On the mixed ionic and electronic conductivity in polarized yttria stabilized zirconia

Solid State Ionics 320 (2018) 239–258

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On the mixed ionic and electronic conductivity in polarized yttria stabilized zirconia Reiner Kirchheim

T

⁎

Institut für Materialphysik, Georg-August-Universität Göttingen, Göttingen, Germany International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan

A B S T R A C T

Electric transport in oxides with mixed conduction, being exposed to dc electric fields and being placed between blocking electrodes is discussed and exemplified for yttria stabilized zirconia (YSZ). This is relevant for cases of large current densities in solid oxide fuel cell (SOFC), solid oxide electrolysis cells (SOEC) and during flash sintering of oxide powders. In the present study equilibrium of defect reactions is not attained in the presence of blocking electrodes leading to a continuous generation of holes in p-regions and electrons in n-regions feeding the external current. In the remaining region between n- and p-region ion conduction is dominant. Thus a p-i-n junction is formed. The electronic species may be continuously generated by reaction with gaseous oxygen or by the creation and annihilation of single-charged vacancies. For both cases current voltage-relations are derived by assuming that the gradient of the chemical potential of double-charged vacancies is reduced below its value given by a zero electrochemical potential. The reduction is introduced as a consequence of the Le Chatelier Principle, which requires reactions to occur, which counteract the accumulation and depletion of doublecharged vacancies by the applied electric field and corresponding deviations from charge neutrality. Scenarios are discussed where electrons and holes recombine with a concomitant emission of light. This explains why the spectrum of the emitted light deviates from that of black body radiation for both examples of flash sintering and the Nernst glower.

1. Introduction Under ambient oxygen pressures most oxides have a large band gap in their electronic structure and, therefore, exhibit negligible electronic conductivity. However at high temperatures vacancies in the oxygen sublattice give rise to ionic conductivity. Yttria doped zirconia or yttria stabilized zirconia (YSZ) is one of the examples containing a large fraction of structural vacancies. Yttria forms a solid solution in zirconia up to high concentrations [1]. In this solid solution of oxides the tetravalent zirconium is partly substituted by trivalent yttrium and vacancies are generated in the oxygen sublattice in order to maintain charge neutrality. Thus the concentration of the vacancies can be very high and allow vacancies to move through the material. Then YSZ becomes a solid electrolyte and is used in solid oxide fuel cells (SOFC) and in solid oxide electrolysis cells (SOEC) [2–4]. Besides ions, electrons and holes contribute to the conductivity of YSZ. Electrons are formed at low oxygen pressures by a point defect reaction giving rise to n-type conduction [5–9], the relevant reaction in Kröger-Vink's notation is

⁎

OOx →

1 O2 + VO•• + 2e′, 2 x

(1) ••

where OO refers to oxygen on anion lattice sites and VO are doublecharged vacancies on the same sublattice and electrons being placed in the conduction band. Eq. (1) being reversed leads to hole generation expressed by the following equation

VO•• +

1 O2 → OOx + 2h•, 2

(2)

where electrons from the valence band are used leaving holes behind. At dynamic equilibrium the individual forward and backward rates in Eqs. (1) and (2) have to be equal with electrons and holes annihilating each other by the reaction

e′ + h• = nil

(3)

until their equilibrium concentrations are reached. With decreasing oxygen pressures Eq. (2) predicts, that less holes are produced with a concomitant increase of the concentration of electrons as the law of mass action requires for reaction in Eq. (3). Finally at a corresponding low pressure the electrons surpass the contributions of vacancies to

Corresponding author at: Institut für Materialphysik, Georg-August-Universität Göttingen, Göttingen, Germany. E-mail address: [email protected].

https://doi.org/10.1016/j.ssi.2018.03.014 Received 22 November 2017; Received in revised form 5 March 2018; Accepted 9 March 2018 0167-2738/ © 2018 Elsevier B.V. All rights reserved.

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μOOx =

1 μ + ηVO•• + 2ηe′, 2 O2

(4)

where for the neutral species the electrochemical potential reduces to the chemical potential. For a constant composition of YSZ the potentials of OOx remains constant and, therefore, changes of oxygen pressure pO2 or μO2, respectively lead to changes of the electrochemical potential of vacancies and electrons. The latter quantity is the Fermi level of electrons, which then can be moved up and down either in the conduction band or in the valence band leading to n- or p-type conduction, respectively. With μO2 depending linearly on lnpO2 and μe′ on lnce (ce being the concentration of electrons in the conduction band) the slope of −1/ 4 in Fig. 1 is explained, if the electronic conductivity is proportional to ce [5,6]. In an electric field the charged species are moving and Hebb [22] and Wagner [21] assumed for the mixed conductors AgBr and Ag2S that a steady state is attained with the gradients of the electrochemical potentials of ions to be zero. Schmalzried [23] pointed out that under these conditions internal reactions or reactions at the electrode/electrolyte-interfaces occur. Maier [24a,b] proposed the general concept of conservative ensembles including internal reactions with valence changes of moving ions and vacancies. In addition, he pointed out how blocking and/or reversible electrodes for either electrons or ions will change the voltage/current-relationship. Choudhury and Patterson [25], Jacobsen and Mogensen [26] and Näfe [27] treated n- and p-type conduction in YSZ based on Eqs. (1) and (2). In all studies a gradient of the partial pressure of oxygen was evaluated. In order to maintain the related changes of pressure in steady state no exchange with oxygen from the surrounding atmosphere is allowed. The molecular oxygen needed to generate electrons and holes has to be ab- or desorbed from reversible electrode. This is different to Wagner's [20,21] and Hebb's [22] approach treating electron and silver ion production by the reactions

Fig. 1. Measured partial conductivities of YSZ as reproduced from Ref. [6] are shown at different temperatures as a function of the surrounding oxygen partial pressures (σ2 is the partial ionic conductivity, σp and σn the one of holes and electrons). Similar values are obtained by others [5]. Open circles mark the partial pressures where electronic and ionic conductivity are equal.

conductivity and the YSZ becomes a n-type conductor. At increasing values of oxygen pressures the concentration of holes increases with a decreasing electron concentration and finally a transition from ionic to p-type conduction occurs. This has been experimentally proven in various studies [5,6]. The corresponding measurements of ionic conductivity σ2 were performed at low dc-electric fields of the order of mV/ cm or low ac-electric fields, in order to minimize or avoid polarization of YSZ. Under these conditions ionic conductivity prevails in a wide range of oxygen pressures as shown in Fig. 1. The term ionic conductivity will be used in this study despite the fact that vacancies are the moving charge carriers. However it has been shown recently by Masao and West [7], that electronic conductivity in YSZ dominates at high dc-electric fields of 5 to 100 V/cm in air and at temperatures of 200 to 700 °C where for low dc-electric fields electronic conductivity is negligible. This different behavior was interpreted by the creation of single-charged oxygen ions. In a different experiment [10] with an electrochemical cell Ni/YSZ/Ni composed of a 2 mm thick disc of single crystalline 10YSZ between two nickel electrodes a constant current of 40 mA/cm2 was passed through the electrolyte at 1200 K in vacuum (residual pressure: 10−3 Pa). The ionic part of the current caused the formation of NiO on the positive electrode whereas Ni5Zr was formed at the negative electrode. The charge necessary for the formation of the NiO and the Ni5Zr was about the same, but corresponded to 3% of the total electric charge only for a dc-field of 10 V/cm [10]. Besides applications in electrochemical devices, YSZ is also widely used as a structural material. For this purpose various shapes of the material are obtained by sintering oxide powders at elevated temperatures. As shown recently [11], the sintering process can be accelerated by applying electric fields during the compaction process. For very large electric fields a sudden increase of the current occurs after an incubation time and under potentiostatic control [12,13]. The current run-away is accompanied by emitting light with high intensity giving rise to the name flash sintering [14–16]. Whether the sudden current increase is caused by the onset of electronic conductivity or simply by Joule heating due to increasing ionic currents, is under debate [17,18]. The analysis of mixed ionic and electronic conduction is often based on the pioneering work of Carl Wagner [19–21] describing equilibrium of defect reactions with the electrochemical potentials ηi of the participating defect i. This was applied to YSZ [5,6] yielding for Eq. (1)

x AgAg → Agi• + e ´,

(5)

where gradients of all species could sustain in the solid electrolyte by an internal reaction. Different to the treatment by Choudhury and Patterson [25] and others [26,27] exchange with environmental oxygen is allowed in the present study. Also different to a study by Reiss [28] for compounds with small deviations from stoichiometry the gradient of the chemical potential of the ions will not be zero. The cells used in references [21] to [28] consist of either two reversible or one reversible with one blocking electrodes. Two blocking electrodes are used in order to determine the intrinsic electronic conductivity of the mixed conductor [22]. It is a major goal of this study to demonstrate that the electronic conductivity can be increased after an incubation time by two types of reactions generating electrons and holes. One of these types are the reactions in Eqs. (1) and (2) with gaseous oxygen provided by the environment. The electrons in n-type regions are generated continuously via Eq. (1) and holes in ptype regions via Eq. (2). Between n- and p-type regions a region with dominant ionic transport is required for avoiding a divergence of current. Molecular oxygen being generated in the n-region according Eq. (1) is consumed in the p-region via Eq. (2). Thus an external reaction at the electrolyte/gas-phase is controlling defect reactions. A second type of reaction may play a role, if molecular oxygen is absent. Then electronic conductivity will be explained by the formation and transport of single-charged vacancies, which in Maier's concept [24a,b] can be formally treated with an individual diffusivity. In order to obtain analytical expressions for current and voltage simplifying assumption were made for the gradients of chemical potential of double and single-charged vacancies. Consequences of the models regarding the attainment of steady state and the emission of light are discussed at the end of this study.

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Fig. 2. Cell geometries with YSZ having a ratio of thickness to length much smaller (left) or much larger (right) than unity. The cells are in a compartment of constant oxygen activity. For the case on the left a larger area is exposed to the environment containing molecular oxygen and, therefore, the electronic conductivity may be controlled by the reaction with molecular oxygen. For the case on the right electrons and holes, if contributing to electronic conductivity, have to be produced inside YSZ for instance by the formation and annihilation of single-charged vacancies as discussed in this study.

If the potential gradient is independent of position and time as depicted in Fig. 3 for the steady state, local charge neutrality has to be present because of the Poisson Equation (cf. Eq. (20)). However, the gradient of the chemical potential of VO•• requires a local change of the concentration of the charged vacancies VO••, which contradicts the required local charge neutrality. Therefore, the potential along the sample length has to be curved, which will be always the case for the models discussed in this study (cf. Fig. 6). In other words the sample becomes polarized containing positive and negative charge densities. The resulting stationary changes are discussed in Section 4. Nevertheless, the characteristic time for reaching steady state has to be known, because before that time the intrinsic electronic conductivity will be measured and after that time the reactions discussed here will come into play. The characteristic time will be calculated in the following for the limiting case of having a gradient of μ2 caused by small changes of the concentration of VO••, i.e. allowing small deviations from charge neutrality not affecting the potential trace. Solutions of Eq. (6) for the transient state are described in a few references only [28–30] assuming further on an ideal dilute solution, i.e. μ2 = μ2o + RT ln c2. The resulting transients of the chemical potential of VO•• are shown schematically in Fig. 3. The characteristic time τ for reaching steady state is [29–31]

Whether the reaction with gaseous oxygen or the formation of single-charged vacancies is controlling the generation of electrons or holes may be determined for instance by the shape of the YSZ. Fig. 2 depicts two shapes with a very different free surface to volume ratio of YSZ. The one on the left is a thin elongated sample like a strip of a thin film on a substrate or a sample as used in flash sintering experiment. Due to the large surface area reaction with gaseous oxygen may control conductivity as described in Sections 3.1 and 4.2. The right hand part of Fig. 2 is meant to be a thin disc of YSZ placed between two disc-shaped electrodes. Thus the free surface of YSZ is small and conductivity may be controlled by internal defect reactions generating single-charged vacancies as described in Sections 3.2 and 4.3. 2. Transient and stationary transport of vacancies YSZ shall be placed between two blocking electrodes as shown in Fig. 2. An applied electric field drives vacancies in YSZ from the positive to the negative electrode setting up a concentration gradient or a gradient of the chemical potential of the vacancies VO••, respectively which drives the vacancies backwards. The corresponding flux J2 is then given as the product of mobility and driving forces

J2 =

∂μ 2 ⎤ ∂ϕ D2 c2 ⎡ D c ∂η σ ∂η −2F − = − 2 2 2 = − 2 2, ⎥ RT ⎢ x x RT x 2 F ∂x ∂ ∂ ∂ ⎣ ⎦

(6)

2

1 π 2 2 ⎛ ⎞ +⎛ ⎞⎤ = D2 ⎡ ⎥ ⎢ τ l l ⎝ ⎠ E⎠ ⎦ ⎝ ⎣

••

where the subscript 2 refers to the double-charged vacancies VO , their diffusion coefficient D2, concentration c2, chemical potential μ2, and electrochemical potential η2 = μ2 + 2Fϕ, R is the gas constant, T is temperature, F is Faraday's constant and ϕ is the electric potential. Boundary conditions in a finite sample of length l shall be J2(0, t) = J2(l, t) = 0 corresponding to blocking electrodes and leading to J2(x,t) = 0. Then only electronic conduction occurs. The case of partly blocking electrodes, where only a limited flux Jlim of oxygen can be accommodated by the electrodes, may be also treated like the one of blocking electrodes by substituting the flux J2 with J͠ 2 = J2 − Jlim . A stationary concentration profile as represented by the bold line in Fig. 3 is attained after a characteristic time yielding

∂η2 =0 ∂x

⎜

(9)

with a characteristic length lE defined as

lE =

RT l, 2FU

(10)

where U/l is the applied electric field. For high fields the attainment of steady state could be remarkable reduced, well below the commonly used value for pure diffusion l2/(D2π2). This corresponds to lE < l or U > RT/(Fπ), i.e. at 1000 K this will be the case for U > 0.27 V. In the following chapter it is assumed that a time of the order of the characteristic time in Eq. (9) has passed, in order to have established pronounced changes of the chemical potential of vacancies with concomitant changes of the reaction rates of Eqs. (1) and (2). These changes and consequences on the mixed conduction will be discussed first qualitatively.

(7)

or

∂μ 2 ∂ϕ = −2F . ∂x ∂x

⎟

(8) 241

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Fig. 3. Schematic presentation of transients of the chemical potential of double-charged vacancies at times t1 < t2 < t∞. The steady state profile at t∞ is given for μ2 by the bold straight line and for ϕ by the dotted-dashed line. Note that this profile is modified in this study to allow currents to flow, i.e. to obtain ∂η2/∂x ≠ 0 within YSZ whereas at x = 0 and x = l blocking electrodes require ∂η2/∂x = 0.

3. Novel treatment of electronic and ionic transport: qualitative aspects

accumulated vacancies may be compensated such that electrons from the electrode are trapped by zirconium. This corresponds to a reduced YSZ which could be a n-conductor different to the scenario discussed before. By the same reasoning the region next to the anode may be pconducting again different to the assumption made before. However, as discussed at the beginning of Section 5 this will not affect the results of the modeling, i.e. the current-voltage relation. Thus the previous assignments of n- and p-regions are maintained. Then the electrons and holes produced in the n- and p-region move to the adjacent electrodes giving rise to an electronic current from the n- and the p-region (left to right in Fig. 2). In order to maintain charge neutrality in a stationary situation the same vacancy charge per time has to pass through the iregion driven by the electric field. A corresponding depletion of vacancies in the n-region and accumulation in the p-region is compensated by reactions with molecular oxygen being generated at the surface of the YSZ in the n-region and absorbed in the p-region (c.f. Eq. (2) and Fig. 4). Thus the reactions give rise to sink and source terms in the equation of continuity with no changes of the local vacancy concentration in steady state, i.e. vacancies are treated as a conservative ensemble [24a,b]. Then a divergence of vacancy flux has to compensate the sink and source activity. The resulting vacancy transport in the YSZ is short circuited externally via the gas phase. In other words, the surfaces in the p- and n-region act like cathode and anode for the ionic current, whereas the blocking metallic electrodes act as current collectors. A corresponding quantitative treatment of these processes is provided in the following chapter after treating the effect of singlecharged vacancies in a qualitative manner first.

3.1. Creation of electrons and holes via reactions including molecular oxygen Before applying an electric field to YSZ the solid electrolyte shall be in equilibrium with molecular oxygen of a constant partial pressure in the surrounding atmosphere. The corresponding initial chemical potential for VO•• in YSZ is labeled μ2o. Temperature and pressure range are such that a certain stationary concentration of electrons and holes are produced via Eqs. (1) and (2) but ionic conductivity prevails (cf. Fig. 1 for the corresponding pressure and temperature ranges). As discussed before applying an electric field increases (decreases) the chemical potential of VO•• at the negative (positive) electrode. Therefore, the rate of hole creation is enlarged at regions near the negative electrode (cf. Eq. (2)), whereas less holes are produced near the positive electrode leading to an excess of electrons. Thus with increasing electric fields electronic contributions to conductivity may outperform the intrinsic electronic one and p- and n-type regions form with dominant electron and hole conductivity. Between these two regions μ2 remains close to its initial value μ2o. Therefore, the ionic conductivity will remain to be dominant in the middle of the sample like before the field was applied (cf. Fig. 4). This interpretation implies that the electronic structure of YSZ is not affected by the generated holes. Only the Fermi-level in the p-region is lowered within the gap of the oxide and raised vice versa in the n-region. However, next to the negative electrode the positive charge of the

Fig. 4. Proposed generation of holes (p-region) and electrons (n-region) by reaction with molecular oxygen surrounding YSZ for samples with a large free surface (left part of Fig. 2). Positive and negative electrodes are blocking electrodes exchanging electrons and holes with YSZ only. In order to establish a stationary electric current, vacancies have to be transported from the n- to the p-region. The resulting deficiency and/or accumulation of vacancies are compensated by a counteracting flux of oxygen molecules in the gas phase. The flux of vacancies is accompanied by a flux of lattice oxygen having the same magnitude but different direction. Note that at each position x the concentrations of both VO•• and OOx are assumed to be the same within the corresponding cross section due to the large aspect ratio of sample length to thickness or diameter of the cross section.

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3.2. Creation of electrons and holes via reactions including single-charged vacancies

4. Novel treatment of electronic and ionic transport: quantitative aspects

Without an electric field the concentration of single-charged vacancies VO• is negligible. However with large changes of the chemical potential of VO•• induced by the electric field the following reaction

4.1. General considerations

VO•• → VO• + h•

With the polarity of an applied dc-voltage as shown in Figs. 2 to 5 VO•• is driven to the left with a change of its chemical potential μ2 depicted in Fig. 2. The resulting deviation from charge neutrality may be – as a consequence of the Le Chatelier Principle – minimized by electron and hole formation inside YSZ. Different to this scenario two defect reactions are proposed in this study, which also work against changes of μ2 due to the Le Chatelier Principle. For these reactions a steady state with the electrochemical potential gradient η2 of VO•• being zero (cf. Eq. (7)) and leading to Eq. (8) will not be attained, as the defect reactions will reduce the gradient of μ2. Thus Eq. (8) has to be rewritten as

(11)

is driven to the right in regions next to the negative electrode, where the activity of VO•• becomes very high. For the very low activity of VO•• at the positive electrode the following reaction

VO• → VO•• + e′,

(12)

is also driven to the right. Then h• and VO• are generated closer to the negative electrode. For the sake of simplicity and for assuming having a minor affect the formation of neutral vacancies [24a] is not considered here. With a blocking electrode the VO• ions are piling up at the negative electrode. A corresponding backward diffusion towards the positive electrode sets in against the driving force of the electric field. When the VO• vacancies are reaching the right regions of the sample, where the chemical potential of VO•• is below its initial value μ2o (cf. Fig. 3), they are annihilated with a concomitant generation of holes (cf. Eq. (11)). Holes are neutralized at the negative electrode contributing to the external current, which in steady state has to have the same magnitude as the one from the electrons at the positive electrode. The presented scenario is depicted schematically in Fig. 5, where the regions are labeled n, p or i according to electron, hole or ion conduction being dominant. The flux of double-charged vacancies is not allowed to pass the YSZ/electrode interface; its accumulation at the negative electrode is circumvented by an internal short circuit with single-charged vacancies. For a stationary overall vacancy concentration in YSZ the fluxes of double and single-charged vacancies have to be of different sign but of the same magnitude. A quantitative treatment of the presented scenario requires to take into account reaction rate theory and transport relations for VO• like the one in Eq. (6) for VO••. Arguments for a separate treatment of fluxes of single and double-charged vacancies are provided in Ref. [24a]. A corresponding attempt is made in the following chapter.

r

∂μ 2 ∂ϕ and r > 1. = −2F ∂x ∂x

(13)

The parameter r describes the deviation from the steady state as presented by the bold straight line in Fig. 3. With the extreme cases of r → 1 corresponding to a total polarization and r→∞ for charge neutrality with ∂μ2/∂x = 0. As a consequence a flux of VO•• from the positive to the negative electrode sets in, which has to become zero at the YSZ/electrode-interfaces because of having blocking electrodes. In order to get a stationary situation, i.e. avoid continuing accumulation of VO•• in the p-region and depletion in the n-region, counter-fluxes have to be present. These counter-fluxes are in the present study either transport of oxygen molecules through the environment or transport of single-charged vacancies in the YSZ (cf. Figs. 4 and 5). Again Eq. (13) is a consequence of the Le Chatelier Principle counteracting the effect of the applied electrical potential by forcing the reactions in Eqs. (1), (2), (11) and (12) to proceed to the right hand side. Note that the Le Chatelier Principle also accounts for a reduction of zirconium, which as well counteracts the accumulation of positive charge due to an increase of VO••-concentration. For the sake of simplicity it is assumed that the parameter r in Eq. (13) is independent of x and, therefore, Eq. (13) can be integrated yielding for the p-region

U r [μ 2p (x ) − μ 2po ] = −2F ⎡ϕp (x ) + ⎤ 2⎦ ⎣

(14)

Fig. 5. Proposed generation of electrons and holes by introducing single-charged vacancies within YSZ. By the reactions in Eqs. (11) and (12) (as shown in the figure) regions of dominant hole conduction (p-regions) and regions of dominant electron conduction (n-region) form. Positive and negative electrodes are blocking electrodes exchanging electrons and holes with YSZ only. A stationary electric current requires doublecharged vacancies to be transported from the nto the p-region and single-charged vacancies in the reverse direction. The latter becomes possible, if the gradient of the chemical potential of VO• has a larger effect on the flux compared to the gradient of the electric potential ∂ϕ/∂x (cf. Eq. (6)).

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Fig. 6. Schematic presentation of the chemical potential of double-charged vacancies μ2(x) and electric potential ϕ(x) within the p-, n- and i-region different from the steady state in Fig. 3 as a consequence of Le Chatelier's principle. The slopes of both potentials have to be zero at the YSZ/electrode-interface.

∂ 2ϕ = −4περ (x ) ∂x 2

and for the n-region

U r [μ 2n (x ) − μ 2no ] = −2F ⎡ϕn (x ) − ⎤, 2⎦ ⎣

relating the curvature of ϕ to the charge density ρ(x) and the dielectric constant ε, the p-region has to be negatively polarized, whereas the nregion is positively charged. The corresponding deviations from charge neutrality are very small for the voltages under consideration. In an atomistic interpretation the changes may be due to electrons from the negative electrode entering the p-region and being trapped by Zr-ions. Thus these trapped electrons compensate the accumulation of VO•• to a somewhat larger extent leading to the overall slight negative charge density in the p-region. This leads to F-centers and changes of the valence of Zr from 4+ to 2+ or even 0, i.e. forming eitherZrZr″ or Zr″Zr″. The interpretation is in agreement with studies on polarized YSZ [32,33]. In the n-region the positive polarization stems from electrons entering the positive electrode with holes being trapped at lattice oxygen changing its valence from 0 to +1, i.e. forming OO•. Note that these defects are immobile and, therefore, do not contribute to the currents as discussed for the steady state in the following. So far the conditions for reaching a stationary state regarding mobile defect concentrations and current density have been discussed qualitatively only by stating that the action of the electric field leading to an accumulation of VO•• at the negative electrode and its depletion at the positive electrode have to be compensated by reactions as given by Eqs. (1), (2), (11) and (12). This can be quantitatively treated by a continuity equations of the type

(15)

where the potentials μ2p, μ2n, ϕp, ϕn in the p- and n-region depend on x and μ2po, μ2no are the boundary values at x = 0 and x = l, respectively. U is the cell voltage excluding potential drops at the YSZ/electrodeinterfaces. At these interfaces the boundary conditions for blocking electrodes J2(0) = J2(l) = 0 apply as well, i.e. Eqs. (6) and (13) require

− 2F

∂μ 2 ∂μ ∂ϕ − = (r − 1) 2 = 0 at x = 0 and x = l. ∂x ∂x ∂x

(16)

As a consequence of r > 1 the gradients of both μ2 and ϕ have to be zero at the YSZ/electrode-interfaces. Thus the trace of both potentials μ2 and ϕ along the sample length will look as shown schematically in Fig. 6 leading to a curvature in the p- and n-region. In the i-region both electrical and chemical potential are changing linearly, where the differences of these quantities are obtained from Eqs. (14) and (15) as

Δμ 2i = μ 2n (l − ln ) − μ 2p (lp) = μ 2no − μ 2po −

2F [ϕ (l − ln ) − ϕp (lp) − U ] r n

(17)

Thus the electric current density ii through the i-region is

ii = −2FJ2 = =

ϕn (l − ln ) − ϕp (lp ) Δμ 2i ⎤ σ2 ⎡ −2F − = −2F ⎢ l li ⎥ i ⎣ ⎦

−σ2 [(r − 1)Δμ 2i − 2FU − r (μ 2no − μ 2po )]. 2Fli

∂ck ∂J = − k + Rk = 0, ∂t ∂x

(18)

(21) ′

•

••

•

where k stands for any of the species (e , h , VO , VO ) which are produced or annihilated by a rate Rk. Depending on the sign of Rk it represents a sink or source term within the continuity equation. With the help of Eqs. (6) and (13) the last equation becomes

Here and in the following current densities are positive for applied voltages as given in Figs. 2 to 6, if negative charge carriers are transported to the right and positive ones to the left. The graphs in Figs. 6 and 3 are different, because the reactions with molecular oxygen or reactions involving single-charged vacancies counteract the changes enforced by the electric field. In other words, the Le Chatelier's principle requires

μ 2po − μ 2no < 2FU .

(20)

∂μ 2 ⎤ ∂ϕ ∂ D2 c2 ⎡ D c ∂ 2μ 2 σ ∂ 2μ 2 − [r − 1] = 22 [r − 1] = R2 . −2F ≈ 2 2 2 ∂x RT ⎢ ∂x ∂x ⎥ RT ∂x 4F ∂x 2 ⎣ ⎦ (22)

(19)

Going from the left hand side of Eq. (22) to the right one it has been tacitly assumed that the concentration of VO•• does not change much

According to the Poisson-Equation 244

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within YSZ. For the proposed two types of reactions involving molecular oxygen and single-charged oxygen the reaction rates R2 are evaluated in the following sections as a function of μ2. Then the resulting partial differential equations based on Eq. (22) can be solved. Eq. (22) also states that the curvature of the chemical potential of VO•• is negative in the p-region, where VO•• is annihilated (R2 < 0) by reactions according to Eqs. (1) and (12), whereas the curvature is positive in the n-region following the same reasoning.

ip = F

∂chp ⎤ dx = −2F ⎦g ⎣ ∂t ⎥

∫ ⎡⎢ 0

lp

∂c2p ⎤ dx . ⎦g ⎣ ∂t ⎥

∫ ⎡⎢ 0

(29)

Here it is tacitly assumed that both in the p- and n-region the mobility of the defect electrons is so large that a small gradient of the electric potential is sufficient to transport them to the corresponding electrodes or annihilate them via Eq. (3) without changing their concentration and the electric potential much. Inserting Eq. (23) into Eq. (29) yields for the low polarization case of Δμ2po < < RT

4.2. Reactions including molecular oxygen

l

Δμ μO + 2μ 2o p ⎡ kh ⎛ 2p ⎞ ⎤ ⎞ exp ⎛ 2 ⎢1 − exp ⎜ RT ⎟ ⎥ dx s 2 RT ⎝ ⎠ 0 ⎣ ⎝ ⎠⎦ μO2 + 2μ 2o Δμ 2po kh ⎞ ≈ 2F exp ⎛ lp 2RT s ⎝ ⎠ RT

The relevant defect reactions are given by Eqs. (1) and (2). Turnover rates for these reactions are derived in Appendix A which become for the p-region (Eq. (A8))

ip = −2F

⎜

⎟

⎜

Rp2

Δμ μO + 2μ 2o ⎡ ∂c2p ⎤ k ⎞ 1 − exp ⎜⎛ 2p ⎟⎞ ⎤ < 0 ≈ h exp ⎛ 2 =⎡ ⎢ ⎥ ⎥ ⎢ ∂ t s 2 RT ⎦g ⎣ ⎝ ⎠⎣ ⎝ RT ⎠ ⎦ ⎜

(30)

or

(23)

ip = σ2

Δμ μO + 2μ 2o ⎡ D2 c2o k 2p ⎞ 1 − exp ⎛⎜ 2p ⎞⎟ ⎤ [r − 1] = h exp ⎛ 2 ⎢ ⎥ s 2 RT RT ∂x 2 ⎝ ⎠⎣ ⎝ RT ⎠ ⎦ ⎜

in = −2F

(24)

⎜

⎟

ii =

(33)

σ2 (Δμ 2no − Δμ 2po + 2FU ) 2Fli

(34)

This has to be equal to the current densities in n- and p-region, i.e. (cf. Eq. (31))

σ2 (r − 1) Δμ 2po (Δμ 2no − Δμ 2po + 2FU ) = σ2 lp . 2Fli 2F λ g2

(26)

By the same reasoning the same form of partial differential equation is obtained for the n-region. Within Appendix C solution are derived for two limiting cases of polarization. For low polarization (|Δμ2|/RT < < 1) and λg < < x ≤ lp or λg < < x − l ≤ ln Eqs. (C5) and (C11) apply

(35)

Rearranging the last equation with the trivial relation li = l − lp − ln and by neglecting quadratic terms of ln/λg and lp/λg again leads to

lp l =

2

2FU + Δμ 2po − Δμ 2no Δμ 2po (r − 1)

λ g2 .

(36)

With the last equation and Eq. (31) as well as the condition of equal current densities leads to

(27)

i = ii = ip = in =

and

Δμ 2po − Δμ 2no μ − μ 2no σ2 ⎡ ⎤ = σ2 ⎡U − 2po ⎤. U− ⎥ 2 2F F l l ⎢ ⎦ ⎣ ⎦ ⎣

2

1 x − l⎞ ⎤ ⎡ Δμ 2n = Δμ 2no ⎢1 − ⎛⎜ ≈ Δμ 2no . ⎟ 2 ⎝ λg ⎠ ⎥ ⎣ ⎦

(32)

In steady state the electronic currents in the p- and n-region have to be equal to the ionic current ii in the i-region which is obtained from Eqs. (18), (27) and (28) for small polarization by neglecting quadratic terms of ln/λg and lp/λg

leads to a simple form of Eq. (24)

1 x ⎤ ⎡ Δμ 2p = Δμ 2po ⎢1 − ⎜⎛ ⎟⎞ ⎥ ≈ Δμ 2po 2 ⎝ λg ⎠ ⎣ ⎦

μO + 2μ 2o Δμ 2no kh (r − 1) Δμ 2no ⎞ exp ⎛ 2 ln = −σ2 ln . s 2 RT RT 2F λ g2 ⎝ ⎠

Δμ 2po lp = −Δμ 2no ln .

(25)

Δμ 2p ⎤ ∂2 ⎛ μ 2p ⎞ ⎡ ⎞⎟ . = 1 − exp ⎛⎜ ⎥ ∂x 2 ⎝ RT ⎠ ⎢ ⎝ RT ⎠ ⎦ ⎣

(31)

Because of Δμ2no < 0 both currents are positive. Steady state requires ip = in or

⎟

By measuring the exchange rate of O-18 and O-16 isotopes between the gas phase and YSZ the parameter kn in Eq. (24) can be replaced by a measurable quantity, called exchange parameter h as shown in Appendix B. Inserting the corresponding Eqs. (B2) and (B3) into Eq. (24) and using a characteristic length λg defined as

s (r − 1) h

(r − 1) Δμ 2po lp 2F λ g2

By the same reasoning the current in the n-region, where the electron production is twice the generation rate of VO••, is obtained by integration over the n-region

∂ 2μ

λ g2

⎟

⎟

where R2p is the reaction rate as introduced in Eq. (22), the subscript g refers to the reaction with gaseous oxygen, kh is a reaction coefficient, Δμ2p = μ2p − μ2o > 0 is the deviation of the chemical potential of VO•• in the p-region from its initial value μ2o, and s is the sample thickness. It is assumed that the sample thickness s is small compared to sample length l, in order to allow homogenization from the surface, where the reaction occurs, over the cross section of the sample (cf. Fig. 2). Inserting Eq. (23) in Eq. (22) yields

λg =

∫

(37) Using Eq. (19) the last equation is approximated by

(28)

The case of high polarization with |Δμ2|/RT > > 1 is treated in Appendix D. In the framework of the present model of stationary reactions involving molecular oxygen the related production of holes and electrons gives rise to a stationary current, where all the holes produced in the pregion are absorbed by the negative electrode and vice versa electrons by the positive electrode. Then the production rate of holes is two times the annihilation rate of VO•• (cf. Eq. (2)) and integration over the pregion yields the current

i ≈ σ2

U . l

(38)

Thus the current with blocking electrodes is the same as if nonblocking electrodes are present. However, one has to keep in mind that Eq. (38) requires significant changes of the chemical potential of VO•• with respect to its initial value μ2o which takes time (cf. Fig. 3). At the beginning the conductivity is the intrinsic electronic one of YSZ, i.e. much smaller than the ionic one, σ2 (cf. Fig. 7 in Section 5). So far the requirement of a steady gradient of the chemical potential 245

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Fig. 7. Schematic presentation of the current voltage relation at different times after applying the voltage. An incubation time τ has to pass, in order to reach the steady state discussed in this study. The incubation time is approximated by Eq. (9) and is shown as a dashed line. For t < < τ the current is the intrinsic electronic current iin as taken from Fig. 1. For t > > τ the current is proportional to the applied voltage U for low polarization (cf. Eq. (38)) and reaches a limit imax for high polarization (cf. Eq. (42)).

μ2 at the p/i- and i/n-interfaces has not been used. With Eqs. (27), (28) and (33) it will lead to

2FU + Δμ 2po − Δμ 2no lp Δμ 2i l . = −Δμ 2po 2 = Δμ 2no n2 = li (r − 1) l λg λg

length, l and thickness, s (cf. Fig. 2) and sufficient supply of molecular oxygen as realized in sintering experiments [11–15]. For the contrary of l < < s the free surface of YSZ - not covered by electrodes - is small and the corresponding reactions with molecular oxygen should not play a role for electron and hole generation. Then reactions as described in Eqs. (11) and (12) may come into play.

(39)

Neglecting quadratic terms of lp/λg and ln/λg as before, the chemical potential difference across the i-region is Δμ2i = μ2n(l − ln) − μ2p(lp) = Δμ2no − Δμ2po (cf. Eqs. (27) and (28)). Thus Eq. (39) simplifies to

l 1 2FU =− − li (r − 1) (r − 1)(Δμ 2po − Δμ 2no )

4.3. Reactions including single-charged vacancies Single-charged vacancies shall be produced and consumed via the reactions given by Eqs. (11) and (12). The rate Rsp corresponding to the consumption of double-charged vacancies by Eq. (11) and a generation by Eq. (12) yields for the p-region

(40)

Because of l/li > 1 the relation 2FU > Δμ2po − Δμ2no = μ2po − μ2no has to be fulfilled, which is a consequence of the Le Chatelier Principle (cf. Eq. (6)). The same procedure for large polarizations Δμ2p/RT > > 1 and large values of λg with

2

μ μ1p ∂c2p ⎤ ⎞ − ksh exp ⎛ 2p ⎞ < 0, = ksb exp ⎛ Rsp = ⎡ ⎥ ⎢ RT ∂ RT t ⎝ ⎠ ⎝ ⎠ ⎦s ⎣

with μ1p being the chemical potential of single-charged vacancies in the p-region. It is tacitly assumed that the chemical potential of holes remains unchanged and the related exponential term is included in the rate constant ksh. For the n-region the corresponding rate is

( ) exp ( ) < < 1 is treated in Appendix D lp

λg

Δμ2po RT

and leads to the same current density as for small polarization (cf. Eq. (D12))

i ≈ σ2

U . l

However under the condition density of (cf. Eqs. (D19) and (A3))

i ≤ σ2

μ μ ∂c Rsn = ⎡ 2n ⎤ = ksb exp ⎛ 1n ⎞ − ksh exp ⎛ 2n ⎞ > 0, ⎝ RT ⎠ ⎝ RT ⎠ ⎣ ∂t ⎦g

(41) 2

( ) exp ( ) > > 1 a current lp

λg

Δμ2po

⎜

⎟

⎜

(44)

with μ1n being the chemical potential of single-charged vacancies in the n-region and the assumption of constant chemical potential of electrons. In equilibrium reaction rates have to be zero, i.e.

RT

μ x μO + 2μ 2o 2F h RT ⎞ = 2F lke exp ⎛ OO ⎞, l= lkn exp ⎛ 2 s 2F s RT s ⎝ ⎠ ⎝ RT ⎠

(43)

⎟

(42)

μ μ ksh exp ⎛ 2o ⎞ = ksb exp ⎛ 1o ⎞, ⎝ RT ⎠ ⎝ RT ⎠

is derived, which is independent of the applied voltage. The last equation has a simple physical meaning. The upper bound of the current is due to the maximum desorption of one mole of lattice oxygen (cf. Eq. (1)). This supply is the largest for the n-region extending over nearly all the sample length. One mole OOx desorbed leaves a maximum electronic charge of 2F and the maximum rate of desorption per volume in the n-region (cf. Appendices A and B) is kes−1 exp (μOOx/RT). So far the derived current densities apply to scenarios, where the reaction with molecular oxygen outside the blocking electrode/YSZinterface are rate controlling. This will be the case for large ratios of

(45)

where the subscript o refers to the equilibrium values which were attained before polarization. Deviations from equilibrium values of the chemical potentials are again described by Δμxx and Eqs. (43) and (44) will change to

Δμ Δμ ∂c2p ⎤ μ ⎛ 1p ⎞ ⎛ 2p ⎞ ⎤ = ksh exp ⎛ 2o ⎞ ⎡ Rsp = ⎡ ⎢exp ⎜ RT ⎟ − exp ⎜ RT ⎟ ⎥ < 0 ⎥ ⎢ ∂ RT t ⎝ ⎠ ⎦s ⎣ ⎝ ⎠ ⎝ ⎠⎦ ⎣ and 246

(46)

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Δμ Δμ μ ∂c Rsn = ⎡ 2n ⎤ = ksh exp ⎛ 2o ⎞ ⎡exp ⎛ 1n ⎞ − exp ⎛ 2n ⎞ ⎤ < 0, ⎝ RT ⎠ ⎢ ⎣ ∂t ⎦s ⎝ RT ⎠ ⎝ RT ⎠ ⎥ ⎣ ⎦

i = ip = in = (47)

(48)

to be valid. The rate of consuming (producing) double-charged vacancies is the same as the rate of producing (consuming) single-charged ones, if the total concentration of vacancies remains to be constant, which may not be the case by pore formation or reactions with grain boundaries and dislocations. Assuming conservation of vacancies in steady state Eq. (21) gives the corresponding flux divergencies for both the n- and pregion

∂J2 ∂J = − 1. ∂x ∂x

i = σ2

(50)

σeff ≡

5. Discussion By assuming in Eq. (13) that the gradient of the chemical potential of vacancies in YSZ is reduced by a factor 1/r, it has been shown in the previous chapter that closed, stationary solutions or related approximate relations can be derived for the current/voltage-relation. This is true for the two scenarios of reaction either by the known one including molecular oxygen or a new one including single-charged vacancies. New at least in the context of YSZ (cf. Ref. [24a,b]). These reactions are initiated as a consequence of the Le Chatelier Principle, in order to counteract the changes of the chemical potential of vacancies driven by the applied electric field. Finally reaction rates and electromigration attain a new steady state by adjusting the parameter r and the related changes of chemical and electrical potentials. The reactions generate holes near the cathode and electrons near the anode. The situation is simplified by defining p-regions (n-regions) with dominant hole-conduction (electron-conduction). In between the two regions a region with dominant ionic conductivity prevails. It is shown in the previous chapter that an ionic current of doublecharged vacancies inside YSZ could be established despite of being blocked at the YSZ/electrode-interfaces. The corresponding divergencies of flux are circumvented by assuming sinks and sources to be present. In one scenario oxygen molecules from the surrounding gas phase are absorbed in vacancies generating holes via Eq. (2), whereas in n-regions lattice oxygen is leaving YSZ generating oxygen molecules and electrons. In order to start and accelerate these reactions the applied voltage has to move the vacancies first to set-up a difference of their chemical potential between anode and cathode (see Fig. 3), i.e. an incubation time for reaching steady state is required, which corresponds to the characteristic time given in Eqs. (9) and (10). Thus the current-voltage time dependence looks like the one shown in Fig. 7. After the incubation time a steady state is attained. For small voltages the current density is proportional to the applied voltage (cf. dotted line in Fig. 7) with a conductivity being equal to the ionic one, although holes in the p-region and electrons in the n-region are the dominant charge carriers. With increasing voltage the length of the n-region growths on the expenses of both i- and p-region. Finally the n-region approaches the sample length and the current density reaches the maximum value imax as given by Eq. (42). Before the incubation time

(52)

Δμ 2n Δμ1no −Δμ 2no + Δμ1no l − x⎞ Mn Δμ 2no cosh ⎛ + − = RT (Mn − 1) RT Mn − 1 RT (Mn − 1) RT ⎝ λs ⎠ (53) ⎟

and

Δμ 2p RT

=

−Δμ 2po + Δμ1po

Δμ1po Mp Δμ 2po x cosh ⎛ ⎞ + − . Mp − 1 RT (Mp − 1) RT ⎝ λs ⎠ ⎜

(Mp − 1) RT

⎟

(54) A partial differential equation for Δμ2n and Δμ2p of the type of Eqs. (22) and (26) is obtained by defining a characteristic length λs (cf. Eq. (E4) in Appendix E)

λs =

(r − 1) c2 D2 ksh exp

μ2o

( ) RT

=

RT (r − 1) σ2 4F 2ksh exp

μ2o

( ) RT

(58)

(51)

by assuming that c1 within both n- and p-region is very different but does not change much with position x. This will be a reasonable assumption because of the rather constant value of μ2 within each the nand the p-region. However because of the different values of c1 in both regions different values Mn and Mp are used in the following. In order to calculate the current densities in both regions, Eqs. (21), (46), (47) and (52) are applied to obtain the chemical potentials. By a rather lengthy derivation in Appendix E the results for small polarization are Eq. (E5) ⎜

il 2l M = σ2 n n > > σ2. U λs

Different to the previous case of λs > > lk the width of the p- and n-region is not narrow but is about half the sample length, i.e. leads to a very narrow i-region.

with M being an abbreviation for the term in brackets, c1 is the concentration of single charge vacancies and D1 their diffusion coefficient. For the sake of simplicity M is considered to be independent of position despite expected variations of c1. M is expected to be much larger than unity, because of its definition r > 1 and c1 < < c2. Integration of Eq. (51) yields for both p- and n-region

[μ1 (x ) − μ1 (0)] = M [μ 2 (x ) − μ 2 (0)]

(57)

This current density corresponds to a very high effective conductivity for ln = l/2 expressed as

The last equation is rewritten by using Eq. (6) and the Le Chatelier Principle (Eq. (13))

∂μ1 ∂μ c D r ∂μ = ⎡ 2 2 (r − 1) + ⎤ 2 ≡ M 2 , ⎢ ⎥ c D 2 ∂ x ∂x ∂x 1 1 ⎣ ⎦

(Mn − 1) U . λs

(49)

Integration with blocking electrodes leads to

J2 (x ) = −J1 (x ).

(56)

Thus the cell with two blocking electrodes delivers half the current when compared to a cell with non-blocking electrodes for the case of single-charged vacancies. The blocking electrodes act as current collectors and the narrow p- and n-region become reversible electrodes allowing both oxygen and electron transfer. The fact that only half the current is present is due to the counteracting flux of single-charged vacancies transporting half of the charge in opposite direction. For small values of λs the other limiting case may be λs < < lk corresponding to fast reaction velocities (large value of ksh exp (μ2o/ RT)) and/or values of r close to unity. Under these conditions the current/voltage-relation is discussed in Appendix E leading to (cf. Eq. (E36))

In order for a dominant hole production in the p-region and electron production in the n-region, Rsp has to be negative and Rsn has to be positive, which requires

|Δμ 2n | > |Δμ1n | and Δμ 2p > Δμ1p

σ2 U . 2 l

. (55)

Large deviations from the stationary case of electromigration (bold solid line in Fig. 3) correspond to r > > 1 which together with small reaction velocities (small value of ksh exp (μ2o/RT)) leads to large values of λs. For the limiting case of λs > > lk with k = n, p it was shown in the Appendix E that the p- and n-regions become very narrow with lk < < l and with a current density i of 247

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of the constant current density of 400 A/m2 at 1200 K was Faradaic, i.e. leading to the formation of NiO at the anode and Ni5Zr at the cathode. The formation of these compounds occurred because the related voltage was above the decomposition value of YSZ (cf. Ref. [35]); but accompanying phase formation is not fast enough to accommodate the charge transfer required by the imposed current density. In the context of the present study this is explained by 97% of the current being electronic, i.e. the Ni-electrodes act as blocking electrodes. With the resulting electric field of 103 V/m the effective conductivity is 0.4 S/m. For this case Eq. (56) predicts a conductivity of σ2/2 or 4 S/m [5]. The lower experimental value may be due to overvoltages for the electronic carriers, which are expected to be large at the anode, where a layer of nickel oxide has formed. Similar experiments with blocking platinum electrodes were conducted reporting the formation of intermetallic ZrPt phases at the cathode [35,38–40], but no Faradaic efficiency was reported. The formation of the intermetallics is indicative for the reduction of YSZ. In the last paragraph the Ni-electrodes were partially blocking with an efficiency of 97%. In SOEFs and SOECs electrodes should not be blocking the oxygen transfer across the electrode/electrolyte-interface, which is achieved by using porous electrodes. This also requires that the rate of oxygen transfer is able to speed up with the external current density. However, there will be a limiting current density ilim above which the interface reaction rate is unable to provide enough charge carriers to feed the ionic current through YSZ. Then the required additional carriers have to be electronic ones as produced by the reactions proposed in this study. Formally the current density in the equations derived in this study has to be replaced by the difference i − ilim, in order to be applicable for SOEFs and SOECs. Actually Fig. 8 shows that the reactions involving molecular oxygen at the porous electrodes are the same as proposed in this study. They occur for instance in SOEFs and SOECs with platinum-net electrodes near triple lines between the three phases YSZ, Pt and oxygen gas, because the contact area between electrode and YSZ is blocking oxygen transfer. The result, that for small polarization the conductivity of cells with blocking and non-blocking electrodes is the same, is explained by considering the p- and n-regions as electrodes. By definition they are electron conducting and at the corresponding interfaces to the i-region, they allow oxygen transfer in different directions to the vacancy transfer. Oxygen is neither accumulated nor depleted in these hypothetical electrodes, because by the reactions in Eqs. (1), (2), (11) and (12) a stationary concentration is maintained. These reactions also generate electrons in the n-region and holes in the p-region of the YSZ, which may recombine with electrons and holes provided by the electrodes as discussed in the following. Electrons may enter YSZ from the negative electrode and meet holes generated within the p-region forming excitons which finally decay and generate two photons. A similar process in the n-region with electrons generated by oxidation of VO• and holes provided by the positive electrode leads to electron/hole-recombination and a concomitant emission of light. This scenario would not affect the electronic currents as calculated in Eqs. (29) and (32). This is also true for the scenario discussed at the beginning of this Section 5, where the p- becomes a n-region and vice versa. The currents in both cases are due to the reaction rates with molecular oxygen over the length of either p- or n-region and independent of the destiny of the created electronic charge carriers, i.e. independent of holes being able to reach the negative electrode or being annihilated by meeting excess electrons. In both cases light is emitted and the related photons have been detected during flash sintering leading to an intensity spectrum which differs from the one of a black body [15–17]. The radiation of YSZ at high temperatures has been used a long time ago in so called Nernst-glowers [41]. The famous physical chemist Walther Nernst invented and patented this concept of producing a lamp at the end of the 18th century in Göttingen. At room temperature the currents were too small to raise temperatures by Joule heating. Thus

the measured dc-conductivity corresponds to the intrinsic electronic one iin which is determined by the ambient oxygen pressure (cf. Fig. 1). The incubation time decreases with the applied electric field (U/l) (cf. Eqs. (9) and (10)) as shown in Fig. 7 by the dashed line. The p-region as discussed in this work corresponds to reduced YSZ, which by Levy et al. [32] was proven to exhibit an electronic conductivity much larger than the ionic one of untreated YSZ. The color should change from transparent to yellow and finally black [33,34] with increasing polarization or voltage respectively. These changes are caused by a reduction of zirconium [33,34] and a reaction with electrode material as discussed for the nickel electrodes before [10] and as observed earlier by Weppner [35]. Different to the present modeling Levy et al. [32] assumed n-type conduction in reduced YSZ. If this will be the case, the phenomenological treatment in this study remains to be true but the labels n and p have to be exchanged. Thus the p-region of this study becomes the n-region and O2 as well as VO• entering this region will get the necessary electrons from either the excess electrons near the Fermi level or from the valence band forming holes. In the latter case holes and electrons recombine afterwards leading to the generation of two photons. Vice versa the n-region of this study becomes the p-region and OOx leaving this region as O2 or VO• leaving as VO•• deposit their electrons in the holes generating again the related photons. The reduction and oxidation of YSZ by producing immobile defects like ZrZr′ near the negative electrode and OO• near the positive one will have an effect on changes of the chemical potentials Δμ2no and Δμ2po at the electrode/YSZ-interface. Thus this type of polarization is also contributing to the inequality given in Eq. (19) as a consequence of the Le Chatelier Principle and, therefore, justify the assumption μ2po − μ2no < < 2FU often made in this study. In addition, the stationary polarization also changes the extension of n- and p-region (cf. Eqs. (33), (36) and (E17)). The present prediction of the i/U-relationship is compared with values obtained during a flash sintering experiment with YSZ-3 mol-% Y2O3 in air at 900 °C furnace temperature [14], where for galvanostatic conditions with i=60 mA/mm2 a steady state with U/l = 35 V/cm was attained. These values correspond to a conductivity of 17 Siemens/m. For the blocking Pt-electrodes used in the experiment this has to be compared with 0.4 Siemens/m at 700 °C [36]. With an activation energy of 0.81 eV [36] this yields 2.1 S/m at 900 °C. The last value is still smaller than the one obtained from sintering, which may be explained by Joule heating leading to a higher temperature and a concomitant higher ionic conductivity in the sintering experiment. Different properties of the samples used in Refs [14,36] due to different preparation routes may also contribute to the difference in conductivity. A maximum current density given by Eq. (42) is calculated to be imax = 6.6 × 105 A/m2 = 660 mA/mm2 using T = 1173 K, σ2 = 2.1 S/ m [36], V = 1 V, l = 10−2 m, s = 10−3 m and h = 300 m−1 [37]. Thus the current density applied during the sintering experiment is still below the maximum value. The comparison is hampered by the fact that the experiment was conducted in air, whereas the value of h was determined in pure oxygen. Then according to Eq. (B2) h is reduced by a factor (0.21)1/4 = 0.68 leading to a maximum current of 4.5 × 105 A/ m2. According to Appendix B and Eq. (B2) the parameter h (describing the oxygen uptake) depends on the oxygen partial pressure pO2 of the surrounding gas. If this pressure decreases but the applied voltage remains constant, Eq. (36) requires that the length of the n-region increases. However, the current density does not change (cf. Eq. (38)), because the reduced oxygen supply per area is compensated by an increased area of the n-region. Thus the measured current is independent of the external oxygen pressure within certain limits. In an experiment [10] mentioned before, thin discs of single crystalline YSZ10 and Ni-Electrodes were used, where an exchange of molecular oxygen is limited due to oxygen partial pressures of < 10−3 Pa and due to a large diameter to thickness ratio of YSZ. Only 3% 248

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Fig. 8. Schematic presentation of a solid oxide fuel cell (SOFC) and a solid oxide electrolysis cell (SOEC) with Pt-net electrodes. The contact area between Pt and YSZ does not allow oxygen transfer. Thus the electrochemical reactions including molecular oxygen have to occur in between the triple lines, where the three phases meet (enlargements at the bottom of the figure). Allowing reactions to take place at the triple lines only, would not allow generating sufficient Faradaic currents. This is equivalent to the proposed reaction in this study, where the oxygen is ab- or desorbed via the YSZ-surface in between two electrodes. In all cases there is a limiting ionic current determined by the maximum area of YSZ between the electrodes, which limits the efficiency of both SOFC and SOEC.

3. In accordance with Refs. [24a,b] steady state currents are able to build up within samples. Then a net positive charge is transported towards the negative electrode with a flux of double-charged vacancies diminished by a counteracting diffusion-dominated flux of single-charged vacancies to the positive electrode. 4. In both models with either oxygen molecules or single charge vacancies being involved a region of high (low) oxygen activity electrons (holes) are generated and this region becomes n-type (p-type) conducting. The transition region between n- and p-type regions was called i-region. Within the i-region ion conduction prevails. The corresponding vacancy flux is short-circuited by a counteracting flux of oxygen molecules in the sample environment or by a counteracting flux of single-charged vacancies within the YSZ. 5. Electrons and holes as generated according to the models of this study and the one stemming from the electrodes may recombine within both n- and p-region. Then the spectrum of emitted light deviates from that of black body radiation. 6. The predictions in this study could be verified experimentally by measuring effective conductivity, distribution of electric potential or extension of p- and n-regions, respectively. These experiments may have been hampered in the past by not knowing when the steady state was reached (cf. Fig. 7 and Eq. (9)). 7. The present study is unable to predict, whether p- or n-regions form next to the negative electrode, where an applied voltage decrease the chemical potential of oxygen, i.e. reduces YSZ. This is also the case for the region next to the positive electrode. It depends on an atomistic interpretation of the band structure; but does not affect the present phenomenological approach based on reaction rates and the related continuity equations. 8. Although cells with two blocking electrodes are usually considered to yield the intrinsic electronic conductivity of YSZ, it is shown that this will be the case before the incubation time (cf. Eq. (9)) only. For later times and for small polarizations the ionic conductivity σ2 is measured (Eq. (38)), if gaseous oxygen is involved, or halve of it σ2/ 2 (Eq. (56)), if single-charged vacancies are playing a major role. These findings are astonishing as only part of the YSZ-sample exhibits dominant ionic conductivity. For ongoing polarization and gaseous oxygen being involved a maximum current is reached (Eq. (42)) due to an upper bound of oxygen supply.

the YSZ in the lamp had to be externally heated to reach a starting temperature. At the time of the invention the mechanisms of the creation of electronic charge carriers was unknown and the current flowing through the lamp was attributed to ions only [41]. Ion conduction alone and the concomitant temperature rise by Joule heating would predict a spectrum of the emitted light to be that of a black body only. However, it has been revealed a short time after the invention that the emitted light from a Nernst glower is considerable different from a black body radiation [42–44]. This is expressed by Coblentz [44] as: “Of course the assumption is made that the emissivity function is similar to that of platinum and of a black body. How far this assumption falls short of the observed facts is brought out in the present paper, in which it is shown that the observed radiation curve of a Nernst filament, which at high temperatures gives an apparently continuous spectrum, is in reality the composite of numerous sharp emission bands, which increase in intensity and broaden out with rise in temperature.” Interestingly this statement is found in the internet at www. forgottenbooks.com. It has to be mentioned that in the experiments with a Nernst glower often ac-currents or ac-voltages were applied, whereas the analysis in this study was made for dc-polarization. With ac-currents polarization of YSZ may occur at high fields, if the changes shown in Fig. 4 are asymmetric and YSZ is becoming a rectifier. 6. Conclusion 1. The interpretation of experimental results for YSZ is applicable to other oxidic materials as well. Doping with aliovalent cations leads to the formation of structural vacancies and Eqs. (1) and (2) remain to be valid. Thus the novel reactions introduced to interpret the behavior of YSZ in applied electric fields by a separation into n-type, p-type and i-regions is general and will be easily extended to other oxides. 2. The known scenario of conduction in YSZ, where the oxygen activity (square root of pressure) in the environment of the sample determines the electronic conductivity (cf. Fig. 1), is different in dc fields. High activities of vacancies near the negative electrode within the sample leads to hole generation, whereas low activities near the negative electrode are accompanied by electron generation.

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List of symbols

l

The symbols are listed in the first column and explained in the second one. The third column lists the equation or figure where the symbol is defined or mentioned first.

lE lp ln li lk λg

Symbol Meaning

Defined after

Agi• AgAgx c1 c2

Eq. Eq. Eq. Eq.

Jk Jis k ke

positive charge silver in interstitial position silver on a silver sub-lattice position concentration of single-charged vacancies (VO•) concentration of double-charged vacancies (VO••) initial concentration of double-charged vacancies (VO••) concentration of double-charged vacancies (VO••) in p-region concentration of double-charged vacancies (VO••) in n-region concentration of electrons concentration of electrons in the n-region equilibrium concentration of oxygen isotope 18 concentration of holes in the p-region concentration of oxygen isotope 18 diffusion coefficient of single-charged vacancies (VO•) diffusion coefficient of double-charged vacancies (VO••) diffusion coefficient of oxygen isotope 18 electrons dielectric constant electrochemical potential of a species i =η2, electrochemical potential of doublecharged vacancies (VO••) electrochemical potential of electrons electrochemical potential of double-charged vacancies (VO••) Faraday's constant fraction of O-18 in the gas phase parameter describing the exchange of gaseous oxygen with YSZ holes current density current density within the i-region current density within the p-region current density within the n-region initial current density within the incubation time τ maximum current density determined by the maximum exchange rate with gaseous oxygen flux of single-charged vacancies (mole/time/ area) flux of double-charged vacancies (mole/time/ area) flux of species k (mole/time/area) flux of oxygen isotope coefficient defined in Eq. (B2) as Jis/ceq reaction coefficient for Eq. (1)

kh

reaction coefficient for Eq. (2)

ksh ksb

reaction coefficient for Eq. (11) reaction coefficient for Eq. (12)

c2o c2p c2n ce cen ceq chp cis D1 D2 Dis e′ ε ηi ηVO•• ηe′ η2 F fis h •

h i ii ip in iin imax J1 J2

λs

(5) (5) (51) (6)

M

sample length or thickness depending on its aspect ratio characteristic length width of the p-region width of the n-region width of the i-region being either lp or ln characteristic length for the rate of oxygen exchange with environment characteristic length defined in Eq. (55)

Eq. (B2) Eq. (B2) Eqs. (25), (B3) Eq. (2)

μ2po

ratio of differences of chemical potentials of single- and double-charged vacancies ratio M in the p-region ratio M in the n-region chemical potential of lattice oxygen chemical potential of gaseous oxygen in the environment chemical potential of single-charged vacancies (VO•) initial chemical potential of single-charged vacancies (VO•) chemical potential of single-charged vacancies (VO•) in the p-region chemical potential of single-charged vacancies (VO•) in the n-region μ1p(x = 0) = μ1(x = 0) μ1n(x = l) = μ1(x = l) μ1p − μ1o μ1n − μ1o Δμ1po = Δμ1p(0) Δμ1no = Δμ1n(l) chemical potential of double-charged vacancies (VO••) initial chemical potential of double-charged vacancies as attained before polarization, e.g. by sample annealing or preparation μ2 within the p-region of the sample, i.e. between x = 0 and x = lp μ2p(x = 0) = μ2(x = 0)

Δμ2p

μ2p − μ2o > 0

μ2n(x)

Eq. (18) Eq. (29) Eq. (32) Fig. 7

μ2no

μ2 within the p-region of the sample, i.e. between x = lp + li and x = l μ2p(x = l) = μ2(x = l)

Mp Mn μOOx μO2

Eq. (24) Eq. (23)

μ1

Eq. (44) Eq. Eq. Eq. Eq. Eq. Eq.

μ1o

(4) (A7) (B1) (A9) (B1) (51)

μ1p(x) μ1n(x) μ1po μ1no Δμ1p Δμ1n Δμ1po Δμ1no μ2

Eq. (6) Eq. Eq. Eq. Eq. Eq.

(B1) (1) (20) (4) (4)

μ2o

Eq. (4) Eq. (6)

μ2p(x)

Δμ2no Δμ2i

Fig. 7

μx μxo

Eqs. (49), (50) Eq. (6)

N2 Nis OOx O2 pO2

Eq. (21) Eq. (B1) Eq. (B3) Eqs. (42), (A1) Eqs. (30), (A2) Eq. (43) Eq. (43)

ϕ ϕp(x) ϕn(x) R

250

μ2n(l) − μ2o difference of μ2 between beginning of n- and end of p-region, i.e. Δμ2i = μ2n(l − ln) − μ2p(lp) chemical potential of lattice oxygen (OOx) initial chemical potential of lattice oxygen (OOx) number of double-charged vacancies number of oxygen isotopes 18 oxygen in the oxygen sublattice of YSZ gaseous oxygen in the environment partial pressure of gaseous oxygen in the environment electric potential electric potential within the p-region, i.e. between x = 0 and x = lp electric potential, i.e. between x = lp + li and x=l universal gas constant

Fig. 2 Eq. (10) Fig. 6 Fig. 6 Fig. 6 Eq. (E23) Eq. (25) Eqs. (55), (E4) Eq. (52) Eq. Eq. Eq. Eq.

(54) (53) (4) (4)

Eq. (51) Eq. (51) Eq. (43) Eq. (44) Eq. Eq. Eq. Eq. Eq. Eq. Eq.

(53) (54) (46) (47) (54) (53) (6)

Fig. 3

Eq. (14), Fig. 6 Eq. (14), Fig. 6 Eqs. (23), (24) Eq. (15), Fig. 6 Eq. (15), Fig. 6 Eq. (32) Eq. (17) Eq. (A1) Eq. (A5) Eq. Eq. Eq. Eq. Eq.

(A1) (B1) (1) (1) (4)

Eq. (6) Eq. (14) Eq. (15) Eq. (6)

Solid State Ionics 320 (2018) 239–258

R. Kirchheim

Rk R2 Rp2 Rsp Rsn r ρ(x) s σ2 σp σn

rate of a source or sink term producing or annihilating species k rate of a source or sink term producing or annihilating double-charge vacancy rate of annihilating double-charge vacancy in the p-region rate of annihilating double-charged vacancies in the p-region rate of annihilating double-charged vacancies in the n-region parameter describing deviation from steady state with r > 1 electric charge density thickness of an elongated sample (lhs of Fig. 2) ionic conductivity (of double-charged vacancies) electronic conductivity due to hole generation electronic conductivity due to electron generation

Eq. (21)

σeff

Eq. (22)

T τ U VO•• VO• ZrZrx

Eq. (23) Eq. (43) Eq. (44)

ZrZr″ Eq. (13) Zr″Zr″ Eq. (20) Eqs. (25), (B2) Fig. 1

effective conductivity for the case of large electronic contributions temperature characteristic time for reaching steady state applied voltage double-charge vacancy single-charged vacancy Zr atom at the Zr-sublattice of YSZ (valence 4+) Zr atom with two trapped electrons at the Zrsublattice of YSZ (valence 2+) Zr atom with four trapped electrons at the Zrsublattice of YSZ, i.e. having valence zero

Eq. (58)

Eq. Eq. Eq. Eq.

(9) (9) (1) (11)

Acknowledgement The author is grateful for financial support by the Deutsche Forschungsgemeinschaft within the Priority Program 1959 under KI230/41-1 and for stimulating discussions with Rishi Raj (University of Colorado at Boulder).

Fig. 1 Fig. 1

Appendix A. Rate theory Reaction rate theory applied to Eq. (1) (the individual forward reaction of oxygen exchange) yields

μ ⎡ ∂N2 ⎤ = s ⎡ ∂c2 ⎤ = ke exp ⎛ x ⎞, ⎝ RT ⎠ ⎣ ∂t ⎦ge ⎣ ∂t ⎦ge

(A1) ••

where [∂N2/∂t]g are the number of VO -vacancies leaving a sample of thickness s per unit of time and area causing a change of concentration over the thickness s of [∂c2/∂t]ge, ke is a rate coefficient and μx is the chemical potential of OOx. It is tacitly assumed that in a slice of the YSZ of thickness dx and width s at position x the homogenization within the slice is fast compared to the characteristic time for attaining steady state along the length l of the YSZ-sample (c.f. Eq. (9)). If diffusion of vacancies controls homogenization this condition is fulfilled for s2/l2 < < 1, i.e. for a sample shape given on the lhs of Fig. 2. Note that instead of concentration the thermodynamic activity (=exp(μ/RT)) has been used after the rate coefficient. The subscript ge is used to refer to the reaction with a gas phase and concomitant generation of electrons. A corresponding treatment of Eq. (2) leads to

μO + 2μ 2 ∂c ⎞. s ⎡ 2 ⎤ = −kh exp ⎛ 2 ⎣ ∂t ⎦gh ⎝ 2RT ⎠ ⎜

⎟

(A2)

where μO2 is the chemical potential of oxygen in the atmosphere. In dynamic equilibrium and without an electric field the reactions in Eqs. (1) and (2) both occur at the same time but most probably at different places on the surface. Nevertheless their rates have to be equal. With a corresponding constant chemical potential μ2o and μxo equal rates lead to

μO + 2μ 2o μ ke k ⎞. exp ⎛ xo ⎞ = h exp ⎛ 2 s RT s 2RT ⎝ ⎠ ⎝ ⎠ ⎜

⎟

(A3)

Electrons and holes recombine by the reaction (A4)

e′ + h• = Nil. ••

With an applied electric field and in the n-region the total change of the VO -concentration by Eqs. (1) and (2) is given as

μ + 2μ 2o + 2Δμ 2n μ + Δμxn ⎞⎟. ⎞ − kh exp ⎛⎜ O2 ⎡ ∂c2n ⎤ = ke exp ⎛ xo s RT s 2RT ⎣ ∂t ⎦g ⎝ ⎠ ⎝ ⎠

(A5)

Electron net production is achieved near the positive electrode by lowering the chemical potentials Δμ2n < 0 of vacancies and a smaller change of the one of lattice oxygen, Δμxn > 0. By assuming that the affinity of lattice oxygen OOx is changed slightly only by a change of the vacancy concentration (Δμxn < < Δμ2n) Eqs. (A5) and (A3) yield

μ + 2μ 2o ⎞ ⎡1 − exp ⎛ Δμ 2n ⎞ ⎤ > 0. ⎡ ∂c2n ⎤ = kh exp ⎛ O2 ∂ t s 2 RT ⎣ ⎦g ⎝ RT ⎠ ⎥ ⎝ ⎠⎢ ⎣ ⎦ ⎜

⎟

(A6)

Because of Δμ2n < 0 the rate in Eq. (A6) is positive. Thus the rate of the net production of electrons will be positive, too with two electrons generated per generation of one vacancy, i.e.

μ + μ 2o Δμ ⎞ ⎡1 − exp ⎛ 2n ⎞ ⎤ > 0. ⎡ ∂cen ⎤ = 2 kh exp ⎛ O2 s RT ⎠ ⎥ ⎣ ∂t ⎦g ⎝ ⎝ 2RT ⎠ ⎢ ⎣ ⎦

(A7)

For the p-region with Δμ2p > 0 the same analysis yields

Δμ μ + μ 2o ⎡ ⎡ ∂c2p ⎤ ≈ kh exp ⎛ O2 ⎞ 1 − exp ⎜⎛ 2p ⎟⎞ ⎤ < 0 ⎢ ⎥ ⎥ ⎢ t s 2 RT ∂ ⎝ ⎠⎣ ⎦g ⎣ ⎝ RT ⎠ ⎦

(A8) 251

Solid State Ionics 320 (2018) 239–258

R. Kirchheim

and

Δμ μ + 2μ 2o ⎡ ⎞ 1 − exp ⎜⎛ 2p ⎟⎞ ⎤. ⎡ ∂chp ⎤ = − 2kh exp ⎛ O2 ⎢ ⎥ ⎥ ⎢ t s 2 RT ∂ ⎦gh ⎣ ⎝ ⎠⎣ ⎝ RT ⎠ ⎦ ⎜

⎟

(A9)

With Eqs. (1) and (2) and the related two reactions occurring at the same time in both n- and p-region leads to electron and hole production. They are annihilated partly by the subsequent reaction in Eq. (A4) with a net increase of e′-concentration in the n-region and a net increase of h•concentration in the p-region. Appendix B. Exchange of molecular oxygen The exchange of molecular oxygen in the gas phase with lattice oxygen in YSZ was studied by using the O-18 isotope of oxygen [37,45]. In equilibrium the ratio of O-18 and O-16 has to be the same in both gas and YSZ phase. In the transient case a concentration profile of O-18 occurs with a monotonic decrease from the surface into the bulk of YSZ. This profile Cis(x) was measured with secondary ion mass spectrometry (SIMS) and it has to obey the following boundary condition for the influx Jis of the isotope [37]

Jis (0, t ) = k [ceq − cis (0, t )] = −Dis

∂cis (0, t ) , ∂x

(B1)

where ceq is the isotope equilibrium concentration, k is a rate constant and Dis is the diffusion coefficient of the oxygen isotope. By fitting measured concentration profiles of O-18 both Dis and k were determined. As the isotope is moving via a vacancy mechanism, its diffusivity is Dis = x2oD2, where x2o is the molar fraction of vacancies in YSZ. Dis is the tracer diffusion coefficient of oxygen in YSZ neglecting small effects stemming from the small mass difference compared to O-16. The incorporation of the isotope is described by Eq. (2) and by Eq. (A2) in Appendix A. The latter corresponds to the flux given by Eq. (B1) at t = 0, where no desorption of isotopes occurs (cis(0, 0) = 0). Thus the following relation holds

μ o + 2μ 2o ⎞ μO + 2μ 2o c ∂Nis ⎞ = f kh (p )1/4exp ⎛⎜ O2 = Jis = kceq = k 2o fis = fis kh exp ⎛ 2 ⎟, O2 is x2o 2RT 2RT ∂t ⎝ ⎠ ⎝ ⎠ ⎜

⎟

(B2)

where Nis is the number of isotopes entering the sample per area and c2o is the concentration of vacancies in YSZ in equilibrium with molecular oxygen in the atmosphere of pressure pO2 containing the fraction fis of O-18 and μO2o is the standard chemical potential of gaseous oxygen. Instead of k the parameter h will be used as in Ref. [37] and as defined by

μO + 2μ 2o k k kh ⎞. = = exp ⎛ 2 Dis x2o D2 D2 c2o 2RT ⎝ ⎠

h≡

⎜

⎟

(B3)

Note that the gas ab- and desorption as described in Appendix A was assumed to be fast across the thickness of the sample. Nevertheless, a small concentration gradient may be involved with a diffusivity corresponding to the chemical diffusion coefficient. This would also affect the rate diffusion coefficient k. Due to the small gradients no remarkable difference of the two diffusion coefficients is assumed. A detailed and rigorous discussion of the related differences is provided in Ref. [46]. Appendix C. Deriving chemical potentials for the case of molecular oxygen Normalizing the chemical potentials in Eqs. (26), (A9), (B2) and (B3) with respect to RT leads to

Δμ h ⎡1 − exp ⎛ Δμ 2p ⎞ ⎤ ⎛ 2p ⎟⎞ = ⎜ ⎜ ⎟ ⎥ s (r − 1) ⎢ ∂x 2 ⎝ RT ⎠ ⎝ RT ⎠ ⎦ ⎣ ∂2

(C1)

where h is the measurable exchange coefficient defined in Appendix B. Introducing a characteristic length

λg =

s (r − 1) h

(C2)

simplifies Eq. (C1)

λ g2

Δμ 2p ⎤ ∂2 ⎛ Δμ 2p ⎞ ⎡ ⎞ 1 − exp ⎜⎛ ⎜ ⎟ = ⎟ . ⎥ ∂x 2 ⎝ RT ⎠ ⎢ ⎝ RT ⎠ ⎦ ⎣

(C3)

∂ (Δμ2p / RT )

= 0 and Δμ2p = Δμ2po at x = 0 solution for two limiting cases are derived. With the boundary conditions ∂x For small deviations from the initial state, i.e. |Δμ2p/RT| < < 1 and a series expansion of the exponential function in Eq. (C3) the last equation has the solution x Δμ 2p = Δμ 2po cos ⎜⎛ ⎞⎟ ⎝ λg ⎠

(C4)

with the approximation 2

1 x ⎤ ⎡ Δμ 2p ≈ Δμ 2po ⎢1 − ⎜⎛ ⎟⎞ ⎥ ≈ Δμ 2po . 2 ⎝ λg ⎠ ⎣ ⎦

(C5)

For large deviations the situation is different, because of Δμ2p/RT > 1 the exponential term in Eq. (24) is no longer negligible and the solution will be 252

Solid State Ionics 320 (2018) 239–258

R. Kirchheim

2

Δμ 2p = Δμ 2po +

Δμ 2po ⎤ ⎫ RT ⎧ ⎡ x ⎞ . ln 1 + tanh ⎢ ⎜⎛ ⎟⎞ exp ⎜⎛ ⎟ ⎨ λg ⎠ RT ⎠ ⎥ ⎬ 2 ⎝ ⎝ ⎦⎭ ⎣ ⎩

(C6)

There are two limiting cases for Eq. (C6). The first one with

2

( ) exp ( ) < < 1 (thin films large reaction rates) yields x λg

Δμ2po RT

2

Δμ 2p = Δμ 2po −

Δμ RT ⎛ x ⎞ ⎛ 2po ⎞⎟ ≈ Δμ ⎜ ⎟ exp ⎜ 2po 2 ⎝ λg ⎠ ⎝ RT ⎠

and the second one with

2

(C7)

( ) exp ( ) > > 1 will become x λg

Δμ2po RT

2

Δμ 2po

x ⎞⎟. Δμ 2p = Δμ 2po − RT ⎛⎜ ⎞⎟ exp ⎛⎜ ⎝ RT ⎠ ⎝ λg ⎠

(C8)

For the two approximations the chemical potential decreases in a parabolic manner from the negative electrode towards the interface of the pand i-region. The analogous derivation for the n-region yields the differential equation

λ g2

Δμ ∂2 ⎛ Δμ 2n ⎞ ⎡ = 1 − exp ⎛ 2n ⎞ ⎤. ∂x 2 ⎝ RT ⎠ ⎢ ⎝ RT ⎠ ⎥ ⎣ ⎦

(C9)

For small deviations from the initial state the solution becomes for the boundary condition Δμ2n = Δμ2no and

x − l⎞ Δμ 2n = Δμ 2no cos ⎜⎛ ⎟, ⎝ λg ⎠

∂ (Δμ2n / RT ) ∂x

= 0 at x = l

(C10)

with the approximation 2

1 x − l⎞ ⎤ ⎡ Δμ 2n ≈ Δμ 2no ⎢1 − ⎜⎛ ≈ Δμ 2no . ⎟ 2 ⎝ λg ⎠ ⎥ ⎣ ⎦

(C11)

For large deviations from the initial state, i.e. |Δμ2n/RT| > > 1 and Δμ2n/RT < 0 the exponential term in Eq. (C9) can be neglected yielding 2

Δμ 2n = Δμ 2no +

RT ⎛ x − l ⎞ ⎜ ⎟ . 2 ⎝ λg ⎠

(C12)

Note that the value Δμ2no may be different in Eqs. (C10) and (C12). For λg > > x a series expansion for Eq. (C10) yields by neglecting quadratic terms of l/λg 2

1 l − x⎞ ⎤ ⎡ Δμ 2n ≈ Δμ 2no ⎢1 − ⎜⎛ ≈ Δμ 2no . ⎟ 2 ⎝ λg ⎠ ⎥ ⎣ ⎦

(C13) ••

Thus for both cases of deviations from the initial state the chemical potential of VO decreases in a parabolic way within the n-region. With small values of the curvature for both limiting cases of λg > > x and |Δμ2n/RT| > > 1. As the curvature of Δμ2 is related to the curvature of the potential via Eq. (13) the last conclusion is equivalent with a low density of negative charges (cf. discussion after Eq. (20)). Appendix D. Current densities for high polarization and reaction with molecular oxygen For high polarization (|Δμ2/RT| > > 1) and λg < < x ≤ lp the solutions for the chemical potentials μ2 have been derived in Appendix C. These are used in the present appendix to derive current densities in the three regions of YSZ. Following the procedure for the case of low polarization (Eqs. (30) and (32)) the current density in the n-region will be

in = 2F

≈ 2F

l

μO + 2μ 2o kh ⎞ exp ⎛ 2 s 2RT ⎝ ⎠ ⎜

kh exp ⎛ s ⎝

∫

⎟

μO2 + 2μ 2o

⎜

2RT

lp + li

⎡1 − exp ⎛ Δμ 2n ⎞ ⎤ dx ≈ ⎢ ⎝ RT ⎠ ⎥ ⎣ ⎦

l

⎞ ⎠

⎟

ln ∫ dx = σ2 (r λ−2 1) RT 2F g

lp + li

For large polarization in the p-region and

ip = −2F

≈ 2F

μO + 2μ 2o kh ⎞ exp ⎛ 2 s 2RT ⎝ ⎠ ⎜

kh exp ⎛ s ⎝ ⎜

⎟

μO2 + 2μ 2o 2RT

lp

⎞ ⎠

(D1) Δμ2po

λg

RT

Δμ 2p ⎤ ⎞ dx ⎟ ⎥ ⎝ RT ⎠ ⎦

∫ ⎡⎢1 − exp ⎛ ⎜

0

⎣

lp ⎟

2

( ) exp ( ) > > 1 the current in the p-region will be lp

∫ exp ⎛

Δμ 2p

⎞ ⎟ dx . ⎝ RT ⎠

⎜

0

2

(D2)

( ) exp ( ) < < 1 (very large value of λ ): In the present context of limitation with ( ) exp ( ) < ( ) exp ( ) < < 1 or Δμ

First limiting case of

lp

λg

Δμ2po

g

RT

x λg

2

Δμ2po

lp

RT

λg

2

Δμ2po RT

253

2p ≈ Δμ2po

(cf. Eq. (C7)) the current density in Eq. (D2)

Solid State Ionics 320 (2018) 239–258

R. Kirchheim

will be lp

ip = 2F

∫ ksh exp ⎛

μO2 + 2Δμ 2o

⎜

⎝

0

Δμ Δμ ⎞ exp ⎜⎛ 2po ⎟⎞ dx = σ2 (r − 1) RT exp ⎜⎛ 2po ⎟⎞ lp, 2 2 RT F λ g ⎠ ⎝ ⎠ ⎝ RT ⎠

⎟

2RT

(D3)

which has to be equal to in (Eq. (D1)), i.e.

Δμ 2po ⎞⎟ l . ln = exp ⎛⎜ p ⎝ RT ⎠

(D4)

Thus in the context of high polarization Δμ2po/RT > > 1 the length of the p-region is much smaller than the one of the n-region. The current density in the i-region is given by Eq. (18). For the limiting case used right now the difference of Δμ2i is obtained from Appendix C (Eqs. (C7) and (C11)) as

Δμ 2i = Δμ 2no − Δμ 2po

(D5)

and inserted in Eq. (18)

ii =

σ2 [2FU − μ 2no + μ 2po ]. 2Fli

(D6)

From the equality of current densities in the three regions of YSZ the following relation is obtained by using Eq. (D1)

2FU − μ 2no + μ 2po =

(r − 1) (r − 1) RTln li = RTln (l − lp − ln ). λ g2 λ g2

(D7)

The last equation is simplified further on by using Eq. (19) and ln > > lp (cf, Eq. (D4)) reading then

2FU ≈

(r − 1) RTln (l − ln ). λ g2

(D8)

Solutions of Eq. (D8) are

ln l ⎡ = ⎢1 − λg 2λ g ⎢ ⎣

1−

8FUλ g2

⎤ ⎥ RT (r − 1) l 2 ⎥ ⎦

(D9)

For getting a positive real value for ln the following condition has to be valid

8FUλ g2 RT (r − 1) l 2

< 1.

(D10)

This condition is used in a series expansion of Eq. (D9) yielding

2FUλ g ln = . λg RT (r − 1) l

(D11)

With this value the current density in Eq. (D1) becomes

i = ii = in = ip = σ2

U , l

(D12) 2

( ) exp ( ) < < 1 is equivalent to the case of low polarization (cf. Eq. (38)). Second limiting case of ( ) exp ( ) > > 1: In the following the reverse limitation with ( ) exp ( ) > > 1 is treated. Under these conditions the situation in the n-region will not

which for the present limitation of lp

2

lp

Δμ2po

λg

RT

Δμ2po

λg

RT

lp

2

Δμ2po

λg

RT

change, but in the p-region the chemical potential μ2p is now given in Appendix C by Eq. (C8) and the related current density becomes

ip = 2F

Δμ μO + 2μ 2o kh ⎞ exp ⎛⎜ 2po ⎞⎟ exp ⎛ 2 s 2RT ⎝ ⎠ ⎝ RT ⎠ ⎜

⎟

lp

2

Δμ 2po ⎤ ⎞⎟ dx exp ⎛⎜ ⎥ ⎝ RT ⎠ ⎦ ⎠

∫ exp ⎡⎢−2 ⎛ λxg ⎞ ⎜

⎣

0

⎝

⎟

(D13)

and after integration

ip = 2F

Δμ Δμ μO + 2μ 2o kh ⎞ exp ⎛⎜ 2po ⎞⎟ λ g π erf ⎛⎜ lp 2 exp ⎛⎜ 2po ⎞⎟ ⎞⎟ exp ⎛ 2 s 2RT ⎝ ⎠ ⎝ 2RT ⎠ 2 2 ⎝ 2RT ⎠ ⎠ ⎝ λg ⎜

⎟

which simplifies due to the condition of

2

(D14)

( ) exp ( ) > > 1 to lp

λg

Δμ2no RT

Δμ Δμ μO + 2μ 2o π kh ⎞ exp ⎜⎛ 2po ⎟⎞ = σ2 RT π (r − 1) exp ⎛⎜ 2po ⎞⎟. ip = Fλ exp ⎛ 2 2 2 8 2 s 2 RT RT Fλ g ⎝ ⎠ ⎝ ⎠ ⎝ 2RT ⎠ ⎜

⎟

(D15)

From its equality with the current density in the n-region (Eq. (D1)) the following relation is obtained

1 4

Δμ 2po π ⎞⎟ = ln . exp ⎛⎜ 2 λg ⎝ 2RT ⎠

(D16) 254

Solid State Ionics 320 (2018) 239–258

R. Kirchheim

Due to the exponential term and Δμ2no/(2RT) > > 1 for high polarization the length of the n-region is large, but it will be limited by the sample length. Thus the values of λg and Δμ2po have to obey the relation

Δμ 2po π ⎞⎟ = 1 exp ⎛⎜ 4 2 ⎝ 2RT ⎠

λg 4

Δμ s (r − 1) π exp ⎛ 2no ⎞ < l. 2h ⎝ 2RT ⎠

(D17)

Then the current density reaches an upper limit for ln ≈ l given by Eq. (D1) as

i = in < σ2

μO + 2μ 2o (r − 1) RT h RT 2F ⎞ l = σ2 l= lkn exp ⎛ 2 s 2F s 2RT λ g2 2F ⎠ ⎝ ⎜

⎟

(D18)

or by using Eq. (A3)

ilim =

μO x 2F lkn exp ⎛ O ⎞ s ⎝ RT ⎠ ⎜

⎟

(D19)

The right hand side of the last expression has a simple physical meaning. The upper bound of the current is due to the maximum desorption of one mole of lattice oxygen (cf. Eq. (1)). This supply is the largest for the n-region extending over nearly all the sample length. One mole OOx desorbed leaves a maximum electronic charge of 2F and the maximum rate of desorption per volume in the n-region (cf. Appendices A and B) is kes−1 exp (μOOx/RT). Appendix E. Deriving chemical potentials and currents for the case of single-charged vacancies For the case of small polarization or Δμ2/RT < 1 and Δμ1/RT < 1 Eqs. (46) and (47) are approximated by

Δμ1p Δμ 2p ∂c2p ⎤ μ ⎤ <0 = ksh exp ⎛ 2o ⎞ ⎡ − Rsp = ⎡ ⎢ ⎥ ⎢ RT RT RT ⎥ ∂ t ⎝ ⎠ ⎦s ⎣ ⎦ ⎣

(E1)

and

Δμ Δμ 2n μ ∂c ⎤ > 0. Rsn = ⎡ 2n ⎤ = ksh exp ⎛ 2o ⎞ ⎡ 1n − RT ⎦ ⎝ RT ⎠ ⎣ RT ⎣ ∂t ⎦s

(E2)

Inserting the last equation into Eq. (22) and using Eq. (52) leads to

(r − 1)

μ Mn (Δμ 2n − Δμ 2no ) + Δμ1no Δμ 2n ⎤ c2 D2 ∂2μ 2n = ksh exp ⎛ 2o ⎞ ⎡ − >0 ⎢ ⎥ RT RT ⎦ RT ∂x 2 ⎝ RT ⎠ ⎣

(E3)

with the boundary values Δμ2no = Δμ2n(l) and Δμ1no = Δμ1n(l). By replacing the subscript n by p and reversing the inequality sign yields the corresponding expression for the p-region. The differential Eq. (E3) is solved with the boundary conditions for blocking electrodes (∂μ2/∂x = 0 and ∂μ1/ ∂x = 0). With a characteristic length defined as

(r − 1) c2 D2

λs =

ksh exp

μ2o

( )

RT (r − 1) σ2

=

4F 2ksh exp

RT

μ2o

( )

(E4)

RT

the solution of Eq. (E3) is

Δμ 2n Δμ1no − Δμ 2no Δμ1no l − x⎞ Mn Δμ 2no cosh ⎛ + − = . RT (Mn − 1) RT λ M − 1 RT ( M n n − 1) RT ⎝ s ⎠ ⎜

⎟

(E5) ••

Eq. (12) requires that the rate of electron generation is equal to the rate of VO generation. Then Eqs. (E2), (E5) and (52) yield

Δμ Δμ 2no μ ⎤ cosh ⎛ l − x ⎞ > 0 ⎡ ∂cen ⎤ = ⎡ ∂c2n ⎤ = ksh exp ⎛ 2o ⎞ ⎡ 1no − RT ⎦ ⎝ RT ⎠ ⎣ RT ⎣ ∂t ⎦s ⎣ ∂t ⎦s ⎝ λs ⎠ ⎜

⎟

(E6)

Thus the total amount of electron generation or the current density in the n-region respectively is given as l

in = F

l

∫ ⎡⎣ ∂∂ctne ⎤⎦ dx = F ∫ ⎡⎣ ∂∂ct2n ⎤⎦ dx s

l − ln

l − ln

s

Δμ Δμ 2no μ ⎞ λs sinh ⎛ ln ⎞. = Fksh exp ⎛ 2o ⎞ ⎛ 1no − RT RT RT ⎠ ⎝ ⎠⎝ ⎝ λs ⎠ ⎜

⎟

(E7)

For the p-region Eq. (11) requires that the rate of hole generation is equal to the rate of region the current density in the p-region becomes

Δμ 2po Δμ1po μ ⎞⎟ λ sinh ⎛ lp ⎞. ip = Fksh exp ⎛ 2o ⎞ ⎛⎜ − s RT ⎠ ⎝ RT ⎠ ⎝ RT ⎝ λs ⎠ ⎜

VO••

consumption. By the same procedure as for the n-

⎟

(E8)

In steady state the two current densities have to be equal, i.e.

⎛⎜

Δμ 2po

⎝ RT

−

Δμ1po

⎞ ⎛ lp ⎞ = ⎛ Δμ1no − Δμ 2no ⎞ sinh ⎛ ln ⎞. ⎟ sinh RT ⎠ RT ⎠ ⎝ λs ⎠ ⎝ λs ⎠ ⎝ RT ⎜

⎟

⎜

⎟

(E9)

In order to get closed solutions for the current voltage relation, two limiting cases of λs > > lk < l and λs < < lk < l with k = n, p are 255

Solid State Ionics 320 (2018) 239–258

R. Kirchheim

considered. Limiting case of λs > > lk small deviations from charge neutrality (r ≫ 1) and small reaction rates: With the limiting case of λs > > lk and neglecting (lk/λs)2-terms in a series expansion of Eq. (E5) yields for the chemical potential

Δμ 2n ≈ Δμ 2no . Δμ 2p ≈ Δμ 2po

(E10) ••

•

Thus in both regions the chemical potential of VO is nearly independent of position. The same is true for VO and the electric potential ϕ because of Eqs. (15) and (52). For λs > > lk the current densities in Eqs. (E7) and (E8) are approximated by

Δμ Δμ 2no μ ⎞ ln = (r − 1) σ2 (Δμ − Δμ ) ln in = −Fksh exp ⎛ 2o ⎞ ⎛ 1no − 2no 1no RT ⎠ 4Fλs2 ⎝ RT ⎠ ⎝ RT Δ μ Δ μ μ (r − 1) σ2 1po 2po ⎞ ip = −Fksh exp ⎛ 2o ⎞ ⎜⎛ − (Δμ1po − Δμ 2po ) lp ⎟ lp = RT ⎠ 4Fλs2 ⎝ RT ⎠ ⎝ RT

(E11)

leading to the following relation for the condition of equal currents

(Δμ1po − Δμ 2po ) lp = (Δμ 2no − Δμ1no ) ln

(E12)

In steady state a steady continuation of the fluxes across the p/i- and i/n-regions will occur and Eq. (50) remains to be valid in the i-region. Thus both single and double-charged vacancies contribute to the electric current density with

ii = −(2FJ2i + FJ1i ) = −FJ2i =

σ2 ⎡ 2F Δϕi − Δμ 2i ⎤ . ⎥ 4F ⎢ li ⎣ ⎦

(E13)

Note that fluxes are counted positive, if they are directing in x-direction, whereas electric currents are positive, for flowing from the positive to the negative electrode (cf. Figs. 2–6). As both Δμ2 and ϕ do not change much in the p- and n-region, the last equation is approximated by

σ2 ⎡ 2FU − Δμ 2po + Δμ 2no ⎤ . ⎥ 4F ⎢ li ⎦ ⎣

ii ≈

(E14)

This current density has to be equal to the one in the n-region, i.e. with Eqs. (E11) and (E14) the following relation holds

2FU − Δμ 2po + Δμ 2no li

=

(r − 1) (Δμ 2no − Δμ1no ) ln . λs2

(E15)

With the trivial relation li = l − lp − ln and Eq. (E12) the last equation leads to a quadratic expression of ln

(2FU − Δμ 2po + Δμ 2no ) (r − 1)(Δμ 2no − Δμ1no )

=

(Δμ 2no − Δμ1no ) ln ⎛ ⎞ ln − ln ⎟ . l− (−Δμ 2po + Δμ1po ) λs2 ⎜ ⎠ ⎝

(E16)

2

Neglecting terms of the order (lk/λ) leads to

ln =

λs2 (2FU − Δμ 2po + Δμ 2no ) l (r − 1)(Δμ 2no − Δμ1no )

.

(E17)

The current density will be obtained from Eq. (E11) as

i = in =

σ2 (2FU − Δμ 2po + Δμ 2no ) . 4F l

(E18)

Comparing with Eq. (E14) reveals that the i-region expands over most of the sample length or li ≈ l and lp < < l as well as ln < < l. Thus the reactions given by Eqs. (11) and (12) in the main text are leading to a dominant electronic conduction within narrow regions of the YSZ only, which are adjacent to the electrodes. If the related changes of the chemical potential of VO•• from its initial value μ2o are small, Eq. (E18) will have the simple form

i = in =

σ2 U . 2 l

(E19)

Thus the cell with two blocking electrodes delivers half the current when compared to a cell with non-blocking electrodes. The blocking electrodes act as current collectors and the narrow p- and n-region become reversible electrodes allowing both oxygen and electron transfer. The fact that only half the current is present is due to the counteracting flux of single-charged vacancies transporting half of the charge in opposite direction. Steady slopes of the chemical potentials and electric potential should be present at the p/i- and i/n-interfaces. Because of Eqs. (14), (15) and (52) in the main text and Eq. (E12) this has to be proven for the chemical potential μ2 at one of the interfaces only. Thus the derivative of μ2n at x = l − ln has to be equal to the slope of μ2i in the i-region. According to Eqs. (E5) and (E14) this corresponds for ln/λs < < 1 to

Δμ 2no − Δμ 2po Δμ 2no − Δμ1no ⎛ ln ⎞ Δμ 2i = ≈ ⎜ 2⎟ (Mn − 1) ⎝ λs ⎠ li l

(E20)

So far ln/λs < < 1 has been assumed to be valid. Then for l ≤ λs the last equation requires that Mn is about unity or r is also about unity (cf. Eq. (51)) or more explicitly obeys the relation

r≈1+

c1 D1 , c2 D2

(E21)

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Solid State Ionics 320 (2018) 239–258

R. Kirchheim

where the second term on the right hand side is small because of c1 < < c2. Inserting the last equation in Eq. (E4) yields

c1 D1

λs =

ksh exp

μ2o

( )

, (E22)

RT

which may be small due to the very low concentration of single charge vacancies. Thus the assumption l ≤ λs may not be valid. For the opposite condition of l > λs both M and r may become larger than unity. As a consequence of l > λs both p- and n-region will become again very narrow because of l > λs > > ln and l > λs > > lp. Limiting case of λs < < lk: In the following the other limiting case with λs < < lk < l will be treated by approximating the hyperbolic functions in Eqs. (E4) to (E9) by the exponential ones with the positive argument. In order to avoid many assumptions about the unknowns and the related cumbersome derivations, the polarizations on both ends of the YSZ shall be of the same magnitude with

− Δμ 2po + Δμ1po = Δμ 2no − Δμ1no and − Δμ 2po = Δμ 2no

(E23)

which gives ln = lp because of Eq. (E9). Then the change of the chemical potential Δμ2i across the i-region is obtained from Eqs. (53) and (54) with the additional simplifying assumption of Mn = Mp = M as

Δμ 2i =

1 ⎡ (−Δμ + Δμ ) exp ⎛ ln ⎞ + 2M Δμ − 2Δμ ⎤. 2no 1no 2no 1no ⎥ (M − 1) ⎢ ⎝ λs ⎠ ⎦ ⎣ ⎜

⎟

(E24)

In order to get a negative difference the condition

2Δμ 2no ⎤ ≥ ln > > 1 ln ⎡2 + (M − 1) ⎢ λs (Δμ 2no − Δμ1no ) ⎥ ⎦ ⎣

(E25)

has to be valid. This will be the case for a value of M much larger than unity. Because of |Δμ2no| > |Δμ1no| the following approximation of Eq. (E25) may be useful

ln[M ] ≥

ln >>1 λs

(E26)

The change of electrical potential across the i-region is evaluated with Eq. (17) from the main text

Δϕi = ϕn (l − ln ) − ϕp (lp) = U +

r r (Δμ 2i − Δμ 2no + Δμ 2po ) = U + (Δμ 2i − 2Δμ 2no ). 2F 2F

(E27)

Inserting Eqs. (E23), (E24) and (E26) in Eq. (E13) gives the current density in the i-region

ii = − = −

σ2 [2FU + (r − 1)Δμ 2i − 2r Δμ 2no ]= 4Fli

(r − 1) ⎡ σ2 ⎡ l 2FU + (−Δμ 2no + Δμ1no ) exp ⎛ n ⎞ + 2M Δμ 2no − 2Δμ1no ⎤ − 2r Δμ 2no ⎤ ⎥, ⎥ 4Fli ⎢ (M − 1) ⎢ ⎝ λs ⎠ ⎦ ⎣ ⎦ ⎣ ⎜

⎟

(E28)

which has to be equal to in given in Eq. (E7), leading to

⎡ (l − 2ln ) (M − 1) − 1⎤ (Δμ − Δμ ) exp ⎛ ln ⎞ = (M − 1) 2FU − 2r Δμ 2no + 2M Δμ − 2Δμ . 1no 2no 2no 1no ⎥ ⎢ 2λs (r − 1) ⎝ λs ⎠ ⎦ ⎣ ⎜

⎟

(E29)

For M > > 1 and r > > 1 the last expression finally becomes

⎡1 − (l − 2ln ) (M − 1) ⎤ (Δμ − Δμ ) exp ⎛ ln ⎞ = M − 1 2FU − M − r 2Δμ − 2Δμ . 2no 1no 2no 1no ⎥ ⎢ 2λs r−1 r−1 ⎝ λs ⎠ ⎦ ⎣ ⎜

⎟

(E30)

Because of the following relation (cf. Eq. (51))

M c D r ⎤ >1 ≈⎡ 2 2 + ⎢ r−1 2(r − 1) ⎥ ⎣ c1 D1 ⎦

(E31)

and because of r > 1 (cf. Eqs. (14) and (15)) the following relation holds

2FU > Δμ 2no − Δμ1no − Δμ 2po + Δμ1po = 2Δμ 2no − 2Δμ1no .

(E32)

Then Eq. (E30) reduces to

⎡1 − (l − 2ln ) (M − 1) ⎤ (Δμ 2no − Δμ1no ) exp ⎛ ln ⎞ = M − 1 2FU 2λ r−1 ⎣ ⎦ ⎝ λs ⎠ ⎜

⎟

(E33)

with a positive term on the right hand side. Thus the one on the left hand side has to be positive as well with

(l − 2ln ) (M − 1) < 1. 2λs

(E34)

In the framework of the present discussion with M > > 1 and ln/λs > > 1 the last condition leads to ln ≈ l/2 and lp = ln ≈ l/2 (cf. text after Eq. (E24)), which allows a further simplification of Eq. (E33) to

257

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⎡1 − (l − 2ln ) (M − 1) ⎤ (Δμ − Δμ ) exp ⎛ ln ⎞ ≈ (Δμ − Δμ ) exp ⎛ ln ⎞ = M − 1 2FU . 2no 1no 2no 1no ⎥ ⎢ 2λs r−1 ⎝ λs ⎠ ⎝ λs ⎠ ⎦ ⎣ ⎜

⎟

⎜

⎟

(E35)

Inserting the last equation into the one for the current density (cf. Eq. E7) yields

Δμ Δμ1no λs μ ⎞ exp ⎛ ln ⎞ = σ2 (Mn − 1) U . in = Fksh exp ⎛ 2o ⎞ ⎛ 2no − RT ⎠ 2 λs ⎝ RT ⎠ ⎝ RT ⎝ λs ⎠ ⎜

⎟

(E36)

This current density has to be the same as the external one i leading to an effective conductivity for ln = l/2 with

σeff ≡

il 2l M = σ2 n > > σ2. U λs

(E37)

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