Modeling and Simulation of Electrochemical Cells under Applied Voltage

Abstract

The behavior of an electrochemical thin film under input voltage (potentiostatic) conditions is numerically investigated. Thin films are used in micro-batteries and proton-exchange-membrane fuel cells: these devices are expected to play a significant role in the next generation energy systems for use in vehicles as a replacement to combustion engines. The electrochemical investigation of thin films is a relevant topic for a wide range of applications such as hydrogels, ionic polymer metal composites, biological membranes, and treatment of tumors.

In this work, a continuum-based model is presented in order to describe the behavior of thin membranes. The electrochemical behavior of thin membranes is usually hard to investigate with experiments. Therefore, numerical simulations are carried out in order to enable a better understanding of the chemical reactions occurring within microscopic regions at the electrode/electrolyte interfaces. Diffusive-migrative ionic fluxes and electric field distribution are considered. A one-dimensional domain is employed. The fully-coupled electrochemical field is given by the Poisson-Nernst-Planck equations. The model involves initial and interface/boundary conditions appropriate for an electrolytic/galvanic cell. The latter are the Stern layer conditions for polarization (or diffuse charge) effects and the Frumkin-Butler-Volmer equations for electrochemical kinetics of chemical reactions. Time-dependent numerical simulations within a finite element framework are performed using the commercial tools MATLAB and COMSOL Multiphysics.

The results are consistent with the physical behavior of electrolytic cells under potentiostatic conditions. The time evolution of the main electrochemical parameters is in accordance with the imposed boundary/interface conditions. Interestingly, the ion flux and the electric field show slight asymmetries at the boundaries. Moreover, the model well predicts the behavior of systems, such as redox flow cells or rechargeable batteries, that can either run under applied voltage or applied current conditions. In fact, the field equations and the boundary conditions, presented here for electrolytic cells under applied voltage, can be applied also for galvanic cells under applied current. Equations and boundary conditions for applied voltage and applied current working conditions are presented in a compact form in order to emphasize differences and similarities.

Introduction

Recently, different research areas have gained attention in the investigation of transducers: systems able to convert one form of energy into another. Among these, electrochemical cells (ECs) can transform energy from chemical to electrical fields (galvanic cells) and vice versa (electrolytic cells). Energy storage and release in the field of electrochemical devices is an attractive topic within the scientific community [1]. Fuel cells and rechargeable batteries are expected to play a significant role in the next-generation-energy-systems for use in vehicles as a replacement to combustion engines [2]. In fact, these ECs can guarantee a minimum environmental impact due to carbon dioxide reduced emissions [3].

In Bard and Faulkner [4], electrochemical cells are defined as two metallic electrodes separated by an electrolyte phase. As illustrated in Fig. 1 , it is possible to identify three main regions in ECs (i) anode electrode (where oxidation reactions occur), (ii) electrolyte phase (e.g. ion-exchange membrane, widely used in fuel cells), and (iii) cathode electrode (where reduction reactions occur). The working principle of electrochemical cells can be explained considering redox heterogeneous reactions involving one chemical species R:

$$\frac{n_{\mathrm{e}}}{z_{\mathrm{R}}}\mathrm{R}\underset{\mathrm{reduction}}{\overset{\mathrm{oxidation}}{\rightleftharpoons }}\frac{n_{\mathrm{e}}}{z_{\mathrm{R}}}{\mathrm{R}}^{+z_{\mathrm{R}}}+n_{\mathrm{e}}{\mathrm{e}}^{-},$$

where z_R is the valence of the chemical species R and n_e are the number of electrons e⁻ involved in the redox couple.

At the anode, ions R^+z_R are released into the electrolyte solution (forward reaction) and transported towards the cathode. Here, neutral species R are produced and ions R^+z_R are consumed (backward reaction). Electrons e⁻ flow in an external circuit which connects the electrodes either through an electric load (galvanic cells) or a power supply (electrolytic cells), see Fig. 1.

In galvanic cells, see Fig. 1(a), chemical reactions occur spontaneously at the electrode/electrolyte interface. The chemical energy stored in the reaction is transformed in an ion/electron flow. In equilibrium conditions, the flux determines the passage of an electric current in the electrolyte and generates a difference in voltage between the cell electrodes: thereby, an electrical power is produced as output. Alkaline batteries and fuel cells (e.g. H₂-O₂ cells) are typical examples of galvanic devices. Galvanic corrosion is a chemoelectrical phenomenon based on the physical behavior of galvanic cells.

In electrolytic cells, see Fig. 1(b), chemical reactions are induced by an external voltage. Electrical power is employed to carry out desired chemical reactions which cannot occur spontaneously. Oxygen and hydrogen production through electrolytic devices is a widely exploited technology on board of the International Space Station (ISS) [5], [6]. These cells are also used in some important commercial processes such as: electrolytic synthesis, electrorefining and electroplating [4].

Finally, there are electrochemical cells able to work both as galvanic and electrolytic cells. For instance, reversible galvanic cells (e.g. redox flow cells and reversible solid oxide fuel cells) and rechargeable batteries (e.g. lithium-ion batteries) are hybrid systems which find many applications as primary transportation alternatives [3], [7].

Usually, investigations on the behavior of ECs are carried out by imposing a certain variable as input, like current or voltage. The external stimuli may disturb the initial equilibrium conditions, bringing the cell to a new equilibrium state. In output, different variables are measured. In Bazant et al. [8], the investigation methods for electrochemical cells are classified according to the initial stimulus as (i) galvanostatic conditions (imposed current), and (ii) potentiostatic conditions (imposed voltage).

As shown in Fig. 1, copper (Cu) is plated in either galvanic or electrolytic cells. Therefore, the behavior of a single electrode and the nature of its reaction are independent of whether the electrode is part of galvanic or electrolytic cells [4]. Starting from this observation, the aim of the proposed work is to show the analogy between models which can describe the behavior of both, galvanic and electrolytic cells. Generally, to the authors’ knowledge there is a lack of transparency and clearness in electrochemical models for thin membranes which employ Poisson-Nernst-Planck (PNP) and Frumkin-Butler-Volmer (FBV) equations. Therefore, the present manuscript will give a comprehensive overview of the bulk equations and boundary conditions (Dirichlet and Neumann) for modeling thin membranes under galvanostatic and potentiostatic working conditions. The chemoelectrical coupling within the field equations and the different ways to arrange boundary conditions (BCs) at the interfaces are key elements for the presented electrochemical model.

In the present work, a time-dependent simulation within a finite element framework is performed. An electrochemical model for electrochemical cells (ECs) is applied under an external electric voltage stimulus (potentiostatic conditions). The data, obtained through the simulations, are compared with the literature results [7] for an electrochemical cell under an external stimulus in electric current (galvanostatic conditions). Numerical simulations are important in order to describe the electrochemical behavior of thin membranes, within which many phenomena are difficult to investigate experimentally.

The one-dimensional domain employed in the proposed electrochemical model consists of a thin electrolyte membrane ($\mathcal{B}:0\le x\le L$), see Fig. 2 . Within the electrolyte, cations (produced and consumed at the reaction planes through chemical reactions) are free to move, while anions (counterions) are fixed. Part of the membrane, named Stern or polarization layer, as well as both electrodes are outside the computational domain. The phenomena which take place in these regions are included in the BCs ($\partial \mathcal{B}:x=0$ and x = L, see Fig. 2).

Cations and anions consist of one chemical species each: e.g. protons, H⁺ (cations), and sulfonate groups, SO${}_3^{-}$ (anions). The latter system is representative of a Nafion membrane sandwiched between flat porous electrodes (mostly made of graphite). In fact, configurations like these are suitable for a wide range of electrochemical applications: fuel cells (e.g. Proton Exchange Membrane Fuel Cells, PEMFCs), chlor-alkali and water electrolysers, surface-treated metals (in batteries, sensors, drug release systems and others), Donnan dialysis cells and electrochromic devices [9], [10].

The main phenomena occurring within the electrolyte membrane $\mathcal{B}$ are (i) charge transport (chemical field), and (ii) electric field evolution (electrostatic field).

The electrochemical behavior of the membrane is well described by these two phenomena. The fully-coupled electrochemical field is given by the Poisson-Nernst-Planck (PNP) theory, without assuming local electroneutrality. PNP equations are thoroughly described in Section 2.

The main phenomena occurring at the reaction planes $\partial \mathcal{B}$ are (i) electrochemical kinetics (chemical field), and (ii) polarization effects (electrostatic field).

Electrochemical kinetics takes into account the speed of the chemical reactions occurring at the reactions planes (x = 0 and x = L). This phenomenon is modeled as a BC by the Frumkin-Butler-Volmer (FBV) equation for reaction kinetics. The FBV equation describes the electron-transfer reactions across the Stern layer.

On the other hand, polarization effects take into account the stacked-structure that ions form at the electrode/electrolyte interface. Due to polarization effects, the planes at which chemical reactions occur (named reaction planes) shift from the electrode/electrolyte interface towards the inside of the field. In fact, reaction planes are placed within the membrane at the edges of the one-dimensional domain: x = 0 and x = L, respectively. The distance between the electrode/electrolyte interface and the reaction plane is called Stern layer, see Fig. 2 . The polarization effects occurring in these regions are modeled through the Stern layer (SL) theory (Stern’s modification to the Gouy-Chapman theory [4], also named Gouy-Chapman-Stern model).

The SL theory and the FBV equation are thoroughly described in Section 2.

For example, for galvanic cells such as proton conducting fuel cells, the chemical reactions occurring at the reaction planes are oxidation at the anode (x = 0) and reduction at the cathode (x = L):

$${\mathrm{H}}_2\to 2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-},$$

$$\frac{1}{2}{\mathrm{O}}_2+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\to {\mathrm{H}}_2\mathrm{O},$$

respectively. These reactions are inverted when the model refers to electrolytic cells, as for example a water electrolysis cell. The latter device involves the same chemical species as for a proton conducting fuel cell. The reactions occurring at the reaction planes are reduction at the cathode (x = 0) and oxidation at the anode (x = L):

$$2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}\to {\mathrm{H}}_2,$$

$${\mathrm{H}}_2\mathrm{O}\to \frac{1}{2}{\mathrm{O}}_2+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}.$$

The electrochemical model for electrolytic cells under applied voltage considers a one-electron reaction for a species R, n_e = 1 and a unity valance number z_R = 1, respectively. According to the general reaction (1), the redox couple related to these assumptions is given by:

$${\mathrm{R}}^{+}+{\mathrm{e}}^{-}\to \mathrm{R},$$

$$\mathrm{R}\to {\mathrm{R}}^{+}+{\mathrm{e}}^{-},$$

at the cathode (x = 0) and anode (x = L), respectively. These reactions well describe the behavior of neutral particles released into the electrolyte from the electrodes (e.g. as in the electro-dissolution of a metal [8], [11]), or absorbed neutral hydrogen atoms in equilibrium with hydrogen molecules in an adjacent gas phase [7], [12]. In the latter systems, named hydrogen concentration cells, mobile protons in the electrolyte are exchanged with gaseous hydrogen at the electrodes through redox reactions:

$${\mathrm{H}}^{+}+{\mathrm{e}}^{-}\to \mathrm{H},$$

$$\mathrm{H}\to {\mathrm{H}}^{+}+{\mathrm{e}}^{-}.$$

Usually, electron-transfer kinetics is represented in common electrochemical models [13], [14], [15], [16] through the Butler-Volmer (BV) equation, in addition to the PNP equations for the description of the chemoelectrical field. In these models, diffuse charge (or polarization) effects are neglected and reaction planes are located at the electrode/electrolyte interface. Diffuse charge effects near the electrodes are fundamentals to describe the behavior of ECs. This phenomenon cannot be neglected, especially in microelectrochemical systems which use thin membranes [8], just like the model presented in this work. Therefore, several authors [7], [8], [11], [12], [17] have recently modeled polarization effects at the boundaries through the SL theory and an extension of the BV equation, called the FBV equation. It should be pointed out that using the FBV equation implies also the SL theory (polarization effects).

In the following, we briefly outline the main models for ECs. Please note that the literature review is sorted by the year of publication. A brief outline of the main models for ECs, which employ either PNP with the Butler-Volmer equation or PNP with the Frumkin-Butler-Volmer equation, is given.

Bernardi and Verbrugge [15] have formulated a model for an ion-exchange membrane bonded to a gas diffusion electrode (simplified model for PEMFCs) in 1991. This work is considered as the first attempt to describe the physical behavior of fuel cells. Proton transport and electrochemical kinetics, referred only to the cathode electrode, are modeled by the PNP and the BV equations, respectively. Other additional phenomena are taken into account such as (i) multicomponent gas diffusion, (ii) water transport within the membrane, (iii) material balances and (iv) current conservation (Ohm’s law).

A finite difference method is used in Murphy et al. [13] to solve the time-dependent PNP with the Frumkin-Butler-Volmer equation under input voltage conditions. Three mobile ion species are considered within the one-dimensional domain, while fixed ions are neglected. This manuscript is recognized as the first work which performs a full numerical simulation on the PNP model coupled with polarization effects [7]. However, the hypothesis of zero Stern layer length reduces FBV to the common BV expression. In this way, the complexity of the model is lowered but, also a non-physical behavior can occur in certain conditions [7].

A steady-state model, consisting of the PNP model with the Butler-Volmer equation, is solved by Sokirko and Barke [14] by neglecting the polarization effects. Three ionic species are transported in the electrolyte by diffusion and migration as driving forces, with or without supporting electrolyte. Analytical expressions for the electric current, as a function of the applied voltage, are obtained. Furthermore, a literature review concerning models for ECs is given.

Proton transport (PNP), electrochemical kinetics (BV) and water transport are the phenomena modeled by Singh et al. [16] for a PEMFC.

In Bonnefont et al. [11], the one-dimensional ion-transport (PNP) is coupled with reaction kinetics (FBV). In the latter phenomenon, polarization effects are taken into account. Using the singular perturbation theory, a steady-state response to an imposed current is computed for an electrochemical cell consisting of a binary electrolyte and a neutral solute molecule. This approach is considered as the first model that uses the Frumkin-Butler-Volmer equation as boundary conditions for the PNP problem [8]. Please note that simplified assumptions such as the “zero-length limit”, used by Murphy et al. [13], are not employed in this model.

In Bazant et al. [8], the steady-state PNP problem and the Frumkin-Butler-Volmer equation are solved both numerically and analytically. The analytical solution is obtained by using the matched asymptotic expansions approach. A binary electrolyte is employed, such as for micro-batteries. Transport occurs in one dimension under either applied electric current or applied voltage.

In Biesheuvel et al. [12], only the Frumkin-Butler-Volmer equation is employed in order to define an analytical theory in steady-state conditions for a solid-state-proton-conducting membrane, namely a hydrogen concentration cell. The electrochemical cell is analyzed under input current conditions. The model considers local electroneutrality in the bulk electrolyte and a quasi-equilibrium structure of the polarization layer (thin diffusive layer limit). In a following work, Biesheuvel et al. [7] have added in the model the charge transport effects (PNP). In the latter work, a reversible galvanic cell under imposed current condition is investigated. The model consists of the PNP problem with the Frumkin-Butler-Volmer equation, considering a one-dimensional domain and steady-state conditions. A generalized formulation is employed: the model does neither a priori assume local equilibrium of the polarization layer nor electroneutrality. The presence of different chemical reactions at each electrode has extended the work of Bazant et al. [8], where the same reaction occurs at both electrodes. Furthermore, Biesheuvel et al. [7] discuss a comprehensive overview of literature, history and merits concerning the FBV equation over the standard BV representation.

A galvanic cell with supporting electrolyte is investigated for both, steady-state conditions by Van Soestbergen [18] and time-dependent conditions by Van Soestbergen et al. [19]. The PNP problem with electrochemical kinetics and polarization effects are solved considering a one-dimensional domain. The solution is obtained numerically through a finite element discretization, and analytically through the singular perturbation theory.

Beyond the classical electrochemical kinetic formulations (BV or FBV), Červenka et al. [1] have proposed a novel kinetics description for a cathodic reaction in an EC.

In Marcicki et al. [20], polarization effects are investigated for lithium-ion batteries. Applied current conditions are employed.

Furthermore, it is useful to cite the recent investigations performed by Bazant and coworkers. Bazant [21] has presented a general theory of chemical kinetics based on nonequilibrium thermodynamics which is capable (i) of unifying and extending the Cahn-Hilliard and Allen-Cahn equations and (ii) of generalizing both Marcus and Butler-Volmer kinetics. In Zeng et al. [22], a simple analytical expression for the Marcus-Hush theory is provided. This model for Faradaic reaction kinetics is considered as predictive alternative to the phenomenological Butler-Volmer equation.

Finally, the paper by Yan et al. [17] provides a general formulation for the PNP and FBV model which extends theories and simulations in a variety of situations in order to develop an understanding for more complicated circumstances. This work shows many common aspects related to the investigations performed in the present research.

Due to the complexity of many models for electrochemical cells (see e.g. Refs. [15], [16], [23], [24]), the current research intends to present the electrochemical model for an electrolyte membrane sandwiched between the electrodes. The field equations and the boundary conditions are formulated in a transparent way. The main phenomena occurring within the membrane (charge transport and electrostatic field) and at the boundary electrodes (electrochemical kinetics and polarization effects) are taken into account. These phenomena have been deeply investigated in common models for ECs: controlling either an input current [7], [8], [11], [12], [18], [19] or an input voltage [8], [13], [14], [17]; and modeling electrochemical kinetics either through FBV [7], [8], [11], [12], [17], [18], [19] or BV [13], [14]. Motivated by the latest investigations on electric double layer [17], polarization effects are here modeled using the Frumkin-Butler-Volmer (FBV) equation and the Stern layer (SL) theory. The electrochemical model is suitable for electrolytic cells under applied voltage conditions. The results obtained by Biesheuvel et al. [7] for galvanic cells under applied current conditions are employed in order to validate the performed simulations for the electrochemical model. Figure 3 shows the concepts of the two models. In fact, a time-dependent analysis within a finite element framework is performed in this paper, while the reference work [7] considers steady-state conditions. Furthermore, the electric voltage is the controlled variable (potentiostatic conditions), while Ref. [7] considers the electric current as input parameter (galvanostatic conditions). It should be pointed out that, in steady-state conditions, electrochemical cells show the same results either under applied current or applied voltage.

The paper is organized as follows: In Section 2, the fundamental aspects of the electrochemical model are presented for a time-dependent analysis. The set of equations and BCs are discussed for both investigated regimes, imposed current and imposed voltage. In Section 3, the results of the numerical simulations are shown considering input voltage conditions: the space and time evolution of the main electrochemical parameters (cation concentration c₊(x, t), potential ϕ(x, t), flux j(x, t) and electric field E(x, t)) is analyzed. Furthermore, the concentration and the electric potential profiles are presented for different input voltages at steady-state conditions. Conclusions and outlook are presented in Section 4.