Steady-state theory of interdigitated array of electrodes in confined spaces: Case of pure diffusion and reversible electrode reactions

TL;DR

This paper derives analytical equations for steady-state current in interdigitated electrodes within confined spaces, considering reversible reactions and pure diffusion, revealing how cell dimensions influence current behavior.

Contribution

It introduces a conformal transformation approach using Jacobian elliptic functions to solve the diffusion equation in confined geometries, providing new analytical expressions for steady-state current.

Findings

Steady-state current depends on cell dimensions and elliptic integrals.

Shallower cells exhibit smaller steady-state currents.

Tall cells' currents approach semi-infinite geometry values.

Abstract

Analytical equations were found for interdigitated electrodes, which considered reversible electrode reactions and pure diffusion within confined spaces. A conformal transformation, obtained by the use of Jacobian elliptic functions, was applied to solve the diffusion equation in steady state. The obtained steady-state current depends on the ratio of elliptic integrals of the first kind, in which their moduli are functions of the relative dimensions of the cell. The current is smaller for shallower cells, but approaches similiar values to those of semi-infinite geometries when the cell is sufficiently tall. Approximations using trigonometric and hyperbolic expressions were also found for the steady-state current in the cases of shallow and tall cells respectively.

Tables

Table 1. Table 1 : Regions where the approximations hold with a relative error less than ± 5   % times \pm5 percent \pm 5\text{\,}\mathrm{\char 37\relax} for tall ( H / W > 1 𝐻 𝑊 1 H/W>1 ) and shallow ( H / W ≤ 1 𝐻 𝑊 1 H/W\leq 1 ) cells.

Approx.	Cell	Electrodes	Domain
Eq. (2.43a)	Tall	Large	$2w_E/W>0.46$
Eq. (2.43b)	Tall	Small
Eq. (2.45a)	Shallow	Large	$2w_E+\left(1-0.28\right)H\gtrsim W$
Eq. (2.45b)	Shallow	Small	$2w_E+\left(1-0.36\right)H\lesssim W$

Equations

$$\text{ce}O+n_{e}e-<=>R$$

$${\bar{c}}_{\sigma ,i}^{\text{whole}}:=\frac{1}{W_T}{\int }_{\text{whole}}{c}_{\sigma ,i}(x,z)\mathrm{d}x$$

$$D_O{c}_{O,i}(x,z)+D_R{c}_{R,i}(x,z)=D_O{\bar{c}}_{O,i}^{\text{whole}}+D_R{\bar{c}}_{R,i}^{\text{whole}}$$

$$\frac{1}{W_T}{\int }_{\text{whole}}{c}_{\sigma ,f}(x,z)\mathrm{d}x={\bar{c}}_{\sigma ,i}^{\text{whole}}$$

$$D_O{c}_{O,f}(x,z)+D_R{c}_{R,f}(x,z)=D_O{\bar{c}}_{O,i}^{\text{whole}}+D_R{\bar{c}}_{R,i}^{\text{whole}}$$

$${\eta }_f^E:=\ln \left(\frac{c_{O,f}^E}{c_{R,f}^E}\right)=\frac{F{n}_e}{RT}(V_f^E-{V^o}^{\prime })$$

$$D_O{c}_{O,f}^E+D_R{c}_{R,f}^E=D_O{\bar{c}}_{O,i}^{\text{whole}}+D_R{\bar{c}}_{R,i}^{\text{whole}}$$

$$c_{O,f}^E$$

$$c_{R,f}^E$$

$${\eta }_{\text{null}}=\ln \left(\frac{{\bar{c}}_{O,i}^{\text{whole}}}{{\bar{c}}_{R,i}^{\text{whole}}}\right)=\frac{n_{e}F}{RT}(V_{\text{null}}-{V^o}^{\prime })$$

$$\frac{{\partial }^2{c}_{\sigma ,f}}{\partial x^2}(x,z)+\frac{{\partial }^2{c}_{\sigma ,f}}{\partial z^2}(x,z)$$

$$\frac{\partial c_{\sigma ,f}}{\partial x}(0,z)=\frac{\partial c_{\sigma ,f}}{\partial x}(W,z)$$

$$\mp D_{\sigma }\frac{\partial c_{\sigma ,f}}{\partial z}(x,H)$$

$$\mp D_{\sigma }\frac{\partial c_{\sigma ,f}}{\partial z}(x,0)$$

$$c_{\sigma ,f}(x,0)$$

$$\mathbf{\rho }=T_r^{\rho }(\mathbf{r})$$

$$\mathbf{\rho }=T_{\omega }^{\rho }(\mathbf{\omega })$$

$$\mathbf{\omega }=T_v^{\omega }(\mathbf{v})$$

$$\mathbf{v}=T_r^v(\mathbf{r})$$

$$k_{\rho }^2$$

$$k_r$$

$${\mathbf{r}}_a$$

$${\mathbf{r}}_b$$

$${\mathbf{v}}_a=T_r^v({\mathbf{r}}_a)$$

$${\mathbf{v}}_b=T_r^v({\mathbf{r}}_b)$$

$${\mathbf{\rho }}_a=T_r^{\rho }({\mathbf{r}}_a)$$

$${\mathbf{\rho }}_b=T_r^{\rho }({\mathbf{r}}_b)$$

$${\gamma }_{\sigma ,f}(\xi ,\zeta )={\gamma }_{\sigma ,f}\circ T_r^{\rho }(x,z)=c_{\sigma ,f}(x,z)$$

$$\frac{{\partial }^2{c}_{\sigma ,f}}{\partial x^2}(x,z)+\frac{{\partial }^2{c}_{\sigma ,f}}{\partial z^2}(x,z)={\left|\frac{\partial \mathbf{r}}{\partial \mathbf{\rho }}(\xi ,\zeta )\right|}^{-2}\left[\frac{{\partial }^2{\gamma }_{\sigma ,f}}{\partial {\xi }^2}(\xi ,\zeta )+\frac{{\partial }^2{\gamma }_{\sigma ,f}}{\partial {\zeta }^2}(\xi ,\zeta )\right]$$

$$\frac{{\partial }^2{\gamma }_{\sigma ,f}}{\partial {\xi }^2}(\xi ,\zeta )+\frac{{\partial }^2{\gamma }_{\sigma ,f}}{\partial {\zeta }^2}(\xi ,\zeta )$$

$$\frac{\partial {\gamma }_{\sigma ,f}}{\partial \zeta }(\xi ,0)=\frac{\partial {\gamma }_{\sigma ,f}}{\partial \zeta }(\xi ,{\zeta }_a)$$

$${\gamma }_{\sigma ,f}(0,\zeta )$$

$${\gamma }_{\sigma ,f}(1,\zeta )$$

$$c_{\sigma ,f}(x,z)=c_{\sigma ,f}^A+[c_{\sigma ,f}^B-c_{\sigma ,f}^A]\xi (x,z)$$

$${\gamma }_{\sigma ,f}(\xi ,\zeta )=c_{\sigma ,f}^A+[c_{\sigma ,f}^B-c_{\sigma ,f}^A]\xi$$

$$\mp \frac{j_f(x)}{F{n}_e{D}_{\sigma }[c_{\sigma ,f}^B-c_{\sigma ,f}^A]}=\frac{\partial \xi }{\partial z}(x,0)=-\Im \frac{\partial \mathbf{\rho }}{\partial \mathbf{r}}(x)$$

$$\frac{\partial \mathbf{\rho }}{\partial \mathbf{r}}(\mathbf{r})=\mathbf{i}\frac{k_r^{\prime }}{W}\frac{K(k_r)}{K(k_{\rho })}\frac{(1-{\mathbf{v}}_{\alpha }{)}^{1/2}(1+{\mathbf{v}}_{\beta }{)}^{1/2}}{(\mathbf{v}-{\mathbf{v}}_{\alpha }{)}^{1/2}(\mathbf{v}-{\mathbf{v}}_{\beta }{)}^{1/2}}\mathrm{nd}\left(K(k_r)\frac{2\mathbf{r}}{W},k_r\right)$$

$$j_f(x)=\mp F{n}_e{D}_{\sigma }\frac{\partial c_{\sigma ,f}}{\partial z}(x,0)=\mp F{n}_e{D}_{\sigma }[c_{\sigma ,f}^B-c_{\sigma ,f}^A]\frac{\partial \xi }{\partial z}(x,0)$$

Full text

Analytical equations were found for interdigitated electrodes, which considered reversible electrode reactions and pure diffusion within confined spaces. A conformal transformation, obtained by the use of Jacobian elliptic functions, was applied to solve the diffusion equation in steady state. The obtained steady-state current depends on the ratio of elliptic integrals of the first kind, in which their moduli are functions of the relative dimensions of the cell. The current is smaller for shallower cells, but approaches similiar values to those of semi-infinite geometries when the cell is sufficiently tall. Approximations using trigonometric and hyperbolic expressions were also found for the steady-state current in the cases of shallow and tall cells respectively.

confined cell, interdigitated array of electrodes, diffusion equation, steady state, voltammogram, current density, limiting current

Steady-state theory of interdigitated array of electrodes in confined spaces:

Case of pure diffusion and reversible electrode reactions

Cristian F. Guajardo Yévenes

Biological Engineering Program and

Pilot Plant Development and Training Institute

King Mongkut’s University of Technology Thonburi, Thailand

cristian.gua@kmutt.ac.th

  

Werasak Surareungchai

School of Bioresources and Technology and

Nanoscience & Nanotechnology Graduate Program

King Mongkut’s University of Technology Thonburi, Thailand

werasak.sur@kmutt.ac.th

(March 6, 2019)

Abstract

Analytical equations were found for interdigitated electrodes, which considered reversible electrode reactions and pure diffusion within confined spaces. A conformal transformation, obtained by the use of Jacobian elliptic functions, was applied to solve the diffusion equation in steady state. The obtained steady-state current depends on the ratio of elliptic integrals of the first kind, in which their moduli are functions of the relative dimensions of the cell. The current is smaller for shallower cells, but approaches similiar values to those of semi-infinite geometries when the cell is sufficiently tall. Approximations using trigonometric and hyperbolic expressions were also found for the steady-state current in the cases of shallow and tall cells respectively.

Keywords: confined cell, interdigitated array of electrodes, diffusion equation, steady state, voltammogram, current density, limiting current

Graphical abstract

Contents

1. 1 Introduction

2. 2 Theory

3. 2.1 Definition of the problem

4. 2.1.1 Description of the cell

5. 2.1.2 Properties of initial and final conditions

6. 2.1.3 Properties of boundary conditions

7. 2.1.4 Problem reduced to diffusion in the unit cell

8. 2.2 Exact solution in steady state

9. 2.2.1 Transformation of the unit-cell domain

10. 2.2.2 Concentration profile

11. 2.2.3 Current density

12. 2.2.4 Current per band

13. 2.2.5 Voltammogram

14. 2.3 Approximations for shallow and tall cells

15. 2.3.1 Case of tall cells

16. 2.3.2 Case of shallow cells

17. 3 Results and discussion

18. 3.1 Concentration profile

19. 3.2 Current density

20. 3.3 Current per band

21. 3.3.1 Effect of the counter electrode in the collection efficiency

22. 3.3.2 Influence of the cell geometry

23. 3.4 Voltammogram shape

24. 3.5 Approximations for shallow and tall cells

25. 4 Conclusions

26. S1 Additional proofs

27. S1.1 Transformation of the unit cell domain

28. S1.1.1 From $\bm{r}$ to $\bm{v}$ domain

29. S1.1.2 From $\bm{v}$ to $\bm{\omega}$ domain

30. S1.1.3 From $\bm{\omega}$ to $\bm{\rho}$ domain

31. S1.1.4 Conformality of the transformation

32. S1.2 Derivative of the domain transformation

33. S1.3 Differences of concentrations when using internal counter electrode

34. S1.3.1 Difference with respect to the average in the unit cell

35. S1.3.2 Difference with respect to the average in the whole cell

36. S1.3.3 Limits for the differences of concentration

37. S1.4 Difference of potential (voltage) when using internal counter electrode

38. S1.5 Approximations for $k_{\rho}$ and $k_{\rho}^{\prime}$

39. S1.5.1 Case of tall cells

40. S1.5.2 Case of shallow cells

41. S2 Numerical calculations

42. S2.1 Simulations in steady state

43. S2.1.1 Implementation

44. S2.1.2 Selection of the parameters for the exponential mesh

45. S2.1.3 Simulation of concentration profile in Fig. 3

46. S2.1.4 Exhaustive simulations

47. S2.2 Direct evaluations and curve fitting

48. S2.2.1 Voltammogram and potentials in Figs. 8(a) and 8(b)

49. S2.2.2 Curve fitting in Fig. 9

50. S2.2.3 Approximated normalized current in Figs. 10 and 11.

1 Introduction

The use of microelectrodes has been in constant increase since approximately the beginning of 1980 [1, 2, 3], mainly due to the development of microfabrication techniques and because of the benefits of microelectrodes towards sensing: reduced ohmic potential drop, fast non-faradaic time constants, shorter diffusion times, fast establishment of steady-state signals and increased signal-to-noise ratio [4, 5].

There are several kinds of microelectrode configurations: disk, cylinder, disk array, microband array111composed of only anodes or only cathodes (all band electrodes at the same potential). , interdigitated array222composed of band electrodes organized as alternating anodes and cathodes. , ring and recessed electrodes [6, 4]. Among all these configurations, interdigitated array of electrodes (IDAE) is a popular choice and has drawn great attention, because it can produce high currents from the redox cycling in between closely arranged generators and collectors [6, 5], apart from all the known advantages inherited from microelectrodes.

The use of simple mathematical models allows understanding, prediction of behavior and design of electrode configurations. In this way, the behavior of an IDAE can be prescribed, when its physical implementation is fabricated according to the constraints and assumptions imposed by its mathematical model. Among these models, pure diffusional transport and Nernstian boundary conditions are probably the most used for IDAEs, since they allow great simplifications.

Using this model, simulations have been performed to understand the time dependence of the current generated at IDAE [7]. Also, theoretical results have been obtained in [8, 9, 10] by analytically solving the diffusion equation in steady state. From these theoretical results, the work of Aoki stands out, due to the obtention of exact expressions for the current-potential curve and limiting current in steady state for reversible [8] and irreversible [9] electrode reactions.

These theoretical results consider that the IDAE is subject to semi-infinite geometry, that is, they consider that the IDAE is in contact with a large amount of solution around it, so that the diffusion layer is much smaller than the total size of the electrochemical cell. In practical terms, this is equivalent to say that the ratio between the “height of the cell” and the “center-to-center separation between adjacent electrodes” is very large.

This semi-infinite condition is in general not valid for every IDAE electrochemical cell. This can be seen particularly in the case of microfluidic devices, where the height of the microchannel is clearly finite, especially when using low-cost fabrication techniques: softlithography, transparency-film masks, paper and screen-print to name a few [11, 12, 13, 14]. Typical heights of microfluidic channels fabricated using softlithography depend on the thickness of the photoresist molds, which can range between $1\text{\,}\mathrm{\SIUnitSymbolMicro m}200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [11]. The width and gap of IDAE bands fabricated using photolithography is commonly constrained by the resolution of transparency-film masks, which can range between $20\text{\,}\mathrm{\SIUnitSymbolMicro m}50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for printers operating between $3380\text{\,}\mathrm{d}\mathrm{p}\mathrm{i}5080\text{\,}\mathrm{d}\mathrm{p}\mathrm{i}$ [11, 13]. Therefore, these fabrication techniques can produce microfluidic electrochemical cells where the ratio between “height of the cell” and “center-to-center separation between adjacent electrodes” is clearly finite and ranges between $\sim$ $0.0110$. This implies that the equations obtained in [8, 9] may not be always applicable to IDAEs operating within microfluidic channels.

Currently, and due to the lack of analytical expressions for current and current-potential curves, experiments involving IDAE in microchannels are normally constrasted against simulations of its ideal behavior, namely, the diffusion equation subject to Nernstian boundary conditions [15, 16]. Despite this fact, these studies, together with pure experimental [17] and theoretical [18, 19] results, have contributed to reveal the behavior of IDAE in confined spaces with stagnant solutions. These results indicate that higher currents are obtained when using electrochemical cells with taller microchannels. In fact, the current approaches similar values to the case of semi-infinite cells, predicted by [8, 9], when the “height of the microchannel” $H$ is larger than the “center-to-center separation between adjancent electrodes” $W$. According to [19], finite-height microfluidic cells can be regarded as semi-infinite cells, within 12% error in bulk concentration, when $H/W\gtrsim 1$.

Therefore, this work intends to elucidate the ideal behavior of IDAE in electrochemical cells within confined spaces, and replace the use of simulations, by obtaining analytical expressions for the current and voltammogram as a function of the dimensions of the IDAE and the geometry of the electrochemical cell. This model would enable the design of electrochemical cells that output the maximum current available given microfabrication constraints, and contrast their actual performance with their ideal behavior.

2 Theory

2.1 Definition of the problem

2.1.1 Description of the cell

Consider an electrochemical cell with an IDAE configuration, as shown in Fig. 1. The cell has a finite height $H$, total width $W_{T}$ and it is surrounded by walls which behave as perfect insulators.

The IDAE is located at the floor of the cell and it is composed of two arrays of bands, $A$ (black) and $B$ (gray), of $N_{A}$ and $N_{B}$ bands respectively. Each band $A$ and $B$ has respectively a width of $2w_{A}$ and $2w_{B}$, a common length $L$, and they are placed alternatingly such that two consecutive bands have a center-to-center separation of $W$. Besides the IDAE, the cell may include a counter electrode $C$ of width $w_{C}$ and length $L$, which is assumed to lay on the same plane as the IDAE.

Inside the electrochemical cell there is an oxidated species $O$ and a reduced species $R$, with diffusion coefficients $D_{O}$ and $D_{R}$ respectively. Both species are transported solely by diffusion and react at the surface of the electrodes according to

[TABLE]

where $n_{e}$ corresponds to the number of exchanged electrons. Here it is assumed that the charge transfer on the electrodes follows reversible electrode reactions, and therefore Nernst equation holds even when current flows [20, Eq. (5:8:6)].

The IDAE is assumed to operate in dual mode, that is, different potentials are applied to each array of bands. If one array is potentiostated and its complementary array performs as counter electrode, then it is said that the IDAE operates with internal counter electrode. On the other hand, if both arrays are potentiostaded independently, an additional counter electrode would be required, and the IDAE is said to operate with external counter electrode.

2.1.2 Properties of initial and final conditions

If the common length $L$ of the electrodes is large enough, then the concentration profile of the electrochemical species $\sigma$ doesn’t change along the depth of the cell, and therefore, it can be reduced to two dimensions $c_{\sigma}(x,z,t)$. Here $x$ and $z$ are the horizontal and vertical coordinates respectively, and $t$ corresponds to time.

In this case, the concentration profile of species $\sigma\in\{O,R\}$ is given initially by $c_{\sigma}(x,z,0^{-})=c_{\sigma,i}(x,z)$, and it is assumed to come from a previous steady state. Under this condition, the initial profile has an average (along any horizontal line spaning the whole cell) that is independent of $z$ [19, Remark 2.2]

[TABLE]

After the cell is potentiostated, the concentration profile is shifted out from its initial state, entering a new steady state $c_{\sigma}(x,z,+\infty)=c_{\sigma,f}(x,z)$ after a sufficiently long time compared with the characteristic time of the cell. In this final steady state, the average of concentration (along any horizontal line spaning the whole cell) remains independent of $z$ and equals its initial counterpart $\bar{c}_{\sigma,i}^{\text{whole}}$ [19, Remark 2.2]

[TABLE]

2.1.3 Properties of boundary conditions

The final concentration $c_{\sigma,f}^{E}$ of species $\sigma\in\{O,R\}$, at the surface $E\in\{A,B,C\}$ of each band and the external counter electrode (if present), is governed by Nernst equation

[TABLE]

However, since the weighted sum of concentrations on the surface of $E$ is related to the initial concentrations, due to Eq. (2.3b)

[TABLE]

then the Nernst equation can be decoupled, leading to uniform concentrations on all electrodes $E\in\{A,B,C\}$

[TABLE]

Here, $\eta_{f}^{E}$ and $V_{f}^{E}$ are the normalized and applied potentials at the electrode $E$ in the final steady state, ${V^{o}}^{\prime}$ is the formal potential of the redox couple, $F$ is the Faraday constant, $R$ is the universal gas constant and $T$ is the temperature of the system.

Note that if the final concentrations on the electrodes satisfy $c_{\sigma,f}^{A}=c_{\sigma,f}^{B}=c_{\sigma,f}^{C}=\bar{c}_{\sigma,i}^{\text{whole}}$, then there is no gradient of concentration generated inside the cell, and therefore the current in the cell must equal zero. This corresponds to the case when the null potential $V_{\text{null}}$ is applied to the electrodes $\eta_{f}^{E}=\eta_{\text{null}}$

[TABLE]

2.1.4 Problem reduced to diffusion in the unit cell

If the IDAE fits exactly within the cell, such that the bands at both ends of the IDAE have half width, then the cell in Fig. 1(a) can be modeled exactly as an assembly of two-dimensional unit cells, like the one shown in Fig. 1(c). In case the IDAE doesn’t fit exactly in the cell, the configurations in Figs. 1(a),1(b) still can be regarded as an assembly of two-dimensional unit cells provided the following conditions: (i) The length $L$ is large enough, so that the problem still can be reduced to two dimensions. (ii) The number of bands is so large that the edge effects at the ends of the IDAE are negligible, and it is still possible to consider symmetry boundary conditions for the unit cell [8, 271].

Under these conditions, the final concentration $c_{\sigma,f}(x,z)$ of the electrochemical species $\sigma\in\{O,R\}$ can be reduced to a problem of steady-state two-dimensional diffusion333 for $\pm$ or $\mp$, the upper sign corresponds to $\sigma=O$, and the lower sign, to $\sigma=R$.

inside a representative unit cell

[TABLE]

where $D_{\sigma}$ is the diffusion coefficient of the species $\sigma\in\{O,R\}$, Eqs. (2.8b–2.8d) are symmetry/insulation boundary conditions, and $c_{\sigma,f}^{A}$ and $c_{\sigma,f}^{B}$ are the final concentrations at the surface of the bands $A$ and $B$ respectively, see Eqs. (2.6).

2.2 Exact solution in steady state

2.2.1 Transformation of the unit-cell domain

Alternated concentration and insulation boundary conditions at the bottom of the unit cell, Eqs. (2.8d–2.8f), make it difficult to obtain an analytical solution to Eqs. (2.8). Nevertheless, through domain transformations, it is possible to arrange these alternated boundary conditions, so they can be placed at different walls in a transformed cell.

One convenient way to obtain such transformation is through complex conformal mappings, since they leave the diffusion equation (in steady state), as well as the concentration and non-flux/insulation boundary conditions, invariant under domain changes [21, §5.7]. In particular, the complex Jacobian elliptic functions $\operatorname{sn}()$ and $\operatorname{cd}()$ are of interest, since they conformally map a square domain into the upper half-plane [21, §2.5] [22, §22.18.ii]. Also the Möbius functions are important, since they are able to reorganize the upper half-plane, by mapping into itself [21, §2.3].

One can now regard the concentration $c_{\sigma,f}(x,z)$ as a function $c_{\sigma,f}(\bm{r})$ of a complex variable $\bm{r}=x+\bm{i}z$, where $\bm{i}^{2}=-1$, and find a complex function $\bm{\rho}=T_{r}^{\rho}(\bm{r})$ that can transform the unit cell from the IDAE domain $\bm{r}=x+\bm{i}z$ to a parallel-plates domain $\bm{\rho}=\xi+\bm{i}\zeta$ as shown in Fig. 2.

Lemma 2.1**.**

The domain $\bm{r}=(x,z)$ of the IDAE unit cell can be conformally transformed into a parallel-plates configuration in the domain $\bm{\rho}=(\xi,\zeta)$, through the use of a complex transformation $\bm{\rho}=T_{r}^{\rho}(\bm{r})$ (see Fig. 2). This domain transformation is given by

[TABLE]

where the moduli and their complements are

[TABLE]

and the following points on the boundary of the IDAE domain $\bm{r}$

[TABLE]

are transformed to the auxiliary domain $\bm{v}$ as

[TABLE]

and to the parallel-plates domain $\bm{\rho}$ as

[TABLE]

For a detailed construction of this transformation see Supplementary information §S1.1.

Several special functions have been used for the definition of the conformal transformation $T_{r}^{\rho}$: $\operatorname{cd}(\bm{u},k)$ and $\operatorname{arcsn}(\bm{u},k)$ correspond to Jacobian elliptic functions [22, Eqs. (22.2.8) and (22.15.12)] analogous to their circular counterparts $\cos()$ and $\arcsin()$ respectively, $K(k)$ and $K^{\prime}(k)$ correspond to the complete elliptic integral of the first kind and its associated function respectively [22, Eqs. (19.2.4), (19.2.8) and (19.2.9)], and $Q(k)$ corresponds to the elliptic nome function [22, Eq. (22.2.1)].

The conformal transformation in Lemma 2.1 agrees with that in [8, Eqs. (9), (10), (14), (15) and (19)]444 Note that in [8] the parameter $p=k_{\rho}^{2}$ is used instead of the modulus $k_{\rho}$.

for semi-infinite geometries, since $k_{r}\to 0^{+}$, $\operatorname{cd}(\cdot,k_{r})\to\cos(\cdot)$ and $K(k_{r})\to\pi/2$ when $H\to+\infty$.

2.2.2 Concentration profile

If the concentration profile is now written in terms of the conformal parallel-plates domain $(\xi,\zeta)=T_{r}^{\rho}(x,z)$ by using Lema 2.1

[TABLE]

then the Laplacian operator in the IDAE and the parallel-plates domains are related [23, §2.1 Problem 7, Eq. (5.4.17)], [21, Eq. (5.20)], [24, Eq. (6.3)]

[TABLE]

meaning that the diffusion equation in steady state is invariant under conformal domain transformations. The same is true for insulation/symmetry boundary conditions, that is, they remain invariant under conformal transformations, since the angles between the iso-concentration lines and the flux lines are preserved.

Therefore, the steady-state diffusion problem in Eqs. (2.8) is transformed to the conformal parallel-plates domain $\bm{\rho}=(\xi,\zeta)$

[TABLE]

where $\zeta_{a}=\Im\bm{\rho}_{a}=K^{\prime}(k_{\rho})/K(k_{\rho})$ corresponds to the imaginary part of $\bm{\rho}_{a}$ defined in Eq. (2.13). This leads to the following solution for the concentration profile in steady state

Theorem 2.1**.**

If the IDAE can be modeled as an assembly of unit cells and the final concentrations on the bands $A$ and $B$ are uniform, such as in §2.1, then the concentration profile in the final steady state is given by

[TABLE]

where $c_{\sigma,f}^{E}$ is the steady-state concentration of the species $\sigma\in\{O,R\}$ on each band $E\in\{A,B\}$, defined in Eq. (2.6), and $\xi(x,z)=\Re\,T_{r}^{\rho}(x,z)$ corresponds to the real part of the conformal transformation $T_{r}^{\rho}(x,z)$, defined in Eq. (2.9).

Proof.

Due to the symmetry boundary conditions in Eq. (2.16b), the parallel-plates cell can be extended infinitely along the $\zeta$-axis. The concentration profile $\gamma_{\sigma,f}(\xi,\zeta)$ for the infinite parallel-plates cell doesn’t depend on the $\zeta$-coordinate, therefore the steady-state solution of Eq. (2.16) is given by a linear interpolation of the concentration at each of the plates

[TABLE]

The desired result is finally obtained by applying the conformal transformation $(\xi,\zeta)=T_{r}^{\rho}(x,z)$ in Lema 2.1 to return to the IDAE domain. ∎

Note that the expression for the concentration profile given by Eqs. (2.17) and (2.6) agrees with that in [8, Eqs. (20) and (21)]555 Note that [8, $H$ in Eq. (21)] has a typo, and the $+$ sign, at the middle of the expression, should be replaced by a $-$ sign. for semi-infinite geometries, which depends directly on the real part of the conformal transformation $T_{r}^{\rho}$ in Lemma 2.1.

2.2.3 Current density

Corollary 2.1**.**

Under the conditions of Theorem 2.1, the current density $j_{f}(x)$ in the final steady state is given by

[TABLE]

where $\Im\{\}$ corresponds to the imaginary part and the complex derivative of $\bm{\rho}=T_{r}^{\rho}(\bm{r})$ is given by

[TABLE]

of which its parameters are defined in Eqs. (2.10) and (2.12), and $\operatorname{nd}()$ corresponds to a Jacobian elliptic function, defined in [22, Eq. (22.2.6)].

Proof.

Applying Fick’s law to Eq. (2.17) leads to

[TABLE]

Later, using the Cauchy-Riemann equations [24, Theorem 3.2]

[TABLE]

leads to Eq. (2.19), which ends the main proof. The rest of the proof concerns about obtaining the complex derivative of $T_{r}^{\rho}$ in Eq. (2.20), which is detailed in Supplementary information §S1.2. ∎

The expression for the current density in Eq. (2.19) agrees with that in [8, Eqs. (26) and (19)] for semi-infinite geometries. Both results depend directly on the imaginary part of the complex derivative $\partial\bm{\rho}(x)/\partial\bm{r}$ and the product [8, $c^{*}H$ given by Eqs. (6) and (21)]666 Note that [8, $H$ in Eq. (21)] has a typo, and the $+$ sign, at the middle of the expression, should be replaced by a $-$ sign.

corresponds to $[c_{\sigma,f}^{B}-c_{\sigma,f}^{A}]$.

The expression for $\Im\partial\bm{\rho}(x)/\partial\bm{r}$ in Eq. (2.20) also agrees with that in [8, Eqs. (27) and (17)] for semi-infinite geometries, which can be seen with the help of the identity

[TABLE]

and because $k_{r}\to 0^{+}$, $k_{r}^{\prime}\to 1^{-}$, $K(k_{r})\to\pi/2$, $\operatorname{cd}(\cdot,k_{r})\to\cos(\cdot)$, $\operatorname{nd}(\cdot,k_{r})\to 1$, when $H\to+\infty$.

2.2.4 Current per band

Corollary 2.2**.**

Under the conditions of Theorem 2.1, the current in the final steady state $i_{f}^{E}$ flowing through one band $E\in\{A,B\}$ is

[TABLE]

where the modulus $k_{\rho}$ and its complement $k_{\rho}^{\prime}$ are defined in Eq. (2.10a).

Proof.

The current $i_{f}^{E}$ can be obtained by integrating the flux through one band $E\in\{A,B\}$

[TABLE]

Using the Cauchy-Riemann identities in Eq. (2.22) the current can be further simplified

[TABLE]

Due to symmetry this integral can be taken in a half band

[TABLE]

Since $\bm{\rho}=T_{r}^{\rho}(\bm{r})$ and due to Eq. (2.13), the result in Eq. (2.24) can be obtained. ∎

Note that Eq. (2.24) agrees with the result in [8, Eqs. (28)]777 Note that in [8] the parameter $p=k_{\rho}^{2}$ is used instead of the modulus $k_{\rho}$. for semi-infinite geometries, where the product [8, $c^{*}H$ given by Eqs. (6) and (21)]888 Note that [8, $H$ in Eq. (21)] has a typo, and the $+$ sign, at the middle of the expression, should be replaced by a $-$ sign.

corresponds to $[c_{\sigma,f}^{B}-c_{\sigma,f}^{A}]$.

2.2.5 Voltammogram

The voltammogram in steady state corresponds just to the expression of current, obtained in Eq. (2.24), as the potential applied to the bands is scanned. During the scan, all parameters of the expression for the current remain unchanged, except for the difference of concentrations $c_{\sigma,f}^{A}-c_{\sigma,f}^{B}$, which varies according to the potential due to Eqs. (2.6).

Therefore, the shape of the voltammogram is proportional to the shape of $c_{\sigma,f}^{A}-c_{\sigma,f}^{B}$ as the potential is scanned. This difference is analyzed in two cases: when the IDAE has an external and an internal counter electrode.

Case of external counter electrode

This case is the simplest to analyze, but its experimental setup is relatively complex, since it requires a bipotentiostat connected to the IDAE and an external counter electrode.

Theorem 2.2**.**

Consider the electrodes $A$ and $B$ with an external counter electrode, which undergo reversible electrode reations and satisfy Eq. (2.5) (like in §2.1 with Fig. 1(b)). If the potential at $E\in\{A,B\}$ is scanned and its complementary electrode $E^{\prime}$ is fixed to an extreme potential, such that $c_{\sigma,f}^{E^{\prime}}=0$, then the difference of final concentrations is given by

[TABLE]

where $\sigma^{\prime}$ is the complementary redox especies of $\sigma\in\{O,R\}$, $D_{\sigma^{\prime}}$ and $D_{\sigma}$ are their diffusion coefficients, $\bar{c}_{\sigma,i}^{\text{whole}}$ and $\bar{c}_{\sigma,^{\prime}i}^{\text{whole}}$ are the average concentrations of the whole cell in the initial steady state, $\eta_{f}^{E}$ is the normalized potential applied to $E$, defined in Eq. (2.4), and $\eta_{\text{null}}$ is the normalized null potential in Eq. (2.7).

Proof.

Eq. (2.28) is obtained directly from Eq. (2.6), when $c_{\sigma,f}^{E^{\prime}}=0$. ∎

Case of internal counter electrode

In this case the experimental setup is simpler, since it requires a conventional potentiostat connected to both arrays of the IDAE. Moreover, when the internal counter electrode serves as reference, no potentiostat would be necessary and a voltage source with a sensitive ammeter would suffice as instrumentation [25, end of p. 33].

However, the analysis is not direct, since the potential at the counter electrode bands is controlled automatically by the potentiostat, and therefore it is unknown before performing the experiment. Moreover, when the internal counter electrode acts as reference, even the potential at the working electrode is unknown and only the voltage (difference of potential) between working and counter electrodes would be known.

One solution is to consider the case of bands of equal width $2w_{A}=2w_{B}$. Here the average in Eq. (2.3a) is reduced to $\bar{c}_{\sigma,i}^{\text{whole}}=(c_{\sigma,f}^{A}+c_{\sigma,f}^{B})/2$ [10, Eq. (15)], due to symmetry, or equivalently

[TABLE]

This allows an a priori estimation of the concentration at the counter electrode bands. However, this also restricts the magnitude of the concentrations on the bands, since they must be non-negative, and therefore, they cannot decrease indefinitely below the average $\bar{c}_{\sigma,i}^{\text{whole}}$

[TABLE]

Due to Eq. (2.3b), a similar situation occurs with the complementary species, and has been graphically illustrated in [19, Fig. 2].

Lemma 2.2**.**

Consider the electrodes $A$ and $B$, which undergo reversible electrode reactions and satisfy Eqs. (2.5) and (2.29) (like in §2.1 with Fig. 1(a) and bands of equal width). If the electrode $E\in\{A,B\}$ and its complementary electrode $E^{\prime}$ perform as working and counter electrodes respectively, then the difference of final concentrations of species $\sigma\in\{O,R\}$ is given by

[TABLE]

Nevertheless, this difference cannot reach its ideal maximum, obtainable from Eq. (2.34a), being limited from above and below by

[TABLE]

where the limiting species $\lambda\in\{O,R\}$ is such that $D_{\lambda}\bar{c}_{\lambda,i}^{\text{whole}}=\min(D_{O}\bar{c}_{O,i}^{\text{whole}},D_{R}\bar{c}_{R,i}^{\text{whole}})$. This last expression determines the limiting current of the cell.

See Supplementary information §S1.3 for a detailed proof.

Remark 2.1**.**

Note that, in the case of external counter electrode, the difference of concentrations is bounded by

[TABLE]

due to Eq. (2.28) when $\eta_{f}^{E}\to\pm\infty$. This determines the limiting current in the case of external counter electrode.

The result in Lemma 2.2 allows us to estimate the unknown concentration on the counter electrode, thus facilitating an analytical expresion for the voltammogram

Theorem 2.3**.**

Under the assumptions of Lemma 2.2, the difference of concentrations (voltammogram) in terms of the normalized potential is given by

[TABLE]

where $\sigma^{\prime}$ is the complementary redox especies of $\sigma$, $D_{\sigma^{\prime}}$ and $D_{\sigma}$ are their diffusion coeficients, $\bar{c}_{\sigma,i}^{\text{whole}}$ and $\bar{c}_{\sigma,^{\prime}i}^{\text{whole}}$ are the average concentrations of the whole cell in the initial steady state, $\eta_{f}^{E}$ and $\eta_{f}^{E^{\prime}}$ are the normalized potentials applied to the electrodes $E$ and $E^{\prime}$, given in Eq. (2.4), and $\eta_{\text{null}}$ is the normalized null potential in Eq. (2.7).

Proof.

Eq. (2.34a) is obtained when substracting Eq. (2.6) with $\bar{c}_{\sigma,i}^{\text{whole}}$, and later by applying $[c_{\sigma,f}^{E}-c_{\sigma,f}^{E^{\prime}}]=2[c_{\sigma,f}^{E}-\bar{c}_{\sigma,f}^{\text{whole}}]$. The proof for Eq. (2.34b) can be found in Supplementary information §S1.4. ∎

2.3 Approximations for shallow and tall cells

For calculating the steady-state current through the cell, one must evaluate the ratio $K^{\prime}(k_{\rho})/K(k_{\rho})$ in Eq. (2.24), which depends on several elliptic functions as seen in Eqs. (2.10) and (2.12). Currently, commercial and free and open source software (FOSS) are available to aid in such calculations: Mathematica, Sage and SciPy999 Jacobian elliptic functions, the nome function and their inverses are available through the library mpmath

to name a few examples [22, §22.22]. However, standard scientific calculators and standard office software (with MS Office and LibreOffice as common examples) are not able to compute such special functions, and therefore, approximations using trigonometric/hyperbolic functions are needed.

One convenient way to find such approximations is by using the nome function $q=Q(k)$ [26, §VI.3 Eq. (16)], [22, Eqs. (19.2.9) and (22.2.1)]

[TABLE]

as a way to compute the ratio101010 this ratio is closely related to the lattice parameter $\tau=\bm{i}K^{\prime}(k)/K(k)$ and the nome $q=\exp(\bm{i}\pi\tau)$ [22, §20.1, §22.1, and Eqs. (22.2.1) and (22.2.12)]. $K^{\prime}(k)/K(k)$. This is because the Taylor expansion of the nome function $Q(k)$ is known and converges relatively fast [27, below Eq. (12)] [22, Eq. (19.5.5)]

[TABLE]

Therefore, for a sufficiently small modulus $k$ or $k^{\prime}$, the ratio $K^{\prime}(k)/K(k)$ can be approximated with enough accuracy by using only the first term of the previous series

[TABLE]

Also the following alternative representations of the moduli $k_{\rho}$ and $k_{\rho}^{\prime}$ are useful for finding the desired approximations

Lemma 2.3**.**

The modulus $k_{\rho}$ and the complementary modulus $k_{\rho}^{\prime}$ have alternative representations to those given in Eq. (2.10a). For $k_{\rho}$ the proposed alternative representation depends on the gap between consecutive bands $g=W-2w_{E}$, when the width of both electrode bands is equal $2w_{E}=2w_{A}=2w_{B}$

[TABLE]

For $k_{\rho}^{\prime}$ the proposed alternative representation is given directly by

[TABLE]

Proof.

Eq. (2.38) is obtained from Eqs. (2.10a) and (2.12) when $2w_{A}=2w_{B}=2w_{E}$, later substituting $\operatorname{cd}()$ by $\operatorname{sn}()$

[TABLE]

(which is obtained from translation by quarter/half period [22, Table 22.4.3] and $\operatorname{sn}(-\bm{z},k)=-\operatorname{sn}(\bm{z},k)$ [22, Table 22.6.1]) and finally by substituting $2w_{E}=W-g$.

Similarly, Eq. (2.39) is obtained from Eqs. (2.10a) and (2.12), and later using the identity [28, Eqs. (1.10) and (4.1)]

[TABLE]

∎

The last step remaining is to find trigonometric/hyperbolic approximations of the moduli $k_{\rho}$ and $k_{\rho}^{\prime}$, so they can be plugged into Eqs. (2.37). This is shown below for the case of tall electrochemical cells with small and large electrode bands.

2.3.1 Case of tall cells

In this case, the Jacobian elliptic functions $\operatorname{sn}()$ and $\operatorname{sd}()$ in Lemma 2.3 can be approximated by their trigonometric counterpart $\sin()$. In the same manner, $\operatorname{cn}()$ can be approximated by $\cos()$. See [22, Eqs. (22.10.4)–(22.10.6)] or [29, Eqs. (10.1)–(10.3)].

Theorem 2.4**.**

In case of tall electrochemical cells ($H$ is large), the modulus $k_{\rho}$ and the complementary modulus $k_{\rho}^{\prime}$ are given by

[TABLE]

Therefore, the ratio $K^{\prime}(k_{\rho})/K(k_{\rho})$ for large and small electrodes is given respectively by

[TABLE]

Here $g=W-2w_{E}$ corresponds to the gap between consecutive bands, when they have equal width $2w_{E}=2w_{A}=2w_{B}$.

The results in Eqs. (2.42a) and (2.43a) were first obtained by [8, Eq. (32)], for the case of large electrodes. Almost two decades later, Eqs. (2.42b) and (2.43b) were obtained by [10, Eqs. (2), (6) and (7)], for the case of small electrodes. See Supplementary information §S1.5.1 for a proof of both results. Also see Table 1 in Results and discussion §3.5 to know the width of the electrode bands for which the approximations hold.

2.3.2 Case of shallow cells

Similarly, hyperbolic approximations [29, Eqs. (11.1)–(11.3)] of Jacobian elliptic functions in Lemma 2.3 can be used for shallow electrochemical cells with small and large electrode bands.

Theorem 2.5**.**

In case of shallow electrochemical cells ($H$ is small), the modulus $k_{\rho}$ and the complementary modulus $k_{\rho}^{\prime}$ are given by

[TABLE]

Therefore, the normalized current in Eq. (2.24) for large and small electrodes is given respectively by

[TABLE]

Here the gap between consecutive bands equals $g=W-2w$, when they have equal width $2w=2w_{A}=2w_{B}$.

See Supplementary information §S1.5.2 for a proof. Also see Table 1 in Results and discussion §3.5 to know the cell dimensions for which the approximations hold.

3 Results and discussion

The scripts111111All python scripts are available online at https://gitlab.com/cfgy/elektrodo/tree/p2019mar. for obtaining the plots and simulations in this section were written in Python using the SciPy stack [30]. In particular, Jacobian elliptic functions (required for plotting the theoretical concentration profile, current density and current) were obtained from the library mpmath [31]. The simulations of the concentration profile and current were computed using the finite element library FiPy [32].

3.1 Concentration profile

The concentration profile in Eq. (2.17) was constrasted against simulations of Eqs. (2.8) for different values of aspect ratio $H/W\in\mathopen{]}0,$1.25$\mathclose{[}$ and electrode widths $2w_{A}/W=2w_{B}/W\in\mathopen{]}0,1\mathclose{[}$. In 43 of 45 analyzed cases, the maximum absolute error was not greater than $\approx$0.0045$$ (which corresponds to two decimal places of accuracy), confirming the correctness of the expression obtained in Eq. (2.17). See Table S2 in Supplementary information §S2.1.4 for more details.

Fig. 3 shows the behavior of the concentration profile under the cases of high and low aspect ratios $H/W$ (simulation errors and other details are shown in Supplementary information §S2.1.3). In case of high aspect ratio $H/W$ (Fig. 3(a)), the concentration becomes uniform far from the surface of the electrodes (near the roof of the cell) and full radial diffusion is present, thus mimicking the behavior of semi-infinite configurations [18, Figs. 6 and 9] [33, p. 6406 and Fig. S3] [34, Fig. 39] [16, 309]. Moreover, according to [19, Theorem 2.6], $H/W\gtrsim 1$ corresponds to the condition under which the concentration of a finite-height cell and its semi-infinite counterpart have similar profiles, which agrees with Fig. 3(a).

In the case of low aspect ratio $H/W$ (Figs. 3(b)), the concentration profile doesn’t exhibit a region of uniform concentration, and radial diffusion is truncated by the low roof the cell. Similar results obtained in [18, Fig. 6] [19, §3.2] [33, p.6406, Fig. S3] [34, Fig. 39] [16, p.309] confirm these facts. Moreover, when the aspect ratio is low enough (compare supplementary Figs. 3(b) and 3(c)), the vertical gradient of concentration is almost inexistent, being the horizontal gradient the only one that is clearly appreciable, which resembles that of a parallel-plates configuration in the gap between consecutive electrode bands.

3.2 Current density

The aspect ratio of the unit cell also affects the shape of the current density on the IDAE.

Fig. 4 shows the normalized current density, at the bottom of the unit cell, for different aspect ratios $H/W$, when the width of the electrode bands equals $2w_{A}=2w_{B}=$0.5$W$. As predicted by Eq. (2.20), the current density becomes singular at the edges of the electrode bands. Also, it can be seen that the current density is highly non-uniform near the edges, and becomes approximately uniform near the center of the electrode bands.

Fig. 4 also shows that the magnitude of the plateau of current density varies according to the aspect ratio $H/W$ of the unit cell. For $H/W\gtrsim 1$ the plateau remains the highest and the current density is approximately independent of $H/W$, mainly because radial diffusion is completely developed (see Fig. 3(a)), as in the case of semi-infinite cells. In the range of approximately 0.3$\lesssim H/W\lesssim 1$, the current density decreases slowly as the aspect ratio $H/W$ decreases, due to truncation of the radial diffusion caused by the lower roof of the cell. In the range of approximately $0\leq H/W\lesssim$0.3, the current density decreases quickly as $H/W$ decreases, with plateaus approaching zero for very low $H/W$, mainly because the vertical diffusion in the cell becomes almost nonexistent, and only horizontal diffusion between consecutive bands takes place (see Fig. 3(b)).

3.3 Current per band

The normalized current in Eq. (2.24) was contrasted against simulations of Eqs. (2.8) for different values of aspect ratio $H/W\in\mathopen{]}0,$1.25$\mathclose{[}$ and electrode widths $2w_{A}/W=2w_{B}/W\in\mathopen{]}0,1\mathclose{[}$, see Fig. 5. In 44 of 45 analyzed cases, the maximum absolute error was not greater than $\approx$0.0078$$, thus confirming the correctness of Eq. (2.24). The theoretical result also agrees with simulation results previously reported in the literature [18, Fig. 7a] [19, Fig. 7a], which are also shown in Fig. 5. For more details on the data see Tables S4, S4 and S5 in Supplementary information §S2.1.4.

3.3.1 Effect of the counter electrode in the collection efficiency

The expression in Eq. (2.24) shows that the currents at bands of both arrays are equal in magnitude but opposite $i_{f}^{A}=-i_{f}^{B}$. This suggests that 100% collection efficiency is a necessary condition for this expression to hold, which is immediately satisfied when the IDAE operates with internal counter electrode [8, start of p. 280] [35, p. 7558 mid col. 2].

However, in the case of external counter electrode, the collection efficiency may not reach 100% [8, start of p. 280] [35, p. 7558 mid col. 2]. In case the roof of the cell is low ($H/W\lesssim 1$), the collection efficiency is close to 100% [18, Fig. 5b] [16, Fig. 5e] and Eq. (2.24) approximately holds. However, when the roof of the cell becomes taller ($H/W\gtrsim 1$), the collection efficiency departs from 100% and decreases [18, Fig. 5a] [16, Fig. 5e]. In this last case, a correction that takes into account the effect of external counter is needed to accurately predict the current through the IDAE. This kind of correction was done semi-empirically in [8, Eq. (33)] and later in [10, Eqs. (13) and (20)], both for the case of semi-infinite cells ($H\to+\infty$).

3.3.2 Influence of the cell geometry

According to Corollary 2.2, the current in steady state through the IDAE depends on the ratio $2K^{\prime}(k_{\rho})/K(k_{\rho})$, which is a purely geometrical factor, since the modulus $k_{\rho}$ depends only on the relative dimensions of the unit cell $2w_{A}/W$, $2w_{B}/W$ and $H/W$, as seen in Eqs. (2.10) and (2.12). This implies that, for known electrochemical species in the cell ($D_{\sigma}$ and $n_{e}$ are known) and for known potentials applied to the IDAE (therefore $c_{\sigma,f}^{A}$ and $c_{\sigma,f}^{B}$ are also known), the performance of the cell can be optimized just by adjusting the widths of the electrode bands and the aspect ratio of the unit cell.

Visualization of the ratio $2K^{\prime}(k_{\rho})/K(k_{\rho})$, as a function of its three geometrical parameters ($2w_{A}/W$, $2w_{B}/W$ and $H/W$), is helpful to find the right combination of cell dimensions that allow an optimal performance. However, this ratio is difficult to visualize, since it corresponds to a three-dimensional scalar field. Therefore, Fig. 6 shows two-dimensional slices and contour lines of the ratio $2K^{\prime}(k_{\rho})/K(k_{\rho})$, as a function of the relative dimensions of the cell. These slices were chosen such that they provide enough information to understand how the normalized current $2K^{\prime}(k_{\rho})/K(k_{\rho})$ behaves for different values of $2w_{A}/W$, $2w_{B}/W$ and $H/W$.

Figs. 5 and 6 show that the current increases as both the width of the electrode bands and the aspect ratio of the unit cell increase (similarly occurs for the case of elevated electrodes in the simulations of [15, Fig. 5] [33, Table 1] [34, Table 2]). For aspect ratios $H/W\gtrsim 1$, the current becomes independent of $H/W$, and only depends on the width of the electrode bands. For 0.3$\lesssim H/W\lesssim 1$ and fixed band widths, the current decreases slowly as the aspect ratio $H/W$ decreases. For $0\leq H/W\lesssim$0.3 and fixed band widths, the current decreases much faster as the ratio $H/W$ decreases. This variation in the magnitude of the current, in relation to the aspect ratio $H/W$, has the same explanation as the one given for the current density in the previous section, and relates to the truncation of the radial diffusion by a low roof of the cell.

Also, Figs. 5 and 6(a) confirm that there is a certain limit about $H/W\gtrsim 1$ where the cell is so high that diffusion is not affected by the roof of the cell, and the current approaches that of semi-infinite diffusion, as predicted earlier in [18, §3.2 and Fig. 7a] [19, §3.3 and Fig. 7a] [16, p. 309 and Fig. 5]. Similar results have been also found through simulations for the case of elevated electrodes [15, p.451 and Fig. 5b].

Finally, note that the amount of material used for fabricating the electrodes can be also optimized for a given current. This amount of material is proportional to $m=2w_{A}/W+2w_{B}/W$ and corresponds to an anti-diagonal line in the domain of Figs. 6(b), 6(c) and 6(d). Therefore, the minimum amount of utilized material that produces a desired current is obtained when $2w_{A}=2w_{B}$. This can be seen by selecting a desired iso-current line, and intersecting it with the anti-diagonal $m=2w_{A}/W+2w_{B}/W$. This produces two intersection points, which are symmetric with respect to the diagonal line $2w_{A}/W=2w_{B}/W$. The minimum of $m$ is obtained by moving the anti-diagonal towards the origin, which makes the two intersection points on the iso-current line converge to a single point, thus obtaining $2w_{A}=2w_{B}$.

3.4 Voltammogram shape

Consider the current121212 for $\pm$ or $\mp$, the upper sign corresponds to $\sigma=O$, and the lower sign, to $\sigma=R$. through one array of the IDAE, consisting of $N_{E}$ bands of length $L$, see Eq. (2.24)

[TABLE]

Here one can distinguish two clear factors with different roles: A geometrical factor $N_{E}L\;2K^{\prime}(k_{\rho})/K(k_{\rho})$ and an electrochemical factor $Fn_{e}D_{\sigma}[c_{\sigma,f}^{E}-c_{\sigma,f}^{E^{\prime}}]$.

The inherent shape of the voltammogram is solely due to the electrochemical factor, which is proportional to the difference of concentrations in Eqs (2.28) and (2.34). This inherent shape can be amplified or atenuated by the geometrical factor, which is shown in Figs. 5 and 6, and expressed approximately in Eqs. (2.43) and (2.45).

Therefore, here we examine the shape of the voltammogram just by looking at the behavior of the difference of concentrations in two cases: With external and internal counter electrodes.

The first, and the most commonly found in the literature, corresponds to the case where the counter electrode is external to the IDAE, which means that each array of the IDAE is potentiostated individually (commonly, the potential of one array is scanned, while the complementary array is fixed to a sufficiently negative potential).

Fig. 7(a) shows the plot of the normalized difference of concentrations when using an external counter electrode, see Eq. (2.28). This normalized difference (and thus the current) is unipolar (its is either always positive or always negative), and it increases with the weighted sum of initial concentrations of both electrochemical species. Moreover, the steady-state current reaches plateaus (limiting current) that are proportional to this weighted (total, when $D_{O}=D_{R}$) sum of concentrations [35, p. 7558 start of col. 2], when the applied potential at the scanning electrode exceeds $|\eta_{f}^{E}-\eta_{\text{null}}|\gtrsim 4$.

The second is the case where the counter electrode is internal to the IDAE, which means that one of the arrays is potentiostated at will, while the complementary array performs as counter electrode (its potential is controlled automatically by the potentiostat).

Fig. 7(b) shows the plot of the normalized difference of concentrations when using an internal counter electrode, see Eq. (2.34a). This normalized difference (and thus the current) is bipolar (it is positive and negative in the same plot), and presents plateaus when the applied potential is sufficiently high $|\eta_{f}^{E}-\eta_{\text{null}}|\gtrsim 4$. Ideally, the magnitude of the difference of concentrations (steady-state current) should increase as the ratio $r_{\lambda}=(D_{\lambda^{\prime}}\bar{c}_{\lambda^{\prime},i})/(D_{\lambda}\bar{c}_{\lambda,i})$ between the electrochemical species increases, but it is actually limited by Eq. (2.32) (dashed line), due to depletion of its limiting species $\lambda$ [19, Fig. 2]. This causes the limiting current to be proportional to the concentration of the limiting species, instead of the weighted sum of concentrations of both species [35, p. 7558 mid col. 2] [36, Fig. 2] [25, Fig. 3.4]. This fact has been also discussed in [10, §2.3] and [19, §2.3] and implies that simultaneous presence of both electrochemical species is a necessary condition to obtain steady state currents.

Note that, in case the internal counter electrode serves as reference electrode, only a voltage (difference of potentials) can be applied to the cell. Fig. 8(a) shows that a sigmoidal shape is maintained under this condition, of which its plateaus (limiting current) are also proportional to the concentration of the limiting species, see Eq. (2.34b). Fig 8(b) shows how the applied voltage is distributed among the potentials of the working and counter electrodes. When the voltage applied to the cell increases in one direction, the potential at one electrode increases indefinitely, whereas the potential at the other electrode saturates in the opposite direction. This is due to depletion of the limiting species $\lambda$ at the electrode with greatest potential (in absolute value). This phenomenon has been observed experimentally in [36, Fig. 3] [25, Fig. 3.6], where figures similar to Fig. 8(b) were obtained. See Supplementay information §S2.2.1 for more details on how Figs. 8(a) and 8(b) were generated.

Finally, we show that the models in Eqs. (2.28) and (2.34) are also applicable to fit experimental data. Combining Eq. (3.1) with Eq. (2.28), for the case of external counter electrode, leads to151515 the upper sign corresponds to the case where the concentration of oxidized species at the complementary array is zero (at very negative potential), and the lower, to the case where the concentration of reduced species is zero (at very positive potential), see Corollary 2.2.

[TABLE]

where the voltammogram is mirrored horizontally and vertically when changing the polatiry at the complementary electrode [37, §3.3]. Fig. 9(a) shows that the model in Eq. (3.2) correctly fits the experimental data for the generator in [8, Fig. 7], whereas the data for the collector is slightly overestimated, thus showing that the collection efficiency is near 100%.

For the case of internal counter electrode with bands of equal width, we combine Eqs. (3.1) with (2.34b) when $D_{O}\bar{c}_{O,i}^{\text{whole}}=D_{R}\bar{c}_{R,i}^{\text{whole}}$, obtaining

[TABLE]

Fig. 9(b) shows that this model also fits correctly the experimental data in [35, Fig. 2] [25, Fig. 3.12], both at the linear and the limiting current regions. For details on the experimental data used in both curve fittings see Supplementary information §S2.2.2.

Remark 3.1**.**

Note that the models in Eqs. (2.28) and (2.34) for the difference of concentrations (shape of the steady-state voltammogram), in particular Eqs. (3.2) and (3.3), are valid for any electrochemical cell satisfying the following conditions: (i) The cell has reversible electrode reactions (Nernstian boundary conditions). (ii) The current is proportional to the difference between concentrations at both electrodes. (iii) The weighted sum of concentrations at each electrode satisfies a relation similar to Eq. (2.5). (iv) In case of internal counter electrode, the concentrations at both electrodes satisfy a relation similar to Eq. (2.29).

This is because the previous conditions are properties that depend only on the boundary conditions at the electrodes, which are independent of the whole domain of the cell.

3.5 Approximations for shallow and tall cells

The approximations of the normalized current $2K^{\prime}(k_{\rho})/K(k_{\rho})$ in Eqs. (2.43) for tall electrochemical cells are already available in the literature, and were obtained first by Aoki and colleagues, for the case of large electrodes [8, Eq. (32) and the equation above it], and later by Morf and colleagues, for the case of small electrodes [10, Eqs. (6) and (7)]. Therefore, these results serve as a validation of the exact results obtained in Corollary 2.2.

Fig. 10 shows the approximations for tall cells $H/W>1$ as well as their relative errors. Here the current becomes independent of $H/W$ and depends only on the relative size of the electrodes. The approximation for large electrodes becomes accurate (error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$) for bands of relative width $>$0.46. In the case of small electrodes, the approximation has an error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$ for bands of relative width $<$0.56. The regions of approximation for both sizes of electrodes overlap, thus covering all cases of $H/W>1$ with a relative error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$ in the current. See Table 1 for a summary and Supplementary information §S2.2.3 for details on the errors.

Approximations of the normalized current $2K^{\prime}(k_{\rho})/K(k_{\rho})$ in Eqs. (2.45) for shallow electrochemical cells are results that have not been published before. Fig. 11 shows the approximations for shallow cells $H/W\leq 1$ as well as their relative errors. The approximation for large electrodes becomes accurate (error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$) for combinations of $(2w_{E}/W,H/W)$ approximately at the right of the line given by the points (1, 0) and ($0.28$, 1), where the line that passes through the points $(1,0)$ and $(p,1)$ is given by

[TABLE]

In the case of small electrodes, the approximation has an error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$ for combinations of $(2w_{E}/W,H/W)$ approximately at the left of the line given by the points (1, 0) and ($0.36$, 1). The regions of approximation for both sizes of electrodes overlap, thus covering all cases of $H/W\leq 1$ with a relative error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$ in the current. See Table 1 for a summary and Supplementary information §S2.2.3 for details on the errors.

4 Conclusions

Thanks to Jacobian elliptic functions, it is possible to transform the unit cell from the IDAE domain into a parallel-plates domain. In this last domain, the solution of the diffusion equation in steady state is simple, and corresponds to a linear interpolation of the concentrations on both plates. This solution is transformed back into the IDAE domain, leading to an analytical result for the concentration profile that depends on elliptic functions and integrals. Both, current density and current were derived from this concentration profile.

The results for the concentration profile, current density and current depend geometrically on the relative dimensions of the cell, not on absolute dimensions. Their behavior approaches that of an IDAE in a semi-infinite cell as it becomes taller (approximately when the cell is taller than the separation between centers of consecutive bands).

The shape obtained for the voltammogram is sigmoidal and can be unipolar (it has either always positive or always negative currents) or bipolar (presenting positive and negative currents) depending on whether the IDAE is bipotentiostated using an external counter electrode, or potentiostated using one of its arrays as internal counter electrode. In case of using an external counter electrode, the plateaus of current (limiting current) are proportional to the weighted sum of initial concentrations (total initial concentration when the diffusion coefficients are equal). Whereas, in case of using an internal counter electrode, the plateaus of current (limiting current) are proportional to the limiting species, which will be the species of least concentration if the diffusion coefficients are equal.

Approximations for the exact results were found. Trigonometric and hyperbolic functions were used to approximate the cases of tall and shallow cells respectively. The approximations are accurate with a relative error smaller than $\pm 5\%$ with respect to the exact values. Finally, when the approximations are used in combination, they cover all possibilities of interest for IDAE in confined cells.

Acknowledgements

The authors deeply appreciate the aid and comments of Dr. Mithran Somasundrum, which helped to improve the quality of this manuscript. The authors would like to thank the financial support provided by King Mongkut’s University of Technology Thonburi through the KMUTT 55th Anniversary Commemorative Fund, and the Petchra Pra Jom Klao Ph. D. scholarship (Grant No. 28/2558) for sponsoring CFGY. Finally, the authors acknowledge the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, Ministry of Education, Thailand.

S1 Additional proofs

S1.1 Transformation of the unit cell domain

S1.1.1 From $\bm{r}$ to $\bm{v}$ domain

Considering the special values of the function $\operatorname{arcsn}()$ [22, Table 22.5.1]

[TABLE]

one can construct a function $(T_{r}^{v})^{-1}$ that maps the upper half-plane of $\bm{v}$ into the IDAE domain $\bm{r}$, as shown in Fig. S1. The scale and translation of the function $\operatorname{arcsn}()$ must be chosen such that $\bm{v}_{a}=-1$ is mapped to $\bm{r}_{a}=0$ and $\bm{v}_{b}=1$ is mapped to $\bm{r}_{b}=W$, which leads to

[TABLE]

which is corresponds to Eq. (2.9d) due to the quarter- and half-period properties [22, Table 22.4.3]

[TABLE]

The appropriate modulus $k_{r}$ can be obtained by forcing $\bm{v}_{m}=-1/k_{r}$ to be mapped to the upper-left corner $\bm{r}_{m}=\bm{i}H$

[TABLE]

which leads to Eq. (2.10b), by applying the inverse nome function Eq. (2.35) [22, Eqs. (19.2.9) and (22.2.1)]. Therefore, the function $T_{r}^{v}$ in Eq. (S1.2b) transforms the IDAE domain $\bm{r}$ into the upper half-plane $\bm{v}$ as shown in Fig. S1.

Finally, the point $\bm{v}_{\alpha}$, in Eq. (2.12a), is obtained directly by evaluating Eq. (S1.2b). Whereas the point $\bm{v}_{\beta}$, in Eq. (2.12b), is obtained by evaluating Eq. (S1.2b), and by combining the fact that $\operatorname{sn}()$ is an odd function and Eq. (S1.3).

S1.1.2 From $\bm{v}$ to $\bm{\omega}$ domain

The function $T_{v}^{\omega}$ in Eq. (2.9c), which is also shown below for convenience,

[TABLE]

corresponds to a Möbius function and it is constructed such that: (i) it maps the upper half-plane of $\bm{v}$ into the upper half-plane of $\bm{\omega}$ and (ii) it maps the half band of $A$ to the negative real axis of $\bm{\omega}$ and the half band of $B$ to the real interval $\bm{\omega}\in[1,\bm{\omega}_{b}]$, see Fig. S1.

Condition (ii) is immediately achieved, since Eq. (S1.5) maps $\bm{v}_{a}$, $\bm{v}_{\alpha}$ and $\bm{v}_{\beta}$ into $\bm{\omega}_{a}=\bm{\infty}$, $\bm{\omega}_{\alpha}=0$ and $\bm{\omega}_{\beta}=1$. Condition (i) can be analyzed by rewritting Eq. (S1.5) as a composition of translations, rotations and scalings, and inversion

[TABLE]

The fact that $\bm{v}_{a}\in\mathds{R}$, $\bm{v}_{a}-\bm{v}_{\alpha}<0$ and $p>0$ ensures that the upper half-plane of $\bm{v}$ is mapped to the upper half-plane of $\bm{\omega}$. See [23, Eq. (5.7.3)], [26, §V.2 Eqs. (6), (7) and (10)] or [24, Examples 5.3 and 5.4] for more details on decomposition and mapping of Möbius functions.

S1.1.3 From $\bm{\omega}$ to $\bm{\rho}$ domain

The function $T_{\omega}^{\rho}$ in Eq. (2.9b), which is written below for convenience,

[TABLE]

is in charge of mapping the upper half-plane of $\bm{\omega}$ into the parallel-plates domain $\bm{\rho}$, see Fig. S1. This is achieved in two stages: (i) The upper half-plane of $\bm{\omega}$ is mapped to the first quadrant of $\sqrt{\bm{\omega}}$, such that the half band of $A$ is mapped to the positive imaginary axis of $\sqrt{\bm{\omega}}$ and the half band of $B$ is mapped to the real interval $\sqrt{\bm{\omega}}\in[1,\sqrt{\bm{\omega}_{b}}]$. (ii) The first quadrant of $\sqrt{\bm{\omega}}$ is mapped to the parallel-plates domain $\bm{\rho}$ by using the special values of $\operatorname{arcsn}()$ in Eqs. (S1.1) or [22, Table 22.5.1].

The scaling and translation of the function $\operatorname{arcsn}()$ are chosen such that $\sqrt{\bm{\omega}_{\alpha}}=0$ is mapped to $\bm{\rho}_{\alpha}=0$ and $\sqrt{\bm{\omega}_{\beta}}=1$ is mapped to $\bm{\rho}_{\beta}=1$, which leads to Eq. (S1.7). The appropriate modulus $k_{\rho}$ is obtained by choosing $\sqrt{\bm{\omega}_{b}}=1/k_{\rho}$, such that $\sqrt{\bm{\omega}_{b}}$ be mapped to the upper right corner $\bm{\rho}_{b}$ of the parallel-plates domain, leading to Eq. (2.10a).

Finally, the point $\bm{\rho}_{a}$, in Eqs. (2.13), can be obtained directly by evaluating $\sqrt{\bm{\omega}_{a}}=\infty$ in Eq. (S1.7), by using the properties in (S1.1).

S1.1.4 Conformality of the transformation

The conformality of $T_{r}^{\rho}$ comes from the fact that $T_{r}^{\rho}$ has non-zero complex derivative in the interior of the IDAE domain. This can be seen from Eq. (2.20) or (S1.14), where the zeros of $\operatorname{nd}(K(k_{r})2\bm{r}/W,k_{r})$

[TABLE]

lay only on the top vertices of the IDAE domain. These zeros are obtained by using [22, Table 22.4.2] and Eq. (S1.4).

S1.2 Derivative of the domain transformation

This proof concerns about obtaining the complex derivative of the domain transformation $T_{r}^{\rho}$ in Eqs. (2.9), which is written below for convenience

[TABLE]

First, the complex derivative of each component of $T_{r}^{\rho}$, in Eq. (S1.9), is taken

[TABLE]

where

[TABLE]

since $1/k_{\rho}^{2}=\bm{\omega}_{b}=T_{v}^{\omega}(\bm{v}_{b})$. Later, two components are combined

[TABLE]

which leads to an expression that consists of two complex branches: $\operatorname{e}^{-\bm{i}(\pi/2+\pi\mathds{Z})}=\pm\bm{i}$. Using the identity $1-\operatorname{cd}(\bm{u},k)^{2}={k^{\prime}}^{2}\operatorname{sd}(\bm{u},k)^{2}$ from [22, Eq. (22.6.4)] [28, Eq. (1.1)] and $\bm{v}_{a}=-1$ and $\bm{v}_{b}=1$ from Fig. S1

[TABLE]

the final expression for the complex derivative can be obtained

[TABLE]

In case $c_{O,f}^{B}>c_{O,f}^{A}$, the current density at each electrode band $B$ must be positive

[TABLE]

In order to achieve this, the $+\bm{i}$ branch of the complex derivative must be chosen as the one carrying physical meaning, such that the current density be positive on each electrode band $B$.

S1.3 Differences of concentrations when using internal counter electrode

S1.3.1 Difference with respect to the average in the unit cell

Inside the unit cell the average concentration satisfies [19, Remark 2.2]

[TABLE]

S1.3.2 Difference with respect to the average in the whole cell

Note that the average in the unit cell may not equal that in the whole cell

[TABLE]

where $x_{o}$ is some value within the diffusion layer, due to the first mean value theorem for integrals [22, Eq. (1.4.29)] (note that the choice of $k$ and $x_{o}$ depends on $z$).

Therefore, the larger the number of bands $N_{E}$, the closer the averages $\bar{c}_{\sigma,i}^{\text{unit}}=\bar{c}_{\sigma,i}^{\text{whole}}$. This leads to the following equality when the bands satisfy $2w_{A}=2w_{B}$

[TABLE]

due to the result in the previous section, which allows to estimate the unknown concentration on the internal counter electrode.

S1.3.3 Limits for the differences of concentration

To obtain the limits for the difference of concentration in steady state, one considers Eq. (S1.18) with non-negative concentrations on all electrodes

[TABLE]

where $E^{\prime}$ are the complementary bands of $E\in\{A,B\}$ and $\sigma^{\prime}$ is the complementary species of $\sigma\in\{O,R\}$.

The limits for the concentration of species $\sigma^{\prime}$ may also affect the limits for the concentration of species $\sigma$. This can be seen by applying the weighted sum of concentrations in Eq. (2.3b) at the bands $E$ and $E^{\prime}$

[TABLE]

which agrees with [10, Eqs. (15) and (16)] when $D_{O}=D_{R}$.

Combining the last four equations leads to

[TABLE]

which can be summarized as

[TABLE]

when defining $D_{\lambda}\bar{c}_{\lambda,i}^{\text{whole}}=\min(D_{O}\bar{c}_{O,i}^{\text{whole}},D_{R}\bar{c}_{R,i}^{\text{whole}})$.

S1.4 Difference of potential (voltage) when using internal counter electrode

Consider Eq. (2.34a) with $c_{\sigma,f}^{E}-c_{\sigma,f}^{E^{\prime}}=2[c_{\sigma,f}^{E}-\bar{c}_{\sigma,i}^{\text{whole}}]=-2[c_{\sigma,f}^{E^{\prime}}-\bar{c}_{\sigma,i}^{\text{whole}}]$, which is written below for convenience

[TABLE]

Now taking their inverses

[TABLE]

and combining both expressions

[TABLE]

leads to

[TABLE]

after reorganizing the factors.

Finally, by using the hyperbolic identity [22, Eq. (4.37.24)]

[TABLE]

one obtains

[TABLE]

which leads to Eq. (2.34b) after applying the relation in Eq. (2.31) $c_{\sigma,f}^{E}-c_{\sigma,f}^{E^{\prime}}=2[c_{\sigma,f}^{E}-\bar{c}_{\sigma,f}^{\text{whole}}]$.

S1.5 Approximations for $k_{\rho}$ and $k_{\rho}^{\prime}$

S1.5.1 Case of tall cells

Consider first the following approximations when the nome $q=Q(k)$ tends to zero [29, (10.1)–(10.3)]

[TABLE]

Since $q_{r}=Q(k_{r})=\exp(-\pi\,2H/W)$ in Eq. (2.10b) tends to zero when $H\to+\infty$, then

[TABLE]

can be approximated by

[TABLE]

which leads to Eqs. (2.42).

S1.5.2 Case of shallow cells

Consider the following approximations when the associated nome $q^{\prime}=Q(k^{\prime})$ is sufficiently small [29, Eqs. (11.1)–(11.3)]

[TABLE]

Since $q_{r}^{\prime}=Q(k_{r}^{\prime})=\exp(-\pi W/2H)$ in Eq. (2.10b) is approximately zero when $H\approx 0$, then $k_{\rho}$ and ${k_{\rho}}^{\prime}$ in Eqs. (S1.30) can be approximated by

[TABLE]

where the last equality was obtained through the identity $2\sinh(\bm{u})\cosh(\bm{u})=\sinh(2\bm{u})$, as shown below for $E\in\{A,B\}$

[TABLE]

Finally, due to Eqs. (2.10b) and (2.35), that is $K(k_{r})/K^{\prime}(k_{r})=-\ln Q(k_{r}^{\prime})/\pi=W/2H$, the desired result in Eqs. (2.44) is obtained.

S2 Numerical calculations

S2.1 Simulations in steady state

S2.1.1 Implementation

A normalized version of the diffusion equation in Eqs. (2.8) was used for the simulations

[TABLE]

where

[TABLE]

The width of each band electrode was taken equal $2w_{A}=2w_{B}=2w_{E}$ for all simulations, and different aspect ratios $H/W$ for the unit cell were considered.

An exponential mesh was used to partition the unit cell, in order to keep the memory usage low while maintaining good resolution near the electrode bands (see [38, Chapter 7] for more details on this kind of mesh). The number of elements of the mesh is $n_{x}\times n_{z}$, of which their growth factors are $r_{x}=r_{z}=r$, and the width and height of its smallest element are $\delta_{x}=\delta_{z}=\delta_{0}$. Additionally, the mesh parameter $R$ is sometimes given, which corresponds to the ratio between the largest $(\delta_{0}\,r^{n_{z}-1}$) and the smallest ($\delta_{0}$) elements in the mesh.

S2.1.2 Selection of the parameters for the exponential mesh

Here four cases were considered: $2w_{E}/W\in\mathopen{\{}$0.20.40.60.8$\mathclose{\}}$ with $H/W=$1.0$$. The mesh for each case was succesively refined until the absolute error between the simulated $\xi_{{}_{\mathrm{sim}}}(x,z)$ and the theoretical concentration profile $\xi(x,z)$ was less than $0.005$ (approximately to two decimal places of accuracy). See Refinement output of Fig. S2 at the end of this section.

The mesh candidates are shown in Fig. S2. The case for $2w_{E}/W=$0.8$$ is the most restrictive of all, and therefore it requires the finest mesh to reach the desired accuracy. This is because the current flowing in this cell is the highest, due to the largest electrode widths, producing the greatest gradient of concentration among all the cases.

See Refinement output of Fig. S2 for values of the normalized current $2K^{\prime}(k_{\rho})/K(k_{\rho})$. Note that the normalized current has been writen in terms of the lattice parameter $\tau$ for brevity, that is $-2\bm{i}\tau=2K^{\prime}(k)/K(k)$ due to Eq. [22, (22.2.12)]. The ratio $K^{\prime}(k_{\rho})/K(k_{\rho})$ obtained in the simulations was computed numerically using the expression

[TABLE]

due to Eqs. (2.26), (2.27) and (2.13).

Finally, the parameters of the finest mesh among the candidates in Fig. S2:

[TABLE]

were selected for all the simulations in this report.

Refinement output of Fig. S2

S2.1.3 Simulation of concentration profile in Fig. 3

Here a validation is done by comparing the theoretical normalized concentration $\xi(x,z)$ in Eq. (2.17) or (S1.9)

[TABLE]

with its simulated counterpart in Eqs. (S2.1) and (S2.2).

The width of each band electrode was taken equal to $2w_{A}=2w_{B}=$0.5$W$ for all simulations and three different aspect ratios for the unit cell $H/W\in\{$\approx 0.33\approx 0.511.0$\}$ were considered. The parameters used to generate each mesh were obtained from Eq. (S2.4).

The results of the simulations are shown in Fig. S3, of which Fig. 3 shows only a part. In all cases the theoretical concentration $\xi(x,z)$ agrees with the simulations $\xi_{\mathrm{sim}}(x,z)$ up to two decimal places, since the absolute error obtained was not greater than $\approx$0.0030$$. The error appears to be greater near the big-sized elements of the mesh, but it actually reaches its extrema at the edges of each electrode band (this is not easy to see from Fig. S3, but was obtained as output data of the simulations, see section Simulation output below).

Simulation output

S2.1.4 Exhaustive simulations

The normalized concentration profile $\xi(x,z)$ and current $2K^{\prime}(k_{\rho})/K(k_{\rho})$ were exhaustively simulated for all combinations of the following electrode widths

[TABLE]

The implementation in Eqs. (S2.1) was used, together with the meshes specified in Table S2 and the selected mesh parameters in Eq. (S2.4). These simulations were constrasted against their theoretical counterparts, and the results are shown in Tables S2, S4 and in Fig. 5.

Table S2 shows that the minimum and maximum errors between the simulated and theoretical concentration profile were not greater than $\approx$0.0045, except for the two first band widths when $H/W\approx$0.05.

Table S4 shows the simulated values of the normalized current and the error with respect to its theoretical counterpart, which was not greater than $\approx$0.0051 for most of the cases (except for the case $H/W\approx$0.05 and $2w_{E}/W=$0.2 with an error of $\approx$0.1279, and for the last six aspect ratios $H/W$ when $2w_{E}/W=$0.8 with an error not greater than $\approx$0.0078).

Table S4 shows exhaustive simulations of the normalized limiting current (case of internal counter electrode) obtained in [19, Fig. 7]. In order to compare the results in Table S4 against the normalized current of this text (Eq. (2.24))

[TABLE]

($D_{\lambda}|c_{\lambda,f}^{E}-c_{\lambda,f}^{E^{\prime}}|=2D_{\lambda}\bar{c}_{\lambda,i}^{\text{whole}}$ for the limiting current case, see Eq. (2.32)) the average flux [19, $\bar{\phi}_{\lambda}^{\lim}$] was related to the current per electrode band $i_{f}^{E}=Fn_{e}\,\bar{\phi}_{\lambda}^{\lim}\,2w_{E}L$, and then replaced in the expression of [19, Fig. 7a]

[TABLE]

showing that both normalized currents are related by a factor of $\pi^{2}\approx 10$. After scaling by this factor, one can see the agreements between Tables S4 and S4. See Fig. 5 for a graphical representation of both tables.

Table S5 shows simulations of the normalized limiting current (case of external counter electrode) obtained in [18, Fig. 7a]. In order to compare against the normalized current of this text (Eq. (2.24))

[TABLE]

($D_{\sigma}|c_{\sigma,f}^{E}-c_{\sigma,f}^{E^{\prime}}|=D_{\sigma}\bar{c}_{\sigma,i}^{\text{whole}}+D_{\sigma^{\prime}}\bar{c}_{\sigma^{\prime},i}^{\text{whole}}$, see Eq. (2.28) when $\pm\eta^{E}\to+\infty$ for the limiting current case) the expressions for [18, $G_{g}$ and $G_{c}$ in p. 171] were rewritten using the notation of this text

[TABLE]

were the normalized current at the generator $G_{g}$ and collector $G_{c}$ have equal magnitude (but opposite sign) in steady state $|G_{ss}|=|G_{g}|=|G_{c}|=:|G_{E}|$. For plotting the data of [18, Fig. 7a] in Fig. 5, the horizontal variable [18, $w_{e}/h_{c}$ in Fig. 7a] was also rewritten according to the notation of this text, that is $2w_{E}/H$ in Table S5, and later related to $H/W$ by

[TABLE]

where $g$ is the gap between electrodes of equal width, such that $W=2w_{E}+g$. Only the data for the case $2w_{E}=g$ ($2w_{E}/W=$0.5$$) was plotted in Fig. 5, which agrees with its theoretical counterpart.

S2.2 Direct evaluations and curve fitting

S2.2.1 Voltammogram and potentials in Figs. 8(a) and 8(b)

Fig. 8(a) was generated from Eq. (2.34b) with $r_{\lambda}=(D_{\lambda^{\prime}}\bar{c}_{\lambda^{\prime},i}^{\text{whole}})/(D_{\lambda}\bar{c}_{\lambda,i}^{\text{whole}})$

[TABLE]

by evaluating several points for $[c_{\lambda,f}^{E}-c_{\lambda,f}^{E^{\prime}}]/2\bar{c}_{\lambda,f}^{\text{whole}}\in\mathopen{]}-1,1\mathclose{[}$, and obtaining a list of points for $\pm(\eta_{f}^{E}-\eta_{f}^{E^{\prime}})$.

The same list of points $[c_{\lambda,f}^{E}-c_{\lambda,f}^{E^{\prime}}]/2\bar{c}_{\lambda,f}^{\text{whole}}\in\mathopen{]}-1,1\mathclose{[}$ was evaluated in Eqs. (S1.24)

[TABLE]

obtaining one list of points for $\mp(\eta_{f}^{E}-\eta_{\text{null}})$ and another list of points for $\mp(\eta_{f}^{E^{\prime}}-\eta_{\text{null}})$. Both of them were plotted against the list of points for $\pm(\eta_{f}^{E}-\eta_{f}^{E^{\prime}})$ resulting in Fig. 8(b).

S2.2.2 Curve fitting in Fig. 9

The data used for curve fitting in Fig. 9(a) was obtained from [8, Fig. 7], from which 16 data points for each curve (generator and collector) were extracted. The extracted data is shown in Table S7.

Similarly, the data used for curve fitting in Fig. 9(b) was obtained from [35, Fig. 2], or equivalently [25, 3.12], from which 16 data points were extracted. The extracted data is shown in Table S7.

S2.2.3 Approximated normalized current in Figs. 10 and 11.

The relative error between the approximated and exact currents, $\tilde{i}_{f}^{E}$ and $i_{f}^{E}$, corresponds to that between its normalized counterparts

[TABLE]

For this reason, it suffices for Figs. 10 and 11 to show the approximated version of the normalized current $2K^{\prime}(k_{\rho})/K(k_{\rho})$, together with its relative error with respect to the exact value, for tall and shallow cells respectively.

The combinations of $(2w_{E}/H,\,H/W)$ that produce a relative error less than \pm 0.05$=$\pm 5\text{\,}\mathrm{\char 37\relax} were considered to define regions where the approximations are valid. These correspond to $2w_{E}/W>$0.46 (large electrodes) and $2w_{E}/W<$0.56 (small electrodes) for the case of tall cells $H/W>1$. In the case of shallow cells $H/W\leq 1$, these regions of validity correspond to the right of the line connecting $(1,0)\to($0.28$,1)$ (large electrodes) and to the left of the line connecting $(1,0)\to($0.36$,1)$ (small electrodes). See Output of Figs. 10 and 11 at the following subsection for numerical results. The combinations of $(2w_{E}/H,\,H/W)$ that produce a relative error of \pm 0.005$=$\pm 0.5\text{\,}\mathrm{\char 37\relax} are also given as a reference in subsection Output of Figs. 10 and 11.

When expressing the ratio $K^{\prime}(k)/K(k)$ (which is involved in the relative error) in terms of the nome function $Q(k)$, or equivalently $Q(k^{\prime})$

[TABLE]

one can realize that the approximations in Eqs. (2.43) and (2.45)

[TABLE]

are due to a two-step approximation process: First, approximation of the nome function by

[TABLE]

and later, the approximations of the moduli $\tilde{k}_{\rho}$ and $\tilde{k}_{\rho}^{\prime}$, which are given in Eqs. (2.42) and (2.44) in the main text.

Therefore, it is possible to analyze the propagation of the relative error, for $k_{\rho}\approx 0$, by looking at the following total ratio

[TABLE]

where a relative error of \pm 0.05$=$\pm 5\text{\,}\mathrm{\char 37\relax} corresponds to the total ratios $0.95$ and $1.05$. The ratios $\ln\tilde{Q}(k_{\rho})/\ln Q(k_{\rho})$ and $\ln\tilde{Q}(\tilde{k}_{\rho})/\ln\tilde{Q}(k_{\rho})$ show the individual contributions of the approximated nome $\tilde{Q}$ and the approximated modulus $\tilde{k}_{\rho}$ to the total ratio $\ln\tilde{Q}(\tilde{k}_{\rho})/\ln Q(k_{\rho})$, and therefore, to the relative error.

This is shown in Fig. S4 for the case of tall cells and large electrodes. The first ratio $\ln\tilde{Q}(k_{\rho})/\ln Q(k_{\rho})<$1.05 (relative error of $5\text{\,}\mathrm{\char 37\relax}$) for widths $2w_{E}/W>$0.94 ($k_{\rho}^{2}\lesssim$0.314), which corresponds to very extreme cases of wide electrodes. This agrees with the relative error of $\approx$4\text{\,}\mathrm{\char 37\relax} in [8, before Eq. (32)] when approximating only the nome function $\tilde{Q}(k_{\rho})$ for gap widths $g/W<$0.059$\Leftrightarrow 2w_{E}/W>$0.941 ($k_{\rho}^{2}<$0.310). Fortunately, the contribution due to the approximation of the modulus $\tilde{k}_{\rho}$ makes the second ratio $\ln\tilde{Q}(\tilde{k}_{\rho})/\ln\tilde{Q}(k_{\rho})<1$ in almost the whole domain $(2w_{E}/W,H/W)$. This helps to extend the region where the total ratio $\ln\tilde{Q}(\tilde{k}_{\rho})/\ln Q(k_{\rho})<$1.05 (relative error less than $5\text{\,}\mathrm{\char 37\relax}$), which corresponds to more reasonable electrode widths $2w_{E}/W<$0.46.

A similar analysis of the propagation of the relative error can be done for $k_{\rho}^{\prime}\approx 0$ when considering the following total ratio

[TABLE]

where the ratios $\ln Q(k_{\rho}^{\prime})/\ln\tilde{Q}(k_{\rho}^{\prime})$ and $\ln\tilde{Q}(k_{\rho}^{\prime})/\ln\tilde{Q}(\tilde{k}_{\rho}^{\prime})$ show the individual contributions of the approximated nome $\tilde{Q}$ and the approximated complementary modulus $\tilde{k}_{\rho}^{\prime}$ to the total ratio $\ln Q(k_{\rho}^{\prime})/\ln\tilde{Q}(\tilde{k}_{\rho}^{\prime})$, and therefore, to the relative error.

This is shown in Fig. S5 for the case of tall cells and small electrodes. Unlike the previous case, the first ratio $\ln Q(k_{\rho}^{\prime})/\ln\tilde{Q}(k_{\rho}^{\prime})>$0.95 (relative error of $-5\text{\,}\mathrm{\char 37\relax}$) for widths $2w_{E}/W<$0.82 (${k_{\rho}^{\prime}}^{2}\lesssim$0.315), which corresponds to a large portion of the parameter domain. This agrees with the statement in [[10](#bib.bib10), before Eq. (5)], which says that the approximation converges rapidly for gap widths $g>$0.1$\cdot 2w_{E}\Leftrightarrow 2w_{E}/W<$0.909 (no quantitative error is specified). Unfortunately, the contribution due to the approximation of the complementary modulus $\tilde{k}_{\rho}^{\prime}$ makes the second ratio $\ln\tilde{Q}(k_{\rho}^{\prime})/\ln\tilde{Q}(\tilde{k}_{\rho}^{\prime})<1$ in most of the domain $(2w_{E}/W,H/W)$. This approximation helps to simplify the expression for the complementary modulus in Eq. (2.42b) at the expense of shrinking the domain where $\ln Q(k_{\rho}^{\prime})/\ln\tilde{Q}(\tilde{k}_{\rho}^{\prime})>$0.95 (relative error of approximation less than $-5\text{\,}\mathrm{\char 37\relax}$), which corresponds to electrode widths $2w_{E}/W\lesssim$0.56.

The propagation of the relative error for cases of shallow cells have not been reported previously in the literature and they are given below as a reference.

Fig. S6 shows the propagation of the total ratio, in Eq. (S2.19), for the case of shallow cells and large electrodes. The first ratio $\ln\tilde{Q}(k_{\rho})/\ln Q(k_{\rho})<$1.05 (relative error of $5\text{\,}\mathrm{\char 37\relax}$) for combinations of $(2w_{E}/W,H/W)$ roughly to the right of the line joining the points $(1,0)\to($0.94$,1)$ ($k_{\rho}^{2}\lesssim$0.316), which again correspond to very wide electrodes. The contribution due to the approximation of the modulus $\tilde{k}_{\rho}$ makes the second ratio $\ln\tilde{Q}(\tilde{k}_{\rho})/\ln\tilde{Q}(k_{\rho})<1$ in most of the domain $(2w_{E}/W,H/W)$. This also extends the region where the total ratio 0.95$<\ln\tilde{Q}(\tilde{k}_{\rho})/\ln Q(k_{\rho})<$1.05 (relative error less than $\pm 5\text{\,}\mathrm{\char 37\relax}$), which corresponds to combinations of $(2w_{E}/W,H/W)$ approximately at the right of the line joining the points $(1,0)\to($0.28$,1)$.

Fig. S7 shows the propagation of the total ratio, in Eq. (S2.20), for the case of shallow cells and small electrodes. The first ratio $\ln Q(k_{\rho}^{\prime})/\ln\tilde{Q}(k_{\rho}^{\prime})>$0.95 (relative error of $-5\text{\,}\mathrm{\char 37\relax}$) for combinations of $(2w_{E}/W,H/W)$ roughly to the left of the line joining the points $(1,0)\to($0.82$,1)$ (${k_{\rho}^{\prime}}^{2}\lesssim$0.315), which corresponds to a large portion of the parameter domain. The approximation of the complementary modulus $\tilde{k}_{\rho}^{\prime}$ makes the second ratio $\ln\tilde{Q}(k_{\rho}^{\prime})/\ln\tilde{Q}(\tilde{k}_{\rho}^{\prime})<1$ in most of the domain $(2w_{E}/W,H/W)$. This approximation helps to simplify the expression for the complementary modulus in Eq. (2.42b) at the expense of shrinking the domain where $\ln Q(k_{\rho}^{\prime})/\ln\tilde{Q}(\tilde{k}_{\rho}^{\prime})>$0.95$$ (relative error of approximation less than $-5\text{\,}\mathrm{\char 37\relax}$), which corresponds to combinations of $(2w_{E}/W,H/W)$ approximately at the left of the line joining the points $(1,0)\to($0.36$,1)$.

Output of Figs. 10 and 11

Output of Figs. S4–S7