TheoretischeTheoretische FestkrperphysikFestkrperphysik::AnwendungsbeispielAnwendungsbeispiel
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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:
Treating long-range interactions:The mixed-space presentation
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
Treating long-range interactions:The mixed-space presentation
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
mean precipitate diameter [nm]0 1 2 3 4 5
0
20
40
60
80
100
E c
hem
mean precipitate diameter [nm]0 1 2 3 4 5
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...
T = 373 K Aging time: 20.0 sec.
END OF REAL TIME-SIMULATION
T = 300 K Aging time: 40.0 sec.
Qualitative comparison with typical TEM-picture
T = 300 K Aging time: 40.0 sec.
ZOOM: [111]-planes
T = 300 K Aging time: 40.0 sec.
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 exampleTop Related