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Ab Initio Thermodynamics

Ab Initio Thermodynamics. Leandro Liborio Computational Materials Science Group MSSC2008 Ab Initio Modelling in Solid State Chemistry. (001). Sr. Ti. O. Double layer model. Castell’s model. Sr-adatom model. c(4x2) surface reconstruction. Experimental Motivation.

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Ab Initio Thermodynamics

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  1. Ab Initio Thermodynamics Leandro Liborio Computational Materials Science Group MSSC2008 Ab Initio Modelling in Solid State Chemistry

  2. (001) Sr Ti O Double layer model Castell’s model Sr-adatom model c(4x2) surface reconstruction Experimental Motivation A great variety of surface reconstructions have been observed, namely: (2x1), c(4x2) [1][2][3], (2x2), c(4x4), (4x4) [1][2], c(2x2), (√5x√5),(√13x√13) [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.

  3. 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)

  4. The systems’ total energy can be linked to the Gibbs free energy, from Thermodynamics G can be used to study the properties of the nanosystem The nanosystems are assumed to be in equilibrium. Ab Initio Thermodynamics General Idea and Considerations Helmholtz free energy: F=U-TS, independent variables (T,V) Enthalpy: H=U+PV, independent variables (S,P) Gibbs Free Energy: G=U-TS+PV, 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

  5. T T const. PO2 reference chemical potential Variation with temperature Variation with pressure Gibbs Free Energy: Gas Phase T > 298 K and PO2< 2 atm

  6. Gibbs Free Energy: Gas Phase NIST-JANAF Thermochemical Tables, Fourth edition Journal of Physical and Chemical Reference Data, Monograph 9 (1998)

  7. P.J.Linstrom. http://webbook.nist.gov/chemistry/guide Gibbs Free Energy: Gas Phase

  8. 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/Docs/liborio-phdthesis.pdf)

  9. T T const. PO2 Gibbs Free Energy: Gas Phase CASTEP, SIESTA: GGA and LDA functionals. Method 2: Calculating the oxygen molecule’s properties from ab initio: (4) W. Li et al., PRB, Vol. 65, pp. 075407-075419, 2002.

  10. Equate the derivatives from the analytical and polynomial expressions Get Gibbs Free Energy: Gas Phase

  11. Ab initio calculations of bulk metals and oxides Polynomial fitting of experimental data Equating the analytical results with the polynomial fitting Ab initio calculations of the oxygen molecule Gibbs Free Energy: Gas Phase Method 1: Using experimental Gibbs formation energies Method 2: Calculating the oxygen molecule’s properties from ab initio:

  12. 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.

  13. (001) Sr Ti O Double layer model Castell’s model Sr-adatom model c(4x2) surface reconstruction M. Castell, Surface Science, 1-13505 (2002) 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. • Sconfig=0 The system configuration is known .

  14. with Lattice Dynamics of Bulk Rutile Calculated using the implementation of Density Functional Perturbation Theory in the CASTEP code (1). The agreement with experimental results is excellent (2). Rutile unit cell 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).

  15. Gibbs Free Energy: Solid Phase (1) K. Refson et al, Phys. Rev. B, 73, 155114, (2006).

  16. C(4x2) reconstruction Gibbs Free Energy: Solid Phase

  17. T=1000 K c(4x2) reconstructions Gibbs Free Energy: Solid Phase Vibrational Helmholtz free energies contribution 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).

  18. Ab initio calculation Calculated, approximated, considered negligible Ab initio calculations of bulk metals and oxides Polynomial fitting of experimental data Equating the analytical results with the polynomial fitting Ab initio calculations of the oxygen molecule Gibbs Free Energy: Summary Solid phases Gas phases

  19. (1) K. Kobayashi et al., Europhysics Lett., Vol. 59, pp. 868-874, 2002. ( 2) W. Masayuki et al., J. of Luminiscence, Vol. 122-123, pp. 393-395, 2007. (3) P. Waldner and G. Eriksson, Calphad Vol. 23, No. 2, pp. 189-218, 1999. Magneli Phases Figure 1b Figure 1a TnO2n-1 composition, .Oxygen defects in {121} planes. Ti4O7 at T<154K insulator with 0.29eV band gap(1). T4O7 Metal-insulator transition at 154K, with sharp decrease of the magnetic susceptibility.

  20. View along the a lattice parameter View of Hexagonal oxygen arrangement Rutile unit cell View of Hexagonal oxygen network Magneli Phases: T4O7 crystalline structure Figure 3c Figure 3b Figure 3a Figure 3e Figure 3d Metal nets in antiphase. (121)r Cristallographic shear plane.

  21. 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 STFC’s e-Science facility. • HPCx, UK’s national high-performance computing service.

  22. Final state Initial state T, pO2 T, pO2 TiO2-x or TinO2n-1 TiO2 bulk + nDef oxygen atoms Phonon contribution pV contribution Defect Formation Energies Figure 5a

  23. Limits for the oxygen chemical potential: Hard limit Soft limit Formation Energies: Oxygen chemical potential CASTEP CRYSTAL

  24. Results for the Magneli phases Figure 8a Isolated defects Magneli phases Figure 8b

  25. Relationship between pO2 and T in the phase equilibrium. Results for the Magneli phases Equilibrium point Ti4O7-TiO9: Equilibrium point Ti3O5-Ti4O7:

  26. Experiment Figure 10c CRYSTAL Figure 10b CASTEP Figure 10a Results for the Magneli phases P. Waldner and G. Eriksson, Calphad Vol. 23, No. 2, pp. 189-218, 1999.

  27. 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.

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

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