Ionic/Electronic Conductivity, Thermal/Chemical Expansion and Oxygen Permeation in Pr and Gd Co-Doped Ceria Pr_xGd_0.1Ce_0.9-xO_1.95-δ

The specimens in this work were synthesized by solid state reaction. Stoichiometric amounts of precursors of Ce(NO₃)₃·6H₂O (Sigma Aldrich 99.99%), Gd₂O₃ (Alfa Aesar 99.9%), Pr₆O₁₁ (Alfa Aesar 99.9%) and Co₃O₄ (Alfa Aesar 99.9%, for cobalt containing samples) were mixed by ball milling in ethanol for 24 hours. The resultant mixed powders were dried at 80°C for 10 hours and subsequently calcined at 1000°C for 5 hours. The calcined powders were then uniaxially cold pressed at 400 MPa followed by isostatic pressing at 600 MPa. Cobalt-containing and cobalt-free pellets were sintered at 1200°C and 1600°C for 10 hours, respectively.

The crystal structures of the material (crushed sintered pellets) were characterized by X-ray powder diffraction (XRD). The XRD patterns were obtained with Cu Kα radiation using a Bruker Robot operating in Bragg-Brentano geometry in the 2θ range from 20° to 90°. The obtained XRD patterns were indexed with the ICCD (International Centre for Diffraction Data) database by means of the software named DIFFRAC plus. Scanning electron microscopy (SEM) was carried out on a FE-SEM Zeiss Supra 35 electron microscope using an acceleration voltage of 15 kV.

Thermogravimetric analysis (TGA) was performed using a Netzsch STA 409CD thermogravimeter and a Netzsch TG 439 thermal balance. The samples were prepared by uniaxially pressing the powder into pellets at a pressure of 400 MPa in a capillary die (ø = 0.5 cm). Porous samples were obtained by sintering the green sample at 1000°C for 3 hours at heating and cooling rate of 2°C min⁻¹. Buoyancy effects were corrected for by measuring the mass change of an Al₂O₃ powder sample with the same volume under the same conditions.

Dilatometry measurements were performed using a differential contact dilatometer (DIL 402CD, NETZSCH GmbH, Germany). The samples were rectangular bars with a dimension of 1 × 1 × 10 mm³. An Al₂O₃ reference with a similar dimension was tested simultaneously with the sample. The heating rate was kept at 3°C min⁻¹ to a final temperature of 1000°C. All the measurements were conducted in air with a total gas flow of 50 ml min⁻¹ controlled by a mass flow controller.

The total conductivity of each specimen was measured by Electrical Impedance Spectroscopy (EIS). Prior to the electrical measurements, symmetric (La_0.6Sr_0.4)_0.99CoO_3-δ (LSC) electrodes were screen printed on both sides of a 1-mm thick polished pellet (ø = 10 mm), followed by sintering at 1000°C for 2 hours to obtain good adhesions between the specimen and the electrodes. The LSC electrodes were then covered with Pt layers (current collector) by hand painting to eliminate any contact resistance. EIS was carried out in the temperature range from 600°C to 900°C using a Solartron 1260 impedance spectrometer within the frequency range from 3 MHz down to 0.1 Hz under a fixed fluctuation voltage of 20 mV. Impedance spectra were plotted and fitted by the software package "Z-view" and "Z-plot". The specimens were placed in a closed alumina chamber in which oxygen partial pressure was varied and monitored by controlling the flow of air and nitrogen by mass flow controllers. The oxygen partial pressure around the samples was measured by a zirconia-based sensor placed in close proximately to the sample.

The electronic conductivity of each sample was measured by means of a microelectrode ion- blocking Hebb-Wagner polarization method. Prior to the measurements, 1-mm thick dense circular samples (ø = 10 mm) were polished by SiC #300 sand papers, followed by further polishing using SiC #1000 and #1600 sand papers. The Hebb-Wagner measurement cell is schematically illustrated in Fig. 1. Similar setups have been used in a number of papers in the past.^13,22–25 A Pt microelectrode with a contact radius of 100–500 μm was placed on the sample under an external load. The periphery of the Pt microelectrode was sealed with a customized glass sealant (MgO/sodium aluminosilicate glass composites, 30/70 vol. %). The Hebb-Wagner cell was sandwiched with two Pt current collectors. The whole cell was finally placed inside a furnace in which the oxygen partial pressure was maintained at 0.21 atm by flowing 100 ml min⁻¹ of air. The cell was initially heated up to 950°C at the rate of 1°C min⁻¹ to soften the glass sealant, followed by cooling down to 750°C at the same rate to achieve a gastight solidified glass. The polarization measurement was then performed by stepwise varying voltage in 25 mV steps in the range from −800 to 200 mV. The steady state was achieved by dwelling for 12 minutes (more than 100 times the theoretical time needed to reach the steady state) in each step. The oxygen partial pressure in the vicinity of the microelectrode can be calculated by the Nernst equation:

where and is the oxygen activity near the microelectrode and reference electrode, respectively. V is the polarization voltage applied on the microelectrode relative to the reference/counter electrode. The sum of electron and electron hole conductivity in the vicinity of the ion-blocking microelectrode can, under a set of assumptions,^13,24,26 be calculated in terms of the derivative of the steady state I-V curve:

where r_c is the radius of the spherical microelectrode. If the micron-sized contact is circular rather than spherical, the factor 2πr_c is replaced by 4r_c.²⁴

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Oxygen permeation measurements were performed in the customized test configuration displayed in Fig. 2. Before the flux measurements, circular samples (ø ≈12 mm) were polished by SiC #500 sand papers to achieve a thickness of approximately 1 mm. To mitigate the effect of slow surface exchange, a highly porous circular La_0.6Sr_0.4CoO_3-δ (LSC)-based catalytic layer (ø ≈5 mm) was applied on the dense samples by screen printing with a LSC-based ink. The as-prepared catalytic layers were then calcined at 900°C for 1 hour to obtain good adhesion with the membrane surface. The area covered with the catalytic layer is ∼0.8 cm².

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The samples were sealed between two alumina tubes using a 30/70 vol.% mixture of a MgO and sodium aluminosilicate glass. The feed side tube below the membrane was supplied with gas from the feed side manifold, which featured a feed gas inlet and outlet. Similarly, the permeate side of the membrane was purged with gas from a "permeate side" manifold. The feed side was flushed with air with a constant flow rate of 100 Nml min⁻¹, and the permeate side was flushed at various flow rates of N₂ (from 30 to 300 Nml min⁻¹). All gas flows were measured by mass flow controllers. The net oxygen permeation flux can be deduced from the pO₂ difference between inlet and outlet gases. The pO₂ was monitored by two customized YSZ-based oxygen sensors. The total leak in the system was assessed by means of a mass flow meter on the permeate stream. When perfect sealing is achieved, the outlet gas flow rate is equivalent to the inlet flow plus the flow of the permeated oxygen. If there are leaks in the membrane or pinholes or cracks in the glass encapsulation, the flow rate of the outlet gas will be lower than that of the inlet gas. Typically, when a significant leak is observed during the permeation measurement, a thermally activated increase in oxygen flux is not observed. In the data reported here, the inlet and outlet flows were identical (within 1% uncertainty) and a thermal activated oxygen permeation flux was observed, indicating close to perfect sealing. The maximum leak originating from pinholes in the sealant and from other sources of oxygen in the permeate flow through the membrane is estimated to be lower than 5% of the measured oxygen flux at low temperature (600°C). Below 600°C a large leak appeared as a result of sealant failure driven by thermal expansion mismatch between membrane and manifold tubes.

Fig. 3 displays the microstructure of thermally etched samples of different composition. Table I summarizes the specific composition, sintering temperature (Ts), theoretical density, measured density, relative density and average grain size of the specimens analyzed. The average grain size of the cobalt oxide free samples sintered at 1600°C lies in the range from 2 to 5 μm, and shows a slightly increasing tendency with increasing Pr content. The cobalt oxide containing samples sintered at 1200°C are completely dense with relatively small average grain size (200–500 nm). This elucidates that cobalt oxide serves as an effective sintering aid in doped ceria, which is well in line with literature.²⁷

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XRD patterns of crushed dense pellets of Pr_xGd_0.1Ce_0.9-xO_1.95-δ (x = 0.02, 0.05, 0.08, 0.15, 0.25, 0.3 and 0.4) and Co_xPr_0.05Gd_0.1Ces_0.85O_1.95-δ (x = 0.02 and 0.05) are shown in Fig. 4. Zero shifts were compensated for using an internal standard (LaB₆ powder). No significant shift of the (111) reflection was detected, indicative of an insignificant influence of the addition of the dopant on the lattice parameter. No indication of secondary phases originating from side reactions or precipitation of insolvable dopants can be detected within the resolution limit of the XRD, indicating that the solubility limit of Pr in CGO is more than 40 at.%, which is in accordance with the solubility limit of 70 at.% reported by Taksu et al.²⁸ for partial Pr-substitution for cerium in Pr_xCe_1-xO_2-0.5x at 1400°C. Co₃O₄ and CoO phases could not be indexed from the diffractograms of Co_xPr_0.05Gd_0.1Ce_0.85O_1.95-δ (x = 0.05). Due to the high temperature (1200°C) used to sinter the cobalt-containing samples, any cobalt oxide not dissolved in the fluorite phase may likely react with trivalent Gd, Pr cations, forming small amounts of GdCoO₃ or PrCoO₃ phases for which the highest intensity peaks (2θ≈32°, PrCoO₃:PDF#25-1069 and GdCoO₃:PDF#25-1057) would be 'hidden' by the peak of the (200) crystal planes from the main phase.

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The lattice parameter obtained from XRD is shown in Fig. 5 along with calculated lattice parameters obtained by Kim's empirical formula²⁹

where a is the unit cell parameter of the fluorite oxide at room temperature, (Δr_k = r_k-r_h) is the difference in ionic radius of the kth dopant (r_k) and the host cation (r_h) in eight-fold coordination from Shannon's compilation,³⁰ (Δz_k = z_k-z_h) is the valence difference between dopant and host, and m_k is the mole fraction of the kth dopant in the form of PrO_x (x = 1.5/2).

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As displayed in Fig. 5, the experimentally observed lattice parameter lies closer to the calculated values for Pr being tetravalent (Pr⁴⁺) in the materials than when assuming Pr³⁺. McCullough reported that praseodymium is readily oxidized to Pr⁴⁺ in the presence of Ce⁴⁺.³¹ This suggests that the concentration of tetravalent Pr prevails at room temperature in the Pr-doped CGO (slow cooling in air). Due to the similar ionic radii between Pr⁴⁺ (CN = VIII 0.96 ų⁰) and Ce⁴⁺ (CN = VIII 0.97 ų⁰), the lattice parameter does not vary significantly with increasing dopant concentration, which is similar to the trend observed in purely Pr-doped ceria.³² The lattice parameter of the Co-containing samples does not deviate from the Co-free comparable specimens. Lewis et al.³³ reported a similar result that the lattice constant of CGO is not influenced by the addition of cobalt oxide because the cobalt is effectively insoluble in the CGO lattice.

Fig. 6 displays the oxygen nonstoichiometry data of the samples along with the best least square fit curves (0.90<R²<0.99) using the ideal defect model (Eq. 11). In the investigated pO₂ regime (blue regime) the oxygen nonstoichiometry (δ) increases with decreasing oxygen partial pressure until it reaches a plateau (at δ = x/2), where all the Pr is trivalent. The equilibrium constants (K_Pr) at each temperature were obtained from the fit of the defect model to the data. The equilibrium enthalpy and entropy were obtained from the slope and intercept of the linear fit to the data of -ln(K_Pr) vs. 1/T according to the equation:

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The thermodynamic parameters thus deduced for each composition are summarized in Table II. It is noteworthy that the enthalpy of reduction of Pr decreases with increasing Pr concentration, indicating a more facile reduction of Pr and release of oxygen with increasing Pr concentration or oxygen nonstoichiometry. Chatzichristodoulou et al.¹⁶ also found that reduction is facilitated with increasing oxygen nonstoichiometry in Pr and Tb co-doped ceria. Stefanik et al.³⁴ also pointed out that the reduction enthalpy of Pr decreases with increasing Pr concentration, indicating higher Pr concentration gives rise to higher reducibility, which is consistent with the general trend observed in th is work.

Thermal expansion curves of Pr_xGd_0.1Ce_0.9-xO_1.95-δmeasured on cooling the samples from 900°C to room temperature in air are shown in Fig. 7a. Below 400°C, all the samples show linear expansion. However, the curves display a non-linear behavior with an inflection point in the range from 500°C to 600°C, particularly for the samples with 15 at.% Pr or more. It is evident that Pr-doped CGO follows a similar tendency as Pr doped ceria where the strain increases with increasing Pr concentration in the specimen. The increased and highly non-linear apparent TEC in Pr-doped ceria is primarily ascribed to the chemical strain originating from the combination of slight contraction of the unit cell upon formation of oxygen vacancies and expansion of the unit cell upon partial reduction of Pr.^16,36 The total expansion (the measured quantity) is the sum of the thermal and the chemical expansion (stoichiometric expansion³⁷), which can be written:

where α is the thermal expansion coefficient (TEC), and β is the stoichiometric expansion coefficient.³⁷ Assuming a constant thermal and stoichiometric expansion coefficient³⁷ at a given oxygen partial pressure, the non-linear total expansion curve can be fitted by a linear thermal expansion curve and a nonlinear chemical expansion curve (Fig. 7b). Values of the thermal expansion and chemical stoichiometric expansion coefficients deduced from this type of fitting are listed in Table III. The chemical expansion coefficient is observed to decrease with increasing dopant concentration. For samples doped with similar Pr concentration (Pr_0.2Ce_0.8O_1.9-δ vs. CPGO25), the chemical expansion coefficients are very close. (0.084 mol⁻¹ for Pr_0.2Ce_0.8O_1.9¹⁶ and 0.08 mol⁻¹ for CPGO25). It should be noted that Eq. 16 applies in general to small perturbations; β is best determined at constant temperature with a change in δ driven by varying pO₂, and α should be determined at fixed δ. We did not conduct such differential measurements (isothermal dilatometry). In the analysis presented, leading to estimates of β (Table III), one assumes that the α value determined at low temperature (where changes δ in are small) is constant in the whole temperature range.

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The contact area of the Pt microelectrode was determined using Newman's formula³⁸

where r_c is the radius of the micro-electrode, R_s is the serial resistance found from the intercept with the real axis at the high frequency of the impedance spectra and σ is the conductivity of the sample. The contact radius thus determined was typically between 100 μm and 500 μm, in good agreement with Ref. 24 where similar conditions were applied. The contact radius evaluated by impedance spectroscopy was as expected not sensitive to the temperature or oxygen partial pressure.

The plots of the steady state current density of CPGO8 measured in the blocking electrode geometry versus the potential at 700°C, 800°C and 900°C are presented in Fig. 8. A current plateau in the range of −0.4–0 V is observed. Excellent coincidence of the branches recorded in the negative and positive polarization direction upon repeated cycles indicate that any hysteresis arising from creep of the Pt micro-electrode and/or sluggish equilibrium between the redox of Pr³⁺ and Pr⁴⁺ is negligible, and accordingly confirms that the I-V curves are obtained in the steady state.

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The oxygen activity dependence of the partial electronic and total conductivity of each sample is presented in Fig. 9. The electronic conductivity of CGO measured in this work agrees well with the electronic conductivity of CGO reported by Chatzichristodoulou et al.²⁴ In general, the electronic conductivity of CPGO is dominated by n-type and p-type under low and high oxygen partial pressures, respectively. The minimum point at intermediate pO₂ corresponds to the transition from n-type to p-type conductivity. It is noteworthy that the minimum point shifts toward lower oxygen partial pressure with increasing Pr concentration.

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In contrast to the electronic conductivity of CGO which follows a typical relationship of , the electronic conductivity of CPGO possesses a more varied pO₂ dependency. The slope of the n-type electronic conductivity of slightly doped samples (CPGO2, CPGO5, CPGO8 and CPGO15) at low pO₂ approaches −¼, whereas for the heavily doped samples (CPGO25, CPGO30 and CPGO40) the slope is depressed especially at low temperature.

Under oxidizing conditions, the slopes vary within the range from to , in agreement with Ref. 13. Evidently, the p-type electronic conductivity strongly increases with increasing Pr concentration. In particular for CPGO40, the p-type electronic conductivity is higher than that of CGO by two orders of magnitude at 700°C. The observed enhancement of p-type conductivity upon Pr substitution is in line with previous literature.^13,16,39,40 It is also seen that the electronic conductivity becomes insensitive to the temperature for these heavily doped samples (>15 at.%), which coincides with previous findings by Chatzichristodoulou et al.²⁴ and Schmale et al.²⁵ This can be explained by the decreased concentration of Pr^x_Ce at elevated temperature which counteracts the increased mobility of the electron holes.²⁴

The total conductivity is insensitive to the change of oxygen partial pressure in the range from 1 × 10⁻⁵ bar to 0.21 bar for Pr substitutions lower than 8 at.% whilst it shows detectable pO₂-dependence for Pr concentrations larger than 15 at.%. The pO₂-dependence becomes more pronounced with increasing Pr concentration because the electronic conductivity is sufficient to influence the total conductivity and the ionic conductivity decreases with decreasing pO₂ (as shown in Fig. 10).

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In Fig. 10, the oxide ion conductivities were obtained by subtracting the electronic conductivities from the total conductivities (see Fig. 9). It can be observed that the oxide ion conductivity is invariant over the full pO₂ range for the samples doped with less than 15 at. % Pr but steadily decreases with decreasing pO₂ for the more heavily doped samples (>15 at. % Pr). As pO₂ decreases, the concentration of acceptor dopants increases because of the increasing Pr^/_Ce concentration. Oxygen vacancy concentration accordingly increases to counterbalance the increased charge of acceptor dopants. The unchanged (CGO, CPGO2, CPGO5 and CPGO8) or decreased conductivity (CPGO15, CPGO25, CPGO30 and CPGO40) with decreasing pO₂ must be ascribed to a decreasing oxide ion mobility with increasing oxygen vacancy concentration and increasing concentration of Pr³⁺. One does, in terms of ionic conductivity not benefit from doping beyond the 10 at.% Gd. This is well in line with literature on optimal conductivity in acceptor doped ceria which typically points to a maximum in ionic conductivity at doping levels between 10 and 20%.⁴¹ The maximum ionic conductivity in this doping regime is due to the trade off between the increased oxygen vacancy concentration and the development of deep vacancy association induced by electrostatic interaction.³

Fig. 11 displays the calculated and measured oxygen flux for CPGO5, Co5CPGO5 and CGO as a function of temperature under a fixed oxygen partial pressure difference (ln(pO₂'/pO₂'') = 7). All samples in the flux measurements were coated on both sides with porous LSC layers (c.f. Oxygen permeation measurements section). The calculated fluxes were obtained from Wagner's equation using the measured electronic and ionic conductivity (see Figs. 9 and 10). It is noteworthy that the calculated flux is consistent with the measured flux. This indicates that the measured oxygen flux is limited by bulk ambipolar diffusion in the 1-mm thick pellet investigated rather than surface exchange. Another noticeable feature is that the oxygen flux of CPGO5 is higher than that of CGO by a factor of five at 900°C. It has previously been reported that cobalt oxide containing ceria based membranes show enhanced oxygen flux compared to the corresponding cobalt oxide free membrane because percolating cobalt oxide forms along the grain boundaries enhancing electronic conductivity without influencing the ionic conductivity.^17,42,43 The enhanced oxygen flux is reported to be due to the enhanced electronic conductivity. However, we did not observe enhanced flux for the cobalt-containing sample, instead the opposite trend was observed. The slightly decreased oxygen flux in cobalt oxide containing samples in this work may be due to the formation of discrete CoO_x-rich segregations at the grain boundaries resulting from the high sintering temperature (1200°C) instead of forming a continuous percolating network as found in other studies.^17,42,43

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Praseodymium oxides (PrO_x) exist in variable compositions (x in the range from 1.5 to 2) at elevated temperatures where the valence state of Pr thus varies between Pr³⁺ and Pr⁴⁺. As the reduction of Pr occurs, the electrical conductivity of PrO_x significantly increases, up to a value of 1.4 Scm⁻¹ at 850°C.⁴⁴ The electrical conductivity of PrO_x at high temperature is assigned to electronic conductivity, which originates from electron hopping between mixed-valence Pr³⁺/Pr⁴⁺ cations present in the lattice. Unlike PrO_x, Gd-doped ceria is an oxide ion conductor with negligible p-type electronic conductivity under oxidizing condition. As observed (Figs. 8 and 9), a combination of PrO_x and Gd-doped ceria; making a Pr/Gd co-doped ceria solid solution gives rise to mixed ionic-electronic conductivity.

The mechanism for the mixed ionic-electronic conductivity of Pr in ceria has been rationalized in many papers in terms of band theory. Stefanik et al. proposed that Pr-substitution for cerium forms discrete acceptor levels (Pr 4f) for the lightly doped ceria (below 10 at.%).^34,35 The discrete states are not capable of giving rise to considerable electronic conductivity. Lübke et al.¹³ proposed that the Pr 4f states lie much closer to the O2p valence edge than the Ce 4f conduction band. At sufficiently high temperature, electrons in the O2p state will be thermally excited and subsequently localized in the Pr 4f states, resulting in formation of holes in the valence band. Furthermore, at high temperatures and/or decrease pO₂, oxygen is released from the lattice increasing the electron occupation of the Pr 4f states. Here, we shall analyze the experimental results in a "chemical picture" relating the conduction to electronic defects associated with the Pr sites.

In several studies,^17,20,25 it has been reported that the electronic conductivity of Pr-doped ceria can be well described by a small polaron mechanism. Hence, mobility and conductivity will scale with temperature according to:³⁴

where μ_polaron is the mobility of the small polaron, [Pr^/_Ce] is the fractional occupancy of electrons trapped on Pr ions. [Pr^{3 +}] is the concentration of trivalent Pr, N is the volumetric density of Pr atoms in the material (mol cm⁻³). v₀ is the jump frequency, E_H is the activation energy for hopping of the electron hole (electron hole migration), and a is the hopping distance. Eq. 20 can be rewritten to a simpler Arrhenius type expression;

where C is (C = Ne²[Pr^/_Pr](1 − [Pr_Pr^/])k^{− 1}a²ν₀). The electronic conductivity is thus influenced by both the concentration of Pr³⁺ and Pr⁴⁺. When the concentration of Pr³⁺ prevails the material is a p-type conductor; where concentration of Pr⁴⁺ prevails it becomes n-type. In the following discussion we concentrate on the p-type electronic conductivity which is typically observed under practically realizable conditions. For comparison of mobilities and activation energies between the compositions we shall compare data at a fixed Pr³⁺/Pr⁴⁺ ratio. The choice of 2.7 for Pr³⁺/Pr⁴⁺ ratio relates the available pO₂ range for achieving a fixed ratio at all temperatures for all compositions. The oxygen partial pressures in equilibrium with a fixed ratio of (Pr^{3 +}/Pr^{4 +} ≈ 2.7 as observed by TGA) in the different samples are listed in Table IV.

Fig. 12A shows Arrhenius plots of electronic conductivity with a fixed Pr³⁺/Pr⁴⁺ ratio and corresponding linear fittings. The activation energies and pre-exponential constants of electronic conductivity obtained by linear fittings in Fig. 12A are shown in Figs. 12C and 12D, respectively. As indicated by Eq. 21, the activation energy in Fig. 12C corresponds to the migration enthalpy of the electron hole whilst the pre-exponential factor is associated with the jump frequency v_0, hopping distance a and doping level. A noticeable feature in Fig. 12B is that there is a marked change in the scaling of the logarithm of the conductivity versus the Pr concentration around 10 at.% Pr especially at low temperature. An abrupt decline of the activation energy (Fig. 12C) and pre-exponential factor (Fig. 12D) is observed in the same x range.

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The Pr-doped CGO solid solutions can be viewed as PrO_x clusters homogenously embedded in the Gd_0.1Ce_0.9O_1.95-δ matrix. Electronic charge carriers (electrons or holes) are localized preferentially in the PrO_x units and migrate via them upon thermal excitation. For samples doped with low Pr concentration, the distance among the discrete PrO_x clusters is relatively long imposing a high energy barrier for electrons to jump across the CeO₂/Gd₂O₃-based zones, which are electronic "insulating". This results in low electronic conductivity and high activation energy for x<0.1 samples, as shown in Fig. 12C. Furthermore, the high value of the pre-exponential constant for the 0<x<0.1 samples is indicative of a long electron hole hopping distance (a in Eq. 19).

As the dopant concentration increases, the distance between the Pr ions decreases, resulting in decreased effective hopping distance and decreased activation energy. The transient behavior in the range 0.08<x<0.15 is akin to the percolation effect that is generally found in composite materials consisting of an electrical conductor and an insulator. That is, the abrupt increase of electrical conductivity occurs when the amount of the electrical conductor surpasses the percolation threshold above which a continuous electrical conduction pathway penetrates all the way through the composite. The same phenomenon is observed also in solid solutions. Swider and Worrell⁴⁵ employed a percolation model to explain the n-type electronic conductivity of Ti-doped YSZ. A theoretical percolation threshold of 12.5% for Ti-YSZ was proposed on the basis of bond percolation in the crystal structure. Kim et al.⁴⁶ recently developed a simple cubic percolation model to interpret the non-linear increment of electrical conductivity in perovskite structured BaZrO₃-BaFeO₃ solutions. Generally, the percolation threshold (Pc) for a close-packed crystal structure e.g. FCC is a function of the site coordination number Z, and can be simply evaluated as⁴⁷

The shortest possible jump distance will be achieved when the electronic defect localized on a given Pr dopant located on a corner of the cubic unit cell has at least one other Pr among the 12 nearest neighbor sites on the cubic face centers. A possible percolating Pr-Pr pathway is illustrated in Fig. 13. The cation coordination number in the FCC structure equals 12 (Z = 12 in Eq. 23). Therefore, by the simple model described in Ref. 47, the percolation threshold is ∼12.5%, which means that the electronic defect may hop continuously among nearest Pr cations when 12.5% of the cation sites are occupied by Pr. The observed transition range (8 at.% −15 at.%) lies close to the one predicted by this simple percolation model. From the conductivity data plotted in Fig. 12 it is evident that a strong correlation exists between the structural connectivity of the Pr dopants and the migration enthalpy for the electron holes.

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Due to the influence of activation energy and the pre-exponential factor, we herein plot the extrapolated conductivity at high temperature (e.g. 800°C) to pinpoint the abrupt change of electrical conductivity arising from the migration enthalpy. In Fig. 14, the extrapolated electronic conductivity at 800°C was fitted by the percolation model proposed by Kim et al.⁴⁶ In this model, the conductivity for x<Pc and x>Pc is phenomenologically described by the expressions:

where s and t are the universal size scaling exponents for the non-percolation matrix and the percolation phase, respectively. σ_GCO and are the electronic conductivities of CGO and PrO_x⁴⁴ at 700°C, respectively. Pc is the percolation threshold being treated as a fitting parameter. s and t are the two fitting parameters where t is theoretically limited to lie in the range from 1.65 to 2 for the simple cubic percolation model. The best fit of the model to the data in Fig. 14 gives t = 1.80, Pc = 0.125 and s = −1.11, with the R² equals to 0.93. The model seems to satisfactorily fit the data in this work, which further elucidates that the electronic behavior can be well explained by the percolation model.

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The chemical expansion of CGPO is related to the volumetric dilation upon changes in oxygen stoichiometry. The cell lattice parameter expands with the increase of the concentration of Pr³⁺. Chatzichristodoulou et al.³⁷ has estimated that when relating unit cell volumes to the ionic radii of the constituting ions the size of an oxide vacancy in fluorites is smaller than that of the oxide ions. The chemical expansion of CPGO is thus in this picture a consequence of lattice dilation due to the increased radius of the cations partially counterbalanced with oxygen vacancy induced lattice contraction. The lattice parameter as calculated by Kim's formula includes the sum of the two effects. The relative chemical expansion can be simply calculated by a Vegard's law type relationship between lattice dimension and fraction of Pr³⁺ or Pr⁴⁺ in CPGO:

where and are the calculated lattice parameters for only Pr³⁺ or Pr⁴⁺in CPGO, respectively (see Fig. 5). a_XRD is the lattice parameter obtained by XRD refinement (cf. Fig. 6), [Pr^/_Pr] is the fraction of Pr³⁺ calculated by the thermodynamic parameters outlined in Table II. Fig. 15 shows the relative chemical expansion (CE) from the experimental dilatometry data and the CE calculated by Eq. 24. Overall, the calculated CE in this simple model reproduces the general trend of the experimentally deduced CE well. Note that Eq. 26 is consistent with a type of expression as Eq. 16 i.e. it predicts proportionality between ɛ_chem and δ.

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A transition is seen experimentally around 12.5% Pr. Below this the chemical expansion is small, above stronger. There are three reasons for this as seen when comparing the experimental data to the curve calculated from Eq. 26 and considering the ΔH, ΔS and β values reported in Tables II and III. Firstly; the overall expansion will increase with the fraction of the cation sites that are occupied by Pr (Fig. 5). Secondly, for a given T, pO₂ condition (900°C, air) the fraction of the Pr that is reduced increases with increasing Pr concentration (see Table II and Fig. 6). Thirdly, there is a weak tendency that above the percolation threshold the lattice responds even stronger to the change in ionic radius than below the threshold (the experimental data lies above the Eq. 26 trend line for x = 15, 25 and 30 but below for x = 8). It should be noted, that the relative chemical expansions listed in Fig. 15 are estimates only and thus encompassed with some uncertainties. They are derived from the measured total expansion (well determined) by subtracting an estimated thermal expansion, which is calculated under assumption of a constant thermal expansion coefficient (α) (see discussion in Thermal expansion section).

Besides the uncertainty introduced by this assumption, the data for the low Pr contents may be encompassed with an uncertainty due to incomplete equilibration with the atmosphere during the temperature sweeps. For these samples full oxygen equilibration, which is limited by electronic conductivity, might not have been achieved at the given cooling rate. This would add to exaggerate the change in behavior around the percolation threshold.

Consistently, all the samples show n-type electronic conductivity under reducing conditions (<1 × 10⁻¹⁵ bar). The n-type regime is associated with partial reduction of Ce⁴⁺ as described in Eq. 4. However, for the samples with high Pr concentration, the n-type electronic conductivity is partly overshadowed by the considerable p-type electronic conductivity (Fig. 9). The "net" n-type electronic conductivity at a given pO₂ (1 × 10⁻¹⁵ bar) is thus obtained by subtracting the extrapolated p-type electronic conductivity based on the slope of the p-type at high pO₂ branch from the total electronic conductivity. It is observed that the apparent activation energy of "net" n-type electronic conductivity decreases with increasing Pr concentration. Lübke et al.¹³ found a slight decrease of the activation energy in CGO doped with 3 at.% Pr relative to that in CGO. Navarro et al.⁴⁹ also found that the 2 at.% Pr-doped CGO20 shows slightly decreased n-type conductivity relative to CGO20, in agreement with our results. In contrast to the trend observed in the pO₂ range dominated by p-type electronic conductivity, the n-type electronic conductivity under reducing conditions steadily decreases with increasing Pr concentration. The decrease of n-type electronic conductivity is more pronounced at high temperature (900°C). It is generally agreed that the n-type electronic conductivity in ceria is correlated with the electron hopping on the ceria sites (Ce³⁺/Ce⁴⁺). The mobility will thus be proportional to the concentration of Ce.²² Here, partial substitution of Pr for Ce decreases the concentration of ceria (Ce³⁺/Ce⁴⁺), accounting for the lower n-type electronic conductivity because of the decreased number of sites. Under strongly reducing conditions, Pr⁴⁺ is completely reduced to Pr³⁺ which increases the oxygen vacancy concentration owing to the summation of now two acceptor dopants ([Pr^/_Ce] and [Gd^/_Ce]). The high oxygen vacancy concentration is further unfavorable to the electron migration,^50,51 resulting from the more severe association between Ce^/_Ce and V^••_O upon a larger oxygen vacancy concentration.⁵² This effect also contributes to the reduction of the n-type conductivity in the Pr-containing samples.

The activation energy of the n-type electronic conductivity was also observed to decrease with increasing Pr concentration. The apparent activation energy is composed of two terms: where H_m is the enthalpy of electron migration and H_Ce is the standard reaction enthalpy. According to literature,⁵² the enthalpy of electron migration increases with increasing oxygen vacancy concentration in ceria. Although H_m of the materials in this work is not known, increased H_m values are expected with increasing Pr concentration. This indicates that Ce will be more readily reduced upon a higher Pr concentration. A similar trend was also observed in purely Pr-doped ceria³⁵ and purely Gd-doped ceria.⁵³ It is also found that the reducibility of Ce⁴⁺ and Pr⁴⁺ in ceria tends to be facilitated by the increasing oxygen nonstoichiometry, leading to a non-ideal reduction behavior.¹⁶ We herein attribute the decreased enthalpy of reduction of Ce to the high oxygen nonstoichiometry in heavily Pr-doped CGO.

For Pr and Gd co-doped ceria, more oxygen vacancies will form to compensate the increased concentration of negatively (compared to Ce⁴⁺) charged aliovalent dopants (Pr^/_Ce and Gd^/_Ce). In analogy with purely Pr-doped or Gd-doped ceria,³ the ionic conductivity of Pr and Gd co-doped ceria does not monotonously increase with increasing oxygen vacancy concentration as illustrated in Fig. 17. Instead, a maximum ionic conductivity occurs around an acceptor dopant concentration of 25 at.% (11.4 at% Pr³⁺, 3.6 at.%Pr⁴⁺, 10 at.% Gd and volumetric oxygen vacancy concentration equals to 1.07 × 10²² cm⁻³ ). According to Dholabhai's DFT calculations,⁵⁴ 20 at. % dopant content in ceria shows the maximum ionic conductivity for 10 at.% Pr and 10 at.% Gd co-doped ceria, in good agreement with the experimental results in this work. Another noticeable feature is that the decrease of oxide ion conductivity above 25 at.% is associated with an increase in the apparent activation energy. The large activation energy of the electrical conductivity in heavily acceptor doped ceria is observed in several studies.^3,55 It is likely due to the association between the extrinsic dopants and oxygen vacancies;⁵⁵

where Acc^/_Ce represents the acceptor dopants Gd^/_Ce and Pr^/_Ce whereby some of the oxygen vacancies are trapped by the immobile dopants. As illustrated in Fig. 17, where the data for several different acceptor doped ceria compounds are compared, the defect association results in an increased apparent activation energy for oxide ion conductivity for the heavily doped samples.

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The maximum achievable oxygen flux occurs when the oxide ion and electronic conductivity in a MIEC are both high and equal (transport number = 0.5). As shown in Fig. 18a where calculated transport numbers are shown, the transport number of CPGO approaches to 0.5 when the Pr concentration is in the range from 30 at.% to 40 at.%. Shuk et al.⁵⁸ also found nearly equal electronic and ionic conductivity for Pr_0.3Ce_0.7O_1.85-δ at 700°C, in good agreement with this work. The maximum achievable theoretical oxygen permeation flux can be calculated in terms of electronic and ionic conductivity using the Wagner's equation:

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The calculated oxygen permeation flux (here any losses at the two surfaces are neglected) follows the trend of t_e (Fig. 18b), due to the fact that the oxygen permeation flux of Pr_xGd_0.1Ce_0.9O_1.95-δ is dominantly limited by the electronic conductivity until a Pr concentration of approximately 40 at.%, where the maximum oxygen permeation flux is predicted for the here considered composition range (0–40 at.% Pr). Despite the reduced ionic conductivity when exceeding 25 at.% doping concentration, considering the bulk diffusion exclusively, the maximum oxygen flux for a 10-μm thick Pr_0.4Gd_0.1Ce_0.9O_1.95-δ based membrane may reach up to 10 Nml cm⁻² min⁻¹ at 800°C under the driving force of 0.21 bar/0.001 bar approaching the required level for being commercially interesting.⁵⁹ However, the considerable chemical expansion of CPGO40 (Fig. 7) is a severe challenge in terms of ensuring mechanical integrity and limits the applicability of the material.⁸ To reduce risks of mechanical failure originating from chemical strain, very low thickness (1–10 μm) would be needed and the oxygen pressure gradient over the materials should be carefully controlled. A discussion of the consequence of chemical strain in supported membrane architectures can be found in Ref. 60–63.