Calculations of phase diagrams using

Thermo-Calc software package

Equilibrium calculation using the Gibbs energy minimisation1. The Gibbs energy for a system2. The Gibbs energy for a phase

Unary system: Sn (calculation of melting temperature, plotting thermodynamic functions)Phase diagram for the Sn-Bi system (Temperature - Composition)Calculation of invariant reaction (T, phase compositions, enthalpy)Calculation of thermodynamic properties of liquid phaseCalculation of phase fraction diagram for Bi concentration 5, 25 and 43 mol.%Scheil solidification simulation for Sn-Bi alloys

Calculation of phase diagram for Fe-C system1. Stable diagram2. Metastable diagram

Content

The Gibbs for a system and for a phase

The Gibbs energy of the system is equal to im

iiGnG ,

where ni is the amount of phase i and imG is the Gibbs energy of phase i depending on

pressure, temperature and composition. To find equilibrium at given temperature, pressure and composition of the system it is necessary to find minimum of its Gibbs energy under composition constraint:

kik

ii cxn ,

where ikx is content of component k in phase i and ck is bulk concentration of component

k.

1+3+5: G135=n1G1+n3G3+n5G5

1+2+4: G124=n1G1+n2G2+n4G4

1+3+4: G134=n1G1+n3G3+n4G4

2+4+5: G245=n2G2+n4G4+n5G5

2+4+6: G246=n2G2+n4G4+n6G6

2+3+5: G235=n2G2+n3G3+n5G5

3+4+6: G346=n3G3+n4G4+n6G6

3+5+6: G356=n3G3+n5G5+n6G6

xA

xB

The Gibbs for a system and for a phase

Elements The Gibbs energy of pure element i, referred to the enthalpy for its stable state at room temperature (298.15 K), is described by following equation

97132ln)15.298()( hTgTfTeTdTTcTbTaKHTGGHSERi ii

For element having magnetic ordering GHSER is referred to para-magnetic state and additional term accounting magnetic contribution is included

)()1ln( fRTG mag , where β is average magnetic moment, τ – is critical temperature i.e Curie or Neel temperature Stoichiometric compounds The Gibbs energy of the compound AaBb is expressed as

)()15.298()15.298( TfKHbKHaG BABA ba ,

where a and b are stoichiometric number and f(T) is identical to equation for element.

The Gibbs for a system and for a phase

Model for solution Substitutional solutions

Emi

iii

iim GxxRTGxG ,ln

xi is mole fraction of component, first term corresponds to mechanical mixture, second one is ideal entropy of mixing contribution and third one is excess energy of mixing. Interaction between two elements is expressed by Redlich-Kister equation

Mixing parameter

ijL can be temperature dependent.

The simplest model is regular solution model

ijji

Em LxxG 0,

Parameter

ijL0 does not depend on temperature.

)(,jiijji

Em xxLxxG

Property diagrams for unary system (Sn)

Tm=505 K (232°C) L=Sn-Bct Htr=-7.029 kJ/mol

Phase diagram of the Sn-Bi phase diagram

Gm-curves for the Sn-Bi phase diagram

Calculation of enthalpy (H) of reaction 1Liq=A(Sn)+B(Bi)

X(Sn) XLiq X(Bi)c=X(Bi) - X(Sn)

a=X(Bi) - XLiq

b=XLiq - X(Sn)

H(Tinv)=AH(Sn)+BH(Bi)-HLiq

H(412K)=-7.717 kJ/mol-at.

c

ba

Stoichiometric coefficients A and B of invariant reaction are calculated by Lever rule

A=a/cB=b/c

Calculation of thermodynamic properties of liquid phase

Thermodynamic functions of mixing(enthalpy, entropy, Gibbs energy) in Liquid phase at 300°C

Activity of Bi and Sn in Liqiud phase at 300°C

Phase fraction diagrams

I II III I

II III

Scheil solidification simulation

Composition T System Liquid

Liquid fraction

T0 X0 X0 1 T1 X0 X1 n1=1·ξ(liq)1 T2 X1 X2 n2= n1·ξ(liq)2 Tn Xn-1 Xn nn= nn-1·ξ (liq)n

ξ(liq)n – fraction of liquid calculated by lever rule at Tn

D.R. Askeland, P.P. Phule„The science and engineering of materials“ p. 370

Scheil solidification simulation for Sn-5Bi alloy

Scheil solidification simulation for Sn-25Bi alloy

Scheil solidification simulation for Sn-43Bi alloy

Phase relations in the Fe-C system

Fig.1. Stable diagram Fig. 2. Metastable diagram