"The topological instability model for metallic glass formation: MD assessment", presented at http://3whpc.lcca.usp.br Cited video available at http://youtu.be/x3XbeN7uJ84

1. 3°Workshop HPC
The topological instability
model for metallic glass
formation: MD assessment
M. F. de Oliveira1 and G. A. Evangelakis2
1 University of São Paulo - Brazil
2 University of Ioannina - Greece

2. 3º WHPC
Outline
Introduction
Volumetric Strain and Topological Instability
model (λ)
Criterion to predict the GFA
Testing the topological instability model
Computational approach
Results with Zr-Cu phases
Conclusions

3. 3º WHPCVolumetric Strain and
Topological Instability (λλλλ)
Egami and Waseda, J. Non-Crist. Sol. (1984)
1.013
3
0 ≅−=
A
B
crit
r
r
f B
λ
f
B
min – critical solute concentration
rB – solute atomic radius
rA – matrix atomic radius
“… upon alloying, the topology is changed in order to keep
the local strains to a minimum. Thus as the solid solution
becomes unstable ... the amorphous state emerges as an
attractive alternative, particularly if another very stable
crystalline structure cannot be found at that composition.”

4. 3º WHPC
Topological Instability

5. 3º WHPC
Extrapolating for any phase
C.S. Kiminami, R.D. Sá Lisboa, M.F. de Oliveira, C. Bolfarini and
W.J. Botta, Mat.Trans. JIM (2007)
de Oliveira, M. F., Journal of Applied Physics (2012)
fi – solute concentration
ri – solute atomic radius
nj – number o j atoms in the formula unit of A
rj – atomic radius of the j element of A
13
3
−=
∑
∑
A jj
ii
AA
rn
r
fλ

6. 3º WHPC
The minimum λλλλ approach
λ
A AB
λmin

7. 3º WHPC
Electronic parameter (∆∆∆∆h)
M. F. de Oliveira - Phil. Mag. Lett. (2011)
( )23/12
.)( wsnkh ∆−∆=∆ φ
∆φ – average work function difference
∆nws
1/3 – average electronic density difference
k – empirical constant
∑ ∑ −=∆ jiji S φφχφ
∑ ∑ −=∆ jwsiwsjiws nnSn 3/13/13/1
χ
∑
= 2
2
jj
jj
j
r
r
S
χ
χ
χ – atomic fraction
S – surface cocentration
φ – work function
nws – electronic density
r – atomic radius

8. 3º WHPC
Criterion to predict the
GFA
de Oliveira, M. F., Journal of Applied Physics (2012)
hGFA ∆+∝ minλ
68 alloys in 30 systems

9. 3º WHPC
Objective of this work
Check Egami and Waseda’s hypothesis
in metallic and intermetallic phases
using Molecular Dynamics Simulations

10. 3º WHPC
Procedure
MD with LAMMPS
S. Plimpton, J. Comp. Phys. 117 (1995) pp 1-19
http://lammps.sandia.gov
EAM for interatomic potentials
H. W. Sheng, M. J. Kramer, A. Cadien, T. Fujita, M. W. Chen,
Phys. Rev. B 83 (2011) pp 134118-134138
https://sites.google.com/site/eampotentials/
Incremental substitutions of Cu, Zr or Al at steps of 0.01 at.%
in Zr-Cu phases followed by equilibration at 300 K and 1 atm
for 10 ps (rate of 1 at%/ns)

11. 3º WHPC
Procedure
16,000 particles – 28,000
Periodic bondary conditions
Verlet time integration style
Time step: 1 fs
Total time: 0.1 ms
Initial minimization: conjugate gradient method
NPT – 300 K and 0 bar
non-orthogonal box
W. Shinoda, M. Shiga, M. Mikami, Phys. Rev. B, 69 (2004) pp
134103-134110
G. J. Martyna, D. J. Tobias, M. L. Klein, J. Chem. Phys., 101 (1994),
pp 4177-4189
M. Parrinello, A. Rahman, J. Appl. Phys., 52 (1981) pp 7182-7190
M. E. Tuckerman, J. Alejandre, R. López-Rendón, A. L. Jochim, G. J.
Martyna, J. Phys. A: Math. Gen., 39 (2006) pp 5629-5651
Calculation of RDF
Calculation of q4 and q6 bond order parameters
P. J. Steinhardt, D.R. Nelson, M. Ronchetti, Phys. Rev. B 28 (1983) pp
784-805
Y. Wang, S. Teitel, C. Dellago J. Chem. Phys. 122 (2005) pp 214722-
214738 http://www.pas.rochester.edu/~wangyt/algorithms/bop/

12. 3º WHPC
Procedure
Phase
Space Group
Formula units
per cell
ICSD file
Cu Fm3m 4 43493
Cu5Zr F43m 4 103165
Cu51Zr14 P6/m 1 629471
Cu10Zr7 Aba2 4 164881
β-CuZr2 I4/mmm 2 103164
α-Zr P63/mmc 2 164572
Simulated Phases
Solutes: Al, Cu and Zr

13. 3º WHPC
Results
Zr in fcc-Cu lattice

14. 3º WHPC
Results
0
1
2
3
4
5
6
2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7
angstrons
g(r)
6 at.% Zr
12 at.% Zr
18 at.% Zr
Zr in fcc-Cu lattice

15. 3º WHPC
Results
Cu in Zr and Zr in Cu

16. 3º WHPC
Results Cu solid solutions

17. 3º WHPC
Results
Al in CuZr2

18. 3º WHPC
Results
Cu in CuZr2

19. 3º WHPC
Results
Zr in CuZr2
β-CuZr2
β-Zr

20. 3º WHPC
Results
Phase Transformation
Phase Solute Onset Strain Offset Strain Final Structure
Cu5Zr
Cu 0.007 0.09 nanocryst. FCC
Zr 0 0.2 nanocryst. FCC
Al 0.01 0.1 amorphous
Cu51Zr14
Cu 0.11 0.14 nanocryst. FCC
Zr 0.04 0.2 amorphous
Al 0.03 0.13 amorphous
Cu10Zr7
Cu 0 0.11 amorphous
Zr 0 0.22 amorphous
Al 0 0.01 amorphous
β-CuZr2
Cu 0.05 0.14 amorphous
Zr none none BCC
Al 0.05 0.07 amorphous

21. 3º WHPC
Results
Cu in Zr and Zr in Cu
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
at.%
vol.strain
Cu in hcp-Zr
Zr in fcc-Cu

22. 3º WHPC
Results
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 10 20 30 40 50 60 70 80 90 100
at.% Al
vol.strain
Al in Cu

23. 3º WHPC
Results
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 10 20 30 40 50 60 70 80 90 100
at.%
vol.strain
Cu5Zr + Zr
CuZr2 + Cu
Dashed lines: original equation

24. 3º WHPC
Results
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
at.% Al
vol.strain
Cu5Zr Cu51Zr14
Solute: Al
Dashed lines: original equation

25. 3º WHPC
Results

26. 3º WHPC
Conclusions
The extended topological instability
hypothesis fails for some solid
solutions
The volumetric strains does not follow
the proposed equation
The collapse of the structure occurs at a
very low level of volumetric strain in
some cases
There is no amorphization in other cases

27. 3º WHPC
Acknowledgements

28. 3°Workshop HPC
Thank you!