Alexandra NavrotskyUniversity of California

at Davis

Complexities of Real Oxide

Solid Solution Systems

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.

Charge Coupled Substitutions in Minerals

Examples of charge-coupled substitutions

• Mg2SiO4-Fe2SiO4, Mg2SiO4-Mg2GeO4

• NaAlSi3O8-CaAl2Si2O8

• ZrO2-YO1.5

• CoO-Li0.5Co0.5O

• LaGaO3-SrGaO2.5

(                   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.

General Reaction xA A + xBB =A(xA)B(xB)

  ΔGºmix = ΔHºmix - TΔSºmix .

 H > 0, clustering and phase separationH < 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  ΔGºmix = xAΔµºT(A) + xBΔµºT(B)  

The thermodynamic activity is defined as

 

Δµº(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) = ∆h¯ºT(A) - TΔs¯ºT(A) , ∆µºT(B) =

∆h¯ºT(B) - TΔs¯ºT(B)

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 = -2R[xA ln xA + xBln xB]

 ∆µ(FeCr2O4) = -2RT ln xA, ∆µ(FeAl2O4) = -2RT ln xB

 a(FeCr2O4) = xA

2 , a(FeAl2O4) = xB2

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 G 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)

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

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:

EmG

r

rr

Em xxxRTxfxfxRTG 21212211 lnln

...53ln 212122110221 xxxxxxxf

...53ln 121221210212 xxxxxxxf

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

121212212211 lnln xWxWxxfxfxRTGE

10211012 ; RTWRTW

322112

2212211 22ln xWWxWWfRT

311221

2121122 22ln xWWxWWfRT

Some more details on configurational entropy, grounded in statistical

mechanics

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.S and G of mixing can be asymmetric because the

configurational entropy term itself departs from symmetrical behavior. A solvus may develop.

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. H, S and G of mixing depend strongly on the degree of order.

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 Vegard’s rule)

Structural Aspects of Solid Solutions in Relation to their

Stability and Miscibility• Fluorite-based structures

• Spinels

• Carbonates

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. H, S and G 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 Vegard’s rule)

Fluorite Structure

Fluorite Derivatives• Parent cubic structure: CeO2, UO2,

PuO2, high T ZrO2 and HfO2

• Tetragonal and monoclinic distortions: ZrO2 and HfO2

• Oxygen excess phases UO2+x, PuO2+x (?)

• Oxygen vacancy phases, M3+ and M2+ doping, e.g. YSZ, YxZr1-x O2-0.5x

• Pyrochlore

Crystal structure of defect-fluorite (Fm3m)

O

Zr+4

Gd+3

VÖ2

Cubic stabilized MOCubic stabilized MO22

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

Local ordering of vacancies in C-type rare earth structure (bixbyite) Y2O3

Enthalpy of formation from m-ZrO2 and C-YO1.5 of c-YxZr1-xO2-0.5x (YSZ)

X-ray diffraction patterns of (a) Disordered X-ray diffraction patterns of (a) Disordered ZrZr0.430.43YY0.570.57OO1.961.96 c-YSZ phase c-YSZ phase (b) Ordered Y(b) Ordered Y44ZrZr33OO1212 δδ-phase,-phase,ΔΔH (ordered – disordered) is 0.4 ± 1.4 kJ/mol = H (ordered – disordered) is 0.4 ± 1.4 kJ/mol = 00short range order rules!!!short range order rules!!!

0250

500750

10001250

1500

10 20 30 40 50 60 70 80 90

2q

Inte

ns

ity

(b) Ordered

0250

500750

10001250

1500

10 20 30 40 50 60 70 80 90

2q

Inte

ns

ity

(a) Disordered

,,

0250

500750

10001250

1500

10 20 30 40 50 60 70 80 90

2q

Inte

ns

ity

(b) Ordered

0250

500750

10001250

1500

10 20 30 40 50 60 70 80 90

2q

Inte

ns

ity

(a) Disordered

Interaction Parameter for Mixing in Fluorite Phase

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

0.0 0.2 0.4 0.6 0.8 1.0

-60

-40

-20

0

20

b)

Ca Y

dopant concentration y in U1-y

CayO

2-y and U

1-yY

yO

2-y/2

0 . 0 0.2 0.4 0.6 0.8 1.0

-60

-40

-20

0

20a)

CaY

H0 f

, kJ/

mo

l

dopant concentration y in U1-y

MyO

z

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.

Conclusions: FluoriteConclusions: 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

Spinel Structure

SPINELS

• AB2O4, two octahedral and one tetrahedral site per formula unit

• Normal A[B2}O4, inverse B[AB]O4, random A1/3B2/3[A2/3B4/3]O4

• Cation distribution varies with T

• Effect on enthalpy and entropy of mixing and thermodynamic properties

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

“ T h e r m o c h e m is t r y o f I r o n M a n g a n e s e O x id e S p in e l s ” , S . G u i l l e m e t - F r i t s c h , A . N a v r o t s k y , P . T a i lh a d e s , H . C o r a d in , M . W a n g , J . S o l id S t a t e C h e m is t r y 1 7 8 , 1 0 6 - 1 1 3 ( 2 0 0 5 )

C a lo r im e t r ic s t u d ie s o f h e a t s o f f o r m a t io n o f i r o n m a n g a n e s e s p in e l s c o n f ir m t h e id e a s o f o x id a t io n - r e d u c t io n e n e r g e t i c s p r e s e n t e d in 1 9 6 8 a n d q u a n t i f y e f f e c t s o f t e t r a g o n a l d i s t o r t io n s .

- 6 0 .0

-4 0 .0

-2 0 .0

0 .0

2 0 .0

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

x = M n / (M n + F e )

H

f (fr

om c

-Fe

3O4 a

nd

t-Mn

3O4) (

kJ/m

ol)

( a )

Calcite and Dolomite Structure

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

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 funLots of fun !! !!