Alexandra Navrotsky University of California at Davis

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Alexandra NavrotskyUniversity of California at
Davis

• Complexities of Real Oxide
• Solid Solution Systems

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The complexity and richness of natural minerals
and multicomponent ceramics (like UO2 spent
nuclear fuel) arises not only from the large
number of phases and crystal structures, but also
from the variety of solid solutions, complex
elemental substitutions, order-disorder
relations, and exsolution phenomena one
encounters. Indeed, a pure mineral or ceramic of
end-member composition is the exception rather
than the rule. Elemental substitutions in
minerals may be classified in terms of the
crystallographic sites on which they occur and
the formal charges of the chemical elements
(ions) which participate. For a correct
description, the thermodynamic punishment must
fit the structural crime. Yet the formalism must
be tractable, transparent, and useful. Herein
lies the challenge.
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Charge Coupled Substitutions in Minerals

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Examples of charge-coupled substitutions

• Mg2SiO4-Fe2SiO4, Mg2SiO4-Mg2GeO4
• NaAlSi3O8-CaAl2Si2O8
• ZrO2-YO1.5
• CoO-Li0.5Co0.5O
• LaGaO3-SrGaO2.5

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(                   Any expression one writes
for a free energy of mixing, an activity, or an
activity coefficient has buried in it some
assumptions about the mixing process on an atomic
scale. Therefore one must make sure those
assumptions are reasonable. There is no escaping
this. (         Equations, whose form is
constrained by theoretical considerations and
whose parameters are physically reasonable, are
more reliably extrapolated than arbitrary
polynomials fit to data in a small P, T, X
range. Models which simultaneously include
constraints imposed by phase equilibria,
calorimetry, and crystal chemistry are more
reliable than those based on any one source
alone.
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General Reaction   xA A xBB A(xA)B(xB)
    ?Gºmix ?Hºmix - T?Sºmix .

 DH gt 0, clustering and phase separation DH lt
0, ordering and compound formation Partial
molar quantities and activities   GºT(solid
solution) xA µºT(A) xBµºT(B) xA?µº(A)
xB?µº(B)  Integral free energy of mixing
  ?Gºmix xA?µºT(A) xB?µºT(B)  
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The thermodynamic activity is defined as   ?µº(A)
RT ln a(A) and ?µº(B) RT ln a(B)
The changes in chemical potential on
mixing can be related to partial molar enthalpies
and entropies of mixing   ?µºT(A) ?hºT(A) -
T?sºT(A) , ?µºT(B) ?hºT(B) - T?sºT(B)
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FeAl2O4-FeCr2O4 One mole of ions
mixed   a(Fei0.5Al O2) xA, where A Fe0.5Al
O2. and a(Fei0.5Cr O2) xB, where B
Fe0.5CrO2   Two moles of ions mixed, entropy an
extensive parameter ?Smix -2RxA ln xA xBln
xB   ?µ(FeCr2O4) -2RT ln xA,
?µ(FeAl2O4) -2RT ln xB   a(FeCr2O4)
xA2 , a(FeAl2O4) xB2

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Multicomponent solid solutions no unique choice
of components Spinel (Fe2,Ni)(Cr Fe3)2O4 a ss
between Fe3O4 and NiCr2O4 or FeCr2O4 and
NiFe2O4.  
Fe3O4 NiCr2O4 Nife2O4 FeCr2O4 DG not
zero   reciprocal term must be
incorporated. Clinoptilolite zeolites,
occurring in the tuffs at Yucca Mountain
(Na,K)6Al6Si30O72.20H2O a(Nax6Al6Si30O72.20H2O)
x(Nax6Al6Si30O72.20H2O)6 a(NaAlSi5O12.3.
67H2O) x (NaAlSi5O12.3.67H2O)
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Regular, Subregular and Generalized Mixing
Models   Starting point useful but inherently
contradictory assumption that, though the heats
of mixing are not zero, the configurational
entropies of mixing are those of random solid
solution.   ?Gºmix, ex ?Hºmix-T?Sºmix, ex
  For a two-component system, the
simplest formulation is  ?Gexcess ?Hmix
?xAxB WH xAxB
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Generalization

• For a binary system, the Guggenheim or
  Redlich-Kister based on a power-series expression
  for the excess molar Gibbs energy of mixing
  which reduces to zero when either x1 or x2
  approach unity
• where the coefficients ?r are called
  interaction parameters. Activity coefficients can
  be obtained by the partial differentiation of
  over the mole fraction x1 or x2

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Generalization

• The subregular model is equivalent to the
  asymmetric Margules model in the
  Thompson-Waldbaum notation
• where the parameters are related to those used
  in the Redlich-Kister model as
• and, in general, depend on temperature and
  pressure. In the Thompson-Waldbaum model, the
  activity coefficients of end-members are
  expressed as

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Some more details on configurational entropy,
grounded in statistical mechanics
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Order- Disorder Phenomena   If ordering slow,
several series of solid solutions, each having a
differing degree of order, can exist.
Thermodynamics of mixing one set of species
(e.g. Na, K or Na, Ca in feldspars) may depend
on the degree of order of another (Al,
Si). Extent of ordering depends on temperature
in a complex fashion. Risky extrapolation.
Enthalpy and entropy of mixing depend strongly
on T. DS and DG of mixing can be asymmetric
because the configurational entropy term itself
departs from symmetrical behavior. A solvus may
develop.
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Observed degree of order is often kinetically
controlled. Low T, can not order. High T, can not
quench Metastable disordered solid solutions may
form. Strong tendency to ordering is manifested
in significant negative heats of mixing and
sometimes leads to compound formation, (e.g.
dolomite. DH, DS and DG of mixing depend strongly
on the degree of order.
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Destabilizing energetics of mixing in the
disordered state may be present at the same time
as stabilizing effect of ordering, leading to
complex interplay between ordering and exsolution
(conditional spinodal). Change in symmetry First
or higher order phase transitions lead to
unconventional phase diagram topologies. Lattice
parameters can vary nonlinearly with composition
(depart from Vegards rule)
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Structural Aspects of Solid Solutions in Relation
to their Stability and Miscibility

• Fluorite-based structures
• Spinels
• Carbonates

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Order- Disorder Phenomena   Extent of ordering
depends on T. Observed degree of order is often
kinetically controlled. Low T, can not order.
High T, can not quench. Metastable disordered
solid solutions may form. Strong tendency to
ordering is manifested in significant negative
heats of mixing and sometimes leads to compound
formation, (e.g. dolomite. DH, DS and DG of
mixing depend strongly on the degree of order.
They can be asymmetric because the
configurational entropy term itself departs from
symmetrical behavior. A solvus may develop.
Order-disorder reactions often involve a change
in the symmetry of the structure. First-order or
higher order phase transitions leading to
unconventional phase diagram topologies. Lattice
parameters can vary nonlinearly with composition
(depart from Vegards rule)
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Fluorite Structure
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Fluorite Derivatives

• Parent cubic structure CeO2, UO2, PuO2, high T
  ZrO2 and HfO2
• Tetragonal and monoclinic distortions ZrO2 and
  HfO2
• Oxygen excess phases UO2x, PuO2x (?)
• Oxygen vacancy phases, M3 and M2 doping, e.g.
  YSZ, YxZr1-x O2-0.5x
• Pyrochlore

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Cubic stabilized MO2

• M4 Ln3 0.5 oxygen vacancy
• High oxide ion conductivity which goes through a
  maximum with increasing doping
• Structural evidence for short range order in
  cubic solid solutions
• Low T long range ordered phases sluggish to form

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Local ordering of vacancies in C-type rare earth
structure (bixbyite) Y2O3
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Enthalpy of formation from m-ZrO2 and C-YO1.5 of
c-YxZr1-xO2-0.5x (YSZ)
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X-ray diffraction patterns of (a) Disordered
Zr0.43Y0.57O1.96 c-YSZ phase (b) Ordered
Y4Zr3O12 d-phase, ?H (ordered disordered) is
0.4 1.4 kJ/mol 0short range order rules!!!
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Interaction Parameter for Mixing in Fluorite
Phase
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Generalizations

• Strong ionic size control on properties
• Negative heats of mixing related to siting of
  bacanciues
• Prediction UO2-LnO1.5 should have little
  stabilization unless U is oxidized

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Enthalpies of formation from oxides (UO2, UO3 and
CaO or YO1.5) of samples with as-synthesized
oxygen contents. (b) Enthalpies of formation of
modeled fully reduced (all uranium tetravalent)
UO2-CaO and UO2-YO1.5 solid solutions relative
to UO2 and CaO or YO1.5. The correction for the
effect of oxidation was made assuming that
oxidation of U4 to U6 in the solid solution
matrix has the same enthalpy as oxidation of UO2
to UO3.
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Conclusions Fluorite

• Short range order dominates the thermodynamics
• Strongly negative heats of mixing are compensated
  by much less than random entropies of mixing
• The m-c transition has a much higher enthalpy for
  HfO2 than ZrO2
• The ceria system is different because of
  different short range order
• Ionic size determines the ordering and enegetics

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Spinel Structure

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SPINELS

• AB2O4, two octahedral and one tetrahedral site
  per formula unit
• Normal AB2O4, inverse BABO4, random
  A1/3B2/3A2/3B4/3O4
• Cation distribution varies with T
• Effect on enthalpy and entropy of mixing and
  thermodynamic properties

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Octahedral Site Preference Energies 
   
Experimental thermodynamic basis of site
preference energies and enthalpies of formation
of spinels. Crystal field effects are but a small
contribution
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Calcite and Dolomite Structure

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Energetics in CaMg(CO3)2-CaFe(CO3)2
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CONCLUSIONS

• Must consider structural details to formulate
  thermodynamics properly
• Many unanswered questions, lots of work for you,
  the students and future generation of
  investigators.
• Lots of fun !!