From Chapter 9 of Saunders and Miodownik ’ s CALPHAD book Computational methods
9
From Chapter 9 of Saunders and Miodownik’s CALPHAD book Computational methods
description
From Chapter 9 of Saunders and Miodownik ’ s CALPHAD book Computational methods. for many years: equil const used to express (the abundance of certain species) in terms of the amounts of other arbitrarily chosen species for one rxn → one equil const needed - PowerPoint PPT Presentation
Transcript of From Chapter 9 of Saunders and Miodownik ’ s CALPHAD book Computational methods
From Chapter 9 of Saunders and Miodownik’s CALPHAD book
Computational methods
• for many years: equil const used to express (the abundance of certain
species) in terms of the amounts of other arbitrarily chosen species
for one rxn → one equil const needed the # of rxn ↑ → many equil const needed but not
available
∴ CALPHAD (CALculation of PHAse Diagram) method: considering the G energy of all phases → true equil calculation by of
mii
ii GNGGNG or )(
component
phase
→ then, the # of unknowns considerably reduced in comparison to equil const approach
• in the case of binary and ternary phase equil
→ analytic method is possible to calculate certain 2-phase equil
ex) binary eutectic, negligible solubility in the terminal solid phase, ideal mixing in the liq, equil btw liq and a(A) :
Fig. 9.1. Simple eutectic system with ideal mixing in the liquid and negl solid solubility in the terminal solid phases, and .
Fig. 9.2. G/x diagram at 850K for eutectic system in Fig. 9.1.
A
lA GG no solubility
lA
lA
lA
lAliq
lA
lA
lA
lA
lA
lA
SSHH
T
STHTSH
GGor
from
this is a special case
not a good method
0
function T from calculated finally exp
ln
solu) ideal assuming(lnln
A
lA
lA
A
lA
lA
lA
lA
lA
lA
lA
x
xRTG
x
GxRT
xRTGaRTGG
)( liq is st standard if lGthenG lAA
• general method: equil btw a solu phase & stoichiometric
according to Hillert,
0 GGxGl
ii
0
GGxG
GxGxGand
GGhere
lii
liiii
lii
BA GGG or
how to calculate T ?
general solution : liq, liquidus of Φ (MTDATA used this methodology)
Newton-Raphson method:
for fixed xiΦ → first choosing arbitrary T
→ iteratively stepping T until ΔG below
a small, defined accuracy limit
• a more general case of both phases with a solubility range
as xiΦ is not fixed, by fixing T and then finding x and xliq
from xo (To), however, it is converted to N in this section)}()({ xGxGT lo
liq
x xlxo
MNNN
eqbalancemassgeneral
GNG
ii
m
and
w.r.t. onminimizati
liq
xi
the alloy first assumed in single-phase → an arbitrary amount of liq introduced → min G reached → fixing liq comp and changing → min G reached again → final minimization of G
if the alloy chosen outside 2-phase equil (xi) →
xiliq calculated to minimize G of +l → by fixing
xiliq and changing the amount of → G ↓ → finally
reaching equil btw and liq
• multi-component sys
calculation must be defined: # of degree of fredom reduced G of the sys calculated some iterative tech to minimize G how to reduce f (= v) : by defining a series of constraints such as balance
in ionic sys composition range of each phase
one of the earliest example of G minimization by White (1958) → chem equil in an ideal gas mixture of O, H, N with the species H, H2, H2O, N, N2, NH, NO, O, O2, OH
jii
ij
iii
iitot
Nxa
aRTGG
GNG
ln
# of atoms of element j (O, H, N only) in species (H, H2, H2O, NH, NO…)
→ presenting 2 methods of G minimization the typical one is to use Lagrange’s method of multipliers: irrespective of # of phases and components
here, finding NΦ (fraction or amount of each phase), xk
Φ
(mole fraction of k in the phase Φ) which minimize G this is the equil calculation
Then, why this method ?
with the constraints of
p
mGNG1
0 ,0111
p
ii
c
kk xNNx
∴ by defining
p c
kk
p
kk
c
kk
p
m xxNNGNagrangianL1 1111
)1()()(
00
adding ∑0·k + ∑0·Φ to the original eq not changing its value
→ minimum condition of L ?
p 1, 0
p 1, and c 1, 0
c 1, 0
p 1, 0
L
kxL
kL
NL
k
k
# of p eq
# of c eq
# of p x c eq
# of p eq
thus, to solve 2p + c (p+1) non-linear eq by the Newton-Raphson method
Germany SOLGASMIX ChemSage FactSage
Canada FACT FactSage
Sweden Thermo-Calc
England MTDATA
Germany, MPI Lukas program
