Theoretical data for assessments

(1)

EPJ Web of Conferences 14, 03003 (2011) DOI: 10.1051/epjconf/20111403003

“ Fundamentals of Thermodynamic Modelling

of Materials ”

November 15-19, 2010 INSTN – CEA Saclay, France

Organized by

Bo SUNDMAN [email protected]

Constantin MEIS [email protected]

PROFESSOR & TOPIC

Pavel KORZHAVYI

MSE, KTH, Sweden

Theorical data for

assesments

(2)

11/17 2010

Theoretical data for

assessments

Pavel A. Korzhavyi

Applied Materials Physics

Department of Materials Science and Engineering

School of Industrial Engineering and Management

Royal Institute of Technology (KTH), Stockholm

phase diagrams

Models for the thermodynamic properties of each phase G(T, P, y)

Database for model parameters

Predictive calculation of: thermodynamic properties

equilibrium states

First-principles theories Empirical

relations Experimental

Information

assessment

s E _m

j s j s s

j end o

m G y RT n y y G

G ( ) () ()ln ( )

A modern CALPHAD flowchart

What data to feed in:

• Lattice stabilities? • Enthalpies?

• Heat capacities? • Models?

(3)

Slide 3

Success example: NIST-JANAF

data for diatomic gases

11/17 2010

Theoretical data for assessments

Internal degrees of freedom taken into account:

• Vibrational (harmonic and anharmonic)

• Rotational

• Rotation-vibration interactions

• Centrifugal stretching

• Electronic (transitions between molecular terms)

H

₂

molecule

Stable state: Paired spins, symmetric

Unstable state: Unpaired spins, antisymmetric

H H

(4)

Slide 5 Theoretical data for assessments 11/17 2010

Vibrational properties of dimer

molecules

Fit:

Spectrum:

Vibrational properties of dimer

molecules

Zero-point energy:

[1] K. K. Irikura, Experimental vibrational zero-point energies: Diatomic molecules , J. Phys. Chem. Ref. Data 36, 389 (2007).

Dimer Data source r₀ (Å) ₀ (cm1₎ _x

0 (cm1) ZPE (cm1) 1_H

2 This work 0.7517 4330.76 155.14 2126.59 Exp. [1] 0.7414 4401.21 121.34 2179.30 16_O

2 This work 1.2294 1560.09 13.22 776.74 Exp. [1] 1.2075 1580.19 11.98 787.38 63_Cu

2 This work 2.2149 264.55 1.02 135.14

Exp. [1] 2.2197 270.89 1.21

(5)

Energy units and accuracy

1000 cm

1

_{0.124 eV}

_{12.0 kJ/mol}

Molecule LDA GGA(PBE) BLYP PKZB

H₂ +0.16 0.21 0.00 +0.22

O₂ +2.36 +1.01 +0.64 +0.47

H₂O +1.49 +0.09 +0.01 0.09

N₂ +1.69 +0.64 +0.49 +0.03

Table 2: Absolute error (eV) of some density functional approximations in

predicting the dissociation energy of small molecules [2].

[2] S. Kurth, J.W. Perdew, and P. Blaha, Molecular and solid-state tests of density functional approximations: LSD, GGAs, and Meta-GGAs, Inlt. J. Quantum Chem. 75, 899 (1999).

Accuracy (continued)

• Accuracy of DFT functionals is routinely checked in molecular and

solid-state tests, whenever a new functional is created.

• It is good practice to find out the accuracy of various DFT approximations for your system.

Some useful references:

[2] S. Kurth, J.W. Perdew, and P. Blaha,Molecular and solid-state tests of density functional approximations: LSD, GGAs, and Meta-GGAs, Inlt. J. Quantum Chem. 75, 899 (1999).

[3] P. Haas, F. Tran, and P. Blaha, Calculation of the lattice constant of solids with semilocal functionals, Phys. Rev B 79, 085104 (2009).

[5] L. Pedroza, A.J.R. da Silva, K. Capelle,Gradient-dependent density functionals of the Perdew-Burke-Ernzerhof type for atoms, molecules, and solids,Phys. Rev. B 79, 201106 (2009).

[4] P. Haas, F Tran, P. Blaha, K. Schwarz, and R. Laskowski,Insight into the

performance of GGA functionals for solid-state calculations, Phys. Rev. B 80, 195109 (2009).

[6] P.J. Feibelman, DFT Versus the ‘‘Real World’’ (or, Waiting for Godft), Top. Catal.

(6)

Phonon spectrum of Cu

₂

O

P.A. Korzhavyi and B. Johansson, Technical Report TR-10-30, (SKB, Stockholm, 2010).

(7)

Thermodynamic properties of Cu

₂

O

+0.5 eV

(8)

Slide 13

Degrees of freedom to be

taken into account in solids

• Atomic defects

(non-stoichiometry, high

temperature)

• Vibrational (harmonic and

anharmonic)

• Magnetic excitations

• Electronic excitations

11/17 2010

> 10

-6

_s

~ 10

-13

_s

~ 10

-14

_s

~ 10

-15

_s

Time scale

Compound energy formalism (CEF)

2-sublattice (Bragg-Williams) model

A B Subl. 1 Subl. 2 AA BA AB BB

(A,B)

₁

(A,B)

₁

BA BA BA BA BA BA BA BA BA BA BA BA

G

SRF ) 2 ( B ) 1 ( B 0 BB ) 2 ( A ) 1 ( B 0 BA ) 2 ( B ) 1 ( A 0 AB ) 2 ( A ) 1 ( A 0 AA 0 SRF

y

G

y

G

y

G

y

G

Energy surface of reference

Ideal mixing entropy

) 2 ( B ) 2 ( B ) 2 ( A ) 2 ( A ) 1 ( B ) 1 ( B ) 1 ( A ) 1 ( A conf

ln

y

S

M. Hillert, The compound energy formalism, J. Alloys and Compounds 320, 161 (2001).

(9)

Slide 15

4-sublattice model, bcc Fe-Ti:

2

4 _{=16 end members}

11/17 2010

A2 A2

B2 B32

D0₃ D0₃

F F F F (1) F F T T (4) T T T T (1)

F F F T (4) FT FT (2) T T T F (4)

(10)

A combined

ab initio

- CALPHAD study

Fe Cr

P.A. Korzhavyi, B. Sundman, M. Selleby, and B. Johansson, Atomic, electronic, and magnetic structure of iron-based sigma-phases, in Integrative and Interdisciplinary Aspects of Intermetallics, MRS Proc., 842, edited by M.J. Mills et al., (Warrendale, 2005), pp. S4.10.1-6.

Icosahedrally coordinated sites

High coordination number (high-CN) sites

M1 (CN=12) M4 (CN=12)

M2 (CN=15) M3 (CN=14) M5 (CN=14)

(11)

(Fe,Cr)

₂

(Fe,Cr)

₄

(Fe,Cr)

₈

(Fe,Cr)

₈

(Fe,Cr)

₈

“end member compound”: FCCFC=Fe

2

Cr

4

Cr

8

Fe

8

Cr

8

2

5

= 32 end member compound energies.

These energies have been calculated from first

principles

-phase model

Energies of the end-member compounds

H at 0 K

H

H – TSconf

(12)

Magnetic moments and entropy contribution

• Magnetic moments are

vanishing on Fe atoms occupying icosahedral sites (1,4).

• _B, on Fe atoms occupying high-CN sites (2,3,5).

• Disorder of Fe magnetic moments gives an important contribution to the entropy:

S_magn=k_B_sa(s)c_Fe(s) M_Fe(s)+1)

3d

4d

5d

N

_d

= 1 2 3 4 5 6 7 8

(13)

C.-S. Oh, H. Murakami, and H. Harada, J. Alloys Compounds 313, 115 (2000).

O. Grånäs, P.A. Korzhavyi, A.E. Kissavos, and I.A. Abrikosov, CALPHAD 32, 171 (2008).

Mo-Ru sigma phase

• TiC_x and TiN_x crystallize in the B1 (NaCl) structure.

• Vacancies only on the non-metal sublattice, (Ti)(X,Va).

Non-stoichiometry in titanium

carbides and nitrides

Ti-C

+ +

L+

graphite+

K. Frisk et al., Calphad (2004)

Ti-N

L

GAS

L

: hcp

: bcc

: fcc

: anti-rutile

(14)

Vacancy-vacancy interactions

Experiment: T. Priem et al. _(1989). Cluster expansion:

Vacancies prefer 3rd _nearest

neighbor distance, <1,½,½>.

Theoretical methods:

Connolly-Williams method (CWM) Generalized perturbation method (SGPM)

Short-range order (SRO)

Monte Carlo modeling of vacancy ordering

Calculated phase diagram

Monte Carlo

•

Simulation box: 32x32x32 (32768 carbon sites)

•

5000 time steps per site

Order of vacancies in the (111) layers of Ti3C2and Ti6C5:

Ti₃C₂ Ti₆C₅

(111)

Ordering pattern

Stacking between layers

(15)

MC simulation of the ground

state structures in TiN

_x

• Ground state ordering is slightly different from TiC_x.

• Stacking of (011)B1

planes instead of (111)B1.

• All structures can be described within a 24 site unit cell (5

sublattices).

• Calculate the total energy of the structures and compare with the previous model. GUS L. W. HART, VOLKER BLUM, MICHAEL J. WALORSKI

AND ALEX ZUNGER (Nature Materials, 2005)

CEF models for Ti-TiC-TiN

(16)

Ti-C phase diagram Calculated and experimental

TiCxstability plot at ~298 K

500 1000 1500 2000 2500 3000 3500 4000 T (K )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x(C) L +graphite + + L+ 1 2 3 4

/+, Wagner

/+, Wagner Isothermally molten, Rudy DTA, Rudy

Incipient melting, Rudy +, Cadoff and Nielsen +, Cadoff and Nielsen , Cadoff and Nielsen , van Loo and Bastin /+, Cadoff and Nielsen Liquidus, Bickerdike and Hughes L

L+

+

+graphite

Prediction of phase diagrams

Ti-C phase diagram

Calculated and experimental TiC_xstability plot at ~298 K

(17)