Les grands axes de communication 2002

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Parametrization of MEAM potentials for the study
of metallic alloys
G. Jomard, C. Bercegeay, S. Bernard, P.M.
Anglade CEA/DIF/DPTA Laboratoire de Structure
Electronique
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Overview

• We are involved in the atomistic description of
  physical problems such that
• irradiation in metals and their alloys
• melting of metallic alloys
• solid state phase transformations
• Molecular dynamics simulations had proven to be a
  powerful technique to treat these kind of
  problems.
• But they require a relevant atomic interaction
  model.
• Our physical systems are composed of either
  f-elements (Pu) or BCC transition metals (Ta,W)
• ? the MEAM formalism of M. Baskes is known to
  correctly describe these metals
• for which there is a strong covalent part in the
  atomic bonding.
• Among the various MEAM formulations we have
  chosen the 2nd NN one which has been proven to
• greatly improve physical description of BCC
  metals compare to the 1st NN one.
• The parametrization of our semi-empirical
  potentials is based on DFT calculations.
• For the ab initio part of this work we used the
  VASP code (PAW pseudopotentials, GGA corrected

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MEAM formalism for one component materials
The total energy of a system of N particles is
approximated as
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MEAM formalism for one component materials
The total energy of a system of N particles is
approximated as
Embedding energy N-body term
based on the Paulings logarithmic relationship
between bond length and number of bonds
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MEAM formalism for one component materials
The total energy of a system of N particles is
approximated as
Embedding energy N-body term
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MEAM formalism for one component materials
The total energy of a system of N particles is
approximated as
Pair interaction between atom i and j
Assuming that for a given reference structure,
the energy per atom follows the URBE of Rose
The pair potential between 2 atoms separated by r
is
1st NN coordinence
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Cutting the interactions in MEAM
Usually we restrict the interactions to 1st NN by
applying angular screening between atoms i and j.
The screening of the interaction ij by atom k is
limited by the introduction of 2 parameters Cmin
and Cmax.
Cmax2.8 and Cmin2.0 ensure that for FCC
structure 1st NN are completely unscreened and
interactions are still 1stNN for other structures
like BCC, HCP,
Note that reducing Cmin allows to take into
account 2NN interactions
This has be proven to improve the physical
description of BCC materials for which d2ndNN is
only 15 grater than d1stNN. In this case, the
pair potential is obtained from a converging
serie
In addition, for computational convenience, a
radial cut-off is also applied to and
.
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Parametrization of 2ndNN MEAM potentials for Ta
and W
As seen previously, there are 14 independent
parameters in the 2nd NN MEAM formalism. The
parametrization procedure is

• Adjusting the URBE of Rose on the DFT calculated
  EOS of BCC Ta and W
• ? E0, r0 and

• Adjusting the , , A, Cmin and
  Cmax on physical properties such as
• the elastic constants (C11, C12 and
  C44), , and
  .
• We use a code which searches a set of parameters
  minimizing a sum of
• qualities defined by

DFT calculated value
relative error allowed
Note that for this study, we have partially used
the potentials published by B.J. Lee and al.
PRB 64 (2000) 184102 since our purpose was to
parametrize the cross potential
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions
The first step of the parametrization procedure
is to choose a reference structure in order to
obtain an analytical formulation for For
convenience, a perfectly ordered binary
intermetallic compound where an atom of type A
has only B-type 1st NN and A-type 2ndNN, may be a
good choice ? B1 (NaCl structure) or B2 (CsCl
structure) We chose the B2 structure for which
we can write
Ta W
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions

• Sequential procedure
• parametrization of ETaW
• determination of STa and SW within B2 compound ?
  STaWTa and SWTaW
• In a first approximation we have tried

pre-determined one-component functions
4 parameters determined by least squares
minimisation
? within this method the screening do not depend
on the type of atom k!
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions
1st attempt the crude one.
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions
1st attempt the crude one.
The agreement with DFT results is not so good!
But there are many crude assumptions in this
parametrization of the cross potential
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions

• Assumption 1
• In the construction of the background density on
  a site i, the weights given to partial densities
  induced by
• the neighbors is only dependent of the type of
  atom i.
• ? we should introduce weights depending on the
  local atomic arrangement around site i

with
Ta W
depends on the types of the neighbors of atom i
depends only of type of atom i
R
we tried
6 new parameters

no new parameters, only an average on local
atomic surrounding
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions

• Assumption 2
• The screening parameters depend only on the type
  of the two atoms which interact and note on the
  type
• of the screening atom.
• ? introduction of more parameters
  in order to calculate Sikj
• this implies to increase the set of physical
  properties entering the parametrization

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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions
2nd attempt
The overall agreement with DFT results is much
more better. The worse agreement is found for the
94at of W composition. But many compounds
scratch with temperature!
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions
3rd attempt
We adjust the parameters of the Roses equation
on the EOS of TaW B2 obtained by first principles
electronic structure calculations. We adjust all
the other parameters and the
on the formation energies of a set of
ideal and relaxed TaW compounds calculated with
VASP Ta15W, Ta3W DO3, Ta3W SQS, TaW B2, TaW
SQS, TaW3 DO3, TaW3 SQS and TaW15. We use the
mean weights for partial densities.
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Parametrization of a 2ndNN MEAM potential for the
Ta-W interactions
The agreement between DFT data and the ones
obtained with the fitted potential is excellent.
All the structures are stable with
temperature. Is this potential transferable to
other crystallographic structures?
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Transferability of the potential?
Blum and Zunger PRB 72 (2005) 020104
Exp _at_ 1200K