2

Outline
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Compound energy formalism
A sublattice phase can be envisaged as being composed of interlocking sublattices on which the various components can mix. It is usually crystalline in nature but the model can also be extended to consider ionic liquids where mixing on particular 'ionic sublattices' is considered.
Simple body-centred cubic structure with preferential occupation of atoms in the body-centre and comer positions.

Compound energy formalism
To work with sublattice models it is first necessary to define what are known as site fractions, y . These are basically the fractional site occupation of each of the components on the various sublattices y i s = n i s
N s
Where n i s is the number of atoms of component i on sublattice s, and N number sites on the sublattice s.
s is total

Compound energy formalism
This can be generalised to include vacancies, which are important to consider in interstitial phases y i s = s n
Va
+ n i s
å n i s i
Mole fractions are directly related to site fractions by the following relationship x i
=
å
å
N s s
(
N s y i s
1
y s
Va
) s

Compound energy formalism
The ideal entropy of mixing is made up of the configurational contributions by components mixing on each of the sublattices. The number of permutations which are possible, assuming ideal interchanges within each sublattice, is given by the following equation
Wp
=
Õ s s N
Õ n i s
!
!
i and the molar Gibbs ideal mixing energy is id
G mix
= -
TS id mix
=
RT
å
N s s i
å y i s ln y i s
Vacancies contribute in that equation!

Compound energy formalism

Compound energy formalism
The Gibbs energy reference state is effectively defined by the 'end members ' generated when only the pure components exist on the sublattice. Envisage a sublattice phase with the following formula (A, B)
1
: (C, D)
1
.
It is possible for four points of 'complete occupation' to exist where pure A exists on sublattice 1 and either pure B or C on sublattice 2 or conversely pure B exists on sublattice 1 with either pure B or C on sublattice 2.

Compound energy formalism

Compound energy formalism

Compound energy formalism

Compound energy formalism

Compound energy formalism

Compound energy formalism

Compound energy formalism

Compound energy formalism

Compound energy formalism

Stoichiometric compound

Stoichiometric compound
Let’s take a look at the sublattice model again: (A,B):(C,D)
If we have components A and C only, then sublattices are occupied: (A):(B) what gives as (in this case) a stoichiometric compound AB
The Gibbs energy of this kind of compound is usually described as follows:
G
A n
B m
=
A
+
B
×
T
+ n
×
GHSERA
+ m
×
GHSERB

Stoichiometric compound
Database file:
PHASE PBTE % 2 1 1 !
CONSTITUENT PBTE :PB : TE : !
PARAMETER G(PBTE,PB:TE;0) 2.98150E+02 -
6.50554752E+04+5.45815447E+00*T+GHSERTE#+GHSERPB#; 3.00000E+03 N REF0 !

Wagner-Shottky model
Variation of the Gibbs energy of formation of compound within a small composition range can be described by Wagner-Schottky model. The model describes homogeneity range as a function of various types of defects
(A,X):(B,Y)
Types of defects:
• Anti-site atoms, i.e. B on sublattice for A and A on sublattice for B
• Vacancies
• Interstitials
• A mixture of the above defects

Wagner-Shottky model
Interstitial defect:
(A) a
:(B) b
:(Va,A,B) c

Wagner-Shottky model
We can find information from the crystal structure. For example, in some phases with B2 structure we have 2 sublattices: one often has anti-site defect, another one vacancies
(A,B)
1
:(B,Va)
1
But since both sublattices are identical from the crystallographic point of view, one has to include all defects on both sublatticies
(A,B,Va)
1
:(B,A,Va)
1

Wagner-Shottky model
Parameters of the model:
G
A:B
G
A:A
– Gibbs energy of formation of pure AB compound and G
B:B
– Gibbs energy of formation of pure A and B, respectively, in the crystal structure of AB compound
G
B:A
L
A,B:A
=L
A,B:B
L
A:B,A
– must not be use
=L
B:A,B
= L
A,B:*
=
L*:A,B
- deviation toward B
- deviation toward A

Wagner-Shottky model
Database file
PHASE PBTE % 2 1 1 !
CONSTITUENT PBTE :PB,TE : PB,TE : !
PARAMETER G(PBTE,PB:PB;0) 2.98150E+02 1.74091200E+05 +2*GHSERPB#; 3.00000E+03 N REF0 !
PARAMETER G(PBTE,TE:PB;0) 2.98150E+02 0.0 ; 3.00000E+03 N REF0 !
PARAMETER G(PBTE,PB:TE;0) 2.98150E+02 -6.50554752E+04+5.45815447E+00*T+GHSERTE#+GHSERPB#;
3.00000E+03 N REF0 !
PARAMETER G(PBTE,TE:TE;0) 2.98150E+02 1.57960355E+05+2*GHSERTE#; 3.00000E+03 N REF0 !
PARAMETER G(PBTE,PB,TE:*;0) 2.98150E+02 -8.68054056E+04-3.26572670E+01*T; 3.00000E+03 N REF0 !
PARAMETER G(PBTE,*:PB,TE;0) 2.98150E+02 0; 3.00000E+03 N REF0 !

Ionic liquid
The ionic liquid model is given by (C i
Vi+ )
P
(A j
Vj,B k
0 ,Va)
Q where P and Q are the number of sites on the cation and anion sublattice, respectively. The stoichiometric coefficients P and Q vary with the composition in order to maintain electroneutrality.
P
=
y j j
( )
+ y v
Q
Q
=
y i i v i where v i is the valency of ion i. The summation over i is made for all anions, summation over j is made for all cations.

Ionic liquid
According to this model, the Gibbs free energy of the liquid phase can be expressed as: srf ionic
G m
= å å y
C i y
A j
0
G
C i
: A j
+
Qy
Va
å y
C i
0
G
C i
+
Q
å y
B k
0
G
B k i j i k cnf ionic
S m xs
G ionic m
=
= -
é
ê
ë
å i y
C i
å å å y i
1 y i
2 ln y
C i y j
L i
1
, i
2
: j
+
Q
æ
å y
A j
+ å j
å y i
1 ln y
A j y i
2 y
Va
+ y
Va
L i
1
, i
2
: Va ln y
Va i
1 i
2 j i
1 i
2
+ å å å y i
1 y j
1 y j
2
L i : j
1
, j
2
+ å å y i y j y
Va
L i : j , Va i j i
+ å j
1 j
2
å å i j k y i y i y k
L i : j , k
+ å i
å k y i y k y
Va
L i : k , Va
+ å k
1
å k
2 y k
1 y k
2
+
L k
1
, k
2
å k
B k ln y
B k ø ú
ù
ú

Ionic liquid
Database:
SPECIES PB+2 PB1/+2!
SPECIES TE-2 TE1/-2!
PHASE IONIC_LIQ:Y % 2 .0247462 2 !
CONSTITUENT IONIC_LIQ:Y :PB+2 : TE-2,VA,TE : !

Ionic liquid
PARAMETER G(IONIC_LIQ,PB+2:TE-2;0) 2.98150E+02 -1.8541625E+04
-2.2751140E+02*T+GHSERPB#+GHSERTE#; 3.00000E+03 N REF0 !
PARAMETER G(IONIC_LIQ,PB+2:VA;0) 2.98150E+02 -2977.961+93.949561*T
-24.5242231*T*LN(T)-.00365895*T**2-2.4395E-07*T**3-6.019E-19*T**7;
6.00610E+02 Y
-5677.958+146.176046*T-32.4913959*T*LN(T)+.00154613*T**2; 1.20000E+03 Y
+9010.753+45.071937*T-18.9640637*T*LN(T)-.002882943*T**2+9.8144E-08*T**3
-2696755*T**(-1); 2.10000E+03 N REF0 !

Ionic liquid
PARAMETER G(IONIC_LIQ,TE;0) 2.98150E+02 -17554.731+685.877639*T
-126.318*T*LN(T)+.2219435*T**2-9.42075E-05*T**3+827930*T**(-1);
6.26490E+02 Y
-3165763.48+46756.357*T-7196.41*T*LN(T)+7.09775*T**2-.00130692833*T**3
+2.58051E+08*T**(-1); 7.22660E+02 Y
+180326.959-1500.57909*T+202.743*T*LN(T)-.142016*T**2+1.6129733E-05*T**3-
24238450*T**(-1); 1.15000E+03 Y
+6328.687+148.708299*T-32.5596*T*LN(T); 1.60000E+03 N REF0 !
PARAMETER G(IONIC_LIQ,PB+2:TE-2,VA;0) 2.98150E+02 3.7254867E+04 -
1.6899525E+01*T; 3.00000E+03 N REF0 !
PARAMETER G(IONIC_LIQ,PB+2:TE-2,TE;0) 2.98150E+02 -1.4689488E+04
+9.2350161E-01*T; 3.00000E+03 N REF0 !