theoretische festkörperphysik: anwendungsbeispiel -...

TheoretischeTheoretische FestkrperphysikFestkrperphysik::AnwendungsbeispielAnwendungsbeispiel

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Vorlesung, Erlangen. WS 2008/2009

First-principles calculations in materials science???

Jaguar XJ with Al-based car frame

-0,66DFTNiAl,B2-0.64Exp.NiAl,B2-0,57DFTCoAl,B2-0,56Exp.CoAl,B2-0.30DFTFeAl,B2-0,26Exp.FeAl,B2

Hf [eV/Atom]System

Formation enthalpy Hf of the B2 Phase for CoAl, NiAl, FeAl

Exp.: P. Villars and M. Calvert, Experimental Handbookof Crystallographic Data (Materials Park, Ohio, 1991)Theo.: S. Mller. J. Phys.: Condens. Matter 15 (2003) R1429.

mass reduced by ~ 200 kg

(thanks to FORD Motor Company,

Michigan, USA)

Modelling materials properties demands the consideration of

huge configuration spaces huge model systems

temperature

T

Al-rich Al-Li: precursor

T. Sato and A. Kamino, Mat. Sci. Eng. A

146 (1991) 161

TEM

PredictionS. Mller, R. Podloucky,and W. Wolf, submitted

time!!!

impossible to handle directly via DFT!!!

Crystallographic atomic structure(relaxation, reconstruction, buckling)

Vibronic properties(phonon spectra)Density Functional Theory

Cluster ExpansionMonte-Carlo Methods

(UNCLE)

Energetics(stability)

Nucleation

Short-rangeorder

Multi-siteadsorption

Electronic structure of materials(band structure, density of states)

Diffusion PrecipitationSegregation

Ground state search in huge configuration spaces

Multi-scale modeling(from atomic to mesoscopic scale)

Activation barriers

Dynamics(diffusion)

Nudge Elastic Band Method,Molecular Dynamics,

Transition State Theory

Precipitation in Al-rich Al-Zn alloys

Quenching a solid solutioninto the two-phase region

Formation of coherentZn-precipitates:

Coherentphase boundarycalculated**experimental*

1000

(R. Ramlau and H. Lffler, phys. stat. sol. (a),79, p.141 (1983))

xZnAl

* J. L. Murray, Bulletin of Alloy Phase Diagrams 4, 55 (1983). ** S. Mller et al., Europhys. Lett. 55, 33 (2001).

Treating long-range interactions:The mixed-space presentation

Problem: Real-space CE fails to predict the energyof long-periodic coherent structures!

Intrinsic fault of any finite Cluster Expansion:

Range of interactions: Hf = 0 for n AnBn-Superlattice

Ansatz: Transform portion of interactions to reciprocal space Easiest to do for pair interactions

H() = J(k) |S(k,)|2 + Df Jf f k 3,4

body

Mixed-space form:


Solving the problem:

J(k) = JCS (k) + JSR (k)

Constituent Strain (CS):Contains the correct long-periodic superlattice limit

Short-Ranged (SR) inter-actions that are ignoredby JCS (chemical part)

can be constructed from the equilibrium constituent strain

Coherency strain energy

ECSeq (a,G)

q(a,G) = EAepi(G,a)EAhydro(a)

Epitaxial Strain Energy:Deform to the substrate lattice

parameter a and relax along G.

Hydrostatic Deformation Energy:Deform hydrostatically to thesubstrate lattice parameter a.

a

G

Substrate

Film (A)

B = 1/3 (C11 + 2C12 ): Bulk Modulus = C44 (C11 C12): Elastic anisotropy parameterharm (G): Geometric function of spherical angles

Bqharm(G) = 1 - C11 + harm (G)

Elasticity theory:

However: Bq(a,G) = 1 -C11 + (a,G)

with (a,G) = harm (G) + l bl (a) Kl (G)ECSeq (x,G) = l Al (x) Kl (G)

Lattice parameter a [a.u.]6,6 6,8 7,0 7,2 7,4 7,6

Epita

xial

sof

teni

ng q

Al

0,1

0,2

0,3

0,4

0,5

0,6

(201)

(110)

(111)

(100)

aAl

fcc-AlEpitaxial Strain Energy:Deform to the substrate lattice

parameter a and relax along G.

Hydrostatic Deformation Energy:Deform hydrostatically to thesubstrate lattice parameter a.

q(a,G) = EAepi(G,a)EAhydro(a)

a

G

Substrate

Film (A)

Elasticity theory:Bqharm(G) = 1 - C11 + harm (G)

B = 1/3 (C11 + 2C12 ): Bulk Modulus = C44 (C11 C12): Elastic anisotropy parameterharm (G): Geometric function of spherical angles

Cu Pd

Coherency strain ofCu1-xPdx-superlattices

xPd


ECS() for any arbitrary structure can be calculated via

This ansatz solves long-periodic superlattice problem!

H() = J(k) |S(k,)|2 + Df Jf f + ECS() k 3,4

body

Mixed-Space Cluster Expansion (MSCE):

A. Zunger, NATO ASI on Statics and Dynamics of Alloy Phase Transformations (Plenum Press, New York, 1994), 361.

Size-shape-relation of precipitates

H() = J(k) |S(k,)|2 + Df Jf f + ECS() k 3,4

body

Separate MSCE-Hamiltonians into two parts:

Strain part:flat (111) layer:

Softest direction in fcc-Zn*

Chemical part:compact shape(NZn = 2175)

H = Echem + ECS

* S. Mller et al., Phys. Rev. B 60, 16448 (1999).

(T 0)

(S. Mller et al., Acta Mater. 48 (2000) 4007)

fcc-Zn precipitate: flattening along [111]

4248 Zn atoms (rpsphere = 25 )

Flattening along [111]: Instability von fcc-Zn

E(eV)

E(eV)

DOS [a.u.]

DOS [a.u.]

EF

EF

-1.0 -0.5 0.5 1.0

-1.0 -0.5 0.5 1.0

Density Of States

(c/a) [%] = 0

(c/a) [%] = 15

fcc-Zn

0,8 0,9 1,0 1,1 1,2

Ener

gy [m

eV/a

tom

]

-20

0

20

40

60

100

111

hcp-Zn

-20 -10 0 10 20

(c/a) [%]

a

G

c

(S. Mller, L.-W. Wang, A. Zunger, C. Wolverton, Phys. Rev. B 60, 16448 (1999) )

Calculated coherent fcc-Zn precipitates in Al-Zn as function of precipitate size and temperature

ac

300K

Number of Zn-atoms

Tem

p. [K

]

30K

200K

918 2175 4248

How do to kinetics in real time ???

Bridging time scales

Idea: Force selected atoms to exchange process Calculate corresponding simulation time afterwards

Prerequisite: Calculation of energy change E(i) for allpossible atomic exchanges i (restriction to NN)

Ene

rgy

A

B

E(i) from MSCE

From DFT calculationsor experiment

(S. Mller, J. Phys.: Condens. Matter 15 (2003) R1429.)

Configuration-dependentactivation barriers*

+: no exp. parameters-: no transformation to

real time becauseE = E(T)Act

ivat

ion

barr

ier [

eV]

0,0

0,1

0,2

0,3

0,4

0,5

Activ

atio

n ba

rrie

r [eV

]

0,0

0,2

0,4

0,6

0,8

Al at Li-site Al at Al-site

Li at Li-site Li at Al-site

(In collaboration with R. Podloucky, Univ. Wien, Austria,and W. Wolf, Materials Design, Le Mans, France) L12 (Al3Li)

calc. of phonon spectraDiffusion coefficients

as function of structureand temperature

Trafo to real time

(* calculated by the Nudge Elastic Band Method; R. Podloucky, Vienna)

Phonon spectra:Al31Li Li migration(Walter Wolf, Materials Science, France)

RelaxedStructure

Al-vacancyFormation Li migration

Al30LiAl31Li Al30Li

Configuration-dependentactivation barriers*

(In collaboration with R. Podloucky, Univ. Wien, Austria,and W. Wolf, Materials Design, Le Mans, France) L12 (Al3Li)

Temperature [K]

1000/T [K -1]

1.0 1.2 1.4 1.6 1.8 2.0

Diff

usio

n co

effic

ient

[m2 /

sec]

1e-16

1e-15

1e-14

1e-13

1e-12

1e-11calculatedBakker et al., 1990Wen et al., 1980Costas, 1963Verlinden and Gijbels, 1980Tmelt =

933K

1000 500 calc. of phonon spectraDiffusion coefficients

as function of structureand temperature

Trafo to real time

+: no exp. parameters-: no transformation to

real time becauseE = E(T)

(* calculated by the Nudge Elastic Band Method; R. Podloucky, Vienna)

Size-shape relation of precipitates(no Al atoms are shown)Al-rich

Al-Li Al-Cu Al-Zn

(T = 373K, t=1.6*105 ks)

5 nm 2 nm

(T = 473K, t = 86.4 ks)

250

(T = 250K, t = 1.2 ks)S. Mller, W. Wolf, R. Podloucky, subm. J. Wang et al., Acta Mat. 53 (2005) 2759

10 nm

T. Sato and A. Kamino, Mat. Sci. Eng. A 146 (1991) 161

T. J. Konno, K. Hiraga, and M. Kawasaki, Scripta Met. 44 (2001) 2303

R. Ramlau and H. Lffler, phys. stat. sol. (a), 79, p.141 (1983))

S. Mller, J. Phys.: Condens. Matter 15 (2003) R1429

mean precipitate diameter [nm]0 1 2 3 4 5

0

20

40

60

80

100

E c

hem

E C

S


0

20

40

60

80

100

E c

hem


0

20

40

60

80

100

E che

m

E C

S

Perc

enta

geof

ene

rgy

part

s

(S. Mller, Advances in Solid State Physics, ed. B. Kramer (Springer, Berlin), Vol. 44, 415 (2004).)

Size vs. shape of precipitates in Al-Zn:Comparison between experiment and prediction

Mean precipitate radius rm [nm]0 1 2 3 4 5 6 7

Axia

l c/a

ratio

0,2

0,4

0,6

0,8

1,0exp.1 (T = 200K)exp.2 (T = 300K)exp.3 (T = 300K)exp.4 (T = 300K)exp.5 (T = 300K)theory (T = 300K)theory (T = 200K)

T = 200K

T = 300K

(x = 0.138)

(x = 0.068)

Ref.1:J. Deguercy et al., ActaMetall. 30, 1921 (1982). Ref.2: G. Laslaz and P. Guyot,Acta Metall. 25 , 277 (1977).Ref.3:E. Bubeck et al., Cryst. Res.Tech. 20, 97 (1985).Ref.4:V. Gerold, W. Siebke, andG. Tempus, Phys. Stat Sol. A 104, 141 (1987).Ref.5: M. Fumeron et al., ScriptaMetall. 14, 189 (1980).

ac

(S. Mller et al., Europhys. Lett. 55 (2001) 33)

Al-6.8% Zn: Simulation of aging process

T = 373 K Aging time: 0.02 sec.

Reduce Temperature to T = 300K...


END OF REAL TIME-SIMULATION


Qualitative comparison with typical TEM-picture


ZOOM: [111]-planes


Al-Zn: Average diameter of Zn-precipitates as function of aging time (T = 250K)

log(t) [sec]

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

log(

d m) [

]

0.8

1.0

1.2

1.4

1.6

1.8

our calc. slope = +1/3

Time t [sec]

10 40 100 250

10

40

25

Mean precipitate diam

eter dm [

]

Power law: dm t

MSCE: = 0.31Ostwald-ripening: = 1/3

(S. Mller, L.-W. Wang, and A. Zunger, Model. Sim. Mater. Sci. Eng. 10 (2002) 131;http://Select.iop.org)

T = 200 K T = 300 K

t = 30 sec

t = 1 min

fcc-Zn precipitates: A multi-scale example

E(eV)EF-1.0 -0.5 0.5 1.0

(c/a) [%] = 15

0,8 0,9 1,0 1,1 1,2

Ener

gy [m

eV/a

tom

]

-20

0

20

40

60

100

111

hcp-Zn

(c/a) [%]

fcc-Zn

-20 -10 0 10 20
Theoretische Festkrperphysik:AnwendungsbeispielFirst-principles calculations in materials science???Modelling materials properties demands the consideration ofPrecipitation in Al-rich Al-Zn alloysTreating long-range interactions:The mixed-space presentationTreating long-range interactions:The mixed-space presentationCoherency strain energyTreating long-range interactions:The mixed-space presentationFlattening along [111]: Instability von fcc-ZnBridging time scalesConfiguration-dependent activation barriers*Phonon spectra:Al31Li Li migrationConfiguration-dependent activation barriers*Size-shape relation of precipitatesSize vs. shape of precipitates in Al-Zn:Comparison between experiment and predictionfcc-Zn precipitates: A multi-scale example

theoretische festkörperphysik: anwendungsbeispiel -...

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