Transcript ILIRIA: preliminary discussion

IV Workshop on Multicale Modelling in Fe-Cr Alloys
Stockholm – May 6-7, 2003
Design of crossed many-body empirical
interatomic potentials for ferritic binary alloys
based on phase diagram information
SCK• CEN contribution to
PERFECT / Physics Modelling Subproject
R. Pasianot
CAC-CNEA, Argentina
and L. Malerba
SCK•CEN, Belgium
With the collaboration of A. Caro (LLNL) and E. Lopasso (CAB-CNEA)
1
No standard procedure currently
exists to fit potentials for alloys
 Potentials for binary alloys are often just superpositions of pure
element potentials
 They are fitted to only few alloy parameters, typically
substitutional energy of one alloying atom, mixing enthalpy for
one concentration, ordered compound energies …
 Only by chance can this procedure guarantee the validity of the
potential for a large range of concentrations and an acceptable
prediction of the alloy thermodynamic functions
 No defined procedure exists to include in the fitting the most
important information about alloys: its phase diagram
2
Two parallel objectives for a task devoted to
improve potentials (PERFECT)
1.
Find alternative formalisms to EAM, more apt to treat,
particularly, Fe and ferritic alloys
2.
Elaborate optimised and possibly standard potential
fitting procedures for binary alloys, including
thermodynamic properties
improved empirical potentials
3
SCK●CEN’s tasks in the PERFECT “ab
initio and potentials” work-package
1. Apply and refine procedure by A.
Caro to validate existing potentials
on binary alloy phase diagrams
2. Elaborate procedure to fit
potentials for binary alloys on the
corresponding real phase diagram
(test case: Fe-Cu)
In collaboration with CNEA via BelgoArgentine bilateral co-operation agreement
4
Summary of method to build phase
diagrams from EAM potentials
• Free energy per particle calculated using Gibbs-Duhem equation
T h( )
T
f (T )  f (T0 )  T 
d
2
T
0
T0

where h(T) is obtained from MD run and fitted to 2nd order
polynomial in T
• f(T0) calculated as difference between the actual system
(described by EAM potential) and a reference system whose free
energy is known (using hamiltonian switching method, via MD
simulation), namely:
 For solid: Einstein solid
 For liquid: ideal gas
• Same method extendable to alloys for different concentrations
• Once f(c,T) is known for all involved phases, the phase diagram is
constructed using the familiar common tangent method
5
Phase diagrams from existing
potentials are far from reality!
2400
Líquido
LIQ
Temperature,
Temperatura [K] K
2200
2400
LIQ
Líquido
2000
1800
BCC
1600
BCC
1400
1200
1000
1800
FCC
BCC
800
1600
FCC
0
20
40
60
80
100
Cuat%
[% atómico]
Cu
1400
FCC
1200
BCC
1000
800
0
20
40
60
80
LIQ
100
Cu [% atómico]
BCC
Temperature,
Temperatura [K]K
2200
2000
FCC
6
Status of method to build phase
diagrams from EAM potentials
• Applied so far to Au-Ni1 and Fe-Cu2,3 systems
• May encounter problems when some phases are not
accessible to the potential, e.g. if they are not predicted
to be stable (phases that do not exist in reality either) /
case of LF potential
• Solutions to this problem are being studied / the
objective is to produce a robust tool for empirical
interatomic potential validation (a posteriori) which
should be extendable to any potential formalism
1E.
Ogando Arregui, M. Caro and A. Caro, Phys. Rev. B66 (2002) 054201
2E. M. Lopasso, M. Caro, A. Caro and P. Turchi, Phys. Rev. B68 (2003) 214205
3A. Caro, P. Turchi, M. Caro and E. M. Lopasso, Phys. Rev. B, in press
7
Methodology to fit potential to PD:
CVM applied to EAM fitting
• The CVM formalism provides a solid theory for the
construction of analytical expressions for the thermodynamic
functions of an alloy, based on configurational considerations,
as a function of a vector of variational variables called cluster
correlation functions: ={1 2 … n}*
• By minimising the relevant thermodynamic potential on the
cluster correlation functions it was possible, applying the
CVM, to build e.g. the phase diagram of even relatively
complex systems (eg Al-Co-Fe)*
• The CVM is otherwise typically used to treat ordered alloys
• The theory is rather involved and for complex alloys becomes
highly involved, but binary random alloys represent the simplest
application case
*
See eg G. Inden and W. Pitsch “Atomic ordering”, in “Phase transformations in materials”, P. Haasen Ed,
Weinheim: VCH (1991), chapter 9, p. 499
8
Methodology to fit potential to PD:
CVM applied to EAM fitting
• The CVM formalism can be used to express not only the
energy of a system, but also the entropy, thereby giving
access to the alloy free energy
• By coupling the CVM formalism to the EAM scheme the
energy of a system can be expressed as a function of:
 The variational variables,

 The parameters of the potential, a
 Species concentrations and alloy volume
• Via a double, iterative minimisation, it is hence expected
that the potential can be fitted to reproduce
experimentally available thermodynamic functions for the
relevant alloy
9
Basic concepts of the CVM
• Cluster = set of lattice points 1, …, r, generally considered to be
representative of a crystalline system with N>>r sites
• Cluster configuration = vector of integral numbers expressing
the configuration of a cluster (each possible value of the site
operator k corresponds to a possible chemical species): r={1
2 … r}
• Cluster probability, p(r) = probability that a cluster of size r is
in a certain configuration r (simplest example: point probability
in random alloy = xi)
• Cluster correlation functions = average values of the product of
the site occupation operators for the corresponding cluster –
used as variational variables on which the termodynamic
potentials can be minimised
10
Basic clusters for bcc and fcc
2
7
1
3
5
4
4
6
3
2
bcc
 irregular tetrahedron
 includes up to 2nn
distance
 can describe large
number of ordered
alloys
1
fcc
 regular tetrahedron and
octahedron
 tetrahedron includes up to
1nn distance only
 octahedron extends to 2nn
11
Cluster probabilities
Cluster of size r, with K chemical species:
 tot  (K 1)r 1
Total number of configurational probas
 indep  K 1
Number of independent probas
r
Binary alloys: probas are linear functions of :
p  M 
(for rules to build M see Inden & Pitsch)
Clusters of relevance:
•Pairs (3p)
●
Tetrahedron (bcc – 6p, fcc – 8p)
•Triangles (8p)
●
Octahedron (fcc - 10p)
12
Example from CVM: expression
for configurational entropy
Sbcc
2
 nptetra (tetra)
( tetra)
( t _ iso _ 112 )
( t _ iso _ 112 ) 
6
p
(
M
,

)

ln
p

12
p
(
M
,

)

ln
p
tit jtk
,t 1 tit jtk
tetra tetra
p
t _ iso _ 112 112
  p

p

1
t
,
t


i j k
 k B N 

2
2
2
(
2
)
(
2
)
(
1
)
(
1
)
 3 p t t ( M ,  )  ln p t t  4

p tit j ( M p , 1 )  ln p tit j   xi ln xi
i j
i j


p
2
 ti ,t j 1

ti ,t j 1
ti 1


nptetra
npocta ( octa)

( octa)
( tetra)
( tetra)
p
(
M
,

)

ln
p

2
p
(
M
,

)

ln
p

octa octa
p
p
tetra tetra
p
 p

p

1
p

1


S fcc  k B N 

2
2
2
 8
p (titt j_tkequi ) ( M t _ equi , t _ equi )  ln p (titt j_tkequi )  6  p (ti1t j) ( M p , 1 )  ln p (ti1t j)   xi ln xi 

 ti ,t j ,tk 1

ti ,t j 1
ti 1


The regular solution theory corresponds to a
zero approximation of the CVM
13
Energy in the EAM/CVM formalism
• The EAM energy can be expressed in the CVM
formalism as a function of the probability of:
 pairs (pair component)
 shell
2
)
E pair  N   m(pair
pt(it j) Vti(tj ) (rij )  E (v, x,  , a)
 1 ti t j 1
 pairs and triangles (many-body component)
[1]
[ 2]
[ 3]
Emb  Emb
 Emb
 Emb
 Emb (v, x,  , (  ))
1


[1]
Emb
 N  pt(i1)  Fti( 0)  Fti( 0 )'' 
2


ti 1
2
 shell
E
[ 2]
mb
E
[ 3]
mb
2nd order Taylor development of
embedding function around equilibrium




1
1

)
 N   m (pair
pt(ivt j)  Fti( 0)'' t2it j (r )  2tit j (r )  Ft (j 0 )'' t2j ti (r )  2t j ti (r ) 
2
2

 1 ti t j 1
N
2
 triangle 123


1






1 ( 0)''
1 ( 0 )''

(v)  1
( 0 )''
p
F

(
r
)

(
r
)

F

(
r
)

(
r
)

Ft j tk t j (r23 )tit j (r12 ) 

ti t j t k 
ti
t j ti 12
ti t j 13
tk
ti t k 13
t j tk
23
2
2
2

ti t j t k 1
2
14
Electronic density and embedding
function in the alloy
Fti (  i )
embedding function depends only on element
 i   t t (rij )
j i
j
t t   t t t
j i
j i
t t 
j i
t
t
j
t
i
electronic density on i is superposition of effects of
surrounding atoms
effect of tj on ti decided by coefficient which can be
fitting parameter or estimated to remain fixed
j
;  ti ti  1
one coefficient is enough, which can be a
fitting parameter or estimated to remain fixed
 shell
i
/
 i( 0)   zt( )t (r0( ) )  1
 1
i
gauge choice
i
15
Thermodynamic functions
• For p=T=0, E=F=G=H  an expression for the
mixing enthalpy can be obtained:
h= h(v,x,,a)
• For p=0, T=T0, G=F=H-TS=E-TS  expression
for the free energy and, by derivation, for the
chemical potential, can be obtained
j=e+xi de/dxj
16
Fitting procedure
• A self-consistent minimisation procedure of
thermodynamic functions on variational variables
(linear) and object function on the potential
parameters (quadratic form or more complex)
must be set up
• Possible thermodynamic fitting “parameters”:
 Mixing enthalpy curve nodes
 Common tangent of the free energy for a certain T,
imposed in correspondence with a certain phase
concentration, taken from the phase diagram
• In addition:
 Classical parameters, such as heat of solution, defect
binding of formation energies, bulk modulus, etc
17
Application to Fe-Cu
• No development on pure metal potential components is
foreseen for the moment
• For pure Fe, Mendeleev potential1 will be used
• For pure Cu, Mishin potential2 will be used
• For crossed potential, fitting on e.g.
 Mixing enthalpy (from ab initio calculations)
 Solubility limit (from phase diagram)
 Cu-V binding energy (from ab initio calculations)
• Validation by building corresponding phase diagram
1M.I.
Mendeleev et al., Phil. Mag. 83(35) (2003) 3977-3994
2Y. Mishin et al., Phys. Rev. B 63 (2001) 224106
18
Summary and Discussion
•
While new formalisms for pure bcc metal potentials are being studied, so
far little effort has been devoted to rationalise the fitting procedure of
alloy potentials
•
The most important information concerning an alloy is its phase diagram
•
A procedure for the validation of an interatomic potential by building the
corresponding phase diagram has been developed and is being refined
•
The CVM theory provides a formalism to express analytically the free
energy of an alloy
•
By introducing the CVM formalism in the EAM scheme, analytical
expressions for the configurational free energy of the alloy can be built
and used to fit the potential to points of the phase diagram, eg imposing
common tangent to the free energy of the two phases
•
The case study is currently Fe-Cu, using existing EAM potentials for pure
Fe and pure Cu
•
If the procedure is successful, it can be in principle extended to both
other systems within the EAM approximation and, with more or less
theoretical difficulties, to other interatomic potential formalisms
19