The Concentration and Diffusivity of Oxygen Interstitials in Niobia

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C: Energy Conversion and Storage; Energy and Charge Transport

The Concentration and Diffusivity of Oxygen Interstitials in Niobia-Doped Ceria Stephan Waldow, Hans Wardenga, Stefan Beschnitt, Andreas Klein, and Roger A. De Souza J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10613 • Publication Date (Web): 20 Feb 2019 Downloaded from http://pubs.acs.org on February 26, 2019

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The Journal of Physical Chemistry

The Concentration and Diusivity of Oxygen Interstitials in Niobia-Doped Ceria Stephan P. Waldow,

†

Hans Wardenga,

‡

Stefan Beschnitt,

Roger A. De Souza

†Institute ‡Dept.

†

Andreas Klein,

‡

and

∗,†

of Physical Chemistry, RWTH Aachen University, 52056 Aachen, Germany

of Materials and Earth Sciences, Surface Science Division, Technische Universität Darmstadt, 64287 Darmstadt, Germany E-mail: [email protected]

Phone: +49 241 8094739. Fax: +49 241 8092128

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We studied the behaviour of oxygen interstitials in donor-doped ceria using equilibrium conductivity measurements, conductivity relaxation experiments and (18 O2 / 16 O ) 2

isotope exchange experiments. Equilibrium conductivities and conductivity re-

laxation experiments were performed in the oxygen activity range 10−6 ≤ aO2 ≤ 10−1 at temperatures of 923 K, 973 K and 1023 K. Oxygen isotope exchanges were carried out at 673 ≤ T /K ≤ 1073 at an oxygen activity of aO2 = 0.2, and oxygen isotope proles in the solid were obtained by Secondary Ion Mass Spectrometry (SIMS). Analysis of the measured equilibrium conductivities with a defect-chemical model yielded the incorporation enthalpy and entropy of oxygen interstitials. Values of the chemical diusion coecient of oxygen (from relaxation experiments) and of the tracer diusion coecient of oxygen (from isotope exchange experiments) were converted into diusion coecients of oxygen interstitials using the defect-chemical model. Based on these data, the activation enthalpy of oxygen-interstitial migration in Ce0.99 Nb0.01 O2+δ was found to be (1.28 ± 0.13) eV. Strong enrichment of Si, Ca and Al at the surfaces of the ceramic samples prevented the determination of surface exchange coecients that refer to donor-doped ceria.

Introduction Solid oxides that exhibit high rates of oxygen transport currently attract substantial attention because of their possible application, for example, as catalysts, 14 and as electrodes or electrolytes in Solid Oxide Fuel Cells (SOFC). 5,6 Two processes often play an important role in determining the overall rate of oxygen transport: oxygen diusion in the bulk phase and the oxygen incorporation at the surface. The latter involves the conversion of oxygen molecules in the gas phase into oxide ions in a crystalline oxide, and it obviously requires the transfer of charge from the solid to oxygen moieties adsorbed on the solid's surface. 7,8 Consequently, one expects the rate of oxygen incorporation at the oxide surface to be related 2


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in some manner to the surface's workfunction (dened as the dierence in electron energy between the Fermi level and the continuum level). In a study on SnO2 , Klein et

al. 9,10 found

that the rate of oxygen incorporation can be increased by raising the Fermi level, i.e., by lowering the workfunction. And recently, De Souza showed 11 that the rate of oxygen exchange for a diverse variety of acceptor-doped oxides can be described quantitatively as a function of temperature and oxygen activity with one common expression, one that includes the concentration of electrons in the conduction band. For a semiconductor, the concentration of electrons in the conduction band is proportional to the energy dierence between the Fermi level and the conduction-band edge. Both studies thus suggest, albeit from dierent perspectives, that by varying the Fermi level one can modify the rate of oxygen exchange. Shifting the Fermi level (EF ) of a semiconducting oxide towards the conduction band can be accomplished by reducing the oxide. In order to produce substantial changes in EF , however, one has to change the oxygen activity by many orders of magnitude. This is not possible whilst keeping molecular oxygen as the ambient environment (since gas mixtures such as H2 /H2 O or CO/CO2 are required). Much larger changes can be achieved more easily through doping, with acceptor doping lowering the Fermi level and donor doping raising it. Therefore, one would expect the surface exchange rate to be enhanced through donor doping, as long as the mechanism of surface exchange remains the same. In order to test this hypothesis, we decided to use CeO2 as a model system. It crystallizes in the (rather simple) cubic uorite-structure and shows phase stability for a wide range of thermodynamic conditions. 12 Furthermore, it is possible to dope ceria with acceptors, such as Gd3+ , or donors, such as Nb5+ , whilst maintaining the cubic symmetry, and in this manner, to lower or raise the Fermi level (see Fig. 1). One known problem of ceria and similar uorite-structured oxides is the segregation of impurities such as Ca, Al or Si towards surfaces. 1318 To date, the vast majority of work concerned with oxygen exchange and diusion in CeO2 based materials has been conducted on acceptor-doped ceria, since such compositions are 3



Acceptor

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

E

F

-

E

CB

/ eV

Donor

0

0

10

10

-2

-2

10

-4

[defect]

VO

-4

10 ' CeCe

''

-6

Oi

10

10

..

' CeCe

10

-6

10

.

-8

10

10

log

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 32

-8

10

VO

-10

-10

x

10

10

VO

-12

10

..

-14

10

VO

VO

-12

10

x

.

VO

''''

VCe

-14

10

-16

10

-16

-1

10

-2

-3

10

10

-3

10

-2

10

10

-1

10

log 10 [acceptor]

log10 [donor]

Figure 1: Calculated variation in the Fermi level EF (relative to the energy of the conductionband minimum, ECB ) and calculated variation in point-defect site fractions, [defect], as a function of donor or acceptor doping in CeO2 at T = 873 K and aO2 = 0.21. Values for the enthalpies and entropies of defect formation were taken from DFT calculations. 19,20 NB: Only data for defects with site fractions above 10−16 are shown. excellent oxide-ion conductors, with negligible electronic conductivity over a wide range of conditions. 2127 There are also many experimental and computational studies on oxide-ion conductivity in ceria-based materials. 2838 Far fewer studies have been carried out on donor-doped ceria and these studies have focussed on the electrical conductivity and the determination of the point-defect structure. 3942 Only one study, by Stratton and Tuller, 43 has examined oxygen diusion in donor-doped ceria. It is worth noting that (see Fig. 1), whereas acceptor doping (for example, with Gd3+ ) leads primarily to the formation of doubly charged oxygen vacancies, that is, the compensation mechanism is ionic; donor doping (for example, with Nb5+ ) leads primarily to the formation 4


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of electrons, that is, the compensation is electronic. Oxygen diusion in donor-doped CeO2 takes place, as we see in Fig. 1, by oxygen interstitials rather than oxygen vacancies. In this study, we fabricated dense ceramic samples of niobia-doped ceria (with the dopant level of 1% Nb); we determined defect-chemical parameters for this material; and we investigated oxygen transport in this material using two complementary methods, conductivity relaxation and isotope exchange, in order to probe the diusion of oxygen interstitials and to examine whether the oxygen exchange rate can be modied by shifting the Fermi level.

Experimental Samples were prepared via a solid-state route: CeO2 [Alfa Aesar, 99.99%] and Nb2 O5 [Sigma - Aldrich, 99.99%] were weighted in the desired molar cation ratio of 1% Nb. Previous experiments with this ceria powder showed that this level of purity was sucient to prevent substantial segregation of impurities to the surface (as we will see later, this is not a sucient condition). The powder then was suspended in deionized water, ball milled with ZrO2 balls (0.5 cm diameter) for 4 hours, and afterwards dried. The ne yellowish powder was ground in a pestle and mortar, and then compacted into pellets with a diameter of 15 mm in a stainless steel die. The pellets were subjected rst to a uniaxial pressure (142 MPa) and then an isostatic pressure (320 MPa), after which they were sintered in air at 1673 K for 36 hours with a heating rate of 250 K per hour and a cooling rate of 200 K per hour. SiC paper (Minimet 1000, Buehler, Illinois, USA) was used to polish the samples, starting with a 180 grid and going up to a 4000 grit. In a second step the samples were polished with diamond-oil suspensions (15 µm to 3 µm) on 4000 grit paper. In the last step the samples were cut into 5 mm × 5 mm big pieces with a thickness of 0.5 mm to t the geometry of the sample holder. RMS surface roughness, according to interference micrographs, was Rq = 12 nm for an area of 1.2 mm × 0.92 mm. XRD diraction pattern conrmed only reections attributable to cubic CeO2 . No indications of a secondary phase or separation of niobia were

5



observed. The lattice constant was calculated to be 5.409 Å by applying Bragg's law, to the indexed reexes of the XRD measurements. The radii of the occurring cations are Ce4+ = 0.97 Å, Ce3+ = 1.143 Å and Nb5+ = 0.74 Å. 44 For doping with 1% Nb and compensation through Ce3+ electron polarons this means one would expect an overall shrinking of the lattice parameter, as the dierence between the radii of Nb5+ and Ce4+ is bigger than between the radii of Ce3+ and Ce4+ . Indeed, the lattice constant we obtain for the niobia-doped ceria is smaller than that of pure ceria (5.411 Å). Scanning electron micrographs (not shown) indicate dense, single phase ceramics (i.e. no visible porosity or second phases) with a grain size of ca. 8 microns. Only samples with a relative density of at least 99% were used in the conductivity and diusion studies. Conductivity measurements were done by means of a Van-der-Pauw set-up. 45,46 Samples were contacted with 4 Pt-electrodes sputtered on the edges of the sample. Conductivity measurements were performed at oxygen activities ranging from 10−6 to 10−1 and at temperatures T = 923, 973 and 1023 K. The change in oxygen activity in the measurement chamber took between 5 min to 10 min. The relaxation experiments took at least one order of magnitude longer. We are aware that the Van-der-Pauw geometry is, strictly speaking, not suitable for relaxation experiments, since the principle requires the material to be of uniform conductivity (and thickness). The changes in conductivity in our case, however, are suciently small that the relaxation is considered to be followed accurately. SIMS tracer diusion experiments 4749 consist of two steps: 1) The actual isotope exchange experiment; 2) The analysis by means of SIMS. In the rst step the chamber of the exchange rig containing the samples was evacuated until the pressure was below 10−7 mbar for an extended time. Oxygen of normal isotope abundance was then introduced into the chamber, and the sample was equilibrated in this atmosphere at the temperature of interest for ten times the exchange time. The sample was then quenched to room temperature. Next molecular oxygen enriched to 50%

18

O, was introduced into the chamber to the same

oxygen activity as previously. The sample was again rapidly heated to the temperature of 6


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interest, and held at that temperature for 20 to 30 minutes. In total, isotope anneal experiments were carried out at various temperatures ranging from 673 K to 1073 K. In the second step, diusion proles were obtained by SIMS depth proling, on a ToF-SIMS IV machine with a ToF-SIMS V analyser (IONTOF GmbH, Münster, Germany). Sequential sputtering and analyzing was used with a cycle time of 55 µs. A 25 kV Ga+ ion beam with a raster of 100 µm × 100 µm, was used to generate secondary ions for analysis, while a 2 kV Cs+ ion beam, with a raster of 400 µm × 400 µm, was used to sputter etch the sample. The Ga+ source was operated in bunched mode. 50 The excess charge on the sample surface was compensated with a high-current, low-energy beam of electrons. The crater depths were determined post-analysis with an interference microscope of the type Veeco NT 1100.

Results 25

10

10

10

2

1

0

10

-1

10

10

-2

aO

2

/ mS cm

-1

20

tot

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-3

15 10

10

10

10 0.0

0.5

1.0

10

-6

-4

-5

-6

1.5

· t / s

Figure 2: Conductivity measured for a Ce0.99 Nb0.01 O2+δ ceramic against time at T = 873 K for a range of oxygen activities. In Fig. 2 we show the variation in the conductivity as the oxygen activity is changed as a function of time at T = 873 K. One sees that an increase in the oxygen activity leads to a decrease in the equilibrium conductivity. This is consistent with the conductivity being due to electrons. In addition, one can extract the chemical diusion coecient Dδ and the chemical surface exchange coecient k δ from the transients. 5154 In the experimental cell only 7



the four small areas and one of the two large areas of the sample are in direct contact with the gas phase. Diusion through the small areas is disregarded and only oxygen incorporation through the large area is considered, which means that the diusion problem reduces to one dimension. As a consequence, the solution to the diusion equation in terms of the normalized conductivity (σnorm =

σ(t)−σ(0) ) σ(∞)−σ(0)

σnorm = 1 −

∞ ∑ n=1

is given by 55 2D t βn δ

2L2a e− a2 . βn2 (βn2 + L2a + La )

(1)

a denotes the sample's thickness. La is dened as La = a · Dkδδ and β as the positive solutions to La = β · tan(β). Fig. 3 shows exemplarily a change in normalised conductivity and the tted curve to Eq. (1).

Exp Fit 1.0

norm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 32

0.5

0.0

0

2

4

10

-4

6

8

· t / s

Figure 3: Normalised total conductivity and corresponding t of Eq. (1) for a jump in oxygen activity from 10−4 to 10−5 at T = 1023 K.

8


Page 9 of 32

= (1.1 ± 0.2) eV

10

-6

H

= (0.6 ± 0.4) eV

Hk

= (1.8 ± 0.1) eV

= (1.2 ± 0.3) eV 10

10

Hk -7

k

/ m

H

-1

-11

/ m s

10

2

s

-1

10

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-8

-12

0.95

1.00

1.05

10

3

·

T

-1

/ K

1.10

0.95

-1

1.00

1.05

10

(a)

3

·

T

-1

/ K

1.10

-1

(b)

Figure 4: Values for Dδ and k δ obtained for ceramics of Ce0.99 Nb0.01 O2+δ plotted against inverse temperature. Red squares refer to jumps in the oxygen activity from 0.01 to 0.1; blue circles to jumps from 0.001 to 0.01. Errors refer to ± 2 standard deviations. Fig. 4 (a) and (b) shows the chemical diusion coecients Dδ and the chemical oxygen exchange coecients k δ , for two dierent jumps in oxygen activity. Dδ increases with increasing temperatures and with increasing oxygen activity; the activation enthalpies are relatively close at (1.1 ± 0.2) eV and (1.2 ± 0.3) eV. Values of k δ also increase with T but there is no overall trend in the aO2 dependence. This unusual behaviour is attributed (see below) to the presence of substantial amounts of impurity phases at the samples' surfaces. The activation enthalpies for k δ are signicantly dierent at (0.6 ± 0.4) eV and (1.8 ± 0.1) eV. Isotope diusion proles obtained for donor-doped ceria ceramics all showed two features. One such prole is plotted in Fig. 5. The appearance of two features suggests that the tracer diusion coecient of oxygen is not constant throughout the sample. It may be that there is a faster parallel path, that is, D∗ is much faster along grain boundaries; 56,57 or it may be that there are two diusion processes in series, with D∗ being much lower in the surface region, either from a surface space-charge zone, 50,5862 a polishing-induced damaged zone, 63 or from a silica-based phase. 18 The detailed analysis revealed that fast grain-boundary diusion, a surface-damaged zone and a surface space-charge zone could be ruled out, because they re9



quire physically unreasonable parameters to describe the isotope proles. Indeed, the best description, with physically reasonable parameters, was obtained with the model of Müller and De Souza 18 for a silica-based phase. In this model an arbitrary low diusion coecient describes diusion through the impurity phase(s) at the sample surface, followed by a region in which the diusion coecient varies strongly, taking account of the inhomogeneous distribution of the impurity phases; the last part refers to the bulk diusion coecient. The bulk tracer diusion coecients and the surface tracer exchange coecients were obtained by varying the parameters until good visual agreement between the measured isotope prole and the calculated one was reached. Describing the prole using a constant D∗ or without the arbitrary diusion coecient for the rst feature of the prole failed for all measured proles. In Fig. 6 we show ToF-SIMS data for the same ceramic sample, focussing on Nb-, Si-, Aland Ca-containing secondary ions. Chemical analyses revealed that the impurities found, originated from the Nb2 O5 powder used. Evidently, even though the dopant concentration was only 1% and the purity of the used Nb2 O5 powder was 99.99%, this was enough to signicantly contaminate the samples. Converting, for instance, the Si− intensity into an Si concentration is not possible at present, since it requires knowledge of the actual phases present at the surface (it is unclear if it is uorite-structured CeO2 ) and of the relative sensitivity factors for the Si− secondary ion in those matrices, because of the SIMS matrix eect. 57 XPS measurements (not shown here) revealed a surface concentration of Si of roughly 10%. Most importantly in Fig. 6, the impurity signals from Si− , CaO− and AlO− are seen to extend over the same length scale as the variation in D∗ needed to describe the isotope prole (see Fig. 5). This strongly suggests that the observed spatial variation in D∗ is indeed due to increased the appearance of a (Ca,Al,Si)Ox phase or phases at the samples' surfaces, which were already reported for ZrO2 , 1417 HfO2 18 and CeO2 . 13 It is also worth noting that the impurity distribution is not laterally homogeneous, but appears in secondary ion images (see Fig. 6 b) to be conned to particles / islands at the surface. As 10


Page 10 of 32

Page 11 of 32

a consequence, values determined for k ∗ (and k δ ) are not characteristic of Ce0.99 Nb0.01 O2+δ because of the presence of the impurity phase(s), but values determined for D∗ (and Dδ ) are characteristic of bulk Ce0.99 Nb0.01 O2+δ since they refer to the region of constant and much lower impurity composition away from the surface. Tracer diusion coecients and surface exchange coecients obtained as a function of temperature are plotted in Fig. 7(a) and (b). For the reasons given above, the data obtained for k ∗ will not be considered further. The data obtained for D∗ shows a surprisingly large amount of scatter, and the reason for this is at present unclear. As a result, the activation enthalpy obtained for tracer diusion is characterised by a large error, ∆HD∗ = (0.73 ± 0.50) eV.

10

-1

-16

2

/ m

r

*

s

-1

10

n -2

*

10

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 10

-17

-3

0.0

0.5

1.0

1.5

2.0

x/

2.5

3.0

3.5

4.0

m

Figure 5: Normalised 18 O isotope fraction n∗r versus prole depth x obtained by ToF-SIMS analysis of a ceramic of Ce0.99 Nb0.01 O2+δ . Isotope anneal performed at T = 973 K and aO2 = 0.2. The black squares refer to the measured 18 O isotope fraction and the red line refers to the numerical description of the prole; the blue line gives the variation in the tracer diusion coecient as a function of the depth that is required to describe the isotope prole. The experiment was conducted at 973 K and at an oxygen activity of 0.20.

11



2

10

1

normalized intensity

10

0

10

NbO

-1

-

AlO

-

10

-2

10

Si

-

-3

10

-4

10

CaO

-

-5

10

0.0

0.2

0.4

0.6

x/

0.8

1.0

m

(a)

(b)

Figure 6: (a) Secondary ion intensities normalized to the CeO− intensity obtained from a ToF-SIMS depth prole of a Ce0.99 Nb0.01 O2+δ ceramic (same sample as data in Fig. 5). An enrichment of Al, Ca and Si secondary ions was observed. The impurity proles extend roughly 350 nm from the surface into the sample. The dopant concentration prole is nearly constant, with a small deviation near the surface. (b) Inhomogeneous Si− -Signal distribution over the measured area.

10

10

-14

10

-1

k/ms

s 2

-9

10

-10

*

/ m

-16

O

*

10

-8

-15

-1

10

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 32

HD 10

10 *

Hk

-17

*

10

10

-11

= (0.73 ± 0.50) eV

-12

= (0.76 ± 0.52) eV

-18

10 0.8

1.0

1.2

10

3

·

T

-1

1.4

/ K

1.6

-13

0.8

-1

1.0

1.2

10

(a)

3

·

T

-1

1.4

/ K

1.6

-1

(b)

Figure 7: Values of D∗ and k ∗ obtained for ceramics of Ce0.99 Nb0.01 O2+δ plotted against inverse temperature. Errors refer to +/- 2 standard deviations

12


Page 13 of 32

Discussion Starting with the equilibrium case we compare in Fig. 8 our measured conductivities with values reported in the literature for comparable donor concentrations. 3941 It should be noted that the oxygen activities are not the same in all the studies; literature studies employed activities between 0.2 to 10−4 , while our values were taken at aO2 = 10−6 . Our data compares well with the reported values and lie between the values reported by De Guire

et al. 41 for

1.35% Ta5+ and 0.8% Nb5+ doped-ceria. Compared with results by Naik and Tien 40 and Yashiro

et al. 39 for a Nb5+ concentration of 0.8% in each and measurements at aO2 of 10−4 ,

our values are slightly higher. Overall the agreement is satisfactory. 7.0

D

B

6.0

B

5.5

A

·

T

/ K S cm

-1

)

6.5

C

5.0

ln(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4.5

4.0

0.7

0.8

0.9

10

3

1.0

·

T

-1

/ K

1.1

1.2

-1

Figure 8: Total conductivity of donor-doped CeO2 plotted against inverse temperature: (A) 1% Nb, this study; (B) De Guire et al., 41 (C) Yashiro et al 39 and (D) Naik and Tien. 40 On the basis of previous defect chemical studies of ceria 3941 and on the basis of the DFT calculations (see Fig. 1), we assume reasonably that the niobium donors are compensated primarily by electrons, localised as small polarons on cerium cations but also by some oxygen interstitials (The concentration of cerium vacancies is neglected here as it is orders of magnitude lower). That is, the electroneutrality condition is given by

[Nb•Ce ] = 2[O′′i ] + [Ce′Ce ]. 13


(2)


Page 14 of 32

The site fractions of the two negative defects may change with T and aO2 because of the incorporation of gaseous oxygen into the sample as oxygen interstitials with concomitant annihilation of electron polarons. This occurs according to

1 O + 2 Ce′Ce ⇀ ↽ Oi′′ + 2Ce× Ce . 2 2

(3)

With increasing oxygen activity this equilibrium shifts to the right, decreasing the concentration of electrons, and thus the conductivity, too. The equilibrium constant of the oxygen interstitial incorporation reaction is

Kinc = e

−

∆Hinc kB T

·e

∆Sinc kB

,

(4)

where ∆Hinc and ∆Sinc are the enthalpy and entropy of reaction (3). kB is the Boltzmann constant. The incorporation reaction, Eq. (3), can be written as a combination of two more fundamental reactions: Lattice reduction,

1 × ⇀ •• ′ 2Ce× Ce + OO ↽ VO + 2 CeCe + O2 ; 2

(5)

′′ OO× ⇀ ↽ V•• O + Oi .

(6)

and anti-Frenkel disorder,

The enthalpies and entropies of these reactions are thus related to those of the incorporation reaction through: ∆Hinc = ∆HaF − ∆Hred and ∆Sinc = ∆SaF − ∆Sred . The measured conductivity is the sum of the partial conductivities of all charged species. We assume that the contribution of ions to the total conductivity is negligible. We will see later that this assumption is justied. We can, therefore, write the conductivity as

σ = [Ce′Ce ][NCat ]µe q.

14


(7)

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The rst two terms on the RHS give the concentration of electron polarons, in terms of the site fraction of polarons and the overall volumetric density of cation sites. µe is the electron mobility and q is the electron charge. The electron mobility is described in terms of a pre-exponential factor B and the activation enthalpy for polaron hopping ∆Hhop . (

µe =

B ·e T

−

∆Hhop kB T

)

(8)

In Fig. 9 the equilibrium conductivities obtained at dierent temperatures are plotted against the oxygen activity. As starting values for ∆Hinc and ∆Sinc we combined the reduction and anti-Frenkel enthalpies and entropies derived by Zacherle et

al. 19 and Grieshammer

et al. 20 from DFT calculations. As starting values for the mobility we took the values for the B and ∆Hhop reported by Tuller and Nowick. 64 Theses parameters were varied until the predicted conductivities could describe the experimental data well. The nal values are given in table 1. Our value of ∆Hhop is slightly lower than Tuller and Nowick's result of 0.4 eV; slightly higher than values by Yashiro

et al. 39 and Naik and Tien 40 who both obtained ∆Hhop = 0.29 eV;

and ts rather well to results of Guire

et al. 41 who obtained ∆Hhop = 0.34 eV. The obtained

value for ∆Hinc is much higher than the -(0.35 ± 0.1) eV by Göbel et al. for 2% Nb-doped ceria thin lms. Compared to values reported by Stratton and Tuller for uranium-doped ceria our results for the incorporation enthalpy are still in agreement if the errors are considered. Our derived value for the entropy of incorporation is 1 eV higher than the predicted value using DFT, which seems consistent with the underestimation of defect formation enthalpies using DFT, as DFT results for ∆Hred are 0.5 eV lower than experimental values.

We compare our derived electron mobility with values reported by Naik and Tien 40 and by Yashiro

et al., 39 shown in Fig. 10. As Naik and Tien assumed a constant value for µe for

15



Table 1: Fit parameters. Reference values for

∆Hinc

and

∆Sinc

were obtained by

subtracting values for the Anti-Frenkel defects from the values for the reduction.

∆Hinc

=

∆HaF

-

∆Hred .

Parameters B / K cm2 V−1 s−1 ∆Hhop / eV ∆Hinc /eV ∆Sinc / kb [Nb•Ce ] µe / cm2 V−1 s−1

This study 419.0 ± 5.0 0.359 ± 0.002 -1.040 ± 0.005 -8.6 ± 0.2 0.010 ± 0.002 0.017(1273 K)

Literature 400.0 0.4; 0.29; 0.29; 0.34 -0.07; (-0.35 ± 0.1); (-0.76 ± 0.5) -7.5 0.0081(1273 K); 0.009(1173 K)

Ref. 64 64; 39; 40; 41 19; 65; 43 20 64; 39

0.28

0.26

0.24

/ mS cm

-1

0.22

tot

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 32

0.2

0.18

0.16

923 K 973 K 0.14

1023 K

-6

-4

-2

aO

log(

0

) 2

Figure 9: Comparison of total conductivities of Ce0.99 Nb0.01 O2+δ as a function of oxygen activity for three dierent temperatures. Symbols refer to experimental data; lines are predicted from a defect-chemical model with the parameters listed in Table 1. dierent temperatures and De Guire

et al. 41 did not give values, we used our defect model

to calculate the charge carrier concentration and with this the electron mobility. Our values compare well to the results by Yashiro

et al. 39 If we assume our defect-chemical model, our

results are in good agreement to De Guire

et al. 41 and Naik and Tien 40 as well.

16


10

-2

/ cm

2

V

-1

s

D

B B

e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-1

Page 17 of 32

C

A

10

-3

0.6

0.7

0.8

0.9

10

3

·

T

1.0 -1

/ K

1.1

1.2

-1

Figure 10: Mobility of electron polarons in donor-doped CeO2 as a function of inverse temperature: (A) This study; (B) De Guire et al., 41 (C) Yashiro et al 39 and (D)Naik and Tien. 40 Since we now can predict the point defect concentrations as a function of T and aO2 , we can convert our chemical and tracer diusion coecients (Dδ and D∗ , respectively) into diusion coecients of oxygen interstitials, Di . Using Eq. (9) we can calculate Di from our tracer diusion coecients. ∗ Di = DO ·

[OxO ] [O′′i ] · fi∗

(9)

The correlation factor for non-colinear interstitialcy migration on the anion lattice of a uorite was taken from Murch 66 with a value of fi∗ = 0.9855. To convert the chemical diusion coecients into diusion coecients for oxygen interstitials, we use

Di =

Dδ · [OxO ] ωO · [O′′i ]

(10)

where ωO is the thermodynamic factor of oxygen. This can be calculated from our defect model. Fig. 11 shows ωO as a function of the oxygen activity for three dierent temperatures. With ωO known, Eq. (10) can be used to calculate Di .

17



4.5

1023 K 4.0

3.5

· 10

-4

3.0

973K

2.5

O

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 32

2.0

923 K

1.5

1.0

0.5 -6

-4

-2

aO

log(

0

) 2

Figure 11: ωO is plotted against the oxygen activity for T = 923, 973, and 1023 K. The results of the analyses are plotted in Fig. 12(a) with Di against inverse temperature. There is acceptable agreement between data obtained from the two diusion experiments (chemical and tracer), with the variation at each temperature being less than one order of magnitude. The activation enthalpy of oxygen-interstitial migration is found to be ∆Hmig,i = (1.28 ± 0.13) eV. In Fig. 12(b) we compare our data with literature data 43,67,68 for Di reported for donordoped ceria and other AO2 uorite-oxides. Using the defect-chemical data of Stratton and Tuller 43 we converted their tracer diusion coecients into oxygen-interstitial diusion coecients. Their Di values for 0.1%U:CeO2 43 are in very good agreement with our results for 1%Nb:CeO2 ; their activation enthalpy of migration, ∆Hmig,i = (1.23 ± 0.03) eV, is close, too. Results for 5%U:CeO2 reported by Tuller and Stratton 43 show signicant deviations, both in terms of the absolute values for Di and the dependence on temperature, ∆Hmig,i = (0.89 ± 0.07) eV; this may indicate that defectdefect interactions are important. We also note that

17

O-NMR studies by Heinzmann

et al. 69 on Ta-doped CeO2 yielded a migration

enthalpy of (0.25 ± 0.05) eV for oxygen interstitials, but as noted by the authors the method is known to underestimate the real value by a factor of 2 to 4. 70,71 Considering all the results for AO2 uorite-structured oxides together, one nds that isothermal values of, Di vary by 18


Page 19 of 32

1-2 orders of magnitude; the activation enthalpies of migration are also relatively similar. For PuO2 Kato

et al. 68 reported a value of ∆Hmig,i = 1.04 eV, while for UO2 Moore et al. 67

determined ∆Hmig,i = 1.11 eV. Lastly, we return to the question of the ionic contribution to the total condutivity of Ce0.99 Nb0.01 O2+δ . Using the NernstEinstein relation, 72

σion =

ci q 2 Di , kB T

(11)

we calculated a ionic conductivity of 2.6 · 10−4 mS cm−1 at 923 K, which rises to 8.2 · 10−4 mS cm−1 at 1023 K. In all cases the ionic conductivity is orders of magnitude smaller than the experimentally measured conductivities (see Fig. 9), which lie in the range of 1.4 · 10−1 mS cm−1 at 923 K to 2.6 · 10−1 mS cm−1 at 1023 K. Our assumption of treating the measured conductivities as solely electronic and disregarding the ionic part, is indeed justied, at least for the temperatures at which our measurements were conducted.

10

10

10

-11

10 10

-9

-10

-11

-12

A

-12

D

-1

s 2

/ m

-13

D

i

10

i

/ m

2

s

-1

10

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

10

10

-14

10

H

A

-15

= (1.28 ± 0.13) eV

mig, i

10

B 10

C 10

10

-16

10 0.9

1.0

1.1

1.2

10

3

T

-1

1.3

/ K

1.4

1.5

-13

-14

B

E

-15

C -16

-17

-18

0.6

-1

0.8

1.0

10

(a)

1.2 3

T

-1

/ K

1.4

1.6

-1

(b)

Figure 12: (a) Diusion coecients of oxygen interstitials in Ce0.99 Nb0.01 O2+δ derived from chemical diusion coecients (A, aO2 = 0.1; B, aO2 = 0.01) or tracer diusion coecients (C, aO2 = 0.2); (b) Comparison of diusivities of oxygen interstitials in AO2 uorite compounds: A (1% Nb:CeO2, this study), B (0.1% U:CeO2 , Ref. 43); C (5% U:CeO2 , Ref. 43); D (UO(2) , Ref. 67); E (PuO2 , Ref. 68)

19



O2

A

Table 2 is a summary of the enthalpies of migration, reduction and anti-Frenkel formation in various AO2 -uorite-type oxides. 19,24,43,65,6769,7388 First, we note that the activation enthalpies of oxygen-interstitial migration for all these materials, 43,6769,75,84,85,87 with the exception of the calculated value for m-HfO2 , are generally around 1 eV. Second, we nd that the reduction enthalpy of weakly donor-doped and nominally undoped ceria lies between 4.4 and 4.7 eV. 19,24,43,73,74 Third, taking a value of ∆Hred = 4.67 eV by Tuller and Nowick, 73 we obtain a value of ∆HaF = 3.63 eV from our value of ∆Hinc = −1.040 eV. Compared with results for donor-doped and undoped ceria, which lie in the range of 3.3 to 4.5 eV, 19,24,43,69,7376 our result seems reasonable. Our result compares well with the results of Stratton and Tuller 43 for weakly donor doped ceria, as well as to results of Charoonsuk

et al. 76 For other AO2 -uorite-type oxides the reported values show a strong scattering with values in the range of 3 to 6 eV, but values in the range of 3 to 4.5 eV seem most reasonable.

20


Page 20 of 32

Page 21 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: Comparison of defect enthalpies for various lated with

∆Hred

AO2 -type

oxides. a calcu-

= 4.67 eV from Tuller and Nowick; 73 c obtained from atomistic

simulations.

Material CeO2

UO2

PuO2 m-HfO2 ThO2 NpO2

Dopant 1% Nb 2% Nb 0.1% U 1% U 5% U undoped undoped undoped undoped undoped 0 to 3% Ta 0.4 to 2.5%Ta undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped undoped

∆Hred / eV ∆HaF / eV ∆Hmig,i / eV Ref. 3.63a 1.28± 0.13 This study a 4.32 65 4.34 3.66 1.23 ± 0.03 43 5.21 4.45 43 5.19 4.47 0.89 ± 0.07 43 4.67 73 4.71 74 4.42 24 c c 4.21 4.14 19 0.80c 75 3.27-4.24 76 0.25 ± 0.05 69 4.76c 77 4.4 78 5.84 1.11 67 4.23c 79 3.3 80 4.6 ± 0.5 81 3.5 ± 0.5 82,83 c 1.00 84 0.88c 85 0.75 ± 0.08 85 4.58c 1.18c 68 1.04 68 2.9 80 5.47c 86 c 1.8 87 4.2 80 3.1 80 c 4.5 88 5.8c 88

Conclusions 1. We obtained comparable values for the diusivity of oxygen interstitials, Di , using conductivity relaxation experiments and SIMS isotope exchange experiments. We de-

21



termined the activation enthalpy of oxygen-interstitial migration to be ∆Hmig,i = (1.28

± 0.13) eV. Both the absolute values for Di as well as ∆Hmig,i are in good agreement with values reported in the literature for CeO2 and other AO2 oxides. 2. We applied a defect-chemical model to our equilibrium conductivity data and obtained a value of ∆Hinc = −1.040 eV for incorporation of oxygen interstitials into ceria. Taking a literature value of 4.67 eV for the enthalpy of reduction we obtained a value of ∆HaF = 3.63 eV for the enthalpy of anti-Frenkel formation, which is again in good agreement with values for ceria and other comparable AO2 oxides. 3. Surface contamination with impurity phase(s) prevented us from examining the eect of donor doping, and thus the eect of a shift in the Fermi level, on the surface exchange coecient. Future studies of the surface exchange coecient k would require control of the surface contamination, so that it is low and comparable across various acceptorand donor-doped samples.

Acknowledgement The authors acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) from projects SO499/7-1 and KL1225/7-1.

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The Concentration and Diffusivity of Oxygen Interstitials in Niobia

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