Upload
ethan-anderson
View
238
Download
4
Tags:
Embed Size (px)
Citation preview
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 nxn ,
where ikx is content of component k in phase i and nk 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
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 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
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)/cb=(XLiq-X(Sn))/c
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
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
Nonequilibrium solidification of Cu-Ni alloy 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