Computational Thermodynamics2

Outline Compound energy formalism

Stoichiometric compound

Wagner-Schottky model

Ionic liquid

Compound energy formalismA sublattice phase can be envisaged as being composed of interlocking sublatticeson which the various components can mix. It is usually crystalline innature but the model can also be extended to consider ionic liquids where mixingon particular 'ionic sublattices' is considered.

Simple body-centred cubic structure with preferential occupation ofatoms in the body-centre and comer positions.

Compound energy formalismTo 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 thecomponents on the various sublattices

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

Mole fractions are directly related to site fractions by the following relationship

Compound energy formalismThe 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

and the molar Gibbs ideal mixing energy is

Vacancies contribute in that equation!

Compound energy formalism

Compound energy formalismThe 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:

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 compoundGA:A and GB:B – Gibbs energy of formation of pure A and B, respectively, in the crystal structure of AB compoundGB:A – must not be use LA,B:A=LA,B:B = LA,B:* - deviation toward BLA: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 liquidThe ionic liquid model is given by (Ci

Vi+)P(AjVj-,Bk

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.

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 liquidAccording to this model, the Gibbs free energy of the liquid phase can be expressed as:

Ionic liquidDatabase:

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 !