operating current density and cell voltage. Recently, it has been demonstrated that alternating current voltammetry (ACV) is a promising method for investigating electrolysis reactions in molten electrolytes. ACV has proven to be a powerful electroanalytical technique in regards to its sensitivity to reaction kinetics [10, 11, 12] and its discrimination against double-layer charging currents [13, 14]. ACV measurements of electrodeposition and gas evolution in molten electrolytes have revealed reproducible waveform extrema in the harmonic currents that are indicative of charge-

2

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transfer reactions [15, 16, 17, 18]. Estimations of the decomposition potential from these data were used to calculate component Gibbs energies and activities. The correlations in these studies are compelling. However, they remain heuristic, for the following reason. Interpretation of the harmonic waveform extrema is done within the framework of conventional ACV theory, which assumes diffusion-controlled transport of both the oxidized and the reduced species taking part in the electrode reaction O + ne− ↔ R. This is appropriate when

of

the electroactive species are soluble redox couples [19] or form amalgams [20]. These types of reactions have historically concerned the majority of experimen-

ro

tal work with ACV. However, in the study of electrolysis reactions, particularly in molten electrolytes, the assumption of diffusion-controlled transport of the

-p

reduced species R is not valid. The theoretical harmonic waveforms fail to accurately model the waveforms of experimental electrolysis reactions. The cathode

re

reaction is typically electrodeposition of a pure metal that is of negligible solubility in the electrolyte. At the anode, the reaction is gas evolution where

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the species R is the common anion of a fused salt electrolyte, and the constraint of electroneutrality requires a spatially uniform concentration of R. In both half-reactions there is the absence of a chemical potential gradient of R

transport.

na

at the electrode surface that serves as the driving force for diffusion-controlled

ur

To date, there is no reported mathematical treatment of ACV under conditions representing electrodeposition and gas evolution, the two half-reactions

Jo

constituting electrolytic decomposition. However, there is evidence from DC voltammetry studies of insoluble reaction products [21, 22, 23, 24] indicating that a constant surface activity of the species R is a key requirement for accurate analytic solutions of the voltammograms. Therefore, it is posited that the following boundary conditions are necessary to model ACV harmonic waveforms of electrolysis reactions: • Rather than assuming linear diffusive transport of both species O and R, R is defined to be insoluble or else its concentration gradient is zero everywhere. The concentration of O is assumed to be sufficiently small that its concentration gradient at the electrode surface remains governed by linear diffusive transport. 3

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• Rather than allowing the surface concentration of R to vary in accordance with the applied potential and charge-transfer kinetics, the activity of R is defined to be constant at the electrode surface. When aR = 1 at the electrode surface, these conditions describe the electrodeposition of R onto an electrode made of R, as well as gas evolution at an inert electrode in a fused-salt electrolyte where R is the common anion. An illustra-

of

tion of the proposed concentration profiles is provided in Fig. 1.

Concentration,

R

1

O

0

x

0,

0

lP

R

x,

,

ro

R

0 R,

R

1

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0 R,

0,

re

R

Concentration,

Position,

0

O

x,

Position,

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Figure 1: Schematic concentration profiles of the species O and R for electrodeposition and gas evolution reactions.

ur

This paper details the derivation of the ACV harmonic waveforms for electrodeposition and gas evolution following the above boundary conditions. In

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doing so, a rigorous connection is made to the thermodynamic and kinetic parameters of electrolysis reactions. The validity of the derivation is confirmed by comparison with experimental ACV measurements of PbCl 2 electrolysis in molten PbCl2 -NaCl-KCl at 700 ◦ C. 2. Derivation Expressions for the current-potential relation of the k th harmonic (k = 0, 1, 2, 3, ...) are derived for electrolysis reactions of the type specified above. The standard state of the species R in the electrodeposition reaction is the pure metal R at temperature T . In the gas evolution reaction, where R is the common anion, the standard state of R is its state in the specific electrolyte composition 4

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at temperature T . Both reversible charge transfer reactions (Section 2.1) and quasi-reversible charge transfer reactions (Section 2.2) are treated. The effects of ohmic drop and double layer charging are included in Section 2.3, and analytic solutions are explicitly given for the case of reversible charge transfer. 2.1. Reversible Charge Transfer The reaction being considered is of the form (1)

of

O + ne− ↔ R.

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It is assumed that the reaction in Eq. 1 is reversible and is therefore governed by the Nernst equation: 

1 0) γO (CO (0, t)/CO

-p

RT E(t) = E − ln nF 0



(2)

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where the surface activity of R is defined to be unity. E(t) is the potential at the working electrode surface, E 0 is the standard potential of Eq. 1, γO is

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0 the activity coefficient of O, CO (0, t) is the surface concentration of O, and CO

is the standard state concentration of O. The applied potential is the sum of a steady-state DC potential, E (time-independent), and a sinusoidal perturbation

be rewritten as:

na

of amplitude ΔE and angular frequency ω: E(t) = E − ΔE sin(ωt). Eq. 2 may     0
nF CO −nF E − E 0 exp ΔE sin(ωt) exp RT RT γO

ur

CO (0, t) =

(3)

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Semi-infinite linear diffusion of O is assumed, and the current is entirely diffusion controlled. The continuity equation (Eq. 4) and the initial (Eq. 5) and boundary conditions (Eqs. 6, 7) are given below. ∂CO ∂ 2 CO = DO ∂t ∂x2

(4)

∗ CO (x, 0) = CO

(5)

∗ CO (x → ∞, t) = CO

(6)

DO



∂CO ∂x



= x=0

5

−i(t) nF A

(7)

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The solution to the continuity equation with these boundary conditions is Z t CO (0, t) ψ(t − u)du = 1 + (8) ∗ CO (πu)1/2 0 where ψ(t) =

i(t) 1/2

∗ nF ADO CO

.

(9)

Substituting for CO (0, t) in Eq. 3 yields     Z t 0 e CO ψ(t − u)du nF = 1+ ∗ exp − RT ΔE sin(ωt) γO C O (πu)1/2 0

of

=


nF E − E0 . RT

(11)

ro

where

(10)

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Substituting the time-dependent exponential term for its Fourier series expan-

na

where

lP

re

sion [25] yields Z t ψ(t − u)du 1+ = (πu)1/2 0 ! ∞ ∞ 0 X X e CO I0 (ξ) − 2 (−1)q I2q+1 (ξ) sin [(2q + 1)ωt] + 2 (−1)q I2q (ξ) cos [(2q)ωt] ∗ γO C O q=0 q=1

ξ=

(12)

nF ΔE RT

(13)

Ik (ξ) =

∞ X (ξ/2)2q+k q=0

Jo

ur

and the Ik are the modified Bessel functions of the first kind. For real-valued ξ: q!(q + k)!

.

(14)

The convolution integral in Eq. 12 may be represented as the sum of the orthogonal harmonic currents that comprise the total current response: Z t Z t Z t Z t ψ(t − u)du ψ0 (t − u)du ψ1 (t − u)du ψ2 (t − u)du = 1 + + + +... 1+ 1/2 1/2 1/2 (πu) (πu) (πu) (πu)1/2 0 0 0 0 | {z } | {z } | {z } 1st

0th

2nd

(15)

These terms can then be equated with their respective harmonic components of

the Fourier series expansion on the right-hand side of Eq. 12. Table 1 shows the terms in the series up to q = 2. I0 is related to the DC term (k = 0), I1 to the fundamental harmonic term (k = 1), I2 to the second harmonic term (k = 2), etc. 6

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Table 1: Terms in the Fourier Series Expansion for the Reversible Case q

Series Terms

0

I0 (ξ) − 2I1 (ξ) sin(ωt)

1

−2I2 (ξ) cos(2ωt) + 2I3 (ξ) sin(3ωt)

2

2I4 (ξ) cos(4ωt) − 2I5 (ξ) sin(5ωt) ...

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2.1.1. DC Component

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The solution to the DC component, including rectification terms, may be

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determined by Laplace transform of Eq. 16: Z t 0 e C O ψ0 (t − u)du = 1+ ∗ I0 (ξ) 1/2 γO CO (πu) 0

(16)

1/2



 ∞ 0 X ξ 2q e CO ∗ I0 (ξ) − CO , I0 (ξ) = 2. 2q γO q=0 2 (q!)

(17)

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nF ADO idc (t) = (πt)1/2

re

which, upon substitution of Eq. 9, gives

The DC component decays following a Cottrell-like time-dependence. Eq. 17

na

may be used to generate theoretical sampled current voltammograms by calculating the current at a time τ for a series of potential steps. An example voltammogram is shown in Fig. 2 (in red) along with the theoretical sampled

ur

∗ = 0 with R soluble (in blue) current voltammograms for the cases where CR

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∗ ∗ and where CR = CO with R soluble (in green).

2.1.2. Fundamental and Higher Harmonics The general solution of the k th current harmonic is of the form: ψk (t) = Ak sin(kωt) + Bk cos(kωt). The unknown coefficients Ak and Bk may be determined by substitution into Eq. 12: Z

t 0

 3k−1 (−1) 2 sin (kωt) 0 Ak sin[kω(t − u)] + Bk cos[kω(t − u)] 2e CO du = ∗ Ik (ξ)  3k γO CO (πu)1/2 (−1) 2 cos (kωt)

(18)

7

k odd k even.

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1

1, R insoluble

R

R

R

0 for all

O

R soluble

0

0, R soluble

1 5

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ro

R

of

d

or

0

5

re

0

Figure 2: Theoretical sampled current voltammograms (τ = 1 s) normalized by

lP

the Cottrell current (id ) for three cases: (in red, this work, Eq. 17) aR = 1 at the electrode surface and R is insoluble or the concentration gradient is zero for

na

∗ ∗ ∗ all x; (in blue) CR = 0, R soluble; and (in green) CR = CO , R soluble. The

waveforms were calculated for the following reaction parameter values: n = 1,

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∗ ∗ T = 25 ◦ C, D = 10−9 m2 s−1 , CO = CR = 1 mol m−3 .

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Eq. 18 may be rewritten using the angle addition identity: Z Z (Ak + Bk ) sin(kωt) t cos(kωu)du (Ak − Bk ) cos(kωt) t sin(kωu)du − π 1/2 u1/2 π 1/2 u1/2 0 0  3k−1 (−1) 2 sin (kωt) k odd 0 2e CO = Ik (ξ) ∗ 3k  γ O CO (−1) 2 cos (kωt) k even.

(19)

In the steady-state limit (t → ∞) the integral expressions in Eq. 19 reduce to (π/2kω)1/2 . Solving for Ak and Bk yields  0 3k−1 e CO   2 Bk = (2kω)1/2  ∗ Ik (ξ)(−1) γO CO Ak =  0  3k  1/2 e CO  Ik (ξ)(−1) 2 +1 −Bk = (2kω) ∗ γO C O 8

k odd (20) k even.

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By applying the trigonometric identity: a sin(x)±b cos(x) =

√

a2 + b2 sin [x ± arccot(a/b)]

and re-dimensionalizing the current (Eq. 9), the solution to the k th harmonic becomes: i(kωt) =

0 (DO kω)1/2 e Ik (ξ) 2nF ACO

γO

  3k−1 π  (−1) 2 sin kωt + 4  π  (−1) 3k 2 +1 sin kωt − 4

for k odd for k even. (21)

The dimensionless fundamental (k = 1), second (k = 2) and third (k = 3)

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harmonic current amplitudes k 1/2 e Ik (ξ) are plotted in Fig. 3 as functions of

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the dimensionless potential, , for ξ = 0.25.

As a result of the condition that aR = 1, all of the harmonic current am-

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plitudes are exponential with increasing positive (anodic) potential. This may be rationalized intuitively by considering that a constant surface activity of R

re

is essentially equivalent to an infinite reservoir of R at the electrode surface. The derivation does not take into account the limiting solubility of the species

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O, which addresses the issue of an infinitely-increasing current density. In this analysis, it is assumed that the positive overpotentials are sufficiently small such

na

that the solubility limit of O can be neglected. 2.2. Quasi-Reversible Charge Transfer

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The Butler-Volmer charge-transfer formalism is adopted for treating the

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quasi-reversible case. The current-potential relation may be written as     

(1 − α)nF −αnF 0 E(t) − E 0 − CO (0, t) exp E(t) − E 0 i(t) = nF Ak 0 CR exp RT RT (22) 0 where CR is the standard state concentration of R. The continuity equation

(Eq. 4) and its initial (Eq. 5) and boundary conditions (Eqs. 6 and 7) are the same as for the reversible case. Substituting for E(t) = E − ΔE sin(ωt) and for CO (0, t) using the general solution to the continuity equation (Eq. 8) with the definitions in Eqs. 9 and 11 yields ψ(t) = g exp [−ξa sin(ωt)] − h(1 + Θ) exp [ξc sin(ωt)] where, for convenience of notation, the following definitions are used: Z t ψ(t − u)du Θ= (πu)1/2 0 9

(23)

(24)

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0.5 0.25

0.0

-p

ro

of

12

e

1

2

0

2

3 2

re

0

lP

Figure 3: Dimensionless fundamental (k = 1), second (k = 2), and third (k = 3) harmonics (see Eq. 21) as functions of the dimensionless potential  for a value of the dimensionless perturbation amplitude ξ of 0.25. All harmonics of the

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ur

applied potential.

na

purely faradaic, reversible reaction exhibit an exponential dependence on the

ξa =

(1 − α)nF ΔE RT

(25)

αnF ΔE RT

(26)

0 k 0 CR eβ 1/2 C ∗ D O

(27)

ξc =

g=

O

h=

k0 1/2 DO

e−α

(28)

Eq. 23 may be rewritten using the Fourier series expansion of the terms exp [−ξa sin(ωt)] and exp [ξc sin(ωt)]. In addition, the dimensionless current ψ(t) can be written as the sum of its harmonic components, ψk (t). Then, terms of like-order in k

10

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may be equated: ∞ X

ψk (t) =

k=0 ∞ X

n=0

− hI0 (ξc ) − 2h ∞ X

k=0

∞ X

(−1)n I2n (ξa ) cos[2nωt]

n=1

∞ X

n=0

(−1)n I2n+1 (ξc ) sin[(2n + 1)ωt] − 2h

Θk − 2h

∞ X ∞ X

n=0 k=0

∞ X

(−1)n I2n (ξc ) cos[2nωt]

n=1

Θk (−1)n I2n+1 (ξc ) sin[(2n + 1)ωt] − 2h

∞ X ∞ X

Θk (−1)n I2n (ξc ) cos[2nωt]

n=0 k=0

(29)

ro

− hI0 (ξc )

(−1)n I2n+1 (ξa ) sin[(2n + 1)ωt] + 2g

of

gI0 (ξa ) − 2g

Table 2 lists terms in Eq. 29 of the same order in k, up to k = 2. Solutions to

-p

ψk (t) may be determined by collecting terms of the k th order and assuming a solution of the form ψk (t) = Ak sin(kωt) + Bk cos(kωt) (see 2.1.2) or by Laplace

re

transform (see 2.1.1).

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Table 2: Terms in the Fourier Series Expansion for the Quasi-Reversible Case Terms of k th Index from Eq. 29

0

gI0 (ξa ) − hI0 (ξc ) − hI0 (ξc )Θ0

1

−2gI1 (ξa ) sin(ωt) − 2hI1 (ξc ) sin(ωt) − hI0 (ξc )Θ1 − 2hI1 (ξc )Θ0 sin(ωt)

2

−2gI2 (ξa ) cos(2ωt) + 2hI2 (ξc ) cos(2ωt) − hI0 (ξc )Θ2 − 2hI1 (ξc )Θ1 sin(ωt) + 2hI2 (ξc )Θ0 cos(2ωt)

na

k

Jo

ur

...

2.2.1. DC Component A derivation of the DC component (k = 0) is provided in Appendix A. The solution is given in Eq. 30: 1/2

idc (t) =

nF ADO (πt)1/2



 0 CO I0 (ξa )  ∗ H(λ) e − CO γO I0 (ξc )

(30)

where λ = hI0 (ξc )t1/2 . This form of the solution closely matches that of the reversible case (Eq. 17), multiplied by a function H(λ) that captures the rate dependence of the reaction. Fig. 4 shows the theoretical sampled current voltammograms (τ = 1 s) for four values of the standard heterogeneous rate constant k 0 as well as for the reversible limit.

11

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108 m s 10

1

7

10

of

0

0

6

10

5

ro

d

1

1 15

10

-p

Reversible

5

0

5

re

0

Figure 4: Theoretical sampled current voltammograms for four values of the

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standard heterogeneous rate constant k 0 , as well as for the reversible case (in red). The currents are normalized by the Cottrell current (id ). The waveforms

na

were calculated for the following reaction parameter values: n = 1, T = 25 ◦ C, ∗ D = 10−9 m2 s−1 , ω = 200π s−1 , ΔE = 10 mV, α = 0.5, CO = 1 mol m−3 .

ur

In the limits α → 0 and H(λ) → 1 (i.e., k 0 → ∞), the reversible solution

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(Eq. 17) is recovered. The condition on the symmetry coefficient α implies completely asymmetric charge transfer, where the Butler-Volmer equation in the reversible limit takes the same form as the ideal diode equation, i = i0 (ef () −1). For positive overpotentials ( > 0) the electrode becomes “forward biased” and the anodic current increases exponentially. For negative overpotentials ( < 0) the electrode becomes “reverse biased” and the current approaches the diffusionlimited current, id . 2.2.2. Fundamental and Higher Harmonics A derivation of the fundamental harmonic is provided in Appendix B. The approach to calculating the fundamental harmonic may also be applied to the higher harmonics, which, for brevity, are not derived here. The solution for the

12

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fundamental harmonic is given in Eq. 31: i(ωt) =

   G(t) 1+y sin ωt + arccot y (1 + 2y + 2y 2 )1/2

(31)

where 1/2

∗ G(t) = −2nF ADO CO (gI1 (ξa ) + hI1 (ξc ) + hI1 (ξc )Θ0 )

(32)

and y=

hI0 (ξc ) . (2ω)1/2

(33)

of

G(t) is a function of both the rate constant and the solution to the DC com-

ro

ponent. Note then that the equation of the k th harmonic for quasi-reversible charge-transfer requires the solutions at k − 1, k − 2, ..., 1, 0. The phase angle

-p

arccot[(1 + y)/y] is also rate-dependent and varies linearly with ω 1/2 . The same

dependence on ω is found for the quasi-reversible fundamental harmonic in the

re

conventional ACV formalism of soluble redox couples. In addition, one can show that Eq. 31 reduces to the expression for the reversible case (Eq. 21) in

lP

the limits of k 0 → ∞ and α → 0.

Fig. 5 shows the calculated fundamental harmonic current amplitudes for several values of k 0 as well as for the reversible limit. With decreasing k 0 , larger

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overpotentials are required to generate the same value of the current amplitude. This “broadening” the waveform shape with slower reaction kinetics is present

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in the higher harmonics as well.

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2.3. Incorporating Ohmic Drop and Double-Layer Charging Double-layer charging and ohmic drop are two non-faradaic phenomena that lead to significant distortion of the ACV waveforms when compared to the purely faradaic case [26]. Experimentally, the effects of ohmic drop may be minimized by employing ultramicroelectrodes (UMEs), small distances between the working electrode and reference electrode, and a supporting electrolyte with high ionic conductivity. The first two approaches can be problematic in molten electrolytes at high temperature. The electrode configuration and size are often dictated by constraints on the electrode material (e.g., liquid metal) or on the cell design, prohibiting the use of UMEs and small distances between electrode. In addition, ohmic drop is rarely ever negligible due to the large current densities of electrolysis reactions [27]. Thus, ACV measurements of electrolysis 13

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5 Reversible 4

mA

0

105 m s

1

3 10

6

of

2 10

05

0

5

-p

ro

1

10

15

7

10

8

20

re

0

Figure 5: Calculated fundamental harmonic current amplitudes as functions

lP

of dimensionless potential for four values of the standard heterogeneous rate constant k 0 , as well as for the reversible case (in red). The waveforms were

na

calculated for the following reaction parameter values: n = 1, T = 25 ◦ C, ∗ D = 10−9 m2 s−1 , ω = 200π s−1 , ΔE = 10 mV, α = 0.5, CO = 1 mol m−3 .

ur

reactions are almost always influenced by non-faradaic effects. These effects must be accounted for in the mathematical description of the ACV waveforms

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if a quantitative connection is to be made between waveform features and the reaction parameters of interest. The following derivation shows the emergence of extrema—peaks and troughs—in the harmonic waveforms of electrolysis reactions when ohmic drop is included, extrema which are notably absent in the purely faradaic case. The incorporation of ohmic drop and double layer charging in analytic solutions of the harmonic waveforms was initially reported by Devay et al. [28, 29], (see also Engblom et al. [30]). An equivalent circuit model of the higher harmonics (k ≥ 2) was proposed in which the elements Ru (uncompensated resistance) and Cdl (double layer capacitance) lie in parallel, rather than in series, to reflect the faradaic element as being a generator of the higher harmonic currents. The 14

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method of Devay, while able to capture the waveform distortion of conventional AC voltammograms, fails to reproduce the correct shape of the higher harmonics for the type of electrode reactions concerned in this work, namely, those with constant surface activity of R. Mathematically, this seems to be the result of the fact that Devay’s solution of the higher harmonics is the expression of the purely faradaic i − E relation multiplied by a factor that captures both the change in the applied perturbation amplitude ΔE due to iRu and the effective shunting

of

of the faradaic element due to Ru in the proposed circuit model. This factor takes the same shape for all higher harmonics. It was shown in Eq. 21 that,

ro

for reactions with constant surface activity, the undistorted solutions have the same exponential form. Therefore, Devay’s approach for treating ohmic drop

-p

and double layer charging yields higher harmonics each with the same waveform shape. This is clearly not correct, as the emergence of waveform extrema in suc-

re

cessive harmonics may be readily observed in experimental ACV measurements of electrolysis reactions (see Fig. 3 in [15]).

lP

Instead, the approach taken here closely follows the work of Diard et al. [31, 32] in the field of nonlinear electrochemical impedance spectroscopy. A Randles circuit model is proposed, consisting of an uncompensated resistance Ru , double

na

layer capacitance Cdl , and a nonlinear faradaic element with an as-yet unspecified phase angle. The electrode reaction is considered reversible. A comment

ur

on the quasi-reversible case is provided in Appendix C. The total current in the

Jo

circuit is defined to be

i(t) = J[E(t)] + K[E(t)]

dE(t) dt

(34)

where J[E(t)] is the nonlinear current-potential relation for the faradaic element and K[E(t)] is the nonlinear current-potential relation for the double layer capacitance. Note that with ohmic drop, Eq. 34 becomes an implicit function:

i(t) = J[V (t) − iR] + K[V (t) − iR]

d(V (t) − iR) . dt

(35)

V is the potential measured by the potentiostat, distinct from the true potential at the working electrode surface, E. The applied potential is the sum of the steady-state potential, V , and a sinusoidal perturbation of amplitude ΔV and angular frequency ω: V (t) = V − ΔV sin(ωt). ΔV is equivalent to ΔE used in 15

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the previous sections of the derivation. The current response Δi(t) due to the perturbation may be represented by a Taylor series expansion of the current i(t) with respect to the applied potential under conditions of steady-state (dV /dt = 0 and i = J[V ]): " # ∞ ∞ p X X 1 dp J 1 d K dΔV (t) Δi(t) = ΔV p (t) + K + ΔV p (t) p p p! dV p! dV dt p=1 p=1

(36)

where

of

ΔV p (t) = (V (t) − V )p = (−ΔV sin(ωt))p .

(37)

ro

The presence of ohmic drop makes Δi(t) an implicit function, and the derivatives dp J/dV p and dp K/dV p in Eq. 36 are not explicitly known. They may

-p

be calculated by sequential differentiation of the lower order derivatives in a recursive manner. The first derivative of J, for example, with respect to V is 

dJ dE



dV dE

−1

dJ = dE



re

dJ = dV

dJ[E] 1 + Ru dE

−1

(38)

lP

where dV /dE follows from V = E + iRu at steady state. The subsequent derivatives in the Taylor series expansion of J are:

na

 p−1   −1  d d J dJ[E] dp J 1 + Ru = , p = 2, 3, ... dV p dE dV p−1 dE

(39)

Eq. 36 can be expressed as the sum of the harmonic components (i(ωt),

Jo

ur

i(2ωt), i(3ωt),...) by expanding sin p (ωt) (see Appendix D):  3k−1 ∞ X ∞  2q+k  (−1) 2 sin(kωt) k odd X J d 2ΔV 2q+k Δi(t) = dV 2q+k 22q+k q!(q + k)! (−1) 3k 2 cos(kωt) k even k=1 q=0   ∞ X d2q J ΔV 2q + 2q 22q (q!)2 dV q=1 (40)  " 3k−1  ∞ X ∞  2q+k  2 2q+k X sin(kωt) k odd (−1) d 2ΔV K + K+ 2q+k 2q+k q!(q + k)!  3k 2 dV (−1) 2 cos(kωt) k even k=1 q=0 #   ∞ X d2q K ΔV 2q + ωΔV cos(ωt) dV 2q 22q (q!)2 q=1 where k is the harmonic index. The rectification terms for k = 0 are also present. J[E(t)] is the steady-state nonlinear current-potential relation of the faradaic element in the proposed circuit model. The form of the steady-state faradaic 16

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current may be represented by the semiintegral [33, 34] of the first-order DC component, the solution to which is given in Eq. 17. The semiintegral Θ 0 is given by Eq. 16, which is reproduced here:   0  Z t i0 (u)du e CO 1/2 ∗ Θ0 = = nF AD C − 1 O O ∗ 1/2 (t − u)1/2 γO CO 0 π

(41)

Semidifferentiation is then required to calculate the harmonic components in Eq. 40. Semidifferentiation-semiintegration tables [35] can be used to convert

of

between the two forms: Semiderivative ↔ Semiintegral

ro

Δi(t) ↔ ΔΘ(t)

(42)

-p

(kω)1/2 sin(kωt + π/4) ↔ sin(kωt)

(kω)1/2 cos(kωt + π/4) ↔ cos(kωt)

re

The lowest order of the perturbation response, given by Eq. 38, is proportional

lP

to sin(ωt), and semidifferentiation may be applied to dΘ0 sin(ωt)/dE to yield 1/2   0  2 2 d dΘ0 e CO ∗ = ω 1/2 n F AD 1/2 CO . (43) sin(ωt) O ∗ dt1/2 dE RT γO CO

na

Subsequent iterations of Eq. 39 operate on Eq. 43. Due to the recursive nature of

this solution, phase angle information is lost. The absence of the phase angle is important only insofar as it affects the magnitude of the total current response.

ur

This is acceptable, as the aim here is to provide a quantitative theory of the ACV waveform shape. Two simplifying assumptions are made regarding the

Jo

phase angle: the phase angles of the faradaic and capacitive elements are their values in the absence of ohmic drop, i.e., π/4 and π/2, respectively; and they are independent of the potential. Under these assumptions, the phase difference between the faradaic element and the double layer capacitance is θ = π/4. Regardless, it may be shown that the value of θ does not have a large influence on the final shape of the theoretical ACV harmonic waveforms. The double layer capacitance is assumed to be independent of potential. From Eq. 35, K = Cdl . Therefore, the recursive term d2q+k K/dV 2q+k in Eq. 40 is zero. Using Eqs. 40 and 43 one can solve for the k th harmonic current amplitude.

17

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First, the fundamental harmonic (k = 1) is considered:  !2 ∞  2q+1  X J d 2ΔV 2q+1 2  + (ωCdl ΔV ) |i(ωt)| = 2q+1 22q+1 q!(q + 1)! dV q=0 +2ωCdl ΔV

∞  2q+1  X J d q=0

2ΔV 2q+1 22q+1 q!(q + 1)!

dV 2q+1

!

cos

π 4

#1/2

(44)

where the recursive term d2q+1 J/dV 2q+1 is calculated using Eqs. 38, 39, and

of

43. It is the phasor sum of the faradaic and capacitive elements for θ = π/4. The double layer capacitance has been defined to be independent of the applied

ro

potential. It remains a linear circuit element and is not present in the higher harmonics. Therefore, the solution to the higher harmonic waveforms is given

|i(kωt)| =

∞  2q+k  X d J q=0

dV

2q+k

2ΔV 2q+k , k = 2, 3, ... + k)!

-p

by:

22q+k q!(q

(45)

re

Fig. 6 shows the fundamental, second, and third harmonic current ampli-

lP

tudes as functions of the dimensionless potential calculated from Eqs. 44 and 45. When compared with the purely faradaic case (Fig. 3), it is clear that ohmic drop causes a characteristic distortion of the waveform shape. The current am-

na

plitudes no longer exhibit an exponential dependence with potential. There is a “plateauing” of the fundamental harmonic at positive overpotentials, as well

ur

as the development of waveform extrema in the second and higher harmonics. These waveforms obey the heuristic—evident in conventional ACV theory of re-

Jo

versible charge transfer reactions—that the shape of the nth harmonic waveform is the same as the derivative of the (n − 1)th harmonic waveform [36]. The limiting current amplitude at positive potentials for the fundamental harmonic is described by a relatively simple closed-form expression: |i(ωt)| = ΔV

"

1 + Ru2

√

2ωCdl + (ωCdl )2 Ru

#1/2

,   0.

(46)

When Cdl is very small, Eq. 46 reduces to ΔV /Ru . The limiting current amplitude at negative potentials for the fundamental harmonic is the double-layer charging current: |i(ωt)| = ΔV ωCdl ,   0.

18

(47)

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Lastly, one can show that Eq. 45 reduces to the reversible solution (Eq. 21) in the absence of Ru and Cdl (see Appendix E).

3. Experimental Methods In Section 2.3, expressions were derived for the ACV harmonic waveforms of reversible electrode reactions with unity surface activity of the species R. The expressions capture the effects of ohmic drop and double layer charging,

of

two nonfaradaic distortions which are always present in ACV measurements. The solutions given by Eqs. 44 and 45 have been verified by comparison with

ro

Fourier-transformed ACV measurements of both Pb electrodeposition on liquid Pb and Cl2 evolution on graphite in molten PbCl 2 -NaCl-KCl at 700 ◦ C. This

-p

section details the experimental conditions of those measurements. Several characteristics of PbCl 2 electrolysis in NaCl-KCl make it ideal for

re

validating the theoretical solutions in Section 2: (1) Pb electrodeposition on liquid Pb in NaCl-KCl at 700 ◦ C is electrochemically reversible [37]; (2) the use of

lP

a liquid Pb electrode prevents electrocrystallization phenomena [38] and ensures unity surface activity of the reduced species on the cathode (negative electrode);

na

(3) on the anode (positive electrode) the use of a fused salt electrolyte with a single common anion ensures that the activity of Cl − is unity at the electrode surface when the standard state for Cl − is its state in equimolar NaCl-KCl at

ur

1 bar pressure [39]; (4) Cl 2 evolution on graphite in molten chlorides is known

Jo

to be kinetically facile [40, 41]; (5) both thermodynamic and transport data for molten NaCl-KCl-PbCl2 are reportedly known; and (6) the electrochemical window of NaCl-KCl is sufficiently large for Pb electrodeposition in the absence of side-reactions. 3.1. Materials The electrolyte was prepared using NaCl and KCl powders (> 99 mol.%, Sigma-Aldrich) and PbCl2 powder (99 mol.%, Fisher Scientific), the amount of which depended on the desired concentration of Pb 2+ . The powder precursors were dried separately in a vacuum oven at 200 ◦ C for a minimum of 48 hours prior to weighing. Pb shot (3 mm, 99.999 % metals basis, Fisher Scientific) was used to form the reference electrode (RE). The counter electrode (CE)

19

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was a graphite rod (> 99.997 % metals basis, 10 mm , 76 mm length, Alfa Aesar). The Pb electrodeposition measurements used a liquid Pb WE of the same type as the RE. The chlorine evolution measurements used a graphite rod WE (> 99.997 % metals basis, 3.0 mm , 38.1 mm length, Alfa Aesar). The graphite rods were washed with anhydrous ethanol then dried in a vacuum oven at 200 ◦ C for a minimum of 48 hours prior to use. Tungsten rods (> 99.95 mol.%, 1.5 mm , 508 mm length, Ed Fagan) were

of

used as electrical leads for the liquid Pb electrodes. Electrical and mechanical

connection to the graphite CE and WE was made using threaded molybdenum

ro

rods (> 99.97 mol.%, Ed Fagan, 3.2 mm , 508 mm length; and 1.5 mm ,

508 mm length, respectively). Prior to electrode assembly the tungsten and

-p

molybdenum rods were washed with acetone, gently abraded with 600 grit silicon carbide paper wetted with anhydrous ethanol, and then dried.

re

The alumina crucibles (> 99.6 mol.%, AdValue Tech: one with 500 mL capacity, 72 mm (outer), 148 mm height; two with 5 mL capacity, 19 mm (outer),

lP

26 mm height) were cleaned by immersion in boiling 3 M nitric acid solution for

15 minutes followed by washing with deionized water and heating at 1000 ◦ C for 3 hours in air. Alumina tubes (> 99.8 mol.%, Coorstek: 6.35 mm (outer),

na

3.96 mm (inner), 457 mm length, and 3.18 mm (outer), 1.6 mm (inner), 457 mm) were used as insulating sheaths for the electrode leads. The tubes were

ur

cleaned by heating at 1000 ◦ C for 3 hours in air.

Jo

3.2. Cell Design and Operation Fig. 7 shows a to-scale schematic of the electrochemical cell. The cell consisted of a primary alumina crucible (500 mL capacity) with two secondary crucibles (5 mL capacity) joined to the bottom of the primary crucible by alumina paste (940HT, Cotronics). The secondary crucibles each contained 30 g of Pb, forming the RE and WE for the Pb electrodeposition measurements. Electrical connection was made using tungsten rods sheathed in alumina tubes. Alumina paste was applied to seal the space between the tungsten rod and sheath near the end that was to be in contact with the liquid Pb pool. Electrical connection to the graphite CE and WE for the chlorine evolution measurements was made using alumina-sheathed threaded molybdenum rods. For each electrical lead, high temperature epoxy was used to seal the space between the rod and sheath 20

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at the end nearest the furnace cap. The electrical leads were held in place by Ultra-Torr fittings welded to the furnace cap. An array of baffles, through which the electrical leads ran, was connected to the underside of the furnace cap, as well as a rigid platform that extended into the furnace tube. The cell was placed in an outer crucible of quartz that rested on this platform and was held in position using a steel alignment ring. A compression clamp and silicone gasket were used to seal the furnace cap to the tube. All furnace tube components

of

were made of 304 stainless steel. Temperature measurements were made using a type B thermocouple (SP30R-010, Omega) in a closed one-end alumina sheath

ro

immersed in the electrolyte. The temperature signals were recorded with a data are accurate to within ±1 ◦ C.

-p

acquisition module (OMB-DAQ-55, Omega). The temperatures reported herein

Melting of the Pb in the secondary alumina crucibles was done prior to the

re

addition of the electrolyte precursor powders. The secondary crucibles were filled with Pb shot, and the cell was placed in the furnace assembly and purged

lP

five times by cycles of gas flow (5 % H 2 in Ar balance, Airgas) and vacuum. The assembly was then heated in a furnace (model CC12-6X12-1Z, Mellen) to 400 ◦ C for 1 hour under 300 sccm 5 % H2 in Ar balance flow.

na

The electrolyte was prepared by mixing 170 g of NaCl and 217 g of KCl with a variable amount of PbCl2 (between 0.07 g and 7.72 g) to yield, at 700 ◦ C,

ur

250 mL of equimolar NaCl-KCl with concentrations of Pb 2+ between 1.2 mM and 111 mM (the density of equimolar NaCl-KCl at 700 ◦ C is 1.58 g cm−3

Jo

[42]). The powders were transferred to the primary alumina crucible, and the assembly was immediately placed in the furnace tube. The assembly was purged five times by cycles of Ar flow (> 99.999 %, Airgas) and vacuum. The Ar was treated for water and oxygen upstream of the furnace assembly by passing it through a column of calcium sulfate (Drierite) followed by a column of copper shot heated to 500 ◦ C in an adjacent furnace (model C1-419 5X24, Mellen). Following purging, an Ar flow of 300 sccm was maintained and the assembly was heated to the setpoint temperature of 700 ◦ C. The internal temperature, as measured by the type B thermocouple, was typically within 10 ◦ C of the setpoint. After holding for one hour at this temperature, the electrical leads were lowered to their appropriate positions. Once the electrochemical measure-

21

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ments were completed, the electrical leads were withdrawn from the cell and the furnace was cooled to room temperature. 3.3. Electrochemical Measurements A three-electrode configuration was employed, consisting of either a liquid Pb or graphite WE, liquid Pb RE, and graphite CE. The reference potential was established by the reaction Pb 2+ + 2e− → Pb(l) , which is known to be

electrochemically reversible in equimolar NaCl-KCl at 700 ◦ C [37]. The bulk

of

concentration of Pb2+ was fixed by the amount of PbCl 2 added to the elec-

ro

trolyte. Post-experiment, all potentials were referenced to the hypothetical reaction Pb2+/0 in 1 M Pb2+ with equimolar NaCl-KCl, following Eq. 48. The

-p

activity data were calculated from the FTsalt database of FactSage 7.3 [43]. aPbCl2 (CPbCl2 = 1 M) RT ln ∗) nF aPbCl2 (CPbCl2 = CO

(48)

re

Edc vs. E1 M Pb2+/0 = Edc,meas −

Electrochemical impedance spectroscopy (EIS) and DC cyclic voltammetry

lP

(CV) measurements were made using a potentiostat (Reference 3000, Gamry). The uncompensated resistance, Ru , of the cell was determined from the Zim = 0 intercept of the impedance spectrum in the complex plane. For the Pb electrode-

na

position measurements, cyclic voltammograms were recorded for scan rates, v, between 10 mV s−1 and 190 mV s−1 for potentials between −0.49 V and

ur

−0.19 V vs. 1 M Pb2+/0 . Fig. 8a shows an example set of voltammograms of

∗ ◦ the Pb electrodeposition reaction for CPb C. The variation 2+ = 12 mM at 707

Jo

of the peak cathodic (negative) current with v 1/2 was used to calculate the WE surface area, following the theory of Berzins and Delahay [21] and applying the reported diffusivity [44] of Pb 2+ of 4.5 × 10−5 cm2 s−1 . The active surface area of the liquid Pb electrode was 1.9 cm2 . For the chlorine evolution measure-

ments, cyclic voltammograms were recorded for scan rates between 10 mV s −1 and 500 mV s−1 for potentials between −0.2 V and 1.3 V vs. 1 M Pb2+/0 . An

example set of voltammograms at 696 ◦ C is shown in Fig. 8b. From the relation derived by Lantelme and Cherrat (see Eqs. 8 and 9 in [23]), the variation

of the maximum anodic (positive) current (the term iM in Eq. 9 of [23]) with 1/2

0 v 1/2 was used to calculate the quantity ACCl DCl2 , where the subscript denotes 2

dissolved chlorine gas. The value of this quantity for the data in Fig. 8b is

22

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5.5 × 10−7 mol s−1/2 . In this calculation, it is assumed that, at positive overpotentials where Cl2 gas bubble nucleation occurs, γCl2 = 1 at the electrode surface. In addition, this calculation relies on a nonlinear fit to the experimental iM -v 1/2 relation due to ohmic drop correction of the positive potential scan limit. In this analysis, the standard state of dissolved Cl 2 is its state in equimolar NaCl-KCl for pCl2 = 1 bar. For the ACV measurements, an analog function generator (DS360, Stanford

of

Research Systems) was used to impose a sinusoidal potential perturbation of fixed amplitude and frequency on the cyclic potential scan of the potentiostat.

ro

The WE potential and current were recorded at 36.125 kHz using an analog-todigital data acquisition module (DT9837, Data Translation). Signal processing

-p

was done using a MATLAB script, which broadly entails the following: Fourier transformation of the current-time data, selection of the power spectrum data

re

corresponding to each current harmonic, and calculation of the inverse Fourier transform for each harmonic. Details of this procedure may be found elsewhere

lP

[45, 46].

The potential axes of the voltammograms shown herein are corrected for

4. Discussion

na

ohmic drop via the relation E = Emeas − idc Ru .

ur

4.1. Comparison Between Theory and Experiment

Jo

Fig. 9 shows a comparison of both Pb electrodeposition and Cl 2 evolution ACV waveforms with waveforms calculated from Eqs. 44 and 45. The fundamental, second, and third harmonic current amplitudes (in black) are plotted as functions of the potential referenced to 1 M Pb 2+/0 (see Eq. 48). These data were collected for an applied excitation frequency of 97 Hz and a perturbation amplitude of 100 mV. Two calculated waveforms are shown in each plot: one using the value of the uncompensated resistance Ru measured at the open circuit potential using EIS (in blue), and one using the value of Ru that closely fits the measured fundamental harmonic amplitude (in orange). Both choices provide quite similar waveforms, and the discrepancy is small. The difference may be due to the simplification of the phase angle relations employed in the derivation (see Section 2.3). For a phase angle θ between 0 and ±π/2, the uncertainty 23

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in the fundamental harmonic amplitude lies within an additive term between 0 and 2ab, where a and b represent the magnitude of the nonlinear response of the faradaic and double layer elements. It may be shown that this term is only relevant for large (relative to E 0 ) positive potentials, and this is indeed where the discrepancy can be most clearly seen in Fig. 9. Despite the relatively large perturbation amplitude used (100 mV), there was difficulty extracting low-noise third harmonic voltammograms, and the agree-

of

ment with theory there is less satisfactory, particularly in the case of Pb electrodeposition. However, the peak positions and order of magnitude of the cur-

ro

rent amplitude are appropriately reproduced. Third harmonic waveforms were not identifiable at 10 mV or 50 mV perturbation amplitudes, though the funda-

-p

mental and second harmonics for those amplitudes also agree well with Eqs 44 and 45, respectively.

re

For chlorine evolution (Fig. 9b), the ACV measurements were limited in the extent of the positive polarization to minimize the effects of bubble nucleation

lP

and detachment from the electrode surface. The results is that the experimental waveforms appear slightly truncated at the positive-going potential scan limit, when compared to those for Pb electrodeposition. However, the shapes of the

na

harmonic waveforms remain consistent with those predicted from theory. Notably, the theoretical waveforms are shifted from the experimental waveforms

ur

by about 50 mV. The reason for the discrepancy is not clear. The shift may be the result of errors in the electrode parameters used to calculate the theoretical

Jo

waveforms.

In addition, it is assumed in the derivation that the double layer capacitance Cdl is independent of potential. The consequences of this assumption with regards to the features of the waveforms are minimal insofar as the faradaic current remains much larger than the capacitive current. In Eq. 44 the double layer charging current serves as a constant background current of the fundamental harmonic and does not affect the peak potentials of the higher harmonics. Values of Cdl between 200 μF cm−2 and 550 μF cm−2 were chosen to coincide with the limiting background current at negative potentials for the fundamental harmonic amplitudes of the Pb electrodeposition reactions in Fig. 9a. These values agree well with previously reported measurements of Cdl for liquid Pb electrodes

24

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in equimolar NaCl-KCl at 700 ◦ C [47]. For the Cl2 evolution reaction in Fig. 9b, the background charging current is negligible compared to the faradaic current. Importantly, the calculated waveforms reasonably match the experimental data and reproduce the features of the fundamental and higher harmonics. In particular, the presence of ohmic drop is correlated with the emergence of current amplitude extrema. It may be said, therefore, that the uncompensated resistance is intimately connected with the characteristic features of these types

ro

∗ 4.2. Insensitivity of the Current Amplitude to CO

of

of ACV waveforms.

In Fig. 9a current harmonics are shown for three values of the bulk concen-

-p

tration of PbCl2 : 1.2 mM, 12.0 mM, and 110.5 mM. Despite there being a two orders of magnitude change in the bulk concentration of the electroactive solu-

re

tion species Pb2+ , the ACV harmonic currents are all of essentially equivalent magnitude, when plotted as functions of a common reference potential, in this

lP

case 1 M Pb2+/0 . This is in contrast to the AC voltammetry of soluble redox couples and amalgams, where the peak current amplitude of the fundamental ∗ harmonic varies linearly with CO . The insensitivity of the harmonic current

na

∗ amplitudes to CO observed in the experimental data is corroborated by the so-

lution derived in Section 2 for reversible charge transfer (Eq. 21), where i(kωt)

ur

∗ ∗ is independent of CO . A more intuitive form of Eq. 21, one which includes CO ,

may be written by defining the nondimensionalized potential  with respect to

Jo

the equilibrium potential at open circuit Eeq rather than the standard state potential of the electrode reaction E 0 :   0 nF |i(kωt)| CO 0 (Edc − E ) exp = RT γO (0, t) 2nF A(DO kω)1/2 Ik (ξ)   ∗ ∗ CO nF γO (Edc − Eeq ) = exp RT γO (0, t)

(49)

∗ is the activity coefficient of species O at its bulk concentration in the where γO

electrolyte and γO (0, t) is the activity coefficient of species O at its concentration at the electrode surface. These quantities are approximately equivalent under the conditions of the PbCl 2 electrolysis measurements. The maximum surface concentration of PbCl2 occurs at the positive potential scan limit and is approx∗ imately 200 mM when CO = 110.5 mM, as determined from the semiintegral of

25

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the DC component [48]. It may be shown that the calculated activity coefficient (FTsalt database, FactSage 7.3) varies by only a few percent between 200 mM ∗ PbCl2 and the infinitely dilute limit [43]. In general, the difference between γO ∗ and γO (0, t) becomes negligible for values of CO much greater than what may be

generated at the electrode surface under positive polarization. In Eq. 49, then, ∗ changes in the magnitude of i(kωt) due to CO are compensated by an equivalent ∗ change in Eeq . The insensitivity of i(kωt) to CO may be seen as a property of

of

the reaction rather than a consequence of nonfaradaic waveform distortion.

ro

4.3. Expression of the Second Harmonic Peak Potential

The utility of ACV harmonic waveforms lies in the ability to mathematically

-p

correlate features such as the current magnitude, potentials of the extrema points, waveform widths, etc., to reaction parameters of interest. Fig. 9 shows

re

that the solutions in Eqs. 44 and 45 may be used for quantitative modeling of electrolysis reactions. In particular, emphasis is placed on the second harmonic,

lP

which has a singular waveform peak that is given by the following linearized expression:

where

na

Epeak − E 0 = − κ=

RT ln(2Ru κ) nF

0 n2 F 2 A(ωDO )1/2 CO . γO RT

(50)

(51)

ur

κ is a term that resembles the prefactor of the fundamental harmonic i(ωt) for

Jo

reversible charge transfer (see Eq. 43). An analytic solution to the peak potential becomes cumbersome to calculate when higher order components of the recursive term in Eq. 45 are included. However, Eq. 50 remains a good approximation even for very large perturbation amplitudes, when linearizability cannot be assumed. This may be seen in Fig. 10, in which the calculated second harmonic is plotted as a function of the nondimensionalized potential for several values of the nondimensionalized perturbation amplitude nF ΔE/RT , from 0.001 (blue) to 2.0 (orange), capturing a range of small to large perturbation amplitudes. For perspective, under the conditions of the PbCl2 electrolysis measurements (n = 2, T = 700 ◦ C), the upper bound of this range is ΔE = 84 mV. The variation of the second harmonic peak potential with perturbation amplitude is shown in the inset plot. 26

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The change in peak potential, from the limit of small perturbation amplitudes to ΔE = 84 mV, is only 4 mV. It shows that the peak potential of the second harmonic is insensitive to ΔE and can be well-approximated by Eq. 50. Eq. 50 provides a path for using ACV in conjunction with a measurement of Ru to quantify electrode reaction parameters. To illustrate this point, one may approach the problem from the opposite perspective and calculate, for example, the diffusivity of Pb 2+ from the second harmonic waveforms of the Pb

of

electrodeposition measurements in Fig. 9. Using the values of Ru measured by EIS and the appropriate values of the other reaction parameters in Eq. 51, one

ro

finds calculated values of DPb2+ (in 10−5 cm2 s−1 ) of 4.9, 6.5, and 6.8, for the measurements with 1.2 mM, 12.0 mM, and 110.5 mM of PbCl 2 , respectively.

-p

These values agree reasonably well with the literature value of 4.5×10−5 cm2 s−1 [44].

re

Another example is the determination of the activity coefficient of the species O. A hypothetical electrochemical cell is proposed in which the WE reaction is

lP

electrodeposition under the constraints proposed in this work. The WE potential is referenced to a known electrode reaction, for example, the standard chlorine electrode (Cl2 /Cl− ) if the electrolyte is a molten chloride. Rearrange  0
aCl2 nF 2Ru n2 F 2 A(ωDO )1/2 CO Epeak − ΔE 0 . exp RT RT

(52)

ur

γO =

na

ment of the terms in Eq. 50 yields:

0 ΔE 0 = −Δgrxn /nF is the standard state decomposition potential of the pure

Jo

0 (T ) metal chloride salt, e.g., MeCl 2 , where Me2+ is the species O. Values of Δgrxn

are generally well-tabulated. For clarity, Eq. 52 includes aCl2 , though its value is unity for the standard chlorine electrode. If instead of the standard chlorine electrode a metal-metal chloride couple was chosen as the reference potential (e.g., Ag/AgCl), ΔE 0 becomes the difference in the standard state decomposition potentials between MeCl 2 and that metal chloride compound. The term aCl2 in Eq. 52 is then replaced with the activity of the metal chloride in the reference electrolyte.

27

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5. Conclusion Solutions of the current-potential response to a sinusoidal perturbation of arbitrary magnitude were derived for both reversible and quasi-reversible electrode reactions of the form O + ne− ↔ R, where the surface activity of R is unity and the concentration gradient of R is zero at the electrode surface. The effects of double layer charging and ohmic drop were also included. These conditions model the electrodeposition of R onto an electrode made of R, as well

of

as gas evolution in a molten electrolyte where R is the common anion (e.g., Cl2 evolution in molten NaCl-KCl). In the absence of nonfaradaic effects, the

ro

functional form of the ACV harmonic current amplitudes is exponential with positive potential. The presence of ohmic drop results in the “plateauing” of

-p

the fundamental harmonic waveform at positive overpotentials and gives rise to the evolution of peaks and troughs in the higher harmonic waveforms. An

re

expression for the second harmonic peak potential was presented that may be used to calculate reaction parameters such as diffusivity, activity coefficient, or

lP

active surface area, provided the other quantities are known. The validity of the analytic solutions was confirmed by comparison with measured fundamen-

na

tal, second, and third harmonic Fourier-transformed AC voltammograms of Pb electrodeposition and Cl 2 evolution in molten PbCl 2 -NaCl-KCl at 700 ◦ C. This work addresses a missing part of the ACV literature by providing a

ur

rigorous analytic description of the harmonic currents of electrolysis reactions. The extension of the investigative capabilities of ACV to this important class of

Jo

electrochemical reactions will benefit the study of electrolytic metal extraction processes, chemical syntheses, and molten state thermodynamics. 6. Acknowledgements This work received partial support from the US National Science Foundation under grant number 1562545, and the US Department of Energy, EERE - AMO, under award number DE-EE0008316. Appendix A. Quasi-Reversible Charge Transfer: DC Component Taking the terms of order k = 0 in Eq. 29 and substituting for g and h using Eqs. 27 and 28, respectively, the integral equation of the DC component 28

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becomes: ψ0 (t) =

k0 1/2

DO



0 CR eβ I0 (ξa ) − ∗ CO



1+

Z

t 0

ψ0 (t − u)du (πu)1/2



e−α I0 (ξc )



(A.2)

of

The Laplace transform is  0  k0 k0 CR ψ (s) β −α ψ 0 (s) = I (ξ )e − I (ξ )e − I0 (ξc )e−α 01/2 0 a 0 c ∗ 1/2 1/2 CO s sDO DO   0 CR k0 β − I0 (ξc )e−α ∗ I0 (ξa )e 1/2 CO sDO   . = k0 1/2 1/2 −α s s + 1/2 I0 (ξc )e

. (A.1)

DO

ro

This is an equation of the form 1/s1/2 (s1/2 + a), the inverse Laplace transform of which is known analytically to be exp(a2 t)erfc(at1/2 ). The inverse Laplace  !2  " #  0 0 0 k k CR β −α −α −α 1/2 − I0 (ξc )e exp  I (ξ )e t erfc I (ξ )e t . ∗ I0 (ξa )e 1/2 0 c 1/2 0 c CO DO DO (A.3)

re

1/2

DO



lP

ψ0 (t) =

k0

-p

transform of Eq. A.2 is

At equilibrium, the following equation is valid for any elementary reaction: (A.4)

na

kb CR (0, t) = kf CO (0, t)

where kf (Eq. A.5) and kb (Eq. A.6) are the forward and backward rate con-

Jo

ur

stants, respectively.

kf = k 0 e−α

(A.5)

kb = k 0 eβ

(A.6)

Combining Eq. A.4 with the Nernst equation gives a relation between the rate constants and the applied potential, as shown in Eq. A.7 [see pg.103 B+F]. 0 CO kb  = 0 e kf γO C R

29

(A.7)

Journal Pre-proof

Applying the definitions in Eqs. 9, 28, A.5, A.6, and A.7 to Eq. A.3 one finds:

of


0 ∗ I0 (ξa ) − kf CO I0 (ξc ) exp[(hI0 (ξc ))2 t]erfc(hI0 (ξc )t1/2 ) idc (t) = nF A kb CR  0  CO I0 (ξa )  1/2 ∗ = nF ADO hI0 (ξc ) exp[(hI0 (ξc ))2 t]erfc(hI0 (ξc )t1/2 ) e − CO γO I0 (ξc )  1/2  0 nF ADO CO I0 (ξa )  ∗ 1/2 − C λ exp(λ2 )erfc(λ) e = O π γO I0 (ξc ) (πt)1/2  1/2  0 nF ADO CO I0 (ξa )  ∗ e − CO H(λ) = γO I0 (ξc ) (πt)1/2 (A.8)

ro

where λ = hI0 (ξc )t1/2 . Fig. A.11 shows the variation of H with λ. Appendix B. Quasi-Reversible Charge Transfer: Fundamental Har-

-p

monic

re

The terms of order k = 1 in Eq. 29 are:

(B.1)

lP

ψ1 (t) = − (2gI1 (ξa ) + 2hI1 (ξc ) + 2hI1 (ξc )Θ0 ) sin(ωt) − hI0 (ξc )Θ1

where Θ0 is the semiintegral of the DC component calculable from Eqs. A.1 and

na

A.3. A solution of the form ψ1 (t) = A1 sin(ωt) + B1 cos(ωt) is assumed: A1 sin(ωt) + B1 cos(ωt)

ur

= − (2gI1 (ξa ) + 2hI1 (ξc ) + 2hI1 (ξc )Θ0 ) sin(ωt) − hI0 (ξc )

Z

t 0

A sin(ω(t − u)) + B cos(ω(t − u))du (πu)1/2

Jo

hI0 (ξc ) = − (2gI1 (ξa ) + 2hI1 (ξc ) + 2hI1 (ξc )Θ0 ) sin(ωt) − [(A + B) sin(ωt) − (A − B) cos(ωt)] (2ω)1/2 (B.2)

Solving for the coefficients A1 and B1 , one finds: A=

−2(1 + y) (gI1 (ξa ) + hI1 (ξc ) + hI1 (ξc )Θ0 ) 1 + 2y + 2y 2

(B.3)

−2y (gI1 (ξa ) + hI1 (ξc ) + hI1 (ξc )Θ0 ) 1 + 2y + 2y 2

(B.4)

B=

where y is given by Eq. 33. Upon re-dimensionalizing (Eq. 9), the solution to the fundamental harmonic is i(ωt) =

   1+y G(t) sin ωt + arccot y (1 + 2y + 2y 2 )1/2

where G(t) is given by Eq. 32. 30

(B.5)

Journal Pre-proof

Appendix C. Note on Quasi-Reversible Charge Transfer Incorporating Ohmic Drop and Double Layer Charging In Section 2.3, expressions for the ACV harmonic currents of reversible charge transfer reactions incorporating ohmic drop and double layer charging are derived. The same approach may be applied to quasi-reversible reactions, where instead of Eq. 41, the steady state current-potential relation J[E(t)] of

ro

of

the faradaic element is given by Eq. C.1. It is the semiintegral of Eq. A.3.   !2  " #  0  0 0 k k CR I0 (ξa )  1 − exp  I (ξ )e−α t erfc I (ξ )e−α t1/2  Θ0,QR = ∗ I (ξ ) e − 1 1/2 0 c 1/2 0 c CO 0 c DO DO (C.1)

-p

Following Eq. 43, numerical semidifferentiation of Θ 0,QR sin(ωt) is then required

re

to calculate the recursive term in Eq. 40.

Appendix D. Harmonic Decomposition

lP

The quantity sinp (ωt) may be written as a sum of linear sine and cosine

ur

na

terms using the following trigonometric identity [49]:    (p−1)/2  X  p−1 1 p   2 −q (−1) sin [(p − 2q)ωt]   q  2p−1 q=0 sinp (ωt) =     (p/2)−1   X p 1 1 p  −q p  2  + p−1 (−1) cos [(p − 2q)ωt]   2p p/2 2 q q=0

p odd

p even (D.1)

Jo

The relation between the power p and the harmonic index k is represented

in Table D.3. Each harmonic is comprised of an infinite number of components. The order of the components for the k th harmonic ascends following p = 2q + k for q = 0, 1, 2, ..., with the first-order component corresponding to q = 0, the second-order component to q = 1, etc. Therefore, the pth component of the k th harmonic will correspond to the particular term, denoted ip,k , in the sum defining i(kωt) for which the following equality is valid: q=

p−k . 2

31

(D.2)

Journal Pre-proof

Combining Eqs. D.1 and D.2 with Eq. 36 gives ip,k , for p ≥ k:  k−1 1     p (−1) 2 +p sin (kωt)  p p p−1 ΔV d K dΔV (t) d J 2       ip,k = + k 1 1 p p−k p+k  dV p dV p dt  ! ! + p−1 (−1) 2 +p cos (kωt)  p 2 2 2 p/2 2 (D.3)

ik=0 (t) =

ΔV p

∞ X

dp J dV p

+

dp K dΔV (t) dV p dt



2p (p/2)!2

.

-p

p=k steps of 2



ro

steps of 2

of

Summing over all p ≥ k in steps of 2, one finds  p  k−1 d J dp K dΔV (t)  ∞ ΔV p dV X (−1) 2 +p sin (kωt) p + dV p dt     i(kωt) = k p+k 2p−1 p−k ! ! (−1) 2 +p cos (kωt) p=k 2 2

k odd k even

k odd k even

(D.4)

ik=0 (t) =

dV

∞  X

K dΔV (t) d dV 2q+k dt

d2q K dΔV (t) d2q J + dV 2q dV 2q dt





2ΔV

 3k−1 (−1) 2 sin (kωt)

2q+k

22q+k q!(q

ΔV 2q+k . 22q (q!)2

+ k)! (−1)

Jo

Table D.3: Relation Between p and k p

i(kωt)

0

idc

X

i(ωt) i(2ωt)

3k 2

cos (kωt)

(D.5)

ur

q=0

+

lP

q=0

2q+k

2q+k

na

i(kωt) =

∞  2q+k X J d

re

Lastly, a more natural summation index may be used:

1

2

3

X X

4 X

X X

i(3ωt)

X

X

X X

X

i(5ωt)

6

X

X

i(4ωt)

5

X X

i(6ωt)

X ...

32

k odd k even

Journal Pre-proof

Appendix E. Recovery of the Reversible Solution in the Absence of Ohmic Drop Without ohmic drop, Eq. 39 becomes d2q+k Θ0 d2q+k Θ0 = = 2q+k dV dE 2q+k



nF RT

2q+k

1/2

0  nF ADO CO e . γO

(E.1)

Substituting for the recursive term in Eq. 40 and observing the definition of Ik

-p

ro

of

(Eq. 14) as well as the fact that ΔE = ΔV , it follows that, for a given k:  3k−1 2q+k ∞  1/2 0  X (−1) 2 sin(kωt) 2q+k nF ADO CO e nF ΔV ΔΘk (t) = RT γO 22q+k−1 q! (q + k)! (−1) 3k 2 cos(kωt) q=0  3k−1 1/2 0  e Ik (ξ) (−1) 2 sin(kωt) k odd 2nF ADO CO = 3k  γO k even (−1) 2 cos(kωt)

re

(E.2)

where ΔΘk (t) denotes the semiintegral form of Δik (t). Upon semidifferentia-

k odd k even.

ur

na

lP

tion, the purely faradaic solution (Eq. 21) is recovered:   3k−1 π  2 1/2 0  sin kωt + (−1) 1/2  2nF ADO CO (kω) e Ik (ξ) 4 i(kωt) =   3k  γO (−1) 2 +1 sin kωt − π 4 List of Symbols

A Cdl

Activity of species O

Jo

aO

Surface area (m2 ) Double layer capacitance (F m−2 )

CO (0, t)

Concentration of species O at the electrode surface (mol m −3 )

0 CO ∗ CO

Concentration of species O in the reference state (mol m −3 )

DO

Diffusivity of species O (m2 s−1 )

Concentration of species O in the bulk solution (mol m −3 )

Continued on next column

33

(E.3)

k odd k even

Journal Pre-proof

Continued from previous column E

Potential at the WE surface (V)

E0

Standard state potential of the electrode reaction (V)

Eeq

Equilibrium potential at open circuit (V)

ΔE

AC perturbation amplitude at the WE surface (V)

F

Faraday constant (96485 C mol −1 )

g

0 eβ /D ∗ k 0 CR O CO (see Eq. 27)

h H(λ)

1/2 k0 e−α /DO (see Eq. 28) π 1/2 λ exp(λ2 )erfc(λ) (see Eq.

i

Total current (A)

Δi

Total AC current (A)

id

Diffusion-limited (Cottrell) current (A)

Ik

Modified Bessel functions of the first kind

J

Nonlinear i-V relation of the faradaic circuit element (see Eq. 34)

k

Harmonic number

kb

Backward reaction rate constant (m s −1 )

kf

Forward reaction rate constant (m s −1 )

k0

Standard heterogeneous rate constant (m s −1 )

K

Nonlinear i-V relation of the capacitive circuit element (see Eq. 34)

n

Moles of electrons per mole of species O in the electrode reaction

R

Gas constant (8.314 J mol−1 K−1 )

Ru

Uncompensated resistance of the cell (Ω)

T

Temperature (K)

V

Potential measured at the potentiostat (V)

ΔV

AC perturbation amplitude at the potentiostat (V)

 a

of

ro

-p

re

lP

na

ur

β γO

A.8)

Greek

Charge transfer coefficient 1−α

Jo

α

1/2

Activity coefficient of species O

nF (E − E 0 )/RT

(1 − α)nF (E − E 0 )/RT Continued on next column

34

Journal Pre-proof

Continued from previous column c

αnF (E − E 0 )/RT

Θ

Semiintegral of the dimensionless current (see Eq. 24)

κ

0 /γ RT (see Eq. 51) n2 F 2 A(ωDO )1/2 CO O

λ

hI0 (ξc )t1/2 (see Eq. A.8)

ν

Potential scan rate (V s−1 )

ξ

Dimensionless perturbation amplitude nF ΔE/RT

φ(t)

∗ (s−1/2 ) i(t)/nF ADO CO

ω

Angular excitation frequency (rad s −1 )

Jo

ur

na

lP

re

-p

ro

of

1/2

35

Journal Pre-proof

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Press, San Diego, 2000.

40

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a

10

2

A

10

5 u

1

of

10

ro

b

-p

5

lP

re

2

10

3

A

0

ur

Jo

3

5

na

10

4

A

0 c 10

0

5

0

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0

Figure 6: Calculated a) fundamental, b) second, and c) third harmonic current amplitudes as functions of dimensionless potential, incorporating ohmic drop and double-layer charging. The separate curves for each harmonic are calculated for a specific value of Ru , from 1 Ω (blue) to 10 Ω (gold) in steps of 1 Ω. The first four components (q = 0, 1, 2, 3) were sufficient for convergence of the recursive terms in Eqs. 44 and 45. The waveforms were calculated for the following reaction parameter values: nF/RT = 25 V−1 , ΔV = 0.1 V, ω = 200π s−1 , 1/2

0 Cdl = 200 μF, nF ADO CO /γO = 1.6 × 10−3 C s−1/2 .

41

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lP

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-p

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Figure 7: To-scale schematic of the apparatus. To the right is an illustration of the electrochemical cell. (1) Tungsten electrical lead for the liquid Pb WE, (2) molybdenum electrical lead for the graphite WE, (3) tungsten electrical lead for the liquid Pb RE, (4) type B thermocouple, (5) molybdenum electrical lead for the graphite CE, (6) furnace cap, (7) high temperature silicone gasket, (8) copper cooling coils, (9) gas inlet tube, (10) baffles, (11) support rod, (12) primary alumina crucible, (13) alignment ring, (14) quartz outer crucible, (15) graphite WE, (16) electrolyte, (17) liquid Pb electrode, (18) graphite CE, (19) gas outlet.

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b

6 6

1

10

2

2

of

mV s

1 12

4

0

0

2

10

20

mV s

1 12

dc

ro

15

A

10

2

5

dc,max

10 0 0

10

5

4

2

0

1

lP

190 mV s

5

0

re

10

0.4

0.2

0

2

1.0

1.2 V vs. 1 M Pb2

na

V vs. 1 M Pb2

10

-p

dc

10

2

A

dc,peak

10

500 mV s

A

4

2

A

a

Figure 8: a) Cyclic voltammograms of the Pb electrodeposition reaction at var-

ur

ious scan rates (v), between 10–190 mV s−1 . The variation of the peak cathodic (negative) current (idc,peak ) with the square root of the scan rate is shown in the

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inset plot. idc,peak is defined for the start of the potential sweep being the open circuit potential. The slope of the linear fit is described by the Berzins-Delahay equation [21], from which the surface area of the WE was calculated. b) Cyclic voltammograms of the chlorine evolution reaction at various scan rates (v) between 10–500 mV s−1 . The inset plot shows the variation of the maximum anodic (idc,max ) current with the square root of the scan rate. The solid curve is the analytic relation derived by Lantelme and Cherrat [23], from which the 1/2

0 unknown parameter ACCl DCl2 was calculated. The curve is slightly less than 2

linear due to ohmic drop correction of the positive potential limit.

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10

1

3

u

12.0 mM PbCl2

EIS fit

u u

0.35 0.39

110.5 mM PbCl2

EIS fit

u u

0.28 0.36

EIS fit

2

u

4

u

3

A

re

2

0 0.4

lP

1

0.2

0

2

1

A

0 4 3 2 1 0 1.0

1.2

1.4

V vs. 1 M Pb2

0

na

V vs. 1 M Pb2

2

3

-p

3

3

10

ro

1

3

of

2

10

2

A 2

10 2 A

EIS fit

0

0 3

0.19 0.22

1

0

10

2Cl

2

1

3

2e

soln

10 mM PbCl2 5 A

A

u

0.29 0.38

Cl2

b

Pbl

1

1.2 mM PbCl2

4

2e

10

Pb2

a

Figure 9: Comparison of the measured and calculated fundamental (top row), second (middle row), and third harmonic

ur

(bottom row) current amplitudes. The experimental data (in black) were recorded for a 97 Hz excitation frequency and a 100 mV perturbation amplitude at a linear potential sweep rate of 10 mV s −1 . The DC potential program

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was initiated at the open circuit potential (OCP), sweeping to the lower potential limit then to the upper potential limit and back, for two cycles, ending at the OCP. Both the forward sweep (positive–to–negative potentials) and the reverse sweep are shown for the first cycle. The waveforms calculated from Eqs. 44 and 45 are obtained either with the value of the uncompensated resistance Ru measured at the open circuit potential using EIS (in blue) or using the value of Ru that best fits the measured fundamental harmonic amplitude (in orange). The first five components (q = 0, 1, 2, 3, 4) were sufficient for convergence of the calculated solutions. a) Pb electrodeposition on liquid Pb with the following concentrations of PbCl 2 : 1.2 mM, 12.0 mM, and 110.5 mM. The analytic waveforms were calculated for the following reaction parameter values: A = 1.9 cm2 , n = 2, DPb2+ = 4.5 × 10−5 cm2 s−1 [44], ΔE = 100 mV,

ω = 194π rad s−1 , Cdl = 480, 230, and 550 μF cm−2 (for the 1.2 mM, 12.0 mM, and 110.5 mM measurements,

respectively), T = 698, 707, and 698 ◦ C, Eeq = −0.289, −0.192, and − 0.097 V vs. 1 M Pb2+/0 (see Eq. 49). b) Cl2 evolution on graphite. The analytic waveforms were calculated for the following reaction parameter values: 1/2

0 ADCl2 CCl = 5.5 × 10−7 mol s−1/2 (see Section 3.3), n = 2, ΔE = 100 mV, ω = 194π rad s−1 , T = 696 ◦ C, 2

E 0 = 1.37 V vs. 1 M Pb2+/0 .

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mV 2, 700 C 20 40 60 80

0

5

2.0

1.18

of

mV

53

0.0 0.5 1.0 1.5 2.0

re

2

-p

2

1.26

ro

0

52

peak

peak

10

51

1.22

3

lP

1

8

6

na

0

0.001

0

4

3

A

50

4

2

0

0

ur

Figure 10: Calculated second harmonic current amplitude (Eq. 45) as a function of nondimensionalized potential for a range of nondimensionalized perturbation

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amplitudes nF ΔE/RT , from 0.001 (blue) to 2.0 (orange). The inset plot shows the peak potential Epeak as a function of the perturbation amplitude. On the inset plot dimensionalized top and right axes for ΔE and Epeak , respectively, are provided for the conditions of the PbCl 2 electrolysis measurements (n = 2, T = 700 ◦ C). The waveforms were calculated to five terms (q = 4) using the following values of the reaction parameters: Ru = 1 Ω, nF/RT = 25 V−1 , 1/2

0 ω = 200π s−1 , nF ADO CO /γO = 3.0 × 10−3 C s−1/2 .

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2

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ro

1.0

re

-p

0.8

lP

0.6

na

0.4

0.2

ur

0.0

3

10

2

10

1

Jo

10

100

101

Figure A.11: Variation of H with λ (see Eq. A.8).

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102

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Highlights • A model of alternating current voltammetry signals for electrochemical reactions involving insoluble species at fixed activity is presented. • The model shows that incorporating the effect of the ohmic effect is essential in order to reproduce characteristic peaks and trough of the current signals • Experimental validation of the model using liquid metal electrodeposition in molten

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ur

na

lP

re

-p

ro

of

salts shows good agreement.

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Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11