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 - Brazil2 University of Ioannina - Greece

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

3º WHPCVolumetric Strain and

Topological Instability (λλλλ)

� Egami and Waseda, J. Non-Crist. Sol. (1984)

1.013

3

0 ≅−=A

B

critr

rf Bλ

f Bmin – critical solute concentration

rB – solute atomic radiusrA – 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.”

3º WHPC

Topological Instability

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 concentrationri – solute atomic radiusnj – number o j atoms in the formula unit of Arj – atomic radius of the j element of A

13

3

−=∑

∑A jj

ii

AArn

rfλ

3º WHPC

The minimum λλλλ approachλ

A AB

λmin

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 differencek – empirical constant

∑ ∑ −=∆ jiji S φφχφ

∑ ∑ −=∆jwsiwsjiws nnSn

3/13/13/1 χ

∑=

2

2

jj

jj

jr

rS

χ

χ

χ – atomic fractionS – surface cocentrationφ – work functionnws – electronic densityr – atomic radius

3º WHPCCriterion to predict the GFA� de Oliveira, M. F., Journal of Applied Physics (2012)

hGFA ∆+∝ minλ

68 alloys in 30 systems

3º WHPC

Objective of this work

� Check Egami and Waseda’s hypothesis

in metallic and intermetallic phases

using Molecular Dynamics Simulations

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)

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/

3º WHPC

Procedure

PhaseSpace Group

Formula units

per cellICSD 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

3º WHPC

ResultsZr in fcc-Cu lattice

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

3º WHPC

Results

� Cu in Zr and Zr in Cu

3º WHPC

ResultsCu solid solutions

3º WHPC

Results� Al in CuZr2

3º WHPC

Results� Cu in CuZr2

3º WHPC

Results� Zr in CuZr2

β-CuZr2

β-Zr

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

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.%

vo

l.s

tra

in

Cu in hcp-Zr

Zr in fcc-Cu

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

v

ol.s

tra

in

Al in Cu

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.%

v

ol.s

tra

in

Cu5Zr + Zr

CuZr2 + Cu

Dashed lines: original equation

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

v

ol.s

tra

in

Cu5Zr Cu51Zr14

Solute: Al

Dashed lines: original equation

3º WHPC

Results

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

3º WHPC

Acknowledgements

3°Workshop HPC

Thank you!