KISSA, A SOFTWARE FOR FAST AND ACCURATE SIMULATION OF COMPLEX ELECTROCHEMICAL PROBLEMS

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KISSA, A SOFTWARE FOR FAST AND ACCURATE SIMULATION OF COMPLEX ELECTROCHEMICAL

PROBLEMS

Irina Svir, Alexander Oleinick, Oleksiy V. Klymenko, Christian Amatore

To cite this version:

Irina Svir, Alexander Oleinick, Oleksiy V. Klymenko, Christian Amatore. KISSA, A SOFTWARE FOR FAST AND ACCURATE SIMULATION OF COMPLEX ELECTROCHEMICAL PROBLEMS.

2018-Sustainable Industrial Processing Summit SIPS2018 Volume 2. Amatore Intl. Symp. / on Electrochemistry for Sustainable Development, 2018, Rio De Janeiro, Brazil. �hal-02368461�

2018 SUSTAINABLE INDUSTRIAL PROCESSING SUMMIT AND EXHIBITION Volume 2: Amatore Intl. Symp. on Electrochemistry for Sustainable Development Editors: F. Kongoli, H. Inufasa, M. G. Boutelle , R. Compton, J.-M. Dubois, F. Murad, FLOGEN, 2018

KISSA, A SOFTWARE FOR FAST AND ACCURATE SIMULATION OF COMPLEX ELECTROCHEMICAL PROBLEMS

Irina Svir¹, Alexander Oleinick¹, Oleksiy V. Klymenko^1,2, Christian Amatore¹

1 PASTEUR, Département de chimie, École normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France;

2 University of Surrey, Department Chemical Process and Engineering, Stag Hill Campus, Guildford GU2 7XH, UK

Keywords: KISSA-1D^©, KISSA-2D^©, simulation, ECL, adsorption

ABSTRACT

KISSA, the software developed in our group, provides a general framework to analyze and rationalize 1D- and 2D-electrochemical problems of any complexity within a user-friendly environment. Results of simulation are produced without any intervention of user into numerical part except for defining the sought reaction mechanism using classical chemical formulations and initial values of the initial concertation, expected and equilibrium rate constants, diffusion coefficients, etc. The accuracy of the numerical solution is guaranteed by KISSA through performing calculations using non-uniform time grids and adaptive space grids constructed on the basis of specific geometric and kinetic criteria. This offers a built-in automatic high dynamic resolution at moving acute reaction front readily detected and tracked by the program. The efficiency of this strategy was proven by addressing such sophisticated problems as i) simulation of reaction mechanisms leading to the emission of electrochemiluminescence (ECL) and ii) solution of electrocatalytic problems involving the reactive dynamic adsorption steps etc.

INTRODUCTION

In 2010-2012, we developed a novel general approach for the numerical simulation of electrochemical problems which was subsequently implemented within KISSA-software [1-7]

for electrochemical systems that can be described in one or two spatial dimensions. The software is capable of accurate and efficient simulation of reaction mechanisms consisting of any sequence of hetero- and homogeneous ET steps and first- or second-order homogeneous reactions with rate constants up to and beyond realistically achievable values (i.e., a few 10¹² s^-1 and 10¹¹ M^-1 s^-1, respectively). Later on, additional features were built in to KISSA enabling it to account for the natural convection [7] and pre-scan conditioning of the electrode [7].

Furthermore, kinetically controlled adsorption-desorption of reactants (‘dynamic’ adsorption) as well as chemical and electrochemical reactions in the adsorbed state (including generalised Laviron’s mechanisms [8]) were also implemented thus significantly expanding the range and potential complexity of mechanisms [9] amenable to the latest version (1.2) of KISSA.

The efficiency of this simulation approach stems from a combination of two principal components. The first one consists in the use of conformal mappings of the space developed for most typical micro- and nanoelectrode geometries, including (hemi)sphere, disk, cylinder, band and double bands [3,4]. The second component of the approach is the built-in automatic adaptation of computational grids driven by a novel and original kinetic criterion, which allows swift concentration changes and/or (travelling) reaction fronts in the bulk of the solution to be

identified for subsequent grid refinement in respective areas of the diffusion layer [1-4]. No user intervention is required at any stage of the solution process, so that only the problem formulation must be supplied to the program (i.e., a reaction mechanism, its kinetic parameters, electrode geometry and electrochemical technique to be simulated). Owing to this approach, both electrochemical currents and concentration distributions are simulated with high accuracy.

Moreover, this is true for concentrations of all species regardless of their spatial localisation so that even extremely minute distributions of highly reactive species restricted to exceedingly narrow reaction fronts or kinetic layers are computed with high precision.

Clearly, the described above complex reaction pathways that lead to ECL emission are fully within the scope of the adaptive simulation approach at the core of KISSA software. A specialised version of this original method aimed at ECL systems is presented below. Its high precision and ability to deal with any numerically challenging situations delineated previously are complemented with outstanding performance in terms of minimal computational times and memory occupation without the need for specific skills on the part of potential users. The validity, precision and accuracy of this approach are established against analytical limits obtained for seminal ECL mechanisms under extreme kinetic conditions putting the numerical simulation under the toughest conditions. The advantages of the approach are then illustrated through typical simulations of the three main ECL generation mechanisms [10, 11].

As mentioned above, there are two strategies that underpin the novel simulation approach neither of which depends on the expertise of or specific intervention from the user which allows them to function in a completely automatic fashion within the software. The first of them relies on such specific (quasi)conformal mappings of the simulation space as to generate computational grids commensurate with diffusion patterns imposed by the geometric configuration of the working electrode(s) [12]. Conformal mappings covering the major types of currently employed metallic solid or liquid micro-/nanoelectrodes such as planar, spherical, cylindrical, disk and band electrodes, including purposely or accidentally recessed or protruding ones and their arrays [13] have been reported by us in recent years. The use of coordinate transformations is coupled with the second strategy designed to automatically identify reaction fronts and rapid concentration variations at each time step by detecting spatial locations where reaction term(s) are over a predefined numerical threshold. Whether such spatial regions are adjacent to the electrode surface (classical kinetic layers) or away from it within the diffusion layer (reaction fronts), the second strategy commands appropriate spatial grid refinement (with respect to the magnitude and width of a kinetic zone/reaction front) in order to resolve the local kinetic terms with sufficient precision as to obtain accurate concentration values at the next time step.

In terms of computational grid adaptation, to the best of our knowledge, the majority of previously proposed strategies rely on local concentration gradients numerically estimated on potentially inappropriate computational grids (although perhaps fitted for the preceding time step) necessitating an iterative process to achieve acceptable grid point distribution which may result in long CPU times. On the other hand, the method developed for KISSA (both 1D and 2D versions) employs the notion of “pure kinetics (KP)” introduced by Saveant’s group [14-16]. In this context, computational grids perfectly suited to a kinetic problem at hand are generated based only on local chemical reaction rate terms. The combination of these two approaches ensures that a low-density finite different grid is used where a conformal map is able to effectively resolve the diffusional the problem at hand while the grid point density is automatically increased in the areas where strong reaction fronts develop [1]. Grid adaptation criteria are checked at each time step and mesh point density adjustment is automatically performed as required. This means that any sharp reaction fronts within the diffusion layer, whether stationary or travelling, are accurately tracked both in space and time with appropriate grid density changes. Finally, to maintain the resulting precision, time grid adaptation may also be necessary with its compression required around sharp temporal variations of, e.g., the electrode potential (that is, at scan direction inversion in voltammetry or potential steps in chronoamperometry) [4]. This helps reduce computational burden and, last but not least, avoid

numerical oscillations when discontinuous (potential steps) or non-smooth (e.g., at inversion potentials in voltammetry) variations of the electrode potential are involved.

In this article we consider the most interesting and complex mechanism leading to ECL since ECL has found a lot of analytical and bioanalytical application in modern electrochemistry.

THEORY

Electrochemiluminescence (ECL) has enjoyed considerable interest for fundamental reasons rooted in photochemistry and physical electrochemistry as well as for various analytical purposes including important biological analyses. ECL belongs to the general class of chemiluminescence in which an electronically excited state, S*, is generated by kinetically shunting a would-be highly exergonic electron transfer (ET) by a much less exergonic one. The excited species then decays to a lower energy level through the emission of a photon the energy of which is close to that of the difference in enthalpies. Therefore ECL is very well suited for probing the Marcus inverted region by invoking competition between ET leading to fundamental states and ET resulting in excited product states followed by radiative decay.

Annihilation ECL was the first type of ECL reaction mechanisms to be investigated since it helped improve the fundamental understanding of highly energetic outer-sphere ET reactions [17]. In this mechanism, an excited state emitting light upon the return to its ground state is formed in solution from an oxidized and a reduced species that are sequentially generated at the same working electrode surface by fast specific potential modulation. Alternatively, the two oxidized/reduced species could be generated at two closely positioned microelectrodes, thus leading to a steady state ECL flux [18].

In the late 1970s, another way of ECL generation was discovered in which an inert parent luminophore molecule (or its reduced or oxidised form) reacts with a (oxidised or reduced, respectively) co-reactant leading ultimately to an emissive decay of the excited state via a cascade of reactions [19]. This quickly found applications for analytical detection of minute quantities of analytes that modulate ECL light intensity through interference in the reaction mechanism. The particularly low limits of ECL detection stem from extremely low noise levels in the optical output (i.e. the ECL yield) because the light signal is practically uncoupled from the electrical input (i.e. the electrochemical trigger) thus precluding the transmission of significant electrical noise. Many reactant/co-reactant couples have been reported for different specific applications. In fact, an excellent review about many possible solutions involving this strategy has been published recently [20-23], and we wish to refer readers to this review for an overview of the various reactant/co-reactant couples that have been investigated. Indeed, in the present chapter we rather wish to focus on the mechanistic aspects of the various general kinetic situations that define ECL generation independently of the very nature of the species involved, though we will exemplify them using actual experimental situations.

Reaction mechanisms of both experimental approaches are characterised by a sequence of very fast second-order reactions taking place in the diffusion layer and generating the extremely short-lived electronically excited state(s), S*, which are deactivated through a rapid first-order emissive decay. Because of this, the emitters are confined to a narrow reaction zone and their concentrations are exceedingly small. This presents severe complications for the mathematical modelling of ECL reaction mechanisms and, in particular, for the evaluation of the ECL light intensity which depends on the small emitter concentration variations in space and time.

Furthermore, other peculiarities of this nontrivial electrochemical problem such as flux discontinuities (arising from rapid stepwise potential changes at the electrode/solution interface) or its intrinsically multiscale nature in both space and time (e.g. locally high reactions rates coupled with smooth diffusion patterns and rates) render its theoretical study rather challenging.

Therefore, with certain exceptions (limiting cases, see below) analytical solutions are not obtainable and numerical methods must be applied. The numerical treatments for such complex

problems require exceptional precision and accuracy in both space and time to allow the prediction of almost negligible emitter concentrations. An overview of simulation approaches and techniques that have been employed over the years to model ECL is presented below including that developed by the authors and implemented in general-purpose electrochemical software (KISSA) [1-7]. The advantages of this latter approach are illustrated by considering the three main reaction mechanisms leading to ECL under typical conditions [24].

Subsequently, we consider recent applications of our simulation approach involving some of the most complex mechanisms leading to ECL through the oxidation of alkyl amines in the presence of transition metal(II) complexes [25]. Following their seminal discovery by Bard et al.

these systems found many analytical and biomedical applications [26]. Yet, the design and optimisation of novel reaction pathways in search of increased efficiency in comparison with the original Ru(bpy)32+

/TPrA (tri-n-propylamine) system typically relies on the well-known and widely accepted mechanism of this classical couple. Variations in ECL intensities are then attributed to expected changes in thermodynamic and kinetic rate constants or to the stability of intermediates, though this is hard or impossible to verify independently. The resulting model may be far from the true case but still provide acceptable agreement with experimental data over a given range of conditions upon fitting based on alleged system parameters. In this context, we have demonstrated on the basis of numerical simulation that one important parameter, namely the ratio of co-reactant diffusivities, had been overlooked in the interpretation of ECL behaviour.

Thus, when the diffusion coefficients of the metal complex species decrease the intensity of the first ECL wave observed at the level of the amine oxidation peak greatly increases with respect to the second wave occurring around the formal potential of the metal(II) complex. This illustrates that the effects of electrochemical reactivity of the amine and the metal complex are sometimes overshadowed by much stronger effects imposed by local transport properties.

RESULTS AND DISCUSSIONS

The discovery of ECL systems involving Ru(bpy)₃²⁺ and substituted alkyl amines such as tripropylamine (TPrA) has been the subject of extensive mechanistic and development studies [17-19] owing to their important applications as a highly sensitive method for immunoassays and DNA analyses. Herein, we do not intend to revisit the question of the validity of the published reaction mechanism. This sequence of reactions is used here to illustrate the ease with which our new approach is able to solve electrochemical problems of such complexity. Thus, we rely hereafter on the mechanism that incorporates all the mechanistic paths considered in a recent study by Bard et al. [26] [Eqs. (1) - (11)]:

TPrAH+ ^TPrA+H+, ^{pKa =10.4} (1)

−e

TPrA TPrA^•+, k_TPrA0/⁰ ₊, E_TPrA0/⁰ ₊ (2)

+

TPrA• TPrA^•+H⁺, ^k_f^TPrA•,K_eq^TPrA• (3)

+

•+Ru(bpy)²₃

TPrA Im⁺+Ru(bpy)₃⁺, ^k_f^1,K1_eq (4)

+ +

• +Ru(bpy)₃

TPrA TPrA+Ru(bpy)₃^2+*, kf², K_eq² (5)

ν +h

→ ⁺

+ 2

3

* 2

3 Ru(bpy)

Ru(bpy) , k_f^ECL (6)

−e

2+

Ru(bpy)3 Ru(bpy)³₃⁺, k_Ru2⁰ _+/3+,E_Ru2⁰ _+/3+ (7)

−e

+

Ru(bpy)3 Ru(bpy)²₃⁺, k_Ru⁰ _+/2+,E_Ru⁰ _+/2+ (8)

* 2 3 2

3 3

3

3 Ru(bpy) Ru(bpy) Ru(bpy)

Ru(bpy)⁺+ ⁺→ ⁺+ ⁺ , k^ET (9)

+

•+Ru(bpy)³₃

TPrA Im⁺+Ru(bpy)²₃^+*, k_f³,K_eq³ (10) DPrA

CO ) (CH

Im⁺⁺ →^H^2O _{3 2} + , ^k_hydrol (11) Scheme 1

In Scheme 1, Im⁺ stands for the iminium product and DPrA for the dipropylamine. Note also that, with respect to ref. [26], we have added reaction (11) which was not considered by the authors of the original work but is essential because Ru(III) is a stronger oxidant than Ru(II).

Figure 1. The KISSA-1D window with the simulated current / ECL-intensity.

This system has a fairly unusual ECL response featuring two distinct “ECL waves” (see Figure 1) originating from three different mechanistic regimes activated as a function of the electrode potential, although in all the cases the only photon emitting species is Ru(bpy)3^2+*. The more anodic ECL wave appears at electrode potentials corresponding to considerable oxidation of Ru(bpy)₃²⁺ (Eq. (7)), which then reacts with Ru(bpy)₃⁺ [Eq. (9)] formed in a reaction between TPrA• and ^Ru(bpy)₃²⁺ [Eq. (4)] as well as with theTPrA^• radical itself [Eq. (10)] yielding the photon emitting species Ru(bpy)²₃^+*[26]. The second less anodic ECL wave owes its existence to two other mechanistic pathways primarily depending on the ratios of the concentrations of

TPrA and • TPrA^•+ radicals to that of Ru(bpy)²₃⁺[26].

The complexity of this reaction mechanism necessitates highly accurate numerical simulation to both precisely compute the concentration distributions of all intermediates regardless of their magnitude and spatial localisation (as illustrated above) and track the interplay between the different mechanistic pathways depending on the electrode potential and bulk concentrations of Ru(bpy)²3⁺ and TPrA. Therefore, this complex system represents a perfect test case for our simulation approach. As illustrated in Figure 4a in our work [10], the results of computations demonstrate all the features of the experimental behaviour reported by Bard et al., i.e., the ECL response with two waves of correct shape and magnitude with the first small- amplitude wave flattening out before Ru(bpy)²₃⁺ can be significantly oxidized and the second one having a CV-like shape at higher potentials. The analysis of concentration variations of the four primary reactants participating in the generation of Ru(bpy)²₃^+* and leading to the first and the second ECL intensity waves (see in Figures 4b, 4c in ref. [10]) also confirms the mechanistic interpretation of this process proposed previously [26]. The ability to accurately compute such concentration distributions in order to validate any tentative mechanistic interpretation is another valuable advantage of our approach.

Figure 2. The KISSA-1D window with the mechanism and parameters of simulation

Using KISSA-1D we varied different parameters (see Figure 2) for getting optimal ECL intensity and discovered very important and strong and unexpected effects of diffusion rates on electrochemiluminescence (ECL) generation by amine/transition metal(II) systems [11].

CONCLUSIONS

The main merit of our novel simulation approach with respect to modelling any electrochemical reaction mechanism resulting in ECL emission lies in the ability to accurately compute

concentration distributions in both time and space of all reacting species regardless of how short- lived they are and how small and spatially localised their concentrations are. Indeed, concentrations of photon emitting species are typically extremely low owing to immense rates of their formation and decay which outstrip most usual reaction rates. Despite this these concentrations are crucial for the evaluation of the ECL intensity which is a key quantity of interest in a variety of modern physicochemical or bioanalytical applications. The new approach owes its remarkable properties to a successful combination of specific conformal maps resolving numerical artefacts due to diffusional transport and automatic adjustment of the computational grid in the conformal space [1, 2] led by a novel criterion tracking any rapid kinetic variations within the diffusion layer in order to resolve them with higher mesh point density and hence accuracy. This ensures that any reactions even with rate constants considerably higher than those physically possible can be successfully treated in a completely automatic fashion without the need for any decisions from the user. The kinetic criterion is formulated on the basis of the magnitude of reaction rate terms and therefore its evaluation does not require a preliminary integration step or iterative estimation typical for common finite difference methods designed to deal with rapid linear or non-linear chemical reactions. This is essential for the efficient numerical treatment of most kinetic situations encountered in ECL systems.

All these features present important advantages to most electrochemical users without any expertise in mathematics. This becomes evident in comparison with other existing software and approaches. Thus, even though general purpose simulation software such as Comsol Multiphysics [27] is capable of solving this type of problem with accuracy similar to that of our approach, this requires considerably longer CPU times while the problem has to be fully defined in the form of rigorous mathematical equations (including reaction terms and specific electrochemical boundary conditions for a particular reaction mechanism) instead of simply a reaction scheme in a conventional chemical format.

ACKNOWLEDGEMENTS

This work was performed in Paris, France (ENS, UMR8640) and supported in parts by CNRS UMR 8640, ENS (Ecole Normale Superieure), PSL and Sorbonne as well as by ANR (grant ANR-AAP-CE06 ChemCatNanoTech).

REFERENCES

[1] C. Amatore, O.V. Klymenko, I. Svir. 2010. A New Strategy for Simulation of Electrochemical Mechanisms Involving Acute Reaction Fronts in Solution: Principle. In:

Electrochemistry Communications. Volume 12. Elsevier (Netherlands). p. 1170-1173.

[2] C. Amatore, O.V. Klymenko, I. Svir. 2010. A New Strategy for Simulation of Electrochemical Mechanisms Involving Acute Reaction Fronts in Solution: Application to Model Mechanisms. In: Electrochemistry Communications. Volume 12. Elsevier (Netherlands), p. 1165-1169.

[3] O.V. Klymenko, A. Oleinick, I. Svir, C. Amatore. 2012. A New Strategy for Simulation of Electrochemical Mechanisms Involving Acute Reaction Fronts in Solution under Spherical or Cylindrical Diffusion. In: Russian J. Electrochemistry. Volume 48. Springer (New York, USA). p. 593-599.

[4] O.V. Klymenko, I. Svir, A. Oleinick, C. Amatore. 2012. A Novel Approach to Simulation of Electrochemical Mechanisms Involving Acute Reaction Fronts at Disk and Band Microelectrodes. In: ChemPhysChem Volume 13. Wiley-VCH (Germany). p. 845-859.

[5] C. Amatore, O.V. Klymenko, I. Svir. 2011. Replies To Comments Contained in “The True History of Adaptive Grids in Electrochemical Simulations” by D. Britz [Electrochim. Acta 56, 2011, 4420]. In: Electrochimica Acta. Volume 56. Elsevier (Netherlands). p. 4422-4423.

[6] O.V. Klymenko, I. Svir, C. Amatore. 2010. Theoretical Study of the EE Reaction Mechanism with Comproportionation and Different Diffusivities of Reactants. In:

Electrochemistry Communications. Volume 12. Elsevier (Netherlands). p. 1378-1382.

[7] C. Amatore, O.V. Klymenko, I. Svir. 2012. Importance of Correct Prediction of Initial Concentrations in Voltammetric Scans: Contrasting Roles of Thermodynamics, Kinetics and Natural Convection. In: Analytical Chemistry. Volume 84. (American Chemical Society) p. 2792-2798.

[8] O.V. Klymenko, I. Svir, C. Amatore. 2013. New Theoretical Insights into the Competitive Roles of Electron Transfers Involving Adsorbed and Homogeneous Phases.

In: J. Electroanalytical Chemistry. Volume 688. Elsevier (Netherlands). p. 320-327.

[9] O.V. Klymenko, I. Svir, C. Amatore. 2014. Molecular Electrochemistry and Electrocatalysis: A Dynamic View. Molecular Physics. Taylor and Francis Group (Oxford, UK). Volume 112. p. 1273-1283.

[10] O.V. Klymenko, I. Svir, C. Amatore. 2013. A New Approach for Simulation of Electrochemiluminescence (ECL). In: ChemPhysChem. Volume 14. Wiley-VCH (Germany). p. 2237-2250.

[11] I. Svir, A. Oleinick, O.V. Klymenko, C. Amatore. 2015. Strong and Unexpected Effects of Diffusion Rates on Electrochemiluminescence (ECL) Generation by Amine/Transition Metal(II) Systems. ChemElectroChem Volume 2. Wiley-VCH (Germany). p. 811-818.

[12] A. Oleinick, C. Amatore, I. Svir. Efficient Quasi-Conformal Map for Simulation of Diffusion at Disk Microelectrodes. In: Electrochemistry Communications. Volume 6.

Elsevier (Netherlands) p. 588-594.

[13] A. Oleinick, O.V. Klymenko, I. Svir, C. Amatore. 2017. Theoretical Insights in ECL. In:

"Luminescence in Electrochemistry" (F. Miomandre & P. Audebert, Eds), Springer (International Publishing AG, Cham, Switzerland). Chapter 7, p. 215-256.

[14] C. Amatore, J.-M. Savéant. 1977. ECE and Disproportionation: Part V. Stationary State General Solution Application to Linear Sweep Voltammetry. In: J. Electroanalytical Chemistry. Volume 85. Elsevier (Netherlands). p. 27-46.

[15] C. Amatore. 1991. In: Principles and Methods. Basic Concepts in Organic Electrochemistry (Eds: M. Baizer, H. Lund). Publisher: M. Dekker (New-York) Chapter.

2, p. 11-119.

[16] C. Amatore. 2016. In: “Organic Electrochemistry”. 5th Edition: Revised and Expanded (Eds: Ole Hammerich & Bernd Speiser) CRC Press (Taylor & Francis Group, Boca Raton). Chapter 1, p. 3-96; Chapter 10, p. 371-394.

[17] K.S.V. Santhanam, A.J. Bard. 1965. Chemiluminescence of Electrogenerated 9,10- Diphenylanthracene Anion Radical. In: J. American Chemical Society. Volume 87.

(American Chemical Society) p. 139-140.

[18] S.W. Feldberg. 1966. Theory of Controlled Potential Electrogeneration of Chemiluminescence In: J. American Chemical Society. Volume. 88. (American Chemical Society) p. 390-393.

[19] L.R. Faulkner, A.J. Bard. 1977. In: Techniques of Electrogenerated Chemiluminescence in Electroanalytical Chemistry (Ed: A. J. Bard). Publisher: Marcel Dekker (New York).

Volume 10. p. 1-95.

[20] A.J. Bard, L.R. Faulkner. 2002. In: Electrochemical Methods: Fundamentals and Applications. John Wiley & Sons (New-York).

[21] W. Miao. 2008. Electrogenerated Chemiluminescence and its Biorelated Applications. In:

Chemical Reviews. Volume 108. (American Chemical Society) p. 2506-2553.

[22] R. Pyati, M.M. Richter. 2007. ECL – Electrochemical luminescence. In: Annual Reports Section "C" (Physical Chemistry). Volume 103. Royal Chemical Society (London, UK) p. 12-78.

[23] Z. Liu, W. Qi, G. Xu. 2015. Recent Advances in Electrochemiluminescence In: Chemical Society Reviews. Volume 44. Royal Chemical Society (London, UK) p. 3117-3142.

[24] C. Amatore, C. Pébay, L. Servant, N. Sojic, S. Szunerits, L. Thouin. 2006. Confocal Microscopy Imaging of Electrochemiluminescence at Double Band Microelectrode Assemblies. Numerical Solution of the Inverse Optical Problem. In: ChemPhysChem.

Volume 7. Wiley-VCH (Germany) p. 1322-1327.

[25] C. Amatore, A. Oleinick, O.V. Klymenko, L. Thouin, L. Servant, I. Svir. 2007. In:

ChemPhysChem. Volume 8. Wiley-VCH (Germany) p. 1664-1676.

[26] W. Miao, J.-P. Choi, A.J. Bard. 2002. Electrogenerated Chemiluminescence. 69:  The Tris(2,2‘-bipyridine)ruthenium(II), (Ru(bpy)32+)/Tri-n-propylamine (TPrA) System RevisitedA New Route Involving TPrA•+ Cation Radicals. In: J. American Chemical Society. Volume 124. (American Chemical Society) p. 14478-14485.

[27] Comsol Multiphysics software, COMSOL (Version 5.3a), Inc., Burlington, MA.