Theoretische Festkörperphysik: Anwendungsbeispiel. (File Bsp_theofkp.pdf unter Vorlesung anklicken!

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1 Theoretische Festkörperphysik: Anwendungsbeispiel (File Bsp_theofkp.pdf unter Vorlesung anklicken!) Vorlesung, Erlangen. WS 2008/2009

2 First-principles calculations in materials science??? Formation enthalpy H f of the B2 Phase for CoAl, NiAl, FeAl Jaguar XJ with Al-based car frame System H f [ev/atom] FeAl,B2 FeAl,B2 CoAl,B2 CoAl,B2 NiAl,B2 NiAl,B2 Exp. DFT Exp. DFT Exp. DFT -0, ,56-0, ,66 Exp.: P. Villars and M. Calvert, Experimental Handbook of Crystallographic Data (Materials Park, Ohio, 1991) Theo.: S. Müller. J. Phys.: Condens. Matter 15 (2003) R1429. mass reduced by ~ 200 kg (thanks to FORD Motor Company, Michigan, USA)

3 Modelling materials properties demands the consideration of huge configuration spaces TEM huge model systems Al-rich Al-Li: precursor δ T. Sato and A. Kamino, Mat. Sci. Eng. A 146 (1991) 161 T Prediction S. Müller, R. Podloucky, and W. Wolf, submitted temperature time!!! impossible to handle directly via DFT!!!

4 Electronic structure of materials (band structure, density of states ) Crystallographic atomic structure (relaxation, reconstruction, buckling ) Energetics (stability ) Activation barriers Density Functional Theory Nudge Elastic Band Method, Molecular Dynamics, Transition State Theory Vibronic properties (phonon spectra ) Dynamics (diffusion ) Ground state search in huge configuration spaces Multi-scale modeling (from atomic to mesoscopic scale) Short-range order Cluster Expansion Monte-Carlo Methods (UNCLE) Multi-site adsorption Segregation Nucleation Diffusion Precipitation

5 Precipitation in Al-rich Al-Zn alloys Quenching a solid solution into the two-phase region Formation of coherent Zn-precipitates: Coherent phase boundary calculated** experimental* 1000 Å Al x Zn (R. Ramlau and H. Löffler, phys. stat. sol. (a), 79, p.141 (1983)) * J. L. Murray, Bulletin of Alloy Phase Diagrams 4, 55 (1983). ** S. Müller et al., Europhys. Lett. 55, 33 (2001).

6 Treating long-range interactions: The mixed-space presentation Problem: Real-space CE fails to predict the energy of long-periodic coherent structures! Intrinsic fault of any finite Cluster Expansion: A n B n -Superlattice Ansatz: Range of interactions: H f = 0 for n Transform portion of interactions to reciprocal space Easiest to do for pair interactions Mixed-space form: H(σ) = Σ J(k) S(k,σ) 2 + Σ D f J f Π f k 3,4 body

7 Treating long-range interactions: The mixed-space presentation Solving the problem : J(k) = J CS (k) + J SR (k) Constituent Strain (CS): Contains the correct longperiodic superlattice limit Short-Ranged (SR) interactions that are ignored by J CS ( chemical part ) can be constructed from the equilibrium constituent strain

8 Coherency strain energy Epitaxial softening qal 0,6 Epitaxial Strain Energy : fcc-al Film (A) Epitaxial Strain Energy : Deform 0,5 (111) to the substrate lattice Deform toathe substrate lattice Film (A) EAepi (G,a) parameter and relax along G. epi (100) 0,4 (G,a) E q(a,g) = G A parameter a and relax along G. Coherency strain of q(a,g) = EAhydro (a) 0,3 EAhydro(a) GPdx-superlattices Cu1-x (201) Hydrostatic Deformation Energy : aal 0,2 Hydrostatic Deformation Energy : (110) Deform hydrostatically to the Substrate Deform hydrostatically to the 0,1 substrate lattice parameter a. Substrate 6,6 6,8 7,0 parameter 7,2 7,4 7,6 substrate lattice a. Elasticity theory: a Lattice parameter a [a.u.] Elasticity theory: a BB However: qharm q(a,g) (G) = 1 B qharm (G) = 1 - CC γγ(a,g) harm (G) Pd Cu C11 + γharm (G) B = 1/3 (C11 + 2C12 ): Bulk Modulus B = 1/ C12 ): Bulk Modulus = C(C 44 ½ (C11 C12): Elastic anisotropy parameter =γ C44 (G): ½ (C x12pd ): function Elastic anisotropy parameter 11 C Geometric of spherical angles harm γharm (G): Geometric function of spherical angles with l bl (a) Kl (G) = l Al (x) Kl (G) ECSeq (a,g) γ (a,g) = γharm (G) + ECSeq (x,g)

9 Treating long-range interactions: The mixed-space presentation E CS (σ) for any arbitrary structure σ can be calculated via This ansatz solves long-periodic superlattice problem! Mixed-Space Cluster Expansion (MSCE): Σ Σ k 3,4 body H(σ) = J(k) S(k,σ) 2 + D f J f Π f + E CS (σ) A. Zunger, NATO ASI on Statics and Dynamics of Alloy Phase Transformations (Plenum Press, New York, 1994), 361.

10 Size-shape-relation of precipitates Separate MSCE-Hamiltonians into two parts: Σ Σ k 3,4 body H(σ) = J(k) S(k,σ) 2 + D f J f Π f + E CS (σ) H = E chem + E CS (T 0) (N Zn = 2175) (S. Müller et al., Acta Mater. 48 (2000) 4007) Chemical part: compact shape Strain part: flat (111) layer: Softest direction in fcc-zn* * S. Müller et al., Phys. Rev. B 60, (1999).

11 fcc-zn precipitate: flattening along [111] 4248 Zn atoms (r p sphere = 25 Å)

12 Flattening along [111]: Instability von fcc-zn fcc-zn Density Of States DOS [a.u.] Energy [mev/atom] (c/a) [%] = ,8 0,9 1,0 1,1 1, a c (c/a) [%] hcp-zn E F DOS [a.u.] (c/a) [%] = 15 E(eV) G (S. Müller, L.-W. Wang, A. Zunger, C. Wolverton, Phys. Rev. B 60, (1999) ) E F E(eV)

13 Calculated coherent fcc-zn precipitates in Al-Zn as function of precipitate size and temperature 300K Temp. [K] 200K 30K a c Number of Zn-atoms

14 How do to kinetics in real time???

15 Bridging time scales Idea: Force selected atoms to exchange process Calculate corresponding simulation time afterwards Prerequisite: Calculation of energy change δe(i) for all possible atomic exchanges i (restriction to NN) From DFT calculations or experiment Energy B A δe(i) from MSCE (S. Müller, J. Phys.: Condens. Matter 15 (2003) R1429.)

16 Configuration-dependent activation barriers* L1 2 (Al 3 Li) (In collaboration with R. Podloucky, Univ. Wien, Austria, and W. Wolf, Materials Design, Le Mans, France) 0,8 Activation barrier [ev] 0,6 0,4 0,2 0,0 Al at Li-site Al at Al-site calc. of phonon spectra Diffusion coefficients as function of structure and temperature Trafo to real time 0,5 Activation barrier [ev] 0,4 0,3 0,2 0,1 +: no exp. parameters -: no transformation to real time because E = E(T) 0,0 Li at Li-site Li at Al-site (* calculated by the Nudge Elastic Band Method; R. Podloucky, Vienna)

17 Phonon spectra:al 31 Li Li migration (Walter Wolf, Materials Science, France) Relaxed Structure Al-vacancy Formation Li migration Al 31 Li Al 30 Li Al 30 Li

18 Configuration-dependent activation barriers* L1 2 (Al 3 Li) (In collaboration with R. Podloucky, Univ. Wien, Austria, and W. Wolf, Materials Design, Le Mans, France) Diffusion coefficient [m 2 /sec] 1e-11 1e-12 1e-13 1e-14 1e-15 Temperature [K] calc. of phonon spectra calculated Diffusion coefficients Bakker et al., 1990 Wen et al., 1980 as function of structure Costas, 1963 and temperature Verlinden and Gijbels, 1980 Trafo to real time T melt = 933K 1e : no exp. parameters -: no transformation to real time because E = E(T) 1000/T [K -1 ] (* calculated by the Nudge Elastic Band Method; R. Podloucky, Vienna)

19 Al-rich Size-shape relation of precipitates (no Al atoms are shown) Al-Li Al-Cu Al-Zn Percentage of energy parts nm 2 nm (T = 473K, t = 86.4 ks) (T = 373K, t=1.6*10 5 ks) S. Müller, W. Wolf, R. Podloucky, subm. J. Wang et al., Acta Mat. 53 (2005) 2759 E chem nm mean precipitate diameter [nm] mean precipitate diameter [nm] E chem E CS Å (T = 250K, t = 1.2 ks) S. Müller, J. Phys.: Condens. Matter 15 (2003) R1429 E chem E CS mean precipitate diameter [nm] (S. T. Müller, Sato and Advances A. Kamino, in Solid State Physics, T. J. Konno, ed. B. K. Kramer Hiraga, and (Springer, Berlin), R. Ramlau Vol. 44, and 415 H. Löffler, (2004).) Mat. Sci. Eng. A 146 (1991) 161 M. Kawasaki, Scripta Met. 44 (2001) 2303 phys. stat. sol. (a), 79, p.141 (1983))

20 Size vs. shape of precipitates in Al-Zn: Comparison between experiment and prediction a c Axial c/a ratio 1,0 0,8 0,6 0,4 0,2 T = 300K T = 200K (x = 0.068) (x = 0.138) exp.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) Ref.1: J. Deguercy et al., Acta Metall. 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, and G. Tempus, Phys. Stat Sol. A 104, 141 (1987). Ref.5: M. Fumeron et al., Scripta Metall. 14, 189 (1980) Mean precipitate radius r m [nm] (S. Müller et al., Europhys. Lett. 55 (2001) 33)

21 Al-6.8% Zn: Simulation of aging process T = 373 K Aging time: 0.02 sec.

22 Reduce Temperature to T = 300K... T = 373 K Aging time: 20.0 sec.

23 END OF REAL TIME -SIMULATION T = 300 K Aging time: 40.0 sec.

24 Qualitative comparison with typical TEM-picture T = 300 K Aging time: 40.0 sec.

25 ZOOM: [111]-planes T = 300 K Aging time: 40.0 sec.

26 Al-Zn: Average diameter of Zn-precipitates as function of aging time (T = 250K) Time t [sec] log(d m ) [Å] our calc. slope = +1/ Mean precipitate diameter d m [Å] Power law: d m t α MSCE: α = 0.31 Ostwald-ripening: α = 1/ log(t) [sec] (S. Müller, L.-W. Wang, and A. Zunger, Model. Sim. Mater. Sci. Eng. 10 (2002) 131;

27 T = 200 K T = 300 K t = 30 sec t = 1 min

28 fcc-zn precipitates: A multi-scale example (c/a) [%] = E F E(eV) 60 Energy [mev/atom] fcc-zn hcp-zn 0,8 0,9 1,0 1,1 1, (c/a) [%]