CT – 3: Equilibrium calculations:
Minimizing of Gibbs energy, equilibrium conditions as
a set of equations, global minimization of Gibbs
energy, driving force for a phase
Equilibrium conditions
dU = T.dS-p.dV + j j.dnj
(see CT-2)
Conjugated properties: T,p,j – intensive, S,V,nj – extensive
At constant entropy, volume, and nj , equilibrium is characterized by minimum of the internal energy.
Most proper conditions: p,T,nj
dG = V dp – S dT + j j dnj
(mole fraction: xi = ni/N, N = j nj)
G may have several minima.
That with the most negative value of G is „global minimum“ which
corresponds to the „stable equlibrium“ and another ones are
„local minima“ and correspond to the „metastable equilibria“
Stable and metastable states
Unstable
state state
Unstable
Metastable
state
Metastable
state
Stable state
Stable state
Equilibrium conditions – cont.
For determination of phases presented in equilibrium:
analytical expression of G needed.
Total Gibbs energy:
G =  m.Gm
(m  0, it is amount of phase )
Amount of components i: Ni = N.xio
Introduce: xi =  m. x  i - lever rule
Equilibrium condition:
min (G) = min ( m. Gm(T,P, x  i or y (l,)k ))
(xi is definite function of yk(l, ); m, x  i are unknowns)
PHASE DIAGRAMS: composition and types of phases
•
Rule 1: If we know T and Co, then we know:
--the composition and types of phases present.
•
Example:
Adapted from Fig. 9.2(a), Callister 6e.
(Fig. 9.2(a) is adapted from Phase
Diagrams of Binary Nickel Alloys, P.
Nash (Ed.), ASM International,
Materials Park, OH, 1991).
Cu-Ni
phase
diagram
6
PHASE DIAGRAMS: composition and amount of phases
•
Rule 2: If we know T and Co, then we know:
--the composition and amount of each phase.
•
Example:
Cu-Ni system
Adapted from Fig. 9.2(b), Callister 6e.
(Fig. 9.2(b) is adapted from Phase Diagrams
of Binary Nickel Alloys, P. Nash (Ed.), ASM
International, Materials Park, OH, 1991.)
Lever rule: mL/m = CoC7 /CoCL
Types of phase diagrams of Cu
Equilibrium conditions as a set of equations
Equilibrium condition using chemical potential:
Constraints relating m, xi and Ni used to eliminate
variables m and Ni :
Gi (T,P,xi) = Gi (T,P,xi)
(i = 1,…,c,  = 1,…p-1,  =  + 1,…,p)
By the definition:
Gi (T,P,xi) = i (i = 1,…,c,  = 1,…,p)
Nonlinear equations – appropriate iteration algorithm
Unknown: xi and i
(for stoichiometric phases, modifications are necessary)
Gm as function of site fractions yk(l, )
instead of mole fractions xi
Lagrange-multiplier method
Constraints:
(1) Total amount of each component Ni
is kept constant
(2) Sum of site fractions in each sublattice
is equal 1
(3) Sum of charge of ionic species in each phase is
equal 0
Constraints mathematically
LFS - CT
Lagrange-multiplier method-cont.
Each constraint is multiplied by „Lagrange multiplier“ and added to
the total Gibbs energy
min (G) = min ( m. Gm(T,P, x  i or y (l,)k ))
to get a sum L.
If all constraints are satisfied, L is equal to G and a minimum of L
is equivalent to a minimum of the total Gibbs energy G.
Newton‘s method
To find x for which y=0: (also for searching the minimum of Gm)
(df/dx)x=xi . xi = -f(xi),
xi+1 = xi + xi
There exists cases, where this method diverges
LFS - CT
There exists cases, where Newton‘s
method diverges
Starting with x1 - diverges
Starting with x3 – solution on the left,
Starting with x4 – solution on the right,
x5, x6 - finally on the left – influence of
starting values on the result of minimization
LFS - CT
Newton-Raphson method
It is extension of Newton‘s method to more than
one variable (n equations for n unknowns).
All iterative techniques like the NewtonRaphson one need an initial constitution for
each phase in order to find the minimum of
Gibbs energy surface for the given conditions.
Thermocalc – starting point tools
Automatic starting values
Set-all-start values
Global minimization of the Gibbs energy
Miscibility gap problem (solution phases only)
LFS - CT
Compounds with fixed compositions
Equilibrium set of phases is given by the tangent
„hyperplane“ defined by the Gibbs energies of a
set of compounds constrained by the given
overall composition and with no compound with
a Gibbs energy below this hyperplane („global“
minimum).
Gibbs energies of a set of compounds
G/kJ.mol-1
A
xB
B
LFS -CT
Minimization techniques to find global
equilibrium
Gibbs energy surface of all solution phases is
approximated with a large number of „compounds“
which Gibbs energy has the same value as the
solution phase at the composition of the compound
(dense grid about 104 (100x100) compounds,
for multicomponent system about 106 such
compounds)
Search for hyperplane representing equlibrium for
the compounds is then carried out.
Minimization techniques to find global
equilibrium
When minimum for these „compounds“ has been
found, the „compound“ in this equilibrium set must
be identified with regard to which solution phase
they belong to.
Each „compound“ --- initial constitution of the
solution phase --- is used in a Newton-Raphson
calculation to find the equilibrium for the solution
phases (correct, not wrong)
Limitation of the method to find the global
equilibrium
T, p and overall composition must be known
For other conditions as starting point (e.g. activity of
components) –-- indirect procedure:
Overall composition calculate first and use it for a new
equilibrium calculation.
Conditions for a single equilibrium
The equilibrium conditions as a set of equations contain
fewer equation than unknowns – the difference = number
of degrees of freedom „f“
Therefore: „f“ extra conditions (equations) must be added to
select definitely single equilibrium
„Unknown state variable“ = „constant value“ :
Example:
For binary system i-j: (f=0)
T = 1273, p = 101352, xi = 0.1, i = -40000
Thermocalc: Fe – W – Cr system
set-condition t=1273 x(W)=0.15 x(Cr)=0.35 p=1E5 n=1
Conditions for a single equilibrium – cont.
For each calculation step: which and how many phases are
present (Gibbs energy description exist only for phases).
Calculation steps with different sets of phases may be
compared
The phases set with lowest Gibbs energy describes the
stable equilibrium
Example:
Thermocalc:
 rej ph *
 res ph liq bcc fcc sigma Chi R Mu
Example
Different starting points may give different sets of
equilibrium phases for the same overall
composition.
Check the total Gibbs energy for global minimum
(In new codes checked automatically.)
Ag-In system
Output from POLY-3, equilibrium number = 1, Ag-In system
Conditions:
T=500, X(IN)=2E-1, P=100000, N=1 DEGREES OF FREEDOM 0
Temperature 500.00, Pressure 1.000000E+05
Number of moles of components 1.00000E+00, Mass 1.09260E+02
Total Gibbs energy -3.15128E+04, Enthalpy -1.91077E+02, Volume 0.00000E+00
Overal composition
Component
Moles W-Fraction Activity
Potential Ref.state
AG
8.0000E-01 7.8982E-01 1.6046E-03 -2.6752E+04 SER
IN
2.0000E-01 2.1018E-01 5.2290E-06 -5.0558E+04 SER
FCC_A1#1
Status ENTERED Driving force 0.0000E+00
Number of moles 5.6253E-01, Mass 6.1400E+01
Mass fractions: AG 8.06153E-01 IN 1.93847E-01
HCP_A3#1
Status ENTERED Driving force 0.0000E+00
Number of moles 4.3747E-01, Mass 4.7860E+01
Mass fractions: AG 7.68872E-01 IN 2.31128E-01
Mapping a phase diagram




2 or 3 variables of the conditions are selected as axis
variables with lower and upper limit and maximal step.
All additional conditions – kept constant throughout
the whole diagram
Start: „initial equilibrium“ for Newton-Raphson
calculation (with all phases „entered“)
All results of calculations usually stored – any phase
diagram may be displayed at the end of calculations
Mapping a phase diagram – cont.
Example (in Thermocalc):
set-axis-variable 1 x(Ag) 0 1 .025
s-a-v
2 t 300 1200 10
map
(T in K)
„Stepping“

By stepping with small decrements of the temperature
(or enthalpy or amount liquid phase-generally one
variable) one can determine the new composition of
the liquid and then remove the amount of solid phase
formed by resetting the overall composition to the new
liquid composition before taking the next step
(Scheil solidification scheme: no diffusion in solid
phase, high diffusion in liquid phase)
Example – Scheil-Gulliver solidification scheme
xNi = 0.1
Azeotropic points
Maxima and minima of binary two-phase fields
Setting additional conditions
For binary: x - x = 0
For ternary: xB - xB = 0
xC - xC = 0
The driving force for a phase
LFS - CT
Driving force-application
Driving force ΔG, GFCC: (Fig.2.5)
(difference in G of paralel tangents for phases and
stability tangent)
-
theory of nucleation of phases
minimization of G (whether another phases set
exists that is more stable than calculated set of
phases)
Conditions for a single equilibrium – cont.
Adding phase to the calculated stable phase set:
Positive „driving force“ of the phase – repeat calculation
Removing phase from the selected set:
Calculation finds negative amount for one of selected
phases
Conditions for a single equilibrium – cont.
Phases with miscibility gaps may have more than
one driving force at different compositions – test
must be performed for each of these
compositions
Test by experiment when some phases appear in
calculations to be stable but experimentally are
found to be not stable
Questions for learning
1.What is a difference between stable and metastable states?
2. What is principle of Lagrange-multiplier method?
3. What is principle of Newton – Raphson method?
4. What conditions must be fulfilled for single equilibrium calculation?
5. What means „mapping“ and „stepping“ in calculations of phase
equilibria?