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Linear sweep and cyclic voltammetry of porous mixed conducting oxygen electrode: Formal study of insertion,
diffusion and chemical reaction model
C. Montella, V. Tezyk, E. Effori, J. Laurencin, E. Siebert
To cite this version:
C. Montella, V. Tezyk, E. Effori, J. Laurencin, E. Siebert. Linear sweep and cyclic voltammetry of porous mixed conducting oxygen electrode: Formal study of insertion, diffusion and chemical reaction model. Solid State Ionics, Elsevier, 2021, 359, pp.115485. �10.1016/j.ssi.2020.115485�. �hal-03054078�
Linear sweep and cyclic voltammetry of porous mixed conducting oxygen electrode: Formal study of insertion, diffusion and chemical reaction model
C. Montella (a) V. Tezyk (a), E. Effori (b), J. Laurencin (b), E. Siebert (a*)
(a) Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, Grenoble INP, LEPMI, 38000 Grenoble, France
(b) Univ. Grenoble Alpes – CEA/LITEN, 17 rue des Martyrs, 38054, Grenoble, France
Abstract
The classical 1D continuum model of oxygen exchange in a porous mixed conductor (bulk path) is presented in terms of Linear Sweep Voltammetry (LSV) and Cyclic Voltammetry (CV). We show that, from a formal point of view, it corresponds to an insertion, diffusion and chemical reaction mechanism. Some general rules for LSV/CV curves are established. Using a dimensionless description, we evidence eight different limiting behaviors of the electrochemical model, which depend on four dimensionless parameters. For an almost full insertion level at equilibrium, these parameters can be reduced to only two, i.e. the dimensionless electrode thickness L and the dimensionless chemical rate constant . We show that the resulting zone diagram, plotted using the representation log L versus log , is useful to predict the possible sequences of model behaviors as a function of the potential sweep rate and the electrode thickness. This theoretical analysis is applied to a porous LSCF oxygen electrode under air at high temperature.
Keywords: Oxygen electrode, Mixed conductor, Solid-state diffusion, Linear sweep voltammetry, Cyclic voltammetry, Zone diagram.
* Corresponding author:
E-mail address: Elisabeth.siebert@lepmi.grenoble-inp.fr
Address: LEPMI – 1130 rue de la piscine – BP 75 - 38402 Saint Martin d’Hères cedex – France
1. Introduction
Solid Oxide Cells (SOCs) are regarded as a promising technology for the reversible gas to electricity conversion as well as for the energy storage. These electrochemical devices, Solid Oxide Fuel Cells (SOFC) or Solid Oxide Electrolysis Cells (SOEC), can reach high conversion efficiencies due to their high operating temperature [1,2].
Recent trends in research and development of SOCs are towards a decrease to lower operation temperature in the range 600°C to 700°C, by using of mixed ionic electronic conductor (MIEC) as the oxygen electrode material. Nowadays, MIEC oxygen deficient perovskite materials such as La0.6Sr0.4Co0.2Fe0.8O3-δ (LSCF) are employed with a good trade-off between electrochemical performances and long-term stability. Indeed, those compounds exhibit high electronic conductivity by holes and a non-negligible ionic conduction by oxygen vacancies [3–6], leading to good electrochemical performances at intermediate temperature [7–11].
Several models for the Oxygen Electrode Reaction (OER) have been proposed for the porous MIEC [12–19] deposited on Ceria doped Gadolinium Oxide (Ce1-xGdxO2-δ - CGO) as electrolyte or electrolyte barrier. Among the most cited works is the so-called ALS model developed by Adler, Lane and Steele in 1996 [12]. This model consists in a 1D continuum approach taking into account the bulk path only, described as oxygen exchange between the gas phase and the MIEC, diffusion of oxygen vacancies in the MIEC electrode and oxygen gas diffusion both in the porous layer and in a stagnant layer above the electrode. Typically, a simplified Gerischer impedance is achieved for the impedance diagram at OCV by considering a porous electrode of infinite thickness and co-limitation by the surface reaction and the solid-state diffusion in the MIEC, with no limitation by oxygen diffusion in the gas phase [12,20]. Using the same model, a Finite- Length-Gerischer (FLG) impedance was derived by Nielsen for a finite-length porous MIEC
electrode [21]. In both cases, the solution corresponds to the formal treatment of the ac response of an insertion, diffusion and chemical reaction process with 1D geometry.
The Linear Sweep Voltammetry (LSV) and Cyclic Voltammetry (CV) methods are extensively used for the study of insertion compounds [22–27]. These electrochemical methods allow studying a material in a large potential and time-scale domain, giving insights on the thermodynamic and kinetic behaviors of the insertion reaction. Moreover, they are very sensitive to structural variation in the insertion compound. Recently, LSV and CV have been proposed to study the OER on MIEC [28–34] as this non-linear response can provide valuable information and complete the ac linear response. However, a proper derivation of the response of an insertion, diffusion and chemical reaction model to LSV/CV is not available in the literature, to the best of our knowledge. The response of such a model can be calculated numerically, e.g. using finite difference (FD) methods [35]. Although FD methods should provide accurate numerical solutions to the problem under consideration, they are not well suited to formal analysis because some limiting situations, for whether diffusion or chemical reaction limits the OER kinetics, would be less easily predictable only looking at numerical current data.
The present work is included in a general trend dealing with the theoretical responses of porous MIEC to LSV and CV. In this paper, analytical LSV/CV models for the insertion, diffusion and chemical reaction mechanism are reported for the first time, under the conditions corresponding to the MIEC oxygen electrode in SOCs. A convolution integral solution that applies to both LSV and CV techniques is presented. It can be easily implemented in classical mathematical software for computation purpose. Then, closed-form expressions for the linear potential sweep voltammograms are derived under many limiting conditions, depending on the model parameter values.
In a general way, analysis of the limiting behaviors of electrochemical reactions is based on the construction of the so-called zone diagram. As far as we are concerned with the LSV/CV techniques in this work, the appropriate zone diagram should describe the operating conditions for observation of the possible voltammogram shapes as a function of kinetic and mass-transport dimensionless parameters. Zone diagrams were popularized by Savéant et al. [36–38], reviewed by Bard and Faulkner [39] and used later by other authors in the eighties [40–42] and more recently [43,44]. Considering the reaction studied in this work, the analytical formulations for the LSV curves will be used first to find the parameters values domains leading to some simplification of the model response, and next to visualize graphically such domains in a zone diagram. Finally, the zone diagram will be applied to predict: (i) the possible sequences of limiting behaviors of the electrochemical model at increasing values of the potential sweep rate or the electrode thickness, and (ii) the behavior of a typical porous LSCF oxygen electrode under conditions similar to those of our previous experimental work [34]. The list of abbreviations and symbols employed in this article is addressed in Appendix A.
2. Mathematical model
MIEC porous electrodes are inherently three-dimension (3D) objects including several interfaces such as MIEC/electrolyte where oxygen ion transfer occurs and MIEC/gas where neutral oxygen is exchanged from the gas phase as depicted in Fig. 1(a). In this microscale description, the oxygen transport within the MIEC is characterized by the chemical diffusion coefficient Dchem, and the global chemical surface exchange rate constant kchem, which reflects heterogeneous chemical kinetics rate in m.s–1. As first suggested by Adler [12] and then used by others [16–18], 3D geometry can be approximated by a 1D simplified model which treats the complex microstructure of a porous electrode in a homogenized way as depicted in Fig. 1(b). In this
macrohomogeneous approach, the electrode can be represented by a homogeneous film containing oxygen ion in the solid phase, denoted , and oxygen in the gas phase, formally denoted O (i.e. ), which is on one side connected to an oxygen conducting solid electrode (x = 0) and on the other side to the ambient, with fixed PO2 at x = l. It can be noticed that the chemical exchange rate constant in the macroscale description, denoted k hereafter, is an effective rate constant, which has the dimension of a homogeneous chemical reaction rate constant (s–1). In the same way, the bulk diffusion is characterized by an effective diffusion coefficient, denoted D hereafter. This point will be developed in Section 6.3. The conditions where the macrohomogeneous approximation applies without loss of accuracy were determined in Ref. [45,46] by comparing with results obtained by finite element analysis employing various 3D representations of the electrode microstructure.
Consequently, from a formal point of view, the OER mechanism for the macrohomogeneous electrode can be simply represented by: (i) an insertion reaction at the electrolyte/electrode interface (x = 0) and (ii) a solid-state diffusion process coupled to a chemical reaction within the MIEC electrode. It is worth noticing that this reaction mechanism could also be applied (i) to an insertion reaction with trapping [47] in Li batteries provided that the occupation level of trapping sites is very low, as well as (ii) to a redox reaction leading to electron injection in a solid oxide electrolyte provided that the electrodes are ionically blocking and the electronic conductivity of the electrolyte is not too high [48].
As depicted in Fig. 1, the electron transfer reaction at the electrode surface (x = 0) is given by:
(1)
In this formal equation, R stands for the reduced species, with unity activity in the solid
electrolyte (i.e. O2-), is the oxidized species, corresponding to the inserted species in the electrode material (i.e. O2-), and – is the free site for insertion (i.e. oxygen vacancies), whose concentrations satisfy the equality , at each abscissa 0 ≤ x ≤ l and any time instant t ≥ 0 , with cmax being the maximum concentration of free sites depending on the material structure.
Figure 1
Because of the high-temperature range explored, the electrochemical reaction can be treated as kinetically reversible (quasi-equilibrium conditions) and, therefore, it can be characterized by its formal potential E°. In addition, Langmuir isotherm conditions are assumed to govern the insertion process, which means that the number of electronic states is high and that the point defects in the MIEC are distributed randomly on each lattice site and are described by the ideal solution approximation (no interaction between inserted ions). It follows from this assumption that the thermodynamic factor is equal to 1, which is a simplification in a solid where the thermodynamic factor can be high and a function of the oxygen partial pressure in the case of a mixed oxygen electronic conductor. Consequently, the electrode potential, E(t), is linked to the interfacial concentration, , through the quasi-equilibrium relationship:
with (2)
Here where F is Faraday's constant, R is the ideal gas constant, Tabs is the absolute temperature and z is the number of electron exchanged in the electrochemical step, Eq. (1).
Diffusion of species , with constant diffusion coefficient D, is restricted within the electrode
material of length l, that is to say that its diffusion flux, , is null at the abscissa x = l. In addition, the diffusion process is coupled to the homogeneous chemical reaction in the electrode material:
(3)
where O is assumed to have a constant concentration, denoted . Hereafter, kf and kb are the forward and backward rate constants of the chemical reaction, Eq. (3), respectively.
In the present mathematical model, the Ohmic drop in the electrolyte and the electrode is omitted and the double-layer charging effects are disregarded. Further simplifications are possible, first introducing the insertion level, , next using the equilibrium condition for the chemical reaction and finally using the notations, for the chemical reaction rate constants, and for deviation of the insertion level from its initial (equilibrium) value. Indeed, equilibrium conditions are assumed to occur at the initial time, i.e. .
Using the above simplifications and notations, the mass transport equation for planar diffusion of species , coupled to the chemical reaction in the electrode, takes on the following formulation:
(4)
Kinetic reversibility for the electron transfer reaction at x = 0 provides the first boundary condition for mass transport:
with (5)
where is the dimensionless notation used for the electrode potential:
(6)
and is the potential-dependent function given by:
(7)
Restricted diffusion at x = l provides the second boundary condition:
with (8)
The Faradaic current density is derived from the interfacial diffusion flux, using the continuity equation between diffusion flux and reaction rate at x = 0, as:
with (9)
3. Steady-state solution
Steady-state conditions, at any constant electrode potential E, and therefore at any constant ξ value in non-dimensional form, are achieved setting , ,
, and removing the time variable in Eqs. (4)-(9). This leads to:
(10)
where m is the mass transport constant under steady-state conditions, given by:
(11)
and and are the steady-state versions of Eqs. (6) and (7), respectively. Derivation of the above equations is addressed in Section B of the additional document joined to this article.
The shape of the steady-state polarization curve is that of the potential-dependent function , with zFmcmax being a scaling factor as illustrated in Fig. 2. The Faradaic current density is null at and two limiting current density values are predicted in the anodic (subscript ‘a’) and cathodic (subscript ‘c’) directions, respectively, as:
and
(12)
Note that these limiting steady-state current densities correspond to a full insertion level ( in the anodic direction and to an empty one ( in the cathodic direction.
Figure 2
4. Solutions for LSV/CV
In this work, we are mainly concerned with the model responses to LSV and CV, both techniques running at the same potential sweep rate v. In what follows, we tacitly assume that each potential sweep starts from the equilibrium potential in the reduction (cathodic) direction. The input signal for LSV experiments is a single potential sweep:
with (13)
The simplest situation for CV experiments results from the use of the triangular potential signal:
with (14)
where denotes the reversal potential and is the time where the potential sweep is reversed. Generalization to bi-triangular potential signals and more complex cyclic signals is straightforward.
4.1. Relevant integral equation
The integral equation (IE) method pioneered by Nicholson and Shain [49], extensively used by Savéant [50], and well formalized in the recent textbook by Bieniasz [51], provides the Faradaic current density j(t) as the solution of the following IE for the electrochemical model studied in this work (see the derivation in Section C of the additional document joined to this article):
(15)
In Eq. (15), (*) is the usual symbol for convolution, , and denotes the inverse Laplace transform operator. The term is the so-called mass transfer function [52-54] defined as , where is the Laplace operator and s is the Laplace variable. The function is the IE kernel in Eq. (15) and the function is given by Eq. (7) where the dimensionless potential signal depends on the controlled-potential technique employed, according to Eqs. (6) and (13) or (14) for LSV/CV, respectively.
In the more general way [52–54], the mass transfer function depends on the electrode geometry (here planar geometry), the mass transport process (here diffusion coupled to the homogeneous chemical reaction) and the boundary condition away from the electroactive interface (here restricted diffusion at the abscissa x = l). The relevant formulation for in this work is:
(16)
Knowing that the inverse Laplace transform of is obtained from the theta-3 function [55] as , and that the inverse Laplace transform of M(s) can be
derived from Eq. (16) using the inverse form of first shift theorem, Eq. (15) can be written as:
(17)
This equation provides the relevant IE formulation for the electrochemical model studied in this work. The IE can be solved numerically to compute as a function of the applied potential, using Eqs. (6) and (7), as depicted in Appendix B. The corresponding algorithms are denoted IE- LSV/CV for the LSV/CV methods, respectively. The accuracy of both algorithms is evaluated in Section E of the additional document joined to this article.
4.2. Explicit solutions
4.2.1. General case (LSV/CV)
Starting from Eq. (15) and using the Laplace transform method, an explicit solution can be derived for the model response to LSV/CV as the following convolution integral (see the derivation in Section D of the additional document joined to this article):
(18)
Here, is obtained from Eq. (16). The relevant convolution integral solution (CIS) for LSV/CV is derived in Appendix C as:
(19)
where . This leads to the general CIS formulation:
(20)
where the function is given by Eq. (7). The theta-2 function in Eq. (20) can be evaluated at any l and t values from two infinite series, at least one of which is always rapidly convergent [55]. Then, the Faradaic current density j(t) is evaluated numerically from Eq. (20) as indicated in Section F of the additional document joined to this article. It can be noticed that this explicit solution applies irrespective of the initial (equilibrium) value of electrode potential, and that it is valid for LSV and CV as well. Moreover, its more attractive feature is that all the limiting behaviors of the electrochemical model investigated by LSV/CV can be readily predicted from simplified expressions of (see Eq. (16)) and of its inverse Laplace transform. This point is addressed in Section 5.3.
4.2.2. Particular case ( )
In classical MIEC electrode materials such as LSCF, the oxygen vacancies concentration is low at equilibrium under air. It means that the initial concentration of species in the electrode material is almost equal to cmax. It follows from Eq. (2) that the dimensionless electrode potential
eq is very large, and infinite at the limit where the initial concentration is equal to cmax.
LSV: An infinite series solution for the Faradaic current density can be derived if considering a single potential sweep in the reduction direction. Infinite series solutions for the LSV responses of electrochemical reactions were first derived by Reinmuth [56], Nicholson and Shain [49], and Keller and Reinmuth [57]. Such solutions apply for example to the reversible one-step electrochemical reaction, so-called Er reaction [39], possibly coupled to first-order (or pseudo- first order) homogeneous chemical reactions in the electrolyte. An extended bibliography is available from Bieniasz textbook [51]. In a recent work of one of the authors [58], it was shown that, provided that the dimensionless initial potential is infinite, the Faradaic current density in
the direction of reduction, denoted in this text by a bullet superscript as , can be derived from Eq. (18) as the following infinite series involving the mass transfer function M(s) where the Laplace variable s is replaced by n times the product zfv:
(21)
Using the relevant expression for given by Eq. (16), we get the infinite series solution (ISS) for the LSV technique:
(22) Main advantage of using Eq. (22) is that the infinite series turns out to analytical expressions of the model response to LSV in many limiting situations, depending on the model parameters values. This point is addressed in Section 5.4. It must be emphasized that numerical computation of the Faradaic current density from Eq. (22) was disregarded in this work due to the high computational time required in its divergence domain (the use of an appropriate non-linear sequence transformation together with high computation accuracy is necessary at < 0), and because of the possibility to use an equivalent integral formulation as shown hereafter.
Indeed, in the previous work [58], three integral representations of Eq. (21) have been derived using respectively Abel, Lindelöf and Euler-Ramanujan summation formulae [59–61]. The three formulae enable us to transform the ISS into complex integrals over an infinite interval. Using Lindelöf summation, Eq. (21) yields the following integral solution (IS) for the Faradaic current density [58]:
(23)
In the above equation, ω is the dummy variable of integration and denotes the potential- dependent function in Eq. (24) where , is the real part of the complex number between square brackets, and the Laplace variable s is replaced by (iω +1/2)zfv in the mass transfer function taken from Eq. (16):
(24)
The highly accurate values of current density calculated from Eqs. (23) and (24) have been used in this work as benchmark data to check the convergence of IE-LSV algorithm as a function of the computational time increment (see Section E of the additional document joined to this article).
CV: The formulation given in this Section concerns the first forward-backward potential sweep for cyclic voltammograms in the special case where the dimensionless reversal potential ξrev tends towards minus infinity. Then, the CV curve is modeled by the bipartite function:
(25)
where the subscripts f and b refer to the forward and backward potential sweeps, respectively, and for CV is the same as for LSV in Eqs. (22) and (23). The bipartite formulation in Eq. (25) is a generalization of Nicholson and Shain conjecture [49], who stated that “by using for the base line the forward sweep curve which would have been obtained if there had been no change in direction of potential sweep, all the anodic curves are the same, independent of switching potential”. Thus, Eq. (25) provides an approximating model for the CV technique.
Starting from the ISS derived for LSV in Eq. (22), and setting , Eq. (25) should
provide an infinite series approximation to the cyclic voltammogram if the dimensionless reversal potential is negative enough. In the same way, the IS derived for LSV in Eq. (23) can be substituted for in Eq. (25) to get an accurate integral formulation for this voltammogram.
5. General and limiting conditions. Zone diagram 5.1. Dimensionless variables and parameters
The goal in this Section is to determine simplified expressions for the model response to LSV/CV resulting from limiting conditions on the parameter values used for {l, D, k, Tabs, v, Eeq, Erev} in the electrochemical model. Because of the number of parameters, i.e. 6 or 7 parameters depending on the LSV or CV technique employed, it is useful to introduce dimensionless notations. In this approach, the variables and parameters t, l, E, k, s are replaced by T, L,
respectively, while the mass transfer function M and the Faradaic current density j are replaced by M* and The relationships between the natural and dimensionless variables/parameters of the electrochemical model are given in Table 1. The dimensionless model responses to LSV/CV take on the formulations reported in Appendix D. Using the change of variables in Table 1, the CIS is expressed from Eq. (18) as follows:
(26)
Table 1. Relationships between the natural and dimensionless variables/parameters of the electrochemical model.
Variable/Parameter Natural Dimensionless Relationship
Time t T
Electrode thickness l L
Electrode potential E
Chemical rate constant k
Laplace variable s
Mass transfer function M
Faradaic current density j
5.2. General rules for LSV/CV curves
Two general rules apply to the LSV/CV curves in the general case: (i) no voltammetric peak is observed when λ ≥ 1, (ii) the current density in the reduction direction tends towards its limiting steady-state value at large enough T values, i.e. at negative enough ξ values, as derived in Appendix E.
A third rule can be established when the dimensionless reversal potential for CV is located on the limiting steady-state current plateau ( ), and the dimensionless initial potential satisfies the condition . This rule results directly from Eq. (D.11) in Appendix D, which simplifies, for the backward potential sweep, to:
(27)
It results from Eq. (27) that: (i) the forward (cathodic) and backward (anodic) voltammetric peak
potentials, observed when λ < 1, have opposite values in dimensionless notation, i.e. . Turning back to the natural variables, the following equation enables the formal potential to be determined as in this case, (ii) the dimensionless anodic and cathodic peak currents satisfy the equality , which can be written in the equivalent form,
, using the natural variables. Thus, if the backward sweep current is measured from the steady-state current plateau chosen as the base line, the ratio of anodic and cathodic peak currents should be equal to unity in absolute value. The above rules are illustrated in Fig. 3 where a dimensionless cyclic voltammogram has been plotted as an example.
Figure 3 5.3. Limiting behaviors for LSV/CV (general case)
Using the non-dimensional description, the different limiting behaviors of the electrochemical model depend only on four parameters, i.e. , L, eq and rev.. Using the Laplace transform method, the dimensionless CIS in Eq. (26) can be rearranged to give the relevant expression:
(28)
where the function g(T) is the following inverse Laplace transform (Appendix D):
(29)
This equation takes on simplified formulations depending on the values of the dimensionless parameters λ and L. Using inverse Laplace transformation of these formulations, limiting expressions of the dimensionless current density ψ(T) can be derived from Eq. (28), as reported in Table 2.
Table 2. Simplified (closed-form or convolution integral) expressions for the dimensionless current density
as a function of the limiting conditions on the dimensionless parameters λ and L. The theoretical formulations apply to both LSV and CV techniques, irrespective of eq and rev values. The zone numbering refers to the zone diagram plotted in Fig. 4. The potential-dependent functions h, h’ and g are defined in the text for LSV or CV respectively.
Zone Conditions
1 2 3 or
4 and L (*) and
and L (*)
and L
( ) At not too small T values
Zone 1 in Table 2 pertains to very low λ values, i.e. no influence of the chemical reaction on the model response due to a very low rate constant k and/or a very high potential sweep rate v. The voltammogram is peak shaped. At the opposite, when λ takes on large values (zone 2), due to a high rate constant k and/or a very low potential sweep rate v, the simplified expression for ψ(T) is the dimensionless analogue of the current expression derived in Section 2 under steady-state conditions. The voltammogram shape is that of the potential-dependent function , with
being a scaling factor. No voltammetric peak is observed in this case (quasi steady-state conditions).
The conditions in zone 3 pertain to diffusion coupled to the chemical reaction in a semi-infinite
electrode material, because of a large electrode thickness l and/or a small diffusion coefficient D and/or a high potential sweep rate v. The voltammogram is peak-shaped at . In contrast, diffusion and chemical reaction in a very thin film electrode, because of a small electrode thickness l and/or a large diffusion coefficient D and/or a low potential sweep rate v, yield the closed-form expression for ψ(T) in zone 4. The voltammogram is also peak-shaped at . Note that the limiting expression for ψ(T) in zone 4 does not satisfy the initial condition at . Hence, it cannot be used to model the voltammetric response at very short times.
Further simplifications are possible if considering the intersections of the primary zones 1-4. The first case pertains to small λ values together with large L values (zone 1 ∩ zone 3). The simplified model response is the same as that for the Er reaction investigated under semi-infinite diffusion conditions for redox species in the electrolyte [39]. In contrast, zone 1 ∩ zone 4 pertains to small values for both parameters λ and L. The simplified model response closely resembles that for a reversible electro-adsorption reaction [62] in this case. Finally, the limiting conditions in zone 2
∩ zone 3 and zone 2 ∩ zone 4 pertain to quasi-steady-state behaviors (no voltammetric peak) for the electrochemical model. The only difference lies in the dimensionless Faradaic current observed, i.e. or , which are the two limits of the model behavior in zone 2 at large or low values, respectively. The above presentation is further developed in Section I of the additional document joined to this article, so-called ‘A survey of linear potential sweep voltammograms’, including many illustrative examples.
5.4. Limiting behaviors for LSV (particular case, )
If we restrict the formal study to the LSV technique in the special case where ξeq is very large (infinite at the limit), the dimensionless voltammogram is modeled by the ISS derived in Eq.
(D.7) of Appendix D:
(30)
Here M*(n) denotes the dimensionless mass transfer function in Eq. (26) where the dimensionless Laplace variable is replaced by the running index n.
Taking the same limiting conditions on the dimensionless parameters and L as in Table 2, simplified expressions can be obtained for M*(n), and closed-form expressions for the LSV curve can be derived from Eq. (26) as reported in Table 3. It can be noticed that the expressions pertaining to zone 1 ∩ 3 and zone 3 in Table 3 involve the use of the polylogarithm function [63]
and Lerch transcendent function [64], respectively. The use of polylogarithm function was first advocated by Lether [65] and next by Mocák and Bound [66] for modeling the LSV response of Er reaction. The use of Lerch transcendent function was first proposed by Mocák and Bound [66]
for modeling the LSV response of ErC’ reaction, which is a reversible electron transfer reaction with semi-infinite diffusion of electroactive species being coupled to a catalytic chemical reaction in the electrolyte.
Table 3. Closed-form expressions for the dimensionless LSV curve in the reduction direction, depending on the same limiting conditions on λ and L as in Table 2. Li is the polylogarithm function, ), [63]
with real argument . is the Lerch transcendent function, , [64] with same argument. The general infinite-series formulation for is also given in the Table. All limiting formulations are special cases of this series. The zone numbering refers to the zone diagram plotted in Fig. 4.
Zone
General Eq. (26) with =n
1 No closed-form expression 2
3 4
Table 4 provides the relevant expressions for the dimensionless voltammetric peak coordinates in the cathodic direction, and , which can be derived from the closed-form expressions of the dimensionless current density given in Table 3. When a steady-state cathodic current plateau is observed, we give the dimensionless half-wave potential . The cathodic limit of the dimensionless current density is also reported in most of the zones. Those relationships will be used hereafter.
Table 4. Theoretical expressions for the dimensionless voltammetric peak coordinates and the half-wave potential where , and the limiting steady-state current density derived in the cathodic direction from the closed-form expressions of given in Table 3. The symbol (---) denotes a meaningless quantity.
Zone
1 No closed-form
expression
--- No closed-form
expression
0
2 --- 0 ---
3 No-closed form
expression
No-closed form expression No-closed form expression
4
---
---
--- 0
0 --- 0
--- 0 ---
--- 0 ---
5.5. Zone diagram (particular case,
The numbering of the different zones used in the previous Tables 2-4 comes from their definition, first of all zones 1 to 4 based on the four primary conditions on model parameters, and next the zones defined by intersection of the primary ones. An alternative notation is possible, which is based on the concept of kinetic and/or mass-transport control of the model response, as employed by Savéant [50]. Indeed, the insertion, diffusion and chemical reaction model developed and studied in the present work can be compared to the ErC’ reaction scheme studied by Savéant and Su [38], which is composed of a reversible electron transfer (Er) reaction, with diffusion of redox
species being coupled to a homogeneous catalytic (C’) reaction. Such a comparison is presented in Section A of the additional document joined to this article. Because of the similarities of both models, we make use hereafter of the nomenclature of Savéant [50], i.e. D, K, KD and KS, for diffusion control, kinetic control, mixed kinetic-diffusion control, and kinetic control with sigmoidal-shaped voltammogram under quasi-steady-state conditions, respectively. However, a complication for the electrochemical model that is the subject of our work results from the boundary condition for mass transport away from the electrode surface, which satisfies a restricted diffusion condition (zero diffusion flux) at the abscissa x = l in the electrode material.
In contrast, the species involved in the ErC’ reaction are moving and reacting in a semi-infinite medium [38]. Hence, the nomenclature of Savéant will be slightly modified in this work by adding a characteristic subscript referring to the dimensionless thickness L of the electrode material. The analogy between the numbering of the different zones in the previous Tables 2-4 and the notation based on the concept of kinetic and/or mass-transport control will be highlighted in the zone diagram.
The zone diagram is plotted for LSV in the special case where . Consequently, the model response is dependent on two dimensionless parameters only, i.e. and L. It means that, for any set of numerical values for the parameters {z, Tabs, l, D, k} and for the potential sweep rate v, the model behavior is well described by the set of characteristic values {, L}. Given each set of and L values, the general voltammogram, which is plotted using the IE-LSV algorithm, can be compared to the voltammograms calculated from the limiting expressions in Table 3.
It may be pointed out that most of the zone diagrams plotted in the electrochemical literature are based on the choice of one (or two) characteristic feature(s) of the model response to the
perturbation signal employed. Using such a local characterization of the model response may cause problems for plotting the zone diagram. Indeed, a voltammetric peak may or may not be observed, depending on the values of the parameters, which is the case of the model we are studying. This explains, for certain reaction mechanisms, the simultaneous use of two criterions for plotting the relevant zone diagram. An alternative approach is to compare the voltammogram plotted from the general model to those plotted from simplified versions of this model, using a quadratic comparison criterion in the whole potential range of interest. A single criterion is then needed to plot the zone diagram, as it is done in this work.
Here, we use a global comparison criterion based on the weighted quadratic deviation:
(31)
where n = (n) and simp,n = simp(nT) are the discrete values of dimensionless Faradaic current for the general model and the simplified model, respectively, T is the dimensionless computational time increment, and nmax is the maximum number of current data computed over the potential range explored. We have considered that the general response of the electrochemical model is well approximated by that of the simplified one when the quadratic deviation, q in Eq.
(31), is less than or equal to 0.025. The resulting zone diagram is plotted in Fig. 4 using the bi- logarithmic representation, log L versus log The description of the model behavior in each individual zone is summarized in Table 5, as well as the corresponding voltammogram features, which are obtained from the closed-form expressions reported in Table 3.
Figure 4
The shaded area in Fig. 4 is the domain of parameters values where no voltammetric peak is
observed on the LSV curve. A singular model behavior pertains to the diagram zone where λ ≥ 1, but λ is of the order of magnitude of unity (not too large λ values). No peak is observed on the LSV curve, but the voltammogram differs from quasi-steady-state conditions because of half- wave potential shift. Indeed, as shown in Table 4, the dimensionless half-wave potential is null under quasi-steady-state conditions (at high values, irrespective of L), while the theoretical expression which is valid in zone 4 of the zone diagram leads to at
= 1, for example.
Table 5. Summary of the different zones and voltammogram features for the insertion, diffusion and chemical reaction model corresponding to the zone diagram of Fig.4.
Zone CV curve Description
Reversible insertion reaction under semi-infinite diffusion conditions ( ). No chemical reaction effect. The current is proportional to the square root of sweep rate and the square root of D.
Mixed kinetic and diffusion control for a semi-infinite electrode material. The sweep rate-dependent current is a function of k and D.
The cathodic current plateau is proportional to the square root of kD.
Quasi-steady-state conditions for a semi-infinite electrode material.
The current is independent of sweep rate and is proportional to the square root of kD.
Reversible insertion reaction under finite-space diffusion conditions in the electrode material. No chemical reaction effect. The sweep rate- dependent current is a function of l and D.
Mixed chemical-kinetic and diffusion control for a finite-space electrode material. The sweep rate-dependent current is a function of l, k and D. The same applies to the cathodic current plateau.
Quasi-steady-state conditions for a finite-space electrode material. The current is independent of sweep rate. It depends on l, k and D.
Reversible insertion reaction with quasi-uniform concentration in a very thin-film electrode material ( ). No chemical reaction effect.
The current is proportional to the sweep rate and to the electrode thickness l.
Reversible insertion reaction with chemical-kinetic effect in a very thin-film electrode material. The sweep rate-dependent current is a function of l and k. The cathodic current plateau is proportional to kl.
Quasi-steady-state conditions in a thin-film electrode material. The current is independent of sweep rate. It is linearly proportional to kl.
Abbreviations: D (diffusion), K (kinetic), S (sigmoidal), G (general). The subscripts 0, L and refer respectively to low, intermediate and large values of the dimensionless length L.
The so-called model trajectory in the zone diagram is the trace left by the characteristic moving
point, with coordinates log and log L, when a parameter is increased, all other parameters being maintained constant. The model trajectories plotted at increasing values of k, l, D and v are represented in Fig. 4. The intersections of the resulting straight lines with the limiting zones of the zone diagram should provide the possible sequences of voltammetric curves observable in practice for the electrochemical model. Any physical parameter has a limited domain of variation, of course, so that the model trajectories are only portions of straight lines in the zone diagram, as illustrated in the next Section.
Because of its construction rules, the zone diagram in Fig. 4 directly applies to the analysis of LSV curves. A similar diagram could be plotted by application of the quadratic criterion in Eq. (31) to cyclic voltammograms. In practice, the zone diagram shape remains nearly the same as in Fig. 4, with only small horizontal or vertical shifts being observed for the different zone boundaries. Hence, both LSV and CV curves can be analysed at the same time using the zone diagram plotted in Fig. 4.
It can also be pointed out that the integral equation method, employed in the present work, does not require prior knowledge of the concentration (or insertion level) profiles of the inserted species in the electrode material, in contrast to finite difference methods for example. However, the change of the insertion level profile with the time-dependent controlled potential could be helpful for a better understanding of the limiting behaviors of the electrochemical model pertaining to the different zones of the zone diagram plotted in Fig. 4. It is the aim of Section L in the additional document joined to this article.
6. Application to the analysis of LSV/CV curves
In this part, we will use the previous theoretical developments to predict the effect of the
