Thermodynamic data
A tutorial course
Session 5: More complicated solution phases
Alan Dinsdale
“Thermochemistry of Materials” SRC
This session
• Thermodynamic models for more complicated
sorts of solution phases
– Chemical ordering
– Gas phases
– Reciprocal systems (eg. Molten salts)
– Oxide phases (eg. Spinels, Halite)
– Liquids with short range ordering
• Oxides, slags, mattes
Chemical Ordering
Chemical ordering
• Phase diagrams with ordered phases are quite
common
• Generally concerned with BCC_B2 (ordered
BCC_A2), FCC_L12 and FCC_L10 (ordered
FCC_A1)
• Ideally we would like to be able to model the
disordered phases and ordered phases within
the same dataset
• These can give rise to first order
transformations (two phase regions) or
second order transformations (dashed line)
Fe-Ni
L12
A ssessed A u - C u p h ase d i ag r am .
Au-Cu
L10
L12
L12
A ssessed A l - N i p h ase d i ag r am .
Al-Ni
FCC_A1
FCC_L12
A ssessed C u - Z n p h ase d i ag r am .
Cu-Zn
BCC_A2
Order – disorder
reaction
BCC_B2
A ssessed T i - F e p h ase d i ag r am .
Fe-Ti
BCC_A2
BCC_B2
Example
• (A, B)0.5 (A, B)0.5
• Assume
– G(A:A) and G(B:B) = 0
– G(A:B) = G(B:A) = -10000
• At low temperature A and B atoms will occupy
different sublattices (ordered)
• As the temperature increases more mixing will
occur until eventually the solution becomes
disordered
A:B
Gibbs energy of ordering
Model for chemical ordering
• We would like to model thermodynamic
properties of the disordered phase from
experimental data (eg phase diagram,
enthalpies) using standard Redlich Kister
expression
• Add on Gibbs energy of ordering to this
disordered data set
• Compound Energy Formalism has the correct
behaviour but gives residual Gibbs energy
which needs to be subtracted off
𝐺𝑜𝑟𝑑 = 𝐺𝑑𝑖𝑠𝑜𝑟𝑑 + 𝐺𝐶𝐸𝐹 − 𝐺𝑐𝑜𝑟𝑟
𝐺𝐶𝐸𝐹 = 1𝑍𝐴 2𝑍𝐵 𝐺𝐴:𝐵 + 1𝑍𝐵 2𝑍𝐴 𝐺𝐵:𝐴
+2𝑅𝑇 𝑍𝐴 ln 𝑍𝐴 + 𝑍𝐵 ln 𝑍𝐵
+𝐺 𝑒𝑥
𝐺𝑐𝑜𝑟𝑟 = 𝑥𝐴 𝑥𝐵 𝐺𝐴:𝐵 + 𝑥𝐵 𝑥𝐴 𝐺𝐵:𝐴
+2𝑅𝑇 𝑥𝐴 ln 𝑥𝐴 + 𝑥𝐵 ln 𝑥𝐵
+𝐺 𝑒𝑥
• The model works well for a wide range of
chemical ordering reactions
• However it does not take into account short
range effects
• For more sophisticated treatments it may be
necessary to use 4 or even 8 sublattices –
difficult to use
Gas phase
Model for the gas phase
• We assume that the gas phase consists of lots of
different chemical species mixing ideally
• Generally this is a good approximation for low
pressures and high temperatures
𝑛
𝐺=
1
𝑛
𝑦𝑖 𝐺𝑖 + 𝑅𝑇 ln 𝑃 +
1
𝑅𝑇 ln 𝑦𝑖
• 𝑦𝑖 and 𝐺𝑖 are the fraction of each species and
their individual Gibbs energies
• 𝑦𝑖 are determined by minimising the Gibbs
energy for a chosen set of 𝑥𝑖 , T and P
Gas phase speciation
Gibbs energy of mixing
Data for gas phase species
• Data for gas phase species usually derived
through mixture of experiment and statistical
mechanics
• Structure of species, bond distances, vibrational
and electronic frequencies deduced from
spectroscopy
• Statistical mechanics then used to calculate heat
capacity and entropy
• Enthalpy of formation derived from enthalpies of
combustion, vapour pressure measurements
• Ab-initio calculations of enthalpy of formation
generally not accurate enough
Isomers
• Ab-initio can be very useful in determining the
relative stability of isomers eg NaCeI4
• Both isomers will be present in the
equilibrium mixture
Reciprocal systems
Reciprocal systems
• This is where mixing occurs
on more than one sublattice
• eg. Fcc (Fe, V)1 (C, Va)1
• The reciprocal reaction
V:Va + Fe:C = V:C + Fe:Va
strongly favours Fe and VC.
• This strong reaction reduces
solubility and creates a
miscibility gap across the
diagonal
Crystalline oxide phases
Crystalline oxide phases
• Halite phase
– “FeO” (wüstite), CaO (lime), MgO (periclase), NiO (bunsenite)
• Spinel phase
– Fe3O4 (magnetite), FeAl2O4 (hercynite), FeCr2O4 (iron chromite),
MgAl2O4 (spinel)
• Corundum
– Al2O3 (sapphire), Cr2O3, Fe2O3 (haematite)
• Olivine
– Mg2SiO4 (forsterite), Fe2SiO4 (fayalite)
• Clinopyroxene
– MgSiO3 (clinoenstatite), pigeonite, CaMgSi2O6 (diopside)
• Orthopyroxene
– MgSiO3 (enstatite), FeSiO3 (ferrosilite)
Halite phase
• (Ca+2, Fe+2, Fe+3, Mg+2, ….., Va)1 (O-2)1
• Wüstite (Fe+2, Fe+3, Va)1 (O-2)1
– Allows the range of homogeneity to extend to
compositions richer in oxygen than FeO
Spinel phase: Fe3O4
• Can be modelled as
(Fe+2, Fe+3)1 (Fe+2, Fe+3)2 (O-2)4
• The first sublattice is tetrahedrally coordinated and the
second octahedrally coordinated
• This is a reciprocal system with the corners of the
square
–
–
–
–
(Fe+2)1 (Fe+2)2 (O-2)4
(Fe+2)1 (Fe+3)2 (O-2)4
(Fe+3)1 (Fe+2)2 (O-2)4
(Fe+3)1 (Fe+3)2 (O-2)4
(net charge -2)
(net charge 0)
(net charge -1)
(net charge +1)
• The charge neutral material is represented the a line on
the reciprocal diagram. The position on the line is
determined by minimum in Gibbs energy
Inverse spinel
“Normal” spinel
More components
Complicated liquids
Complicated liquids
• A number of system show very negative Gibbs
energies of mixing in the liquid phase
• Not possible to use the standard RedlichKister model to represent the thermodynamic
properties
• Examples
– Cu-S, Fe-O, CaO-SiO2
Miscibility gaps
Three main approaches to assess data
for these systems
• Associated solution model
– Assume the liquid contains species (like the gas) but they
interact in a non-ideal way
• Ionic Liquid model
– Uses the two sublattice model with ions, charged species
and charge vacancies. Essentially it in one big reciprocal
system
• Modified quasichemical model focusing on bonds
rather than chemical entities
• All models have their strong points and failings. We do
need a universal model for the liquid. We don’t have
one yet !
Associated solution model
• In a gas phase there will be a number of species mixing
ideally
– C, C2, C3, C4, C5, CO, CO2, C2O, C3O2, O, O2, O3
• The amount of each species will be determined
through obtaining a minimum in the Gibbs energy for a
given temperature and pressure
• The associated solution model postulates that there
are similar species in the liquid, or at least they might
correspond to compositions of stable crystalline phases
– CaO, Ca2SiO4, CaSiO3, SiO2
• Again the amount of these species will be obtained my
minimising the Gibbs energy. Note that these species
interact in a complicated way
Good points and bad points
• It works !
• Tends to underestimate miscibility gaps
• Has various parameters which can be used to
improve agreement with experimental data
• Not really very predictive
• No unique set of parameters eg. play off between
species Gibbs energies and interactions between
the species
• Not really physically based
• Does have capability to model other sorts of
properties eg. viscosities
Ionic Liquid model
• Based on Compound Energy Model
– eg. Liquid salt system could be represented by
(Li+, Cs+)1 (F-, I-)1
– This is a reciprocal system and there is a very
strong tendency for the small ions to bond
together and for the big ions to bond together
– The combination LiF+CsI is strongly favoured over
LiI+CsF
– This leads to big miscibility gap across the diagonal
of the reciprocal system ….. which is found
experimentally, although this model does tend to
over predict miscibility gaps
• Liquids with different valency ions
(Li+, Ca+2)p (F-, Br-)q
– Ratio of sites p:q between the two sublattices
varies from 1:1 to 1:2
– The model defines q to be the average charge on
the cation sublattice and p to adopt a value to
maintain electroneutrality
– Can be extended to model silicates
(Ca+2)p (O-2, SiO4-4, SiO3-2, SiO2)q
– Looks rather different from associate model but
under certain circumstances the data are
interchangeable
• Ionic liquid model can be used for metal –
oxygen systems. Here it is necessary to
introduce charged vacancies
(Fe+2, Fe+3)p (O-2, Va-q)q
• Again q is set to be average charge on the
cation sublattice. Here though the vacancies
also have a charge of –q
• For simple systems the model works well,
however it always tends to over-predict
miscibility gaps. This can make it very difficult
to model multicomponent systems
Good points and Bad points
• It works well for simple systems
• Tends to over-estimate miscibility gaps and
doesn’t really have parameters available for
controlling them
• Makes modelling of multicomponent systems
difficult
Modified Quasi-chemical model
• Discussed regular solutions during last session
• Total lattice energy given by
𝐸 = 𝑛𝐴𝐴 𝐸𝐴𝐴 + 𝑛𝐴𝐵 𝐸𝐴𝐵 + 𝑛𝐵𝐵 𝐸𝐵𝐵
• where 𝑛𝐴𝐴 , 𝑛𝐴𝐵 and 𝑛𝐵𝐵 are the number of AA, A-B and BB bonds respectively
• Assumed that the interaction has no effect on
the order within the solution ie. there is
random mixing of the atoms
• Quasi-chemical model doesn’t make this
assumption
(A-A)pair + (B-B)pair = 2(A-B)pair
;ω
𝑍𝑁𝑥𝐴 = 2𝑛𝐴𝐴 + 𝑛𝐴𝐵
𝑍𝑁𝑥𝐵 = 2𝑛𝐵𝐵 + 𝑛𝐴𝐵
Where Z is the coordination number, 𝑛𝐴𝐴 , 𝑛𝐵𝐵 and 𝑛𝐴𝐵
the number of bond pairs per mole
𝑥𝑖𝑗 = 𝑛𝑖𝑗 𝑛𝐴𝐴 + 𝑛𝐵𝐵 + 𝑛𝐴𝐵
𝑥𝑖𝑗 = pair fraction
In the quasi-chemical model it is these pairs that are
distributed randomly over pair sites
𝑥𝐴𝐴
𝑥𝐵𝐵
𝑥𝐴𝐵
𝑖𝑑
∆𝑆 = −𝑅 𝑥𝐴𝐴 ln 2 + 𝑥𝐵𝐵 ln 2 + 𝑥𝐴𝐵 ln
𝑥𝐴
𝑥𝐵
2𝑥𝐴 𝑥𝐵
−𝑅(𝑥𝐴 ln 𝑥𝐴 + 𝑥𝐵 ln 𝑥𝐵 )
𝑥𝐴𝐵
∆𝐻 =
𝜔
2
Often the minimum is not at 50%
0
[30Kaw] Calorimetry 800°C
[71Nay2] Calorimetry 800°C
[80Som] Calorimetry 800°C
[80Som] Calorimetry 860°C
Enthalpy of Mixing (kJ/mol)
-5
[80Som] Calorimetry 940°C
-10
-15
Calculated at 800°C
-20
0.0
0.2
0.4
0.6
Mole fraction Sn
0.8
1.0
• Requires different coordination numbers for
the components
𝑍𝑆𝑛 = 2𝑍𝑀𝑔
• 𝜔 may be expanded as a power series in
either 𝑥𝑖 or 𝑥𝑖𝑖
• Same approach can be used for oxide systems
– Is often necessary to introduce ternary
parameters or even associated species
– For some systems the model does not appear to
work well eg. K2O-SiO2
Quasi-chemical approach to molten salts
LiF
o
0.4 492
0.2
(848 )
o
0
75
55
0
0.8
0.8
o
(857 )
0
80
478
o
0.8
0
75
0
70
0
65 0
70
800
KF
0.6
0.6
0.6
60
0
65
0
605
o
500
0.2
70
0
718
o
0
60
LiCl
o
(610 )
0.2
0
75
0
70
0
50
0
55
345
o
0.2
65
60 0
0
0.4 353
0.4
0.4
o
70
0
75
0
o
0.6
Mole fraction
0.8
KCl
o
(771 )
LiF-KCl diagonal
900
o
o
Temperature ( C)
850 848
Gabcova et al
Berezina et al
Margheritis et al
800
Liquid
771
750
o
LiF + Liquid
718
700
o
LiF + KCl
650
0
0.2
0.4
0.6
Mole fraction KCl
0.8
1.0
• The quasi-chemical model is really the only one
which works for molten salts
• It is horrendously complicated
• The associate model predicts complete mixing
between the pure salts – incorrect entropy of
mixing
• The ionic liquid model overestimates the
tendency towards formation of miscibility gaps
• The quasichemical model reflects to tendency for
preferential bonding between pairs of cations and
anions (first nearest neighbour interactions)
• Second nearest neighbour interactions require
use of “quadruplets”
Good points and bad points
• Seems to have better predictability for high
affinity metallic systems
• Appears to work well for oxide systems
(according to the authors) but beware of small
ternary interactions which have a big effect,
the possible need for associates, and systems
which are just not well represented
• The only really good model for molten salts ….
but it is a nightmare mathematically
General thoughts about liquids
• We do not have a good model
– Function of temperature (strange cp behaviour,
metallic glasses)
– Function of composition (metals, slags, salts, aqueous)
• Lots of miscibility gaps
– Cu and Cu2S, Cu2S and S, CaSiO3 and SiO2, Cu and Cu2O
• We would like a single model to represent the
thermodynamic properties across all composition
and temperature space
• Challenge for the next generation !