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Thermodynamic assessment of the Ce–O system in solid state from 60 to 67 mol.% O

  • Hans Jürgen Seifert EMAIL logo , Pankaj Nerikar and Hans Leo Lukas
Published/Copyright: January 11, 2022
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International Journal of Materials Research
From the journal International Journal of Materials Research Volume 97 Issue 6

Abstract

A thermodynamic dataset was developed for the central part of the Ce–O system covering the range between Ce2O3 and CeO2 up to about 2000 K. All literature data for the thermodynamic functions, the phase diagram and crystallography were critically assessed. The modeling of the phases takes into account their crystal chemistry including order – disorder behavior. From the dataset, diagrams were calculated for the (1) heat capacities of CeO2 and Ce2O3, (2) enthalpy increment of CeO2, (3) chemical potentials of oxygen in CeO2–x, (4) partial enthalpies of oxygen in CeO2–x, (5) the partial phase diagram of Ce–O and (6) chemical potentials of oxygen in the two-phase areas. These diagrams reproduce fairly well the experimental data from literature.

Keywords: Cerium oxides; CALPHAD; Oxides; Thermodynamic modeling
Hans Jürgen Seifert, Associate Professor University of Florida Department of Materials Science and Engineering PO Box 116400, Gainesville FL 32611-6400 Tel.: +1 352 846 3779 Fax: +1 352 846 3355

Dedicated to Professor Dr. Fritz Aldinger on the occasion of his 65th birthday

  1. We are grateful for financial support by U.S. Department of Energy (DE-FC07-051D14649).

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Transformation of polynomial terms of the Gibbs energy description of the ordered D53 phase into parameters of the compound energy formalism (CEF)

This part uses matrix algebra. It is written only to document how the transformation used in Appendix 2 was derived.

In the D53 type the second sublattice of the C1 type splits into two sublattices in the ratio 3 : 1. The site fractions on the three sublattices are:

(4) y C e + 4 ′ = 1 − 4 u y O − 2 ′ ′ = 1 − u + v y O − 2 ′ ′ = 1 − u − 3 v y C e + 3 ′ = 4 u y V a ′ ′ = u − v y V a ′ ′ ′ = u + 3 v

where for simplification y of the C1 description is replaced by 4u (that is: u equals x=2 of the formula CeO2–x). v is an ordering parameter, where v = 0 describes the disordered state.

For this numerical treatment of ordering Ansara et al. [44] pointed out:

  • In the disordered state (for v = 0) the G description must reproduce that of the C1 type.

  • In the disordered state, the derivative of the Gibbs energy with respect to v must be zero as the difference between sublattices two and three disappears and, thus, no driving force can exist to initiate ordering.

(5) ∂G∂vv=0=0

These conditions can be used to express the parameters of D53 with those of C1 and an additional ordering contribution ΔGord to the Gibbs energy. The factors of the parameters of C1 do not contain v and those of ΔGord at least v2, because terms with v in the 1st power contradict condition (5). We write in matrix formulation

(6) exGD53=((GD53))⋅((XD53))exGC1=((GC1))⋅((XC1))

where ((GD53)) and ((GC1)) are line vectors of CEF parameters, ((XD53)) and ((XC1)) are column vectors of the corresponding products of site-fractions. The elements of ((XD53)) and ((XC1)) are expanded into power series of u and v, written as matrix expressions:

(7) ((XD53))=((MD53))⋅((XP))((XC1))=((MC1))⋅((XP))

The elements of column vector ((XC1)) do not contain v and its powers. To use identical vectors ((XP)) in both parts of (8), we augment vector ((XC1)) by all terms of ((XP)) containing v and call the augmented column vector XC1A. The line vector ((GC1A)) is correspondingly augmented. Related to v2+μuν contributions to ΔGord are added and related to vuν zeroes are added to satisfy condition (5) (0≤ μ ≤2, sum of powers ≤4).

Inversion of the matrix ((MD53)) in (7) allows the expression of the elements of vector ((XP)) as functions of the elements of ((XD53)). Finally, vector ((XC1A)) can be expressed by the elements of ((XD53)):

(8) ((XP))=((MD53))−1⋅((XD53))((XC1A))=((MC1A))⋅((XP))=((MC1A))⋅((MD53))−1⋅((XD53))

Using the same transformation, the line vector ((GD53)) is a function of ((GC1A)):

(9) ((GD53))=((GC1A))⋅((MC1A))⋅((MD53))−1

or, in other words, the CEF parameters of the D53 phase can be composed of the parameters of the C1 phase and those of an ordering contribution. The zeros in ((GC1A)) provide that in the resulting Gibbs energy expression of the D53 phase all polynomial terms cancel, which contain v in 1st power.

To construct the matrices in detail we have to set a limit for the maximal powers of u and v. Selecting 4 as the maximal power, the column vector ((XP)) contains these 15 elements: 1, u, v, u2, uv, v2, u3, u2v, uv2, v3, u4, u3v, u2v2, uv3, and v4. The line vector ((GCEF)) must consist of 15 CEF parameters, where the corresponding elements of ((XCEF)) all are linearly independent and the powers of the site-fraction products do not exceed four. We use the arbitrariness due to the electroneutrality condition to set all parameters for Ce+4 equal the corresponding ones with Ce+3 on the first sublattice and use only their sums. By this treatment the occupation of the first sublattice cancels in ((XD53)). The 15 elements thus selected for ((GD53)) are GOO, GOV, GVO, GVV, L0OL, L0VL, L0LO, L0LV, L1OL, L1VL, L1LO, L1LV, L0LL L2*L, L2L*, where the abbreviations represent G for end-member parameter, and L0, L1, L2 for interaction parameters of 0th, 1st or 2nd order, respectively. The two following letters belong to the 2nd and 3rd sublattice, respectively, and mean: O = occupation by O–2, V = vacant site, L = interaction on this sublattice, * = equal values for O–2 occupation and vacant site, sum only is used (The site fractions of this sublattice sum up to unity and the power is diminished by one). The elements of vector ((GC1A)) are G1, G2, L0, L1, L2, the end members and interaction parameters of the binary Redlich –Kister expression introduced in section 3 for the C1 type phase, and A0, B1, A1, C2, B2, A2, which are multiplied with the terms 16v2, 128uv2, 128v3, 1024u2v2, 1024uv3 and 1024v4 of ((XC1A)), respectively, and form the contribution ΔGord. The numerical factors 16, 128, 1024 are arbitrarily chosen in order to get integers, not fractions, in the transformation matrix ((MC1A)) · ((MD53))–1. They appear in matrix ((MC1A)) at the section points of lines corresponding to A0, B1, A1, C2, B2 or A2 in ((XC1A)) and columns corresponding to v2, uv2, v3, u2v2, uv3 and v4 in ((XC1A)), respectively. All other elements in these columns and lines of ((MC1A)) are zero.

Matrices ((MD53)) and ((MC1A)) can be easily constructed by expanding the elements of ((XD53)) and ((XC1A)) into powers of u and v. It was proved that ((MD53)) is not singular and, thus, ((XC1A))· ((XD53))–1 exists. It is applied in Appendix 2.

Assessed Parameter Values Phase description covering C1-type and D53-type

The names of the CEF-parameters are abbreviated. The correspondence to the conventional parameter names is explained by the following three examples:

GOV=GCe+4:O−2:VaD53=GCe+3:O−2:VaD53
L0LO=0LCe+4:O−2,Va:O−2D53=0LCe+3:O−2,Va:O−2D53
L 2 ∗ L = 2 L C e + 4 : O − 2 : O − 2 , V a D 5 3 = 2 L C e + 3 : O − 2 : O − 2 , V a D 5 3 = 2 L C e + 4 : V a : O − 2 , V a D 5 3 = 2 L C e + 3 : V a : O − 2 , V a D 5 3
GOO = G1
GOV = G2 +A0 +2 B1
GVO = –2 G1 +3 G2 –6 L0 –30 L1 –150 L2 +A0 +6 B1
GVV = –3 G1 +4 G2 –12 L0 –84 L1 –588 L2
L0OL = L0 –A0 –3 B1
L0VL = L0 +18 L1 +216 L2 –A0 –5 B1
L0LO = 9 L0 +54 L1 +288 L2 –A0 –9 B1
L0LV = 9 L0 +108 L1 +936 L2 –A0 +B1
L1OL = –L1 +B1
L1VL = –L1 –24 L2 +B1
L1LO = –27 L1 –216 L2 +3 B1
L1LV = –27 L1 –432 L2 +3 B1
L0LL = –216 L2
L2*L = L2
L2L* = 81 L2

with

G 1 − 2 H C e S E R ( 298.15 K ) − 4 H O S E R ( 298.15 K ) = − 2234276.02 + 896.23274 T − 149.856582 T ln ⁡ ( T ) − 0.004995164 T 2 + 1260000 T − 1
G 2 − 2 H C e S E R ( 298.15 K ) − 3 H O S E R ( 298.15 K ) = − 1762786.32 + 785.90292 T − 137.445298 T ln ⁡ ( T ) − 0.007678898 T 2 + 1050000 T − 1

L0 = –126584.84 + 34.57038 T

L1 = –105949.66 + 34.30324 T

L2 = 15103.74

A0 = 19000.00 –13.00000 T

B1 = –40800.00 + 9.50000 T

The functions G1 and G2 need a reference state for definition, given as enthalpies of the pure elements at 298.15 K in the “stable element reference (SER)” state.

Stoichiometric phases

The following G(T)-functions are given in J · mol–1, referred to 1 mole of formula units. G and G2 are the same functions as already used in the description of the C1-type – D53-type phase, they represent 6 (2CeO2) or 5 (Ce2O3) moles of atoms, respectively.

C e 2 O 3 : G − 2 H C e S E R ( 298   K ) − 3 H O S E R ( 298   K ) = − 85664.20 + 10.97351 T + G 2
C e 7 O 12 : G − 7 H C e S E R ( 298   K ) − 12 H O S E R ( 298   K ) = − 151683.46 − 31.11896 T + 1.5 G 1 + 2 G 2
Ce19O34:G−19HCeSER(298 K)−34HOSER(298 K)=−286581.6−110.33063T+5.5G1+4G2
C e 5 O 9 : G − 5 H C e S E R ( 298   K ) − 9 H O S E R ( 298   K ) = − 71315.72 − 28.71204 T + 1.5 G 1 + G 2
Ce26O47:G−26HCeSER(298 K)−47HOSER(298 K)=−356389.65−144.92325T+8G1+5G2
Ce11O20:G−11HCeSER(298 K)−20HOSER(298 K)=−142441.59−58.43438T+3.5G1+2G2
Received: 2005-11-22
Accepted: 2006-02-25
Published Online: 2022-01-11

© 2006 Carl Hanser Verlag, München

Articles in the same Issue

  1. Frontmatter
  2. Microstructure and mechanical behavior of Pt-modified NiAl diffusion coatings
  3. Evolution of C-rich SiOC ceramics
  4. Evolution of C-rich SiOC ceramics
  5. Nanostructured SiC/BN/C ceramics derived from mixtures of B3N3H6 and [HSi(Me)C≡C]n
  6. Thermodynamic analysis of structural transformations induced by annealing of amorphous Si–C–N ceramics derived from polymer precursors
  7. Thermodynamic modelling of the Ce–Ni system
  8. Thermodynamic assessment of the Ce–O system in solid state from 60 to 67 mol.% O
  9. Phase transformations of iron nitrides at low temperatures (< 700 K) – application of mechanical mixtures of powders of nitrides and iron
  10. Effect of organic self-assembled monolayers on the deposition and adhesion of hydroxyapatite coatings on titanium
  11. Reconstruction and structural transition at metal/diamond interfaces
  12. Microstructure, hardness, and fracture toughness evolution of hot-pressed SiC/Si3N4 nano/micro composite after high-temperature treatment
  13. High-temperature plasticity of SiC sintered with Lu2O3-AlN additives
  14. Interaction of functionalised surfaces on silica with dissolved metal cations in aqueous solutions
  15. XRD and TEM study of NiO–LSGM reactivity
  16. Microstructure and dielectric properties of nanoscale oxide layers on sintered capacitor-grade niobium and V-doped niobium powder compacts
  17. Knudsen effusion mass spectrometric studies of the Al–Ni system: Thermodynamic properties over {AlNi + Al3Ni2} and {Al3Ni2 + Al3Ni}
  18. Aqueous solution deposition of indium hydroxide and indium oxide columnar type thin films
  19. Thermodynamic properties of B2-AlFeNi alloys: modelling of the B2-AlFe and B2-AlNi phases
  20. Kinetics of precipitate formation in (TixWyCrz)B2 solid solutions: influence of Cr concentration and Co impurities
  21. On the mechanisms governing the texture and microstructure evolution during static recrystallization and grain growth of low alloyed zirconium sheets (Zr702)
  22. Out-of-pile chemical compatibility of Pb–Bi eutectic alloy with Graphite
  23. Microstructural characterisation of a Co–Cr–Mo laser clad applied on railway wheels
  24. The Na–H system: from first-principles calculations to thermodynamic modeling
  25. Personal
  26. Conferences
  27. Frontmatter
  28. Basic
  29. Microstructure and mechanical behavior of Pt-modified NiAl diffusion coatings
  30. Evolution of C-rich SiOC ceramics
  31. Evolution of C-rich SiOC ceramics
  32. Nanostructured SiC/BN/C ceramics derived from mixtures of B3N3H6 and [HSi(Me)C≡C]n
  33. Thermodynamic analysis of structural transformations induced by annealing of amorphous Si–C–N ceramics derived from polymer precursors
  34. Thermodynamic modelling of the Ce–Ni system
  35. Thermodynamic assessment of the Ce–O system in solid state from 60 to 67 mol.% O
  36. Phase transformations of iron nitrides at low temperatures (< 700 K) – application of mechanical mixtures of powders of nitrides and iron
  37. Effect of organic self-assembled monolayers on the deposition and adhesion of hydroxyapatite coatings on titanium
  38. Reconstruction and structural transition at metal/diamond interfaces
  39. Applied
  40. Microstructure, hardness, and fracture toughness evolution of hot-pressed SiC/Si3N4 nano/micro composite after high-temperature treatment
  41. High-temperature plasticity of SiC sintered with Lu2O3-AlN additives
  42. Interaction of functionalised surfaces on silica with dissolved metal cations in aqueous solutions
  43. XRD and TEM study of NiO–LSGM reactivity
  44. Microstructure and dielectric properties of nanoscale oxide layers on sintered capacitor-grade niobium and V-doped niobium powder compacts
  45. Knudsen effusion mass spectrometric studies of the Al–Ni system: Thermodynamic properties over {AlNi + Al3Ni2} and {Al3Ni2 + Al3Ni}
  46. Aqueous solution deposition of indium hydroxide and indium oxide columnar type thin films
  47. Thermodynamic properties of B2-AlFeNi alloys: modelling of the B2-AlFe and B2-AlNi phases
  48. Regular Articles
  49. Kinetics of precipitate formation in (TixWyCrz)B2 solid solutions: influence of Cr concentration and Co impurities
  50. On the mechanisms governing the texture and microstructure evolution during static recrystallization and grain growth of low alloyed zirconium sheets (Zr702)
  51. Out-of-pile chemical compatibility of Pb–Bi eutectic alloy with Graphite
  52. Microstructural characterisation of a Co–Cr–Mo laser clad applied on railway wheels
  53. The Na–H system: from first-principles calculations to thermodynamic modeling
  54. Notifications
  55. Personal
  56. Conferences
https://doi.org/10.3139/ijmr-2006-0120
Keywords for this article
Cerium oxides; CALPHAD; Oxides; Thermodynamic modeling

Articles in the same Issue

  1. Frontmatter
  2. Microstructure and mechanical behavior of Pt-modified NiAl diffusion coatings
  3. Evolution of C-rich SiOC ceramics
  4. Evolution of C-rich SiOC ceramics
  5. Nanostructured SiC/BN/C ceramics derived from mixtures of B3N3H6 and [HSi(Me)C≡C]n
  6. Thermodynamic analysis of structural transformations induced by annealing of amorphous Si–C–N ceramics derived from polymer precursors
  7. Thermodynamic modelling of the Ce–Ni system
  8. Thermodynamic assessment of the Ce–O system in solid state from 60 to 67 mol.% O
  9. Phase transformations of iron nitrides at low temperatures (< 700 K) – application of mechanical mixtures of powders of nitrides and iron
  10. Effect of organic self-assembled monolayers on the deposition and adhesion of hydroxyapatite coatings on titanium
  11. Reconstruction and structural transition at metal/diamond interfaces
  12. Microstructure, hardness, and fracture toughness evolution of hot-pressed SiC/Si3N4 nano/micro composite after high-temperature treatment
  13. High-temperature plasticity of SiC sintered with Lu2O3-AlN additives
  14. Interaction of functionalised surfaces on silica with dissolved metal cations in aqueous solutions
  15. XRD and TEM study of NiO–LSGM reactivity
  16. Microstructure and dielectric properties of nanoscale oxide layers on sintered capacitor-grade niobium and V-doped niobium powder compacts
  17. Knudsen effusion mass spectrometric studies of the Al–Ni system: Thermodynamic properties over {AlNi + Al3Ni2} and {Al3Ni2 + Al3Ni}
  18. Aqueous solution deposition of indium hydroxide and indium oxide columnar type thin films
  19. Thermodynamic properties of B2-AlFeNi alloys: modelling of the B2-AlFe and B2-AlNi phases
  20. Kinetics of precipitate formation in (TixWyCrz)B2 solid solutions: influence of Cr concentration and Co impurities
  21. On the mechanisms governing the texture and microstructure evolution during static recrystallization and grain growth of low alloyed zirconium sheets (Zr702)
  22. Out-of-pile chemical compatibility of Pb–Bi eutectic alloy with Graphite
  23. Microstructural characterisation of a Co–Cr–Mo laser clad applied on railway wheels
  24. The Na–H system: from first-principles calculations to thermodynamic modeling
  25. Personal
  26. Conferences
  27. Frontmatter
  28. Basic
  29. Microstructure and mechanical behavior of Pt-modified NiAl diffusion coatings
  30. Evolution of C-rich SiOC ceramics
  31. Evolution of C-rich SiOC ceramics
  32. Nanostructured SiC/BN/C ceramics derived from mixtures of B3N3H6 and [HSi(Me)C≡C]n
  33. Thermodynamic analysis of structural transformations induced by annealing of amorphous Si–C–N ceramics derived from polymer precursors
  34. Thermodynamic modelling of the Ce–Ni system
  35. Thermodynamic assessment of the Ce–O system in solid state from 60 to 67 mol.% O
  36. Phase transformations of iron nitrides at low temperatures (< 700 K) – application of mechanical mixtures of powders of nitrides and iron
  37. Effect of organic self-assembled monolayers on the deposition and adhesion of hydroxyapatite coatings on titanium
  38. Reconstruction and structural transition at metal/diamond interfaces
  39. Applied
  40. Microstructure, hardness, and fracture toughness evolution of hot-pressed SiC/Si3N4 nano/micro composite after high-temperature treatment
  41. High-temperature plasticity of SiC sintered with Lu2O3-AlN additives
  42. Interaction of functionalised surfaces on silica with dissolved metal cations in aqueous solutions
  43. XRD and TEM study of NiO–LSGM reactivity
  44. Microstructure and dielectric properties of nanoscale oxide layers on sintered capacitor-grade niobium and V-doped niobium powder compacts
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