Ab Initio Thermodynamics

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Ab Initio Thermodynamics
Leandro Liborio Computational Materials Science
Group MSSC2008 Ab Initio Modelling in Solid State
Chemistry
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Experimental Motivation
A great variety of surface reconstructions have
been observed, namely (2x1), c(4x2) 123,
(2x2), c(4x4), (4x4) 12, c(2x2),
(v5xv5),(v13xv13) 1. And several structural
models have been proposed. Under which
circumstances are any of these models
representing the observed surface reconstructions?
1 T.Kubo and H.Nozoye, Surf. Sci. 542 (2003)
177-191. 2 M.Castell, Surf. Sci. 505 (2002)
1-13. 3 N. Erdman et al, J. Am. Chem. Soc. 125
(2003) 10050-10056.
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General Idea and Considerations

• DFT provides the 0K Total energy E(RI).
• Classical thermodynamics studies real systems.
• The systems are assumed to be in equilibrium.
  For the nanosystems considered here surfaces and
  defective systems- this approximation is good
  enough.
• We want to calculate appropriate thermodynamic
  potentials F, G, U, etc.
• Ab initio thermodynamics might have a different
  flavour depending on the first principles code
  we are using CRYSTAL, CASTEP, SIESTA, VASP, etc.

Ab initio atomistic thermodynamics and
statistical mechanics of surface properties and
functions. K. Reuter, C. Stampfl and M.
Scheffler, in Handbook of Materials Modeling
Vol. 1, (Ed.) S. Yip, Springer (Berlin, 2005). (
http//www.fhi-berlin.mpg.de/th/paper.html)
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General Idea and Considerations
Helmholtz free energy FU-TS, independent
variables (T,V) Enthalpy HUPV, independent
variables (S,P) Gibbs Free Energy GU-TSPV,
independent variables (T,P)
If, for a given P and T, G(T,P) is a minimum,
then the system is said to be in a stable
equilibrium.
DFT allow for the calculation of the total energy
of a nanosystem
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Gibbs Free Energy Gas Phase
T gt 298 K and PO2lt 2 atm
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Gibbs Free Energy Gas Phase
NIST-JANAF Thermochemical Tables, Fourth edition
Journal of Physical and Chemical Reference Data,
Monograph 9 (1998)
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Gibbs Free Energy Gas Phase
Parameter Value
A 29,659x10-3 KJ/(mol.K)
B 6,1373x10-6 KJ/(mol.K2)
C -1,1865x10-9 KJ/(mol.K3)
D 0,09578x10-12 KJ/(mol.K4)
E 0,2197x103 KJ.K/mol
F -9,8614 KJ/mol
G 237,948x10-3 KJ/(mol.K)
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Gibbs Free Energy Gas Phase

• Experimental errors
• Neglect of the thermal contributions to the
  Gibbs free energies of solids.
• DFT exchange and correlation approximations
• Presence of pseudopotentials (depends on the code)

Ab initio atomistic thermodynamics of the (001)
surface of SrTiO3. L. Liborio, PhD Thesis.
(http//www.ch.ic.ac.uk/harrison/Group/Liborio/Doc
s/liborio-phdthesis.pdf)
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Gibbs Free Energy Gas Phase
CASTEP, SIESTA GGA and LDA functionals.
Method 2 Calculating the oxygen molecules
properties from ab initio
Exp. PW-GGA (4) CRYSTAL (B3LYP)
Binding energy eV 2.56 3.6 2.53
Bond length ang 1.21 1.22 1.23
(4) W. Li et al., PRB, Vol. 65, pp.
075407-075419, 2002.
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Gibbs Free Energy Gas Phase
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Gibbs Free Energy Gas Phase
Method 1 Using experimental Gibbs formation
energies
Method 2 Calculating the oxygen molecules
properties from ab initio
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Gibbs Free Energy Solid Phase
Helmholtz vibrational energy
E(0K) Total ab initio energy. Sconfig
Configurational entropy. pV Related with the
systems volume, (0.005 J/m2 in the SrTiO3
surfaces.) Fvib(T) Helmholtz vibrational
energy.
The quantities of interest to us, namely surface
energies and defect formation energies, depend on
differences of Gibbs free energies.
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Gibbs Free Energy Solid Phase

• E(0K) total energy of the system calculated ab
  initio. This is the dominant term and the
  difficulty in calculating it depends essentially
  on the type of system and the chosen ab initio
  code.

• Sconfig0 The system configuration is known .

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Gibbs Free Energy Solid Phase
(1) K. Refson et al, Phys. Rev. B, 73, 155114,
(2006). (2) J. G. Taylor et al, Phys. Rev. B, 3,
3457, (1971). (3) N. Ashcroft and D. Mermin,
Solid State Physics, (1976).
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Gibbs Free Energy Solid Phase
(1) K. Refson et al, Phys. Rev. B, 73, 155114,
(2006).
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Gibbs Free Energy Solid Phase
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Gibbs Free Energy Solid Phase
Vibrational Helmholtz free energies contribution
Compound Vib. Cont. J/m2
PdO (1) 0.2
RuO2 (2) 0.15
aAl2O3 (3) 0.18
NiO100 (4) 0.2
SrTiO3001 0.2
The quantities of interest to us, namely surface
energies and defect formation energies, depend on
differences of Gibbs free energies.
(1) J. Rogal et al, PRB, 69, 075421, (2004). (2)
K. Reuter et al, PRB, 68, 045407, (2003). (3) A.
Marmier et al, J. Eur. Cer. Soc, 23, 2729,
(2003). (4) M.B. Taylor et al, PRB, 59, 6742,
(1999).
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Gibbs Free Energy Summary
Solid phases
Gas phases
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Magneli Phases
Figure 1a
Figure 1b
TnO2n-1 composition, .Oxygen
defects in 121 planes. Ti4O7 at Tlt154K
insulator with 0.29eV band gap(1). T4O7
Metal-insulator transition at 154K, with sharp
decrease of the magnetic susceptibility.
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Magneli Phases T4O7 crystalline structure
Figure 3c
Figure 3b
Figure 3a
Figure 3e
Figure 3d
Metal nets in antiphase. (121)r Cristallographic
shear plane.
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Technical details of the calculations
CASTEP
CRYSTAL
Local density functional LDA Ultrasoft
pseudopotentials replacing core electrons Plane
waves code Supercell approach
Hybrid density functional B3LYP,
GGA Exchange
GGA Correlation
20 Exact
Exchange All electron code. No
pseudopotentials Local basis functions atom
centred Gaussian type functions. Ti 27 atomic
orbitals, O 18 atomic orbitals Supercell
approach

• SCARF cluster. Facility provided by STFCs
  e-Science facility.
• HPCx, UKs national high-performance computing
  service.

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Defect Formation Energies
Figure 5a
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Formation Energies Oxygen chemical potential
CASTEP
CRYSTAL
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Results for the Magneli phases
Isolated defects
Figure 8a
Magneli phases
Figure 8b
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Results for the Magneli phases
Equilibrium point Ti4O7-TiO9
Equilibrium point Ti3O5-Ti4O7
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Results for the Magneli phases
Log(pO2) T(K)
Exper. -15 1538.5
CASTEP -15 1515.2
CRYSTAL -15 1379.3
P. Waldner and G. Eriksson, Calphad Vol. 23, No.
2, pp. 189-218, 1999.
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Conclusions

• Ab initio thermodynamics uses DFT to estimate
  Gibbs free energies.
• Ab inito thermodynamics allows general
  thermodynamic reasoning with nanosystems and it
  can be implemented using different ab initio
  codes.
• It can be used to simulate systems under real
  environmental conditions.
• For the Magneli phases, ab initio thermodynamics
  reproduce the experimental observations
  reasonably well.
• The equilibrium experimental (P,T) diagrams were
  reproduced from first principles.
• At a high concentration of oxygen defects and low
  oxygen chemical potential, oxygen defects prefer
  to form Magneli phases.
• But, at low concentration of oxygen defects and
  low oxygen chemical potential, titanium
  interstitials proved to be the stable point
  defects.

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Acknowledgements
Prof. Nic Harrison Dr. Giuseppe Mallia Dr.
Barbara Montanari Dr. Keith Refson