### Presentation2

```Computational Thermodynamics
2
Outline

Compound energy formalism

Stoichiometric compound

Wagner-Schottky model

Ionic liquid
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
s
i
s
n
y =
N
s
i
Where nis is the number of atoms of component i on sublattice s, and Ns is total
number sites on the sublattice s.
Compound energy formalism
This can be generalised to include vacancies, which are important to consider in
interstitial phases
s
n
yis = s i s
nVa + å ni
i
Mole fractions are directly related to site fractions by the following relationship
åN y
s s
i
xi =
s
s
s
N
1y
å ( 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
N s!
Wp = Õ
s
Õn !
s
i
i
and the molar Gibbs ideal mixing energy is
id
id
Gmix
= -TSmix
= RT å N s å yis lnyis
s
Vacancies contribute in that equation!
i
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:
GAnBm = 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: an extra sublattice !
(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:
GA:B – Gibbs energy of formation of pure AB compound
GA:A and GB:B – Gibbs energy of formation of pure A and B, respectively, in the
crystal structure of AB compound
GB:A – must not be use
LA,B:A=LA,B:B = LA,B:* - deviation toward B
LA:B,A=LB:A,B = L*:A,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 (CiVi+)P(AjVj-,Bk0,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 = å y j (-v j ) + yvQ
j
Q = å yi vi
i
where vi 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
Gmionic = åå yCi yA j 0GCi :A j + QyVa å yCi 0GCi + Qå yBk 0GBk
i
cnf
xs
ionic
m
S
j
i
k
é
æ
öù
= -R êPå yCi ln yCi + Q ççå yA j ln yA j + yVa ln yVa + å Bk ln yBk ÷÷ú
êë i
è j
øúû
k
Gmionic = ååå yi1 yi2 y j Li1,i2 : j + åå yi1 yi2 yVa Li1,i2 :Va
i1
i2
j
i1
i2
+ååå yi1 y j1 y j2 Li:j1, j2 + åå yi y j yVa Li: j,Va
i
j1
j2
i
j
+ååå yi yi yk Li: j,k + åå yi yk yVa Li:k,Va + åå yk1 yk2 Lk1,k2
i
j
k
i
k
k1
k2
Ionic liquid
Database:
SPECIES PB+2
SPECIES TE-2
PB1/+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**324238450*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 !
```