2D diffusion of oxygen in Ln10Mo2O21 (Ln = Nd, Ho) oxides

Solid State Ionics 346 (2020) 115229

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2D diffusion of oxygen in Ln10Mo2O21 (Ln = Nd, Ho) oxides a,b,⁎

c

Vladislav Sadykov , Anna Shlyakhtina , Ekaterina Sadovskaya Valeriy Skazkab,d, Vladimir Goncharova

a,b

a

, Nikita Eremeev ,

T

a

Federal Research Center Boreskov Institute of Catalysis, Akad. Lavrentieva ave. 5, Novosibirsk 630090, Russia Novosibirsk State University, Pirogova str. 2, Novosibirsk 630090, Russia Semenov Institute of Chemical Physics RAS, Kosygina st. 4, Moscow 119991, Russia d Sobolev Institute of Mathematics SB RAS, Acad. Koptyug ave. 4, Novosibirsk 630090, Russia b c

ARTICLE INFO

ABSTRACT

Classification codes: 6610M 6170N 6630D 8220P 8220W 8610BKeywords: Ln molybdates Oxygen diffusion Isotope exchange of oxygen Mathematical modeling 2D diffusion

Ln molybdates are promising materials for hydrogen/oxygen separation membranes. This work aims at elucidating features of oxygen transport in Ln10Mo2O21 (Ln = Nd, Ho) oxides using novel 2D diffusion models. Nd10Mo2O21 and Ho10Mo2O21 were synthesized by the mechanical activation followed by sintering in the 1600–1650 °C temperature range and characterized by XRD as a complex rhombohedral phase and fluorite one, respectively. Oxygen transport features were studied by the oxygen isotope heteroexchange with C18O2 in a flow reactor using temperature-programmed and isothermal modes. According to numerical analysis, isotope exchange in Ln10Mo2O21 cannot be described by a single diffusion coefficient, which is explained by nonuniformity of the oxygen diffusion pathways. The mathematical model including equations for a faster diffusion along grain boundaries and a slower diffusion within grain bulk (2D diffusion model) gives the best fit. The same accuracy was achieved using the model including 2D diffusion and exchange between grain bulk oxygen forms with different M-O bonds strength. The values of oxygen tracer diffusion coefficient are ~10−7–10−6 cm2/s and ~10−11–10−8 cm2/s at 700 °C along grain boundaries and within grain bulk, respectively. Hence, new 2D models were developed to describe oxygen diffusion in polycrystalline oxides. A fast oxygen diffusion demonstrated for Ln10Mo2O21 oxides makes them promising for design of hydrogen/oxygen separation membranes.

1. Introduction Materials with a high ionic (oxide ionic, protonic) and mixed ionicelectronic conductivity are used in catalytic reactors for biofuels transformation based on hydrogen/oxygen separation membranes [1–6]. Such devices contain a ceramic membrane with selective permeability of oxygen or/ and hydrogen, or asymmetric supported membrane with permselective (functional) layers built from such materials. Membrane allows to oxidize biofuels with oxygen separated from air to obtain syngas or separate pure hydrogen as product of catalytic reactions of biofuels reforming. A high oxygen mobility is required for these materials. For oxygen separation membranes, a high oxygen mobility along with the electronic conductivity allow to reach a high permeation flux required for efficient catalytic performance of the membrane reactor [5,6]. For hydrogen separation membranes, the oxygen mobility is recommended to be high due to two reasons: at first, the vehicle mechanism involving structure hydroxyls migration is

one of the ways of protonic transport [7]; at second, additional hydrogen yield can be achieved by water splitting reaction at the membrane purge side followed by the oxide ion migration across the membrane [8]. Thus, oxygen transport properties are to be studied for both oxygen and hydrogen separation membranes' materials. Studies of a new class of R10Mo2O21 compounds as potential mixed oxide-ionic – electronic (in dry atmosphere) and protonic – electronic (in wet atmosphere) conductors have been started recently, however, for rare-earth tungstates with the same composition R10W2O21 (R = La, Ho, Y, Er) the structural data have been already reported [9–13]. Moreover, the total conductivity of La10W2O21 measured in air is quite high (~5 × 10−5 S/cm at 600 °C) [9]. The existence of Ln10Mo2O21 compounds (Ln = La, Nd, Gd) was mentioned in works [9,14,15]. Similar to earlier known class of Ln6WO12-based solid solutions, the most promising mixed ionic-electronic conductors are light rare-earth elements' molybdates Ln10Mo2O21 (Ln = La, Nd, Sm). Differing from

Abbreviations: IIE, isothermal isotope exchange; SEM, scanning electron microscopy; SSITKA, steady-state isotope transient kinetic analysis; TPIE, temperature programmed isotope exchange; XRD, X-ray diffraction ⁎ Corresponding author at: Federal Research Center Boreskov Institute of Catalysis, Akad. Lavrentieva ave. 5, Novosibirsk 630090, Russia. E-mail address: [email protected] (V. Sadykov). https://doi.org/10.1016/j.ssi.2020.115229 Received 17 July 2019; Received in revised form 30 December 2019; Accepted 16 January 2020 Available online 21 January 2020 0167-2738/ © 2020 Elsevier B.V. All rights reserved.

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Nomenclature

L2 α αBULK αS

oxygen tracer diffusion coefficient DO Dgb along grain boundaries within grain bulk Dbulk Dfast, Dslow fast and slow diffusion channels, respectively f16-18 molecular fraction of C16O18O L1 the characteristic particle size

ατ

La6-xWO12-δ (x = 0–0.8) and La10W2O21 tungstates [9], Ln10Mo2O21 (Ln = La, Nd) molybdates crystallize not in the structural class of doubled fluorite, but have a more complex rhombohedral structure being the most stable in reducing conditions [16–18]. Hence, it is interesting to study oxygen diffusion processes for rhombohedral Nd10Mo2O21 and to compare with the same processes for the hightemperature Ho10Mo2O21 polymorph with the fluorite structure [18]. It is to be noted that Ln10Mo2O21 compounds contain 3.5 mol% more MoO3 compared to Ln6MoO12 [17,19]. Light and intermediate rare-earth elements' molybdates Ln6MoO12 (Ln = La, Gd, Dy) contained Ln2O3 admixture even after high-temperature calcination at 1600 °C [20] differing from Ln10Mo2O21 (Ln = La, Gd, Dy) being single-phase compounds [17–19]. Nd molybdate Nd6MoO12 contained Nd2O3 admixture after sintering at 1400 °C, while it became single-phased after sintering at 1600 °C [17]. Thus, single-phase compositions' synthesis temperatures are decreased for Ln10Mo2O21 compared to those for Ln6MoO12. In Ln2O3 – MoO3 phase diagrams light rare-earth elements' molybdates Ln10Mo2O21 (Ln = La, Nd), are single-phase materials at least in studied temperature range 1600–1650 °C differing from La6MoO12 [20]. Note, also, that heavy rareearth elements' molybdates Ln6MoO12 (Ln = Er-Yb) are single-phased at the same temperatures [20,21]. This work aims at elucidating features of oxygen transport in Ln10Mo2O21 (Ln = Nd, Ho) using novel 2D diffusion model in relationship with structural and textural characteristics. Two-dimensional (2D) diffusion model used in the current work includes a fast diffusion along grain boundaries with subsequent slow diffusion within grain bulk [22,23]. This means that additional “dimension” with related equations for diffusion should be added. The first studies devoted to fast diffusion along grain boundary were performed by Fisher [24], Whipple [25], Le Claire [26] and other authors [27–29]. The model proposed by Fisher and LeClaire includes equations for diffusion in two half-spaces with diffusion coefficient D and thin slab between them with diffusion coefficient D′ > D. Fast ionic transport along grain boundaries is also a feature of perovskites [30–34], fluorites [35–38] and nanocomposites [30–35]. In this work, 2D diffusion model was used for analysis of oxygen isotope exchange kinetics in the gas – solid system (Ln2Mo2O21 oxides). The oxides were studied by isotope exchange of oxygen with C18O2 in a flow reactor. Differing from isotope exchange with 18O2, this technique allows more accurate calculation of oxygen tracer diffusion coefficient since the surface exchange with CO2 proceeds 2–5 orders of magnitude faster compared to the surface exchange with O2 [39–43] solving issues of surface exchange limitations [43]. The mathematical model of 2D diffusion used in this work is novel in application for oxygen isotope exchange kinetics analysis in the gas phase – solid oxide system.

the average crystallite size atomic isotope fraction of 18О in CO2 average value of 18O fraction in the oxide bulk average value of 18O fraction at the surface over the reactor length jump value of 18O atomic fraction outlet after isotopic switch

mixture 3 Ln2O3 + MoO3 was uniaxially pressed at 914 MPa and sintered under air in the temperature range of 1600–1650 °C for 3 h. 2.2. Characterization Ho10Mo2O21 and Nd10Mo2O21 samples as dense ceramic pellets were characterized by X-ray diffraction (XRD) on a DRON-3M diffractometer (filtered Cukα radiation, step scan mode with a step of 0.1 or 0.05°, the angular range 2θ = 10–65°). The microstructure of the sintered ceramics was examined by scanning electron microscopy (SEM) using a JEOL JSM-6390LA microscope. Au spraying was preliminary carried out to increase samples conductivity for obtaining high resolution SEM images of the samples' surface. 2.3. Oxygen transport features studies Studies of oxygen isotope exchange of Ho10Mo2O21 and Nd10Mo2O21 samples (0.25–0.5 mm fraction prepared from powdered samples by pressing at 500 MPa into pellets with isopropanol binder following by drying at room temperature, grinding and sieving between meshes 0.25 and 0.5 mm, then calcination under air at 700 °C for 3 h) with C18O2 were carried out in the plug flow reactor (a quartz tube, the inner diameter 3 mm, length 120 mm) in the temperature programmed (TPIE) and isothermal (IIE) modes. The samples were pretreated at 700 °C in 1 vol% O2 + He flow for 30 min. IIE experiments were carried out at 520 °C by technique similar to SSITKA [44–46]. When the steady state was achieved under 1 vol% C16O2 + He flow over oxide loaded into a reactor, the gas mixture was replaced stepwise by the same one but containing C18O2. Ar was added into the isotope molecules containing feed gas to determine the gas-phase input signal. The isotope content in C18O2 used for mixtures preparation was 91%. In TPIE experiments, 1% C18O2 + He was fed into the reactor inlet. A gradual increase of temperature from 50 to 700 °C was conducted with a ramp of 5 °C/min. The sample weight in all experiments was 0.050 g, and the flow rate was 23 Nml/min. Transient changes in the gas isotopic composition (С16O2, С16O18O and С18O2 concentrations) were continuously monitored by a UGA-200 mass spectrometer (Stanford Research Systems, USA). Atomic isotope fraction of 18О in CO2 (α) and molecular fraction of C16O18O (f16-18) were calculated from the mass-spectrometry data

=

I48 + 0.5 × I46 ,f I44 + I46 + I48 16

18

=

I46 . I44 + I46 + I48

Here I44, I46 and I48 are parent peaks at m/e = 44, 46 and 48, respectively. α and f16-18 responses were processed using mathematical model considering two-dimensional mode of oxygen diffusion (Appendix A) including diffusion along the grain boundaries and within the grain bulk for calculation of oxygen surface exchange constant (kex) and tracer diffusion coefficient along grain boundaries (Dgb) and within grain bulk (Dbulk). The error of these parameters calculation does not exceed ± 15%.

2. Materials and methods 2.1. Synthesis of samples

3. Results and discussion

Ho10Mo2O21 and Nd10Mo2O21 were synthesized via the mechanical activation route. The starting Nd2O3 and Ho2O3 oxides were initially calcined under air at 1000 °C for 2 h. MoO3 was preliminary activated in the high energy Aronov ball mill for 4 min. Then Ln2O3 (Ln = Nd or Ho) oxide was mixed with MoO3 in stoichiometric ratio followed by comilling in the SPEX8000 ball mill for 1 h. The mechanically activated

3.1. Structural and textural features Ho10Mo2O21 is single-phase oxide with fluorite structure (space group Fm3m, Fig. 1(a)). Nd10Mo2O21 has more complex structure than 2

Solid State Ionics 346 (2020) 115229

V. Sadykov, et al.

(220)

(125) (413) (006) (241) (422)

(410) *

*

30

40

50

Fig. 2. XRD patterns of La5.5MoO11.25-δ samples sintered at 1600 (a) and 1650 °C (b) compared to that for Nd10Mo2O21 sample sintered at 1600 °C (c). Asterisks show additional lines absent for pure rhombohedral (R3 ) phase.

3.2. Oxygen mobility and surface reactivity Dynamics of α(t) and f16-18(t) isotope responses after switching from C16O2 to C18O2 in the isothermal mode is demonstrated in Fig. 4. As follows

b

(222)

(200)

(100) 20

*

(311)

*

(214)

(122)

Intensity, [arb. un.]

(003) (211)

that of Ho10Mo2O21 or Nd6MoO12 oxides [17]. The structure type is close to R3, however, XRD patterns cannot be described fully by this space group due to presence of additional peaks with a low intensity (Fig. 1(b)). The average scattering region size is very large (3.6–11 μm) due to a high sintering temperature. The unit cell parameters, true density and other characteristics of samples are given in Table 1. Earlier for mixed protonic-electronic conductor La5.5MoO11.25-δ a complex rhombohedral structure was observed as well [16]. Differing from R3 one, this structure with additional peaks in XRD pattern (Fig. 2) is stable in reducing conditions. This structure was also observed in other works for La5.8Zr0.2MoO12.1 and similar materials [16,17,47]. Its cell parameters were shown to be easily derived from parameters of simple R3 rhombohedral structure using the following equations [16]: c = cR/ 3 ; a = aR × 2/ 14 . XRD patterns for La5.5MoO11.25-δ samples after sintering at 1600 and 1650 °C [16,17] and Nd10Mo2O21 sample after similar sintering at 1600 °C are compared in Fig. 2. It is obvious that complex rhombohedral structure forms in all cases, being stable in the temperature range of 1600–1650 °C for both La5.5MoO11.25-δ and Nd10Mo2O21 oxides. Hence, Nd10Mo2O21 cell parameters can be also calculated using the equations given above and parameters of a simple rhombohedral structure, which main lines are present in XRD patterns (Figs. 1(b) and 2(a,b)). SEM micrographs are given in Fig. 3. Very large grains (~3–20 μm) with a good stacking are present along with smaller domains (~0.5 μm), thus forming non-linear grain boundaries. The average grain size matches the scattering region size estimated from XRD data (Table 1). As mentioned above, such a huge size is explained by a high sintering temperature. SEM images demonstrate a seldom presence of pores with sizes ~1–3 μm. No cracking was detected for both samples. It is interesting that Nd10Mo2O21 complex oxide which can be considered as a salt of a strong base of neodymium hydroxide and molybdic acid H2MoO4 is characterized by the geometric density close to the crystallographic one, while Ho10Mo2O21 complex oxide formed from a weak base and a strong acid has geometric density much lower than the crystallographic one. This implies a higher porosity for the latter oxide.

a

60

2 , [°] Fig. 1. XRD patterns for Ln10Mo2O21 samples: Ln = Ho (a) and Nd (b). For Nd10Mo2O21 sample, asterisks show additional lines, which are not observed for pure rhombohedral phase (R3 ). Table 1 Cell parameters (a, c), crystallographic (ρ) and geometric density (δ), specific surface area (Ssp) and average particle size ( < d > ) of Ln10Mo2O21 samples. Ln

a, [Å]

c, [Å]

ρ, [g/cm3]

δ, [g/cm3]

Ssp, [m2/g]

< d > , [μm]

Ho Nd

5.2986(2) 5.481(2) (R3)

– 5.558(2) (R3)

8.09 ≈6 (R3)

6.69 5.55

0.5 0.3

3.6 11

Fig. 3. SEM micrographs for Ln10Mo2O21 samples: Ln = Ho (a) and Nd (b). 3

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1.0 0.9

2, 3 O atoms fraction

0.6 4 0.4

2

Experimental Model (1D diffusion) Model (2D diffusion)

0.7

1

3

0.2

0.8

18

Mole fractions

0.8

0.6

0.0

200

0 100

1000

2000

3000

4000

5000

6000

300

400

7000

500

600

700

800

T, [°C]

t, [s]

Fig. 5. 18O atoms fraction in the gas phase during temperature-programmed isotope exchange with C18O2 in the flow reactor for Ho10Mo2O21 sample. Points – experiment, lines – model.

18

Fig. 4. Dynamics of isothermal isotope exchange with C O2 at 520 °C for Ho10Mo2O21 sample. C16O18O molecules fraction in the gas phase (1) and 18O atoms fraction in the gas phase (2), on the oxide surface (3) and in the bulk (4).

from this figure, the isotope fraction of 18O at the reactor outlet at the initial contact time (αoτ ) is close to 0. As demonstrated in the work [46], if ατ ≅ 0, it means that the surface exchange rate is very high with the isotopic equilibrium between the surface and the gas phase being achieved. In this case, in the first approximation 18O isotope fraction at the surface can be taken as equal to that in the gas phase observed at the reactor outlet. The average integral fraction of 18O in the bulk ( BULK (t )) can be calculated as follows: BULK

(t ) =

2CCO2 UNA WNOX

t

( 0

input

) dt =

1 b

t

(

input

) dt ,

0

where CCO2 is concentration of CO2 in the gas phase, U is the flowrate, NA is Avogadro number, W is the sample weight, NOX is the number of oxide anions per sample gram, αinput is 18O atoms fraction in the input gas mixture, τ is contact time, b ratio of total oxide oxygen to the number of oxygen atoms in the gas phase [46]. While calculating 18 O in the bulk according to BULK (t ) the average integral fraction of this equation, it is demonstrated to be significantly lower compared to that at the surface. Thus, qualitative analysis definitely showed that substitution of oxygen in the sample is strictly limited by diffusion in the bulk. More detailed information on the rate of oxygen substitution in the sample's bulk can be acquired from the temperature-programmed isotope exchange (TPIE) data. According to these data, the exchange process starts at T = 250 °C demonstrating a high oxygen mobility and surface reactivity. Temperatures of 18O atoms fraction minimum on TPIE curves (Fig. 5) are 620 and 550 °C for samples with Ln = Ho and Nd, respectively, which qualitatively shows that the oxygen mobility for sample with Ln = Nd is higher. According to numerical analysis of TPIE data, for Ho10Mo2O21 sample the isotope exchange could not be described by a single diffusion coefficient (Fig. 5, black line), which is explained by nonuniformity of the oxygen migration paths. This non-uniformity may be caused by many factors. At first, the bulk oxygen non-uniformity can be caused by difference in M-O bonds strength as well as different grain size effect, which results in formation of faster and slower diffusion channels. Moreover, such features for similar systems (Nd5.5WO11.25-δ based fluorites) can be explained by the effect of structural and/or defect features [48]. It is to be also noted that extended defects such as grain boundaries, stacking faults etc. can be present. It is a feature of Ln tungstates and tungstates-molybdates revealed by high-resolution transmission electron microscopy [48–50]. Such defects (especially grain boundaries) can promote the oxygen diffusivity. Moreover, samples used in the

Fig. 6. Scheme of 2D diffusion in Ln10Mo2O21. 1 – grains, 2 – diffusion along grain boundaries with diffusion coefficient Dgb, 3 – diffusion within grain bulk with diffusion coefficient Dbulk.

0.9

Mole fractions

0.8 0.7 0.6 0.5 0.4 0.3

f16-18

0.2 0.1 100

200

300

400

500

600

700

800

T, [°C] Fig. 7. The temperature-programmed isotope exchange with C18O2 in the flow reactor for Nd10Mo2O21 sample. Points – experiment, lines – model (2D diffusion).

4

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Table 2 The values of oxygen tracer diffusion coefficient (D⁎/L2), its effective activation energy (Ea) and fraction of each oxygen type with respect to overall oxygen in the oxide (θ) for Ln10Mo2O21 samples at 520 °C. Ln

Grain boundaries

Ho Nd

Grain bulk

Dgb/L2, [s−1]

Ea, [kJ/mol]

θ, [%]

Dfast/L2, [s−1]

Ea, [kJ/mol]

θ, [%]

Dslow/L2, [s−1]

Ea, [kJ/mol]

θ, [%]

3.3 · 10−2 1.1 · 10−1

100 100

5 5

8 · 10−5 7.5 · 10−4

60 60

67 33

2 · 10−5 1.2 · 10−4

60 60

28 62

and a slower diffusion within grain bulk (2D diffusion model) [22,23] was proposed for description of the oxygen isotope exchange. For description of oxygen diffusion along grain boundaries, additional “dimension” with related equations for diffusion (Appendix A, Eqs. (A.1)–(A.7)) should be added with remoteness from grains bulk being added as a new variable differing from the model of “one-dimensional” diffusion [44,45] using only remoteness from the sample surface as variable [23]. This model schematically presented in Fig. 6 gives the best fit (Fig. 5, olive line) for the experimental data. Similar tendencies were demonstrated for Nd10Mo2O21 sample (Fig. 7). The values of oxygen tracer diffusion coefficient and its effective activation energy are given in Table 2 and plotted in Arrhenius coordinates in Figs. 8 and 9. In Table 2, L1 = L2 = L, which is considered to be equal to < d > (Table 1). Very fast diffusion (Dgb up to ~10−6 cm2/s at 700 °C) along grain boundaries involving around 5% of the overall oxygen was demonstrated. This can be explained by some features of the particles' surface and near-surface monolayers (e.g., extended defects presence can be proposed). There is a nonuniformity in diffusion within the grain bulk: two oxygen forms differing by diffusivity can be distinguished (fast and slow diffusion channels). This effect can be explained by different MeO (MoeO, LneO) bonds strength and/or it can be attributed to effect of different grain sizes as well. Fast and slow oxygen diffusion channels were also found for Nd5.5WO11.25-δ fluorites [48] being assigned to effect of structural and/ or defect features which are to be clarified in the future. Higher diffusion coefficients are observed for the Nd10Mo2O21 sample with a larger Nd cation [54], which crystallizes in the structure class based on R3 space group, while Ho10Mo2O21 oxide structure is described by Fm3m space group (vide supra). It is obvious that a larger volume of Nd10Mo2O21 unit cell (V = 483.47 Å3) compared to that of Ho10Mo2O21 fluorite (V = 148.76 Å3) [54] would provide a higher oxygen diffusivity for Nd10Mo2O21. For asymmetric rhombohedral Nd10Mo2O21 structure a difference in values of bulk diffusion coefficient within the grains bulk is also higher, which is predictable.

-4

log(Dgb, [cm2/s])

-6 -8 -10

b a

-12 -14 -16 -18 1.0

1.5

2.0

2.5

3.0

3.5

-1

1000/T, [K ] Fig. 8. Arrhenius plots for the oxygen tracer diffusion coefficient along grain boundaries according to TPIE data for Ln10Mo2O21 samples: Ln = Ho (a) and Nd (b).

log(Dbulk, [cm2/s])

-8

-10

b -12

Dfast -14

Dslow

b a

-16

a 4. Conclusions

-18 1.0

1.5

2.0

2.5

3.0

3.5

Oxygen transport features in relationship with structural and textural characteristics were studied for Ln10Mo2O21 (Ln = Nd, Ho) complex oxides. It was demonstrated that sample with Ln = Ho is singlephase oxide with fluorite structure (space group Fm3m), while the one with Ln = Nd has more complex rhombohedral -like structure R3. Very fast oxygen diffusion along grain boundaries (diffusion coefficient up to ~10−6 cm2/s at 700 °C) involving around 5% of the overall oxygen was demonstrated for Ln10Mo2O21 samples making these oxides promising in design of catalytic membranes and electrochemical devices.

-1

1000/T, [K ] Fig. 9. Arrhenius plots for oxygen tracer diffusion coefficient within grain bulk according to TPIE data for Ln10Mo2O21 samples: Ln = Ho (a) and Nd (b).

current work are ceramics sintered at high temperatures. It was reported that, according to electrochemical impedance spectroscopy analysis, increasing sintering temperature leads to increasing grain boundary conductivity contribution for, e.g., yttria-stabilized zirconia [51] and Sr-doped La zirconate [52,53]. In addition, as mentioned in introduction, fast ionic transport along grain boundaries is also a feature of perovskites [30–34], fluorites [35–38] and nanocomposites [30–35]. Hence, the most reasonable description of isotope exchange data for Ln10Mo2O21 is considered to be a faster diffusion along grain boundaries and a slower diffusion within grains bulk. The mathematical model including equations for a faster diffusion along grain boundaries

CRediT authorship contribution statement Vladislav Sadykov: Writing - review & editing. Anna Shlyakhtina: Investigation. Ekaterina Sadovskaya: Software, Validation. Nikita Eremeev: Writing - oiginal draft. Valeriy Skazkab: Formal analysis. Vladimir Goncharov: Investigation.

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Acknowledgements

gratefully acknowledged.

Authors would like to give acknowledgement to 22nd International Conference on Solid State Ionics (SSI-22) Organization Committee.

Declaration of competing interest

Funding

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Support by Russian Science Foundation (Project 16-13-00112) is Appendix A. 2D diffusion model of isotope exchange

Two-dimensional diffusion and surface exchange of oxygen are described by the set of Eqs. (A.1)–(A.7) [22,23]. The model is similar to that described in authors' previous works [44,45], however, it takes into account a fast diffusion along grain boundaries and a slow diffusion within grain bulk. The dynamics of 18O atoms fraction in the gas phase obeys the equation for the surface exchange according to which it is related to the surface exchange rate and 18O fraction at the oxide surface:

1

+

t

= b1 R (

),

s

(A.1)

18

where α and αs are O fractions in the gas phase and at the oxide surface, respectively; τ is the contact time (s), ξ is dimensionless reactor length; b1 is ratio of the number of oxygen atoms in the gas phase to that at the oxide surface; R – the surface exchange rate. 18 O fraction at the oxide surface depends on the surface exchange rate R (similar to α) and fast diffusion along grain boundaries. The equation for this follows from exchange equation and Fick's first law for diffusion along grain boundaries: s

= R(

t

s)

Dgb

+ b2

gb

,

L1

(A.2)

=0

where b2 is ratio of the number of oxygen atoms at the oxide surface to that in grain boundaries; Dgb is oxygen tracer diffusion coefficients along grain boundaries; L1 is the characteristic size of grain; αgb is 18O fraction within grain boundaries; η is a dimensionless variable characterizing oxygen remoteness from the surface. Dynamics of 18O fraction within grain boundaries add a new “dimension” to the model. The equation for αgb follows from Fick's second law for diffusion along grain boundaries and Fick's first law for diffusion within grain bulk: gb

=

t

2

Dgb

gb 2

L12

+ b3

Dbulk L2

bulk

,

(A.3)

=0

where b3 is ratio of the number of oxygen atoms in grain boundaries to that in grains bulk; Dbulk is oxygen tracer diffusion coefficient within grains bulk; L2 is the average grain size (in particular case of the samples involved, L1 = L2 = L); αbulk is 18O fraction within grains bulk; ζ is a dimensionless variable characterizing oxygen remoteness from the grain boundary. The equation for αbulk is derived from Fick's second law for diffusion within grains bulk: bulk

t

=

2

Dbulk L 22

bulk 2

(A.4)

The equations for fraction of asymmetric isotope molecules chanism according to the Muzykantov's classification [55].

f16

18(I)

t

f16

18(II)

t

+

1 f16

18(I)

1 f16

+

= b1 2R ( (1

18(II)

= b1 R (2

s)

s (1

+

s)

s (1

)

16

18

16

18

O O (or C O O) in the gas phase (f16-18) follow from different types of me-

f34 )

(A.5)

f34 )

(A.6)

Initial and boundary conditions are:

t = 0:

=

s

= 0:

=

input ,

= 0:

gb

= 0:

bulk

= =

=

s, gb,

gb

=

f16

bulk

18

= f16

18

=0

input = f16 18

= 1: = 1:

gb

=0

bulk

=0

(A.7)

Problem (A.1–7) was solved in the following way. Differential operators with respect to ξ, η, ζ were replaced by their difference analogues with the second order of approximation. Further system of ordinary differential equations was solved using the Runge–Kutta–Merson method. Control calculations with 10-time decreasing step of discretization in representation of differential operators by their difference analogues was carried out to control stability of calculations scheme obtained.

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