A continuum model for yttriastabilized zirconia incorporating triple phase boundary, lattice structure and immobile oxide ions
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Abstract
A continuum model for yttriastabilized zirconia (YSZ) in the framework of nonequilibrium thermodynamics is developed. Particular attention is given to (i) modeling of the YSZmetalgas triple phase boundary, (ii) incorporation of the lattice structure and immobile oxide ions within the free energy model and (iii) surface reactions. A finite volume discretization method based on modified ScharfetterGummel fluxes is derived in order to perform numerical simulations. The model is used to study the impact of yttria and immobile oxide ions on the structure of the charged boundary layer and the double layer capacitance. Cyclic voltammograms of an airhalf cell are simulated to study the effect of parameter variations on surface reactions, adsorption and anion diffusion.
Keywords
Solid oxide Double layer Interface Thermodynamics Finite volume methodIntroduction
Detailed continuum models of high temperature solid oxide electrochemical cells (SOEC)^{1} describe the underlying chemistry with spatially distinguished phases (oxide ion conductor, electric conductor, gas) of the triple phase boundary [1, 2, 3, 4]. Surface physics processes such as tangential diffusion and surface chemical reactions of the surface species are employed. In particular, the electrontransfer reaction at the triple phase boundary is usually modelled with ButlerVolmertype kinetics containing overpotential, the difference of the electric potential between the metal and the bulk of the YSZ, as the driving force. The ionically or electrically conductive parts of a solid oxide cell are electroneutral in the respective bulks. The overpotential, appearing at the phase interface, is caused by formation of a charged double layer of oxide ions in YSZ and electrons in the electrode. Although the overpotential correlates with the excess concentration of oxide ions available for the electrontransfer reaction in steadystate scenarios, it cannot capture the dynamics of the double layer. Therefore, if such a model is compared to the results of a dynamic currentvoltage measurement, e.g., electrochemical impedance spectroscopy or linearsweep voltammetry, the dynamics of the double layer is underrepresented.
To determine the structure and dynamics of the spacecharge layer of oxide ions in the YSZ, a generalized PoissonNernstPlanck (PNP) system can employed. By the generalization it is possible to account for the effect of the finite density of available lattice sites for oxide ions at the continuum level.
Such an approach was already used to capture the formation and behavior of the electrochemical double layers at electrodeelectrolyte interfaces [5, 6]. The PNP system was successfully applied to the solidstate electrochemical systems, e.g., lithium batteries [7, 8, 9]. In [10], the PNP equations were already applied for proton ceramic fuel cells; however, the thermodynamics of the crystalline lattice and of the surface were not taken into account.
In this work, we apply a modeling approach for charged bulksurface interfaces based on first principles of nonequilibrium thermodynamics [11, 12]. The resulting generalized PoissonNernstPlanck system is used to formulate the model of dynamics of the spacecharged layer at the YSZmetalair triple interface. The main advantage of this approach is its consistency between the free energy (equilibrium) and fluxes (dynamics).
The paper is organized as follows. The free energy model of the bulk YSZ, capturing the crystalline structure, immobile oxide ions and elastic deformation, is developed in the “Bulk YSZ” section. The resulting chemical potentials are introduced into the gPNP model [11] after its modification for the description of the lattice velocity. In the following section, bulk metal and bulk gas phases are treated under the assumption of diffusional equilibrium. The free energy of the surface and the surface dynamics are described and developed in the the “Surface—triple phase boundary” section. The modeling approach results in a coupled system of evolution equation describing the transport of oxide ions in the bulk of electrolyte, adsorption of oxide ions from bulk to the surface and electrontransfer reaction alongside with the Poisson equation.
Using a finite volume based discretization, double layer capacitance and linearsweep voltammetry simulations are performed in the “Simulation of a SOC halfcell” section. The performed simulations study the effects of the newly introduced concept of immobile oxide ions, the free energy parameters and the kinetic rates on the current response.
The novelty of the approach lies in the synthesis of the crystalline lattice bulksurface free energy description and the coupled bulksurface dynamics in nonequilibrium thermodynamics framework. Owing to this, it is possible to simulate the equilibrium behavior, e.g., the double layer capacitance, and dynamic behavior, e.g., the cyclic voltammetry, using a single model. Notable contribution to the state of the art models of YSZ is the thermodynamic treatment of the surface dynamics.
Bulk YSZ
We consider the charge transport exclusively in the isothermal electrostatic setting, therefore the temperature T is assumed to be constant and the electric field is given as E = −∇φ. Moreover, a simple material model for polarization based on a constant susceptibility χ is chosen.
General mixture and crystalline structure
Mixture quantities
Crystalline structure
The crystalline structure of pure ZrO_{2} is well known (see e.g. [13]) and might be described conveniently in terms of unit crystal cells. Unit crystal cells of yttriadoped zirconia are, due to the yttria doping, difficult to be described systematically [14].
Free energy and chemical potentials
The free energy density^{2}ρψ of YSZ is assumed to be a function of temperature T, partial mass densities ρ_{α} and the electric field E. We suppose that the free energy density ρψ(T,ρ_{α},E) can be split into four additive parts: reference energy, entropy of mixing, elastic energy and polarization energy,
Reference energy
Polarization energy
Elastic energy
Entropy of mixing
Chemical potentials
Bulk governing equations and constitutive modeling
The electrothermodynamic state of YSZ, occupying an interval \({\Omega }_{\text {YSZ}}\subset \mathbb {R}\) at any time t, is described by the number densities n_{α} (\(\alpha \in \mathcal {I}_{\text {YSZ}}\)), the barycentric velocity υ and the electrostatic potential φ, which all are functions of time and position. In the isothermal electrostatic setting with a constant susceptibility, the evolution equations for the electrothermodynamic state variables in the bulk are given by the Poisson equation, partial mass balances and the quasistatic momentum balance [11, 16],
The diffusion flux
The constraint (2.4)_{right} and the constitutive equations (2.8) imply that the diffusion fluxes have to be pairwise linear dependent. We chose J_{Om} as the independent flux and obtain
Here, mobility coefficient M may be a function of the thermodynamic variables and their derivatives, as long as it is guaranteed to be nonnegative.
Incompressibility
Vanishing lattice velocity
The assumptions of incompressibility and vanishing lattice velocity may be also viewed alternatively as a description of the charge transport in the reference frame of the cation lattice which does not undergo any deformation.
Summary of the bulk YSZ model
The constitutive modeling above motivates to change the set of variables from the number densities \((n_{\alpha })_{\alpha \in \mathcal {I}}\) to \(\{ n_{\text {C}}^{\#},\nu ^{\#},x^{{\#}},y \}\). Due to the vanishing lattice velocity, the quantities \(n_{\text {C}}^{\#}\), x^{#} and ν^{#} are constant in time and are further considered as model parameters. Therefore, the thermodynamic state of the bulk YSZ is described by three quantities: filling ratio y, electrostatic potential φ and pressure p. In addition, we define the lattice volume V^{#}, lattice mass m^{#} and lattice charge number z^{#} as
The evolution of the thermodynamic state is then described by
Characteristic values. Perparticle masses m_{α} are used in the calculations
Temperature  T  800 ^{∘}C 
YSZ dielectric susceptibility  χ  27 
Zr cation charge number  z _{Zr}  + 4 
Y cation charge number  z _{Y}  + 3 
Oxide ion charge number  z_{Om},z_{Oi}  − 2 
Zr molar mass  M _{Zr}  91.22 g mol^{− 1} 
Y molar mass  M _{Y}  88.91 g mol^{− 1} 
O molar mass  M _{O}  16 g mol^{− 1} 
Ratio of C/A lattices  m  2 
YSZ molar fraction  x ^{ #}  0.08 
Ratio of immobile O^{2−}  ν ^{ #}  \([0, \frac {1}{m}\frac {2+x^{{\#}}}{1+x^{{\#}}}]\) 
Specific lattice volume of YSZ  V ^{ #}  3.35 × 29^{− 29}m^{3} 
Lattice cation number density  \(n_{\text {C}}^{\#}\)  (V^{#})^{− 1} 
Diffusion coefficient  D  1× 10^{− 11} m^{2}/s 
Bulk metal and gas phase
In order to act as an electrolyte in a SOEC, the YSZ has to be connected to two different materials: a gas phase and some electric conductor. In this paper, we do no consider the internal structure of these parts of the SOEC. Therefore, we assume the gas to be equilibrated such that boundary conditions at the gasYSZ surface can be determined easily. Although not appropriate for the use in real SOEC, we will treat the conductor as a pure metal, since this way the conductor can be almost completely removed from the model.
Bulk gas
The gas in the bulk is assumed to behave as an ideal mixture of ideal gases. We introduce the index set \(\mathcal {I}_{\text {gas}}\) of the constituents of the gas phase. For each constituent, the partial pressure is p_{α} = c_{α}RT. The chemical potential of a gaseous species reads
In the bulk domain \({\mathrm {\Omega }}_{\text {gas}}\subset \mathbb {R}^{3}\), we assume that the diffusion is fast such that the chemical potentials are homogeneous in space, i.e., ∇μ_{α} = 0 for \(\alpha \in \mathcal {I}_{\text {gas}}\). Since there are no charge carriers in the gas, we assume that the electric potential φ is also homogeneous in the gas phase.
Bulk metal
Surface—triple phase boundary

i) The YSZ surface is endowed with a thin layer of metal ions and their corresponding free electrons.

ii) The tangential transport of electrons along the surface is assumed to be fast compared to all the other treated kinetic processes.

iii) Apart from the reference free energy, the metal ions and electrons do not further contribute to the free energy and entropy of the surface.
A more detailed derivation of this reduction of a triple phase line into a 1D model can be found in the context of intercalation electrodes in [20].
The following derivation of the YSZ surface model is based on the general approach developed in [11, 16].
Surface constituents and basic quantities
As in the bulk, we describe the YSZ surface as a mixture of different surface constituents and apply for the surface quantities analogous notation with an underset “s” added. In the isothermal case, the surface temperature \(\underset {\text {s}}{T}\) is identical to the constant bulk temperature T and appears in the equations only as a parameter. In addition to the constituents from the metal and the bulk phases of the gas and YSZ bulk, surface reaction products may be present on the surface. Thus, the index set of all surface constituents is of the form \(\mathcal {I}_{S}=\mathcal {I}_{\text {YSZ}}\cup \mathcal {I}_{\text {gas}}\cup \mathcal {I}_{\text {metal}}\cup \mathcal {I}_{\text {react}}\), where \(\mathcal {I}_{\text {react}}\) is the index set of surface reaction products.
Each surface constituent is characterized by its surface number density \(\underset {\mathrm {s}}{n}{}_{\alpha }\), atomic mass m_{α} and electric charge number z_{α}. The partial mass densities \(\underset {\text {s}}{\rho }{}_{\alpha }\), the total mass density \(\underset {\text {s}}{\rho }\) and the free electric charge density for the surface are defined by
On the YSZ surface gaseous species may adsorb and some reaction products may be formed. The admissible adsorption sites for gaseous species and reaction products in general depend on the lattice sites of the YSZ crystal. We assume that the density of the adsorption sites is proportional to the density of the anion surface lattice sites of YSZ. Several chemical reactions may occur. Denoting the constituents by A_{α} for \(\alpha \in \mathcal {I}_{S}\), the reactions can be written in the form
Surface free energy
The surface free energy can in general be assumed to be independent of the electric field. Here, we also assume that there is no elastic energy contribution and we distinguish two different entropic contributions to the free energy density. One takes into account the entropy of mixing of the mobile oxide ions on the anion lattice and the other is due to for the mixing of adsorbed gas species and reaction products on the adsorption sites. The metal ions and electrons only contribute to the reference energy. The free energy density for the surface is of the form
In general an elastic energy contribution has to be taken into account. The derivation of the energy is quite similar to the bulk. In [6] an example for a metalelectrolyte interface can be found. It turns out that if the constitutive equation of the surface tension depends only on the immobile YSZ species, and the lattice velocity υ^{#} is equal to the surface velocity, then the remaining equations for the adsorption and surface reaction are independent of the elastic contribution. Therefore, for simplicity, we ignore the surface elasticity.
Surface mixing of oxide ions
Surface mixing of gaseous adsorbates and reaction products
Reference surface energy
Surface chemical potentials.
The surface chemical potentials are given in terms of the surface number densities according to definition (4.7)_{right} as
Governing equations, constitutive modeling and coupling to the bulk
In the planar onedimensional approximation of the general surface mass balance equation (cf. [11, 16]), the tangential transport and the curvature related terms vanish. Only the surface chemical reactions (4.4) and mass transport normal to the surface can change the surface densities of the constituents. The surface mass balances and the remaining surface equation for the electric field in the electrostatic approximation read
Constitutive modeling
To derive constitutive equations for the normal mass fluxes and surface reaction rates, we apply the entropy principle according to [11].
At first, we reduce the entropy production \(\underset {\mathrm {s}}{\xi }{}\) derived in [11, eqn. (6.14)] to the isothermal electrostatic onedimensional setting^{4}, viz.,
Adsorption from YSZ bulk
Adsorption from gas phase
In the bulk gas phase, the fluxes are restricted by the constraint \({\sum }_{\alpha \in \mathcal {I}_{\text {gas}}} J_{\alpha } = 0\) and on the surface, (4.3) has to be satisfied. Therefore, we reformulate the entropy production due to the gas adsorption, as
Surface reactions
with \({R_{0}^{i}} \ge 0\). The constants β^{i} ∈ (0,1) are called symmetry factors.
Note that the arguments of the exponentials in (4.25) only depend on the surfacerelated quantities like surf. temperature and the chemical potentials of the surface species. Equation (4.25) is a surface generalization of the mass action kinetics. The adopted modeling approach implies the electric potential to be continuous, the overpotential^{5} appearing in the ButlerVolmer equation can be seen as an accounting for the potential drop due to the (unresolved) charge double layer. The presented model resolves the structure of the charged layer in the bulk YSZ and, in this sense, represents a more detailed description than the ButlerVolmer equation. In an asymptotic limit of vanishing double layer width the constitutive equation (4.25) allows to derive generalized ButlerVolmer equations for the surface reactions (see [21]).
Summary of the surface model
On the surface, we consider a single surface net reaction (5.2)_{right} with β = 1/2. From the YSZ phase only the mobile oxide ions and from the conductor only the surface electrons are allowed to participate in this reaction. We assume fast adsorption from the gas phase, i.e., \(\mu _{\alpha }_{S} = \underset {s}{\mu }{}_{\alpha }\) for \(\alpha \in \mathcal {I}_{\text {gas}}\).
Characteristic values and parameters for the surface part of the model
Reaction kin. coef.  \(\underset {\mathrm {s}}{R}{}_{0}\)  1×10^{10}/m^{2}/s 
Oxide ion adsorption coef.  \(\underset {\mathrm {s}}{A}{}_{0}\)  1×10^{17}/m^{2}/s 
Surface density of cations  \(\underset {\mathrm {s}}{a}{}^{{\#}}\)  \(\sqrt [\frac {3}{2}]{V^{{\#}}}\approx {1.04\times 10^{19\ }}\text {m}^{2}\) 
Surface ratio of imm. ox. ions  \(\underset {\mathrm {s}}{\nu }{}^{{\#}}\)  0.9 
Surface anion lattice num.  \(\underset {\mathrm {s}}{m}{}\)  [0,4] 
Gibbs energy of adsorption  ΔG_{A}  0.2e V 
Gibbs energy of reaction  ΔG_{R}  0.2 eV 
Partial pressure of O_{2}  \(p_{\text {O}_{2}}\)  21 kPa 
Standard pressure  p ^{ref}  100 kPa 
Simulation of a SOC halfcell
Cell potential
Electric current
Double layer capacitance of blocking electrode
We fix χ = 27 and \(\nu ^{{\#}}=\underset {s}\nu ^{{\#}}=0.9\) for all the following numerical simulations if not stated otherwise.
On the surface, we have
Comparison to experiment
We do not attempt to systematically adjust the model parameters to the data due to the polycrystalline nature of the YSZ studied in the experiment, instead, we try to illustrate the possible temperature dependence and the effect of the fitted parameters. As the temperature dependencies would need additional modeling efforts, as a first step, we performed the fit separately for each temperature.
It is difficult to assert that a particular oxide ion is mobile or immobile in the microscopic picture. It is suitable to consider the parameters ν^{#} and \(\underset {\mathrm {s}}{\nu }{}^{{\#}}\) determining certain (dynamic) equilibrium between the admissible and occupied vacancies in state with vanishing macroscopic free charge density. As this is usually an effect of thermal excitations, the values of ν^{#} and \(\underset {s}{\nu }^{{\#}}\) should depend on temperature. Also ΔG_{A} presumably depends on the temperature.
To this end also \(\underset {\mathrm {s}}{m}{}\) was treated as a fitting parameter shared for the three cases.
Capacitive currents
Fitted parameters, see Fig. 5
Temperature  T  475 ^{∘}C  525 ^{∘}C  575 ^{∘}C 
Gibbs adsorption energy  ΔG_{A}  0.14 eV  0.16 eV  0.18 eV 
Bulk immobiles ratio  ν ^{ #}  0.85  0.57  0.07 
Surf. immobiles ratio  \(\underset {\mathrm {s}}{\nu }{}^{{\#}}\)  0.85  0.64  0.44 
Surf. lattice ratio  \(\underset {\mathrm {s}}{m}{}\)  0.26  0.26  0.26 
Kinetic coefficients
Length of domain and sweep rate
The bulk diffusion limitation depends also on the domain length (see Fig. 9_{right}).
Currents of full half cell
Let us now investigate a scenario where the electrochemical reaction (5.2)_{right} proceeds on the surface .
In the constitutive relation for the reaction rate according to (4.25), we choose the symmetry factor \(\beta = \frac {1}{2}\), yielding
Cyclic voltammetry with realistic sweep rate r_{volt} = 1mV s^{− 1} is fixed in further demonstration of the basic features of the investigated system with the reaction.
Free energy parameters
Reaction rate
The effects of D and \(\underset {\mathrm {s}}{A}{}_{0}\) are for the open system similar as for the blocking electrode case. Small values would lead to surface charging and consequently to bulk charging limitations thus hindering the reaction.
Discussion
The representation of the interface was chosen as uncomplicated as possible so that the behavior of oxide ions double layer dynamics remains unobscured. This was achieved, however, let us discuss the drawbacks of the treatment. First, in a real electrode two distinguished surfaces (YSZ, metal) are present and the electrontransfer reaction occurs near their intersection. Hence, tangential diffusion of the surface species comes into play together with the particular geometrical realization. To this end a two or three dimensional model would be required including the inplane transport of the species. A question that naturally follows is: where exactly does the electrontransfer reaction occur, at the contact line or on one of the surfaces? Second, behavior of the metal electrons may in the close vicinity of the contact line start to display quantum effects that may result in richer behavior of the electrontransfer reaction. Third, the adsorption of gaseous species may under some circumstances limit the supply of gaseous species to the surface. Fourth, the appearing surface species depend on the particular electrode material. In particular, the nature and amount of the surface species will be different for Pt, Au or LSM electrodes. Also an additional phase of surface oxide ions with different adsorption energy might be present. Finally, one might consider production of surface oxygen O(s) for the blocking electrode (although no desorption to the gas phase is possible) and investigate the mechanical strain to the interfaces due to this.
Summary and Conclusions
A generalized PoissonNernstPlanck system describing YSZgasmetalinterface has been derived from first principles of nonequilibrium thermodynamics and numerically solved for simulating double layer capacitance and cyclic voltammetry measurements.
The core of the gPNP system is due to carefully derived free energy densities for the bulk YSZ and the YSZmetalgas surface capturing the main features of the YSZ crystalline nature. It is assumed that the described species, except for mobile oxide ions, are bound to the crystalline lattice. These assumptions result, using the entropy principle, in a novel form of the mobile oxide ion flux, which is a certain combination of the electrochemical potentials of all species. The charged layer in the metal is assumed to be in a diffusional equilibrium, since no transport limitations of the electrons is assumed. Finally, the formula for the electric current measured in the apparatus is derived.
A numerical model for the system has been derived and implemented in one spatial dimension using a finite volume method, specifically a variant of the ScharfetterGummel scheme, in the Julia programming language [25] for the details see Appendix A.
Although the model is strictly developed as isothermal, most of its parameters may depend on the temperature. Therefore, the parametric study is also aimed to demonstrate the scenarios where some of the parameters become limiting to the charge transfer of the system. Finally, the capacitance of blocking YSZ electrode taken from literature [22] is fitted with the model, the quality of the fit relies heavily on the newly introduced ratios of immobile oxide ions ν^{#} and \(\underset {s}{\nu }^{{\#}}\). For each temperature these can be fitted alongside with ΔG_{A} to the measured data. While the derivation of the model assumed a single crystal, the measurements had been obtained for polycrystalline YSZ. Therefore, the presented fitting results can be seen only as a first step towards a model for polycrystalline YSZ which ideally should be derived from the presented model using homogenization techniques. Moreover, the presented model can serve as a starting point for further extensions containing more sophisticated surface chemistry capable of describing the anodic and cathodic within one kinetic model.
Footnotes
 1.
Either fuel cells, or electrolysis cells.
 2.
The free energy function is defined here as ρψ = ρu −P ⋅E − Tρs, where ρu is the density of internal energy.
 3.
The electrochemical potential is defined as \(\mu _{\alpha }^{e} = \mu _{\alpha } + \frac {z_{\alpha } e_{0}}{m_{\alpha }} \varphi \)
 4.
For the representation of the entropy production, we assumed that the kinetic term \(\frac {1}{2}\rho (\underset {s}{{\upsilon }}{{\upsilon }})^{2}\) is small and can be ignored.
 5.
Discontinuity in the electric potential between the neighboring phases.
 6.
In general, the total stress has to specified, but due to electroneutrality assumption at x_{B} and the one dimensional approximation, the total stress and material pressure p coincide.
 7.
We chose nitrogen as the reference species for the gas phase, i.e., A_{0} = N_{2}.
Notes
Acknowledgments
The publication of this article was funded by the Open Access fund of the Weierstrass Institute.
Funding information
This work was supported by the German Research Foundation, DFG project no. FU 316/141, and by the Czech Science Foundation, GAČR project no. 1914244J. VM received funding from the Fuel Cells and Hydrogen 2 Joint Undertaking under grant agreement no. 671481. This Joint Undertaking receives support from the European Union’s Horizon 2020 research and innovation programme, Hydrogen Europe and Hydrogen Europe research. PV and VM received partial support by grant SVV2017260455.
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