e E k T BOS Z B - Max Planck Societybemod12/Slides/Neugebauer_Bemod12.pdf · MPIE, Dept. Computational Materials Design BEMOD, Dresden, March 26 - 29, 2012 Do we have to go beyond

MPIE, Dept. Computational Materials Design BEMOD, Dresden, March 26 - 29, 2012

Long time scale simulations to determine accurate ab initio free energies

Jörg Neugebauer, Blazej Grabowski, and Tilmann Hickel

Max-Planck-Institut für Eisenforschung, Düsseldorf, Germany Department of Computational Materials Design

?

{ }∑ −=

I

BBOS

R

TkEeZ

/

T [°

C]

C [wt.%] 0 0

400

800

1200

1 2 3

+

+

+

+

Partition function

Phase diagram


Do we have to go beyond experiment? Calorimetric measurements

fcc Al fcc/bcc Mg

Fundamental input for all thermodynamic databases → But: Scatter of ~0.3 … 1 kB

Point defects (vacancies): Formation energies and entropies Al

Exp. DFT Ef (eV) 0.7 0.6 S (kB) 2.4 0.2


Do we have to go beyond experiment?

Even chemical trends are hard to derive from existing data

Key quantity to design novel high-strength steels Additional complication → magnetism

Stacking fault energies (fcc Fe-Mn)


Accuracy

Free energy difference between bcc and fcc iron

empirical potentials experiment

fcc bcc bcc

~1meV

Energy resolution better 1 meV! → Can we achieve such accuracy with present day ab initio techniques?

Comp. Mat. Sci. 41, 297 (2008)


Ab initio Thermodynamics

Statistical averages over coordinates, magnetic moments, occupations, chemical compositions

{ }( )xTV

TkfZRE BiIIIBOS

exTVZ,,

/,...,,,),,( σ

−=

{ }( ){ } xTVfZR

TkfZRE

iIII

BiIIIBOS

e,,,...,,,

/,...,,,∑ −=σ

σ

{ }{ }{ }

xTVR

TkifIIZRE

I

Bfixed

IBOS

e,,

/,...,,;

∑

−

=

σ

Vibrational excitations

{ }{ }{ }

xTVf

TkIIZIRfE

i

Bfixed

iBOS

e,,

/,...,,;

∑

−

×σ

Electronic excitations

Adiabatic approximation

+ cross terms e.g. electron-phonon interactions


Statistical averages with 1meV accuracy

Accuracy considerations

adiabnonmagT FFFFFFETVF −= ++++++= vacelahqh

K0),(

~0.1 meV/atom ~0.1 meV/atom

challenging < 1 meV/atom

( ) ( ){ }( )dttREt

TEtt

ttI

BOS

t∫∆+

=∞→∆ ∆=

0

0

1lim

a few hours/days 107 configurations (~ 10 temperature steps)


Are contributions beyond quasiharmonic

approximation relevant?


Free energy

Phonons

Expan- sion

Calphad GGA LDA Exp.

20

10

0

ω [

meV

] 0

-80

-160

-240

-320

-400

-480

F [m

eV/a

tom

]

200 400 600 800 T[K]

1.0

0.5

0

C P [

meV

/(K

atom

)]

0 300 600 T[K] 0 300 600

2

1

0

∆ x

[%]

Γ X K Γ L

Al

Example: Bulk (fcc) Aluminum

Heat capacity

Tmelting


Can we use empirical potentials to describe

anharmonic contributions

How to sample over 107 configurations with

ab initio accuracy?


Heat capacity of bulk Aluminum

200 400 600 800 T[K]

3.9

3.6

3.3

3.0

Hea

t ca

paci

ty [

k B]

Ab initio qh,el + Mishin et al.[1] ah

Ab initio qh,el + Ercolessi-Adams[2] ah

Ab initio qh,el + Mei-Davenport[3] ah

Ab initio quasiharmonic + electronic

Experiment[4]

None of the available empirical potentials is able to describe the anharmonic contribution

[1] Y. Mishin, et al., Phys. Rev. B 59, 3393 (1999). [2] F. Ercolessi and J. B. Adams, Europhys. Lett. 26, 583 (1994). [3] J. Mei and J. Davenport, Phys. Rev. B 46, 21 (1992). [4] D. A. Ditmars, et al., Int. J. of Thermophys. 6, 499 (1985).

Tmelting


Coarse graining configuration space

Thermodynamic integration

( ) ( ){ }( )∑=N

iiI

BOS tREN

TU1

Main challenge: Reduce number of (ab inition) configurations by several orders of magnitude

from MD or MC

Two major concepts

Free energy perturbation


Thermodynamic Integration

Key idea: Compute free energy change between reference system A and real system B

( ) ( )∫ ∂

∂=→∆

1

0

λλλ

λ

dUBAF ( ) ( )ABA UUUU −+= λλwith

0 2000 4000

12

11

10

9

8

Ener

gy [

meV

/ato

m]

MD steps

MD simulation in TI-step Application straightforward if good reference is available Typically the number of configurations can be reduced by 1-2 orders of magnitude → several 104 configurations Not affordable on highest ab initio level


Free energy perturbation

Key idea: Compute free energy change between reference system A and real system B

( )AB

ABB Tk

EETkBAF

−−−=→∆ expln

Performance increases with quality of reference For large differences to reference the method becomes inefficient/ fails For the targeted accuracy less efficient than thermodynamic integration

P

{ }IR

A

B


How to boost coarse graining?

Cycles in thermodynamic integration ( ) ( )∫ ∂

∂=→∆

1

0

λλλ

λ

dUBAF

λ integration (2-10 steps)

Thermodynamic average (103…104 MD steps)

→ Improved ref. reduces step number

0 2000 4000

12

11

10

9

8

Ener

gy

MD steps

Performance bottleneck


Solution

λ integration (2-10 steps)

Thermodynamic average (103…104 MD steps)

→ Improved reference reduces step number → Here: Quasiharmonic reference

0 2000 4000

12

11

10

9

8

Ener

gy

MD steps

→ Use free energy perturbation approach → Use low/medium converged DFT as reference

→ Typically reduces number of configurations by 102


Performance of the new approach • Wavefunction extrapolation • Thermodyn. integr. (TI) with Langevin dynamics • Ensemble average instead of time average (parallelization) • Up-sampling in λ−ΤΙ step • etc.

0 2000 4000

12

11

10

9

8

Ener

gy [

meV

/ato

m]

MD steps

Low convergence parameters efficient calculation

MD simulation in TI-step

0 40 80 MD steps

16

15

14

13

12 Ener

gy [

meV

/ato

m]

High convergence parameters few MD steps

Up-sampling

uncorrelated

Grabowski et al., Phys. Rev. B79, 134106 (2009).

Required CPU time (logarithmic)

Molecular dynamics

Thermodyn. Integration UP-TILD

110

year

s

1.1

year

s

20 d

ays

Efficient coarse graining reduces number of configurations from 107 to a few 102 !

→ Opens the way to a fully ab initio determination of free energies


Benchmark against experiment


Thermal expansion coefficient

Excellent agreement with experiment Systematic trend: LDA and GGA provide approximate measure of error bars

Temperature (K)

Experiment

Thermal expansion coefficient of Al

F (V,T) = Fel(V,T)+Fqh(V,T) + Fqh,el(V,T) + Fah(V,T) + Fvac(V,T) + Fint(V,T)

Grabowski,Ismer, Hickel, Neugebauer, Phys. Rev. B 79, 134106 (2009)


Heat Capacity

DFT gives lower bound to all recent experiments

F (V,T) = Fel(V,T)+Fqh(V,T) + Fqh,el(V,T) + Fah(V,T) + Fvac(V,T) + Fint(V,T)

Heat capacity of Al


Applications


Assessment of experimental data


Calcium: Heat capacity


Calcium: Heat capacity


Newly developed approaches allow to systematically improve performance of DFT to describe finite temperature properties

Ab initio thermodynamics (Conclusions)

Accuracy often exceeds experimental data even of stable phases

→ Provide excellent basis to compute thermodynamic data


Thanks to the department

CM Department (2011)


Thanks for your attention

Documents

e E k T BOS Z B - Max Planck Societybemod12/Slides/Neugebauer_Bemod12.pdf · MPIE, Dept. Computational Materials Design BEMOD, Dresden, March 26 - 29, 2012 Do we have to go beyond