Thermodynamics and thermophysical
properties of liquid Fe-Cr alloys
Rada Novakovic
National Research
Council (CNR–IENI)
Genoa, Italy
Mixing behaviour of liquid binary alloys:
energetic & structural factors
Observable indicators:
 Phase diagrams.
 Empirical factors – physical, chemical & structural
properties of alloy constituents (liquid metals), melting
points, volume, first shell coordination, radius size,
valence difference, electronegativity difference...
 Thermodynamic functions – heat capacity, enthalpy,
activity, excess Gibbs energy.
 Microscopic functions – concentration fluctuations in
the long wavelength limit & CSRO (Warren-Cowley short
range order) parameter.
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
What kind of input data are necessary
for modelling?
1. Thermodynamic data
on mixing: heat capacity; enthalpy; entropy; Gibbs
energies (integral & excess).
partial quantities: activities (or chem. potentials).
2. Phase diagram information
type of alloy system: segregating or compound
forming
3. Thermophysical data: molar volume, surface
tension, viscosity of pure components.
4. Structural data: coordination number; neutron
diffraction data to be transformed into the
microscopic functions
5. Experimental data on Thermo-Physical properties of
alloys: for a comparison with theoretical results
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
The Fe-Cr system
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Thermodynamic data of the Fe-Cr liquid phase
[11Xiong] An improved thermodynamic modeling of the Fe–Cr system
down to zero kelvin coupled with key experiments
[86Mas] The Fe-Cr phase diagram
[76Hul;81AB;82HS;87AS] previous assessments of the p.d.
[93BLee] The reassessment of the Fe-Cr phase diagram
[93BLee] T=1873K: The optimised term of the excess Gibbs free energy;
the enthalpy of mixing [84Bat];
the activities [80Mar;69Fru;69Gil;98Zai].
[06Vre] The presence of interm. - phase
[06Ter] The melting, the enthalpy of mixing, thermal diffusivity - by atomic
simulations
Comment: The Cr-Fe phase diagram can be considered as COMPLETE (although
some measurements in the liquid phase are necessary).
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
- phase energ.
favoured ( AB )

Weak influence
on the energetics
of the Fe-Cr liquid
phase.
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
SURFACE TENSION
 SURFACE TENSION MODELS
Binary systems
Ternary systems
Geometric models
EXAMPLES: Fe-Cr, Al-Nb-Ti
Surface properties of liquid binary alloys:
surface segregation & surface tension
Butler(1932) published the paper proposing his well known
equation:
s
 
i

k BT
i
ln
ai
b
ai
( i = A, B), that gives the relation between the surface
tension and thermodynamics of liquids in which the bulk
and surface phases are in equilibrium.
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Subtracting Butler’s equ. for both components,
s
s
   A  ( RT / S A ){ln  A  ln x  (ln  A  ln x )}
s
s
   B  ( RT / S B ){[ln  B  ln( 1  x )]  [ln  B  ln( 1  x )]}
and combining with
 G M  RT ( x A ln  A  x B ln  B )
xs
and taking into account the bulk (surface) phase activity coefficients
obtained by Fowler_Guggenheim method as
and
s
z
s
s
s
s
s
ln  B  {ln(   1  2 x )  ln(   1)  ln( 1  x )}
2
ln  A 
s
z
s
{ln(   1  2 x )  ln(   1)  ln x }
s
s
s
s
2
the  and xs can be calculated. The surface tension can be calculated
inserting xs into the Butler’s equation.
MATGEN-IV.3 summer school on “Materials for Generation IV reactors:
Fundamentals, ongoing research and open questions”, September 19-23, 2011,
Lerici (SP), Italy
Surface tension calculations
of binary systems
* Models based on Butler’s
equation
- Regular solution
& * Probabilistic Models
Singh et al.
- Subregular solution
- “Central” atom
- Compound Formation Model (CFM)
- Self Aggregating Model (SAM)
An interface Liquid / Gas :
Monolayer or Multilayers
Surface tension calculations
of ternary systems
* Models based on Butler’s
equation
- Regular solution
& * Geometric Models
(from thermodynamic
calculations of mixing
properties in the bulk)
SYMMETRIC
- Subregular solution
- Kohler; Colinet; Muggianu
- “Central” Atom
- Compound Formation Model (CFM)
ASYMMETRIC
- Self Aggregating Model (SAM)
- Toop; Bonnier; Hillert;
GENERALIZED
- Chou
An interface Liquid / Gas :
Monolayer
Geometric models
Kohler
xs
  ( X 1  X 2 ) (  12 ) X 1 / X 2  ( X 1  X 3 ) (  13 ) X 1 / X 3  ( X 2  X 3 ) (  23 ) X 2 / X 3
2 xs
2 xs
2 xs
Toop
xs
 
X2
1 X1
  12 ( X 1 ;1  X 1 ) 
xs
X3
  13 ( X 1 ;1  X 1 )  ( X 2  X 3 )   23 (
xs
1 X1
2 xs
X2
;
X2  X3 X2  X3
Chou
xs
 
X1X
X 1 (12 ) X
2
2 (12 )
  12 
X2X3
xs
X
X 3 ( 23 )
2 ( 23 )
 
xs
23

X3
X3X1
X 3 ( 31 ) X 1 ( 31 )
  31
xs
)
Iso-surface tension lines of liquid Al-Ti-Nb alloys calculated by the Butler
equation for the regular solution model at 2073 K. The square symbol
represents the composition location of the Ti46Al46Nb8 (at.%) in the Gibbs
triangle and the corresponding surface tension calculated value
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Surface tension reference data of Cr
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Surface tension reference data of Fe
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Microscopic functions (B-T) &
Thermodynamics
  GM
S cc ( 0 )  RT 
2

C
A

2
1

 a A



C
a
B A

 C A
 T ,P ,N
1

 a B



C
a
A B

 T ,P ,N
 C B
1



 T ,P ,N
For ideal solution the SCC(0) becomes
S CC ( 0 )  c A c B
id
The CSRO parameter and SCC(0) are related to
each other by
S cc ( 0 )
c AcB

1 1
1  ( Z  1 ) 1
where Z is the coordination number.
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Microscopic functions & local
arrangements of atoms in the melt
SCC(0) and CSRO parameter indicate chemical order &
segregation (phase separation):
SCC(0) – the nature of mixing
CSRO parameter – the degree of order
Criteria for mixing behaviour
1. SCC(0) < SCC(0, id) presence of chemical order
SCC(0) > SCC(0, id) segregation
2. -1 < CSRO < 0 ordering in the melt
CSRO = -1 complete ordering
0 < CSRO < 1 segregation
CSRO = 1 phase separation
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
The interdiffusion coefficient (Dm)
can be given in terms of the SCC(0) by
id
D m  (c A D B  c B D A )
*
*
S CC ( 0 )
S CC ( 0 )
For “ideal” alloys, SCC(0)= SCC(0,id)= cAcB,
then
D m  c A D B  c B D A  ( c A D B  c B D A )  D id
*
*
and finally combining the last two eqs. it is obtained,
Dm
D id
id

S CC ( 0 )
S CC ( 0 )
The criteria for mixing behaviour:
SCC(0) > SCC (0, id) segregation  Dm < Did
SCC(0) < SCC (0, id) presence of chem. order  Dm > Did
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Results of calculations
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Viscosity
Viscosity () of liquid alloys - the atomic level
structure and interactions.
The composition dependence of  of liquid alloys in respect
to the linear low (ideal mixture):
0
0
 id  c A A  c B  B
- a linear variation (simple liquids, e.g. Ag-Au, Sn-Pb, Bi-Pb)
- positive deviations (compound forming alloys, H <<0)
- negative deviations (segregating alloys, H >>0).
Sometimes the viscosity of binary liquid alloys exhibits
“strange” behaviour (Bi-Ga, Bi-Cu, Ga-Hg..), i.e. the same
behaviour as their thermodynamic functions (according to
the theory should be opposite!)
In the framework of the QLT the viscosity, , is related to the SCC(0) and diffusion by:
 
k BT
(
D id
CA
B

CB
A
)
C AC B
S CC ( 0 )
For a thermodynamically ideal mixture,SCC(0)=SCC(0,id)=c(1-c) previous equ. becomes:
 id 
with
k
T
B
D
m
id
(
C
A
B

C
Dm  cA DB  cB D A
id
B
A
)
id
id
and for the viscosity of pure components (Stokes-Einstein)

A

k BT
3 rD
id
A

k BT
 A D Aid
and  B 
k BT
3 rD
id
B

k BT
 B D Bid
Assuming 1 = 2 = =1, it is obtained the Stokes-Einstein type relation for diffusion and
viscosity:
k BT
id
id
DA  DB 

Recently, we proposed the following viscosity
equation:
 
k BT
D id
(
CA
 A exp(  m A C B )

CB
 B exp(  m B C A )
)
C AC B
S CC ( 0 )
where mi and i (i=A,B) are parameters that can be
calculated from the experimental data.
Results of calculations
viscosity of
some binary systems
Modelling of the interfacial properties of molten Pb /
FeCr substrate system: Application of the
Phase Field Method
 Study of thermodynamics and thermophysical
properties of the Fe-Cr, Fe-O, Pb-O, Fe-Cr-Pb, Fe-CrPb-O systems
 Model formulation and implementation
 Collection of input parameters for the Pb-Fe and Pb-Cr systems
 Simulations, analysis of model parameters and validation with
experimental micrographs for the Pb-Fe and Pb-Cr systems
 Extension of the model and implementation towards ternary
system Pb-Fe-Cr
 Collection of input parameters for the Pb-Fe-Cr system
 Simulations for the interface between molten Pb / FeCr substrate system
 Comparison with experimental micrographs for Pb / Fe-Cr
diffusion couples
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy
Thank you for your attention!
MATGEN-IV.3 summer school on “Materials for Generation IV reactors: Fundamentals,
ongoing research and open questions”, September 19-23, 2011, Lerici (SP), Italy