Modelling the Phase Diagram of Carbon

1. Modelling the Phase Diagram of Carbon Luca M. Ghiringhelli, Van ’t Hoff Institute for Molecular Sciences, Universiteit van Amsterdam with Jan H. Los and Annalisa Fasolino, Evert Jan Meijer and Daan Frenkel Lyon, 17th October 2005

2. Outline: • Bond order potential(s) • LCBOP(I) • Calculating the phase diagram of carbon: method • Calculating the phase diagram of carbon: results • On the nature of liquid carbon • Improving LCBOP(I): LCBOPII • Summary and perspectives

3. Bond order • Bij depends on: • # neighbours • Angles (bending) • Conjugation • Dihedrals (torsion) VR Bij VA Carbon potential A Bond Order Potential:

4. Bond order = 1 for second and further neighbours Bonded interactions Non-bonded interactions = 1 for first neighbours • Bij depends on: • # neighbours • Angles (bending) • Conjugation • Dihedrals (torsion) Carbon potential LCBOP(I): Carbon Bond Order Potential with intrinsic non-bonded interactions

5. In depth: angular interactions Coordination three or more Coordination two or less G

6. LCBOP(I) In depth: torsional interactions

7. Calculating Phase equilibria Thermodynamic integration for free energy evaluation Reference system for the liquid: Lennard-Jonesliquid. Reference system for the solids: harmonic (Einstein) solid.

8. From Helmoltz to Gibbs free-energy

9. The calculated carbon phase diagram

10. T = 6000 K Liquid structure: no LLPT T = 6000 K T = 6000 K DF-MD (PBE) (Wu et al., PRL 2002) Brenner potential (Glosli and Ree, PRL 1999)

11. T = 6000 K 5000 K, 150 GPa Unsatisfactory features of LCBOP(I). Diamond (LCBOP(I)) Liquid (LCBOP(I)) Liquid (DF-MD)

12. LCBOPII • Extension of the angular dependency via a middle range (2.2 - 4 Å) potential, to account for dangling bonds 3HC CH3 C C 3HC CH3

13. LCBOPII • Extension of the angular dependency via a middle range (2.2 - 4 Å) potential, to account for dangling bonds • Angular correlations flexibly coordination dependent

14. LCBOPII • Extension of the angular dependency via a middle range (2.2 - 4 Å) potential, to account for dangling bonds • Angular correlations flexibly coordination dependent • New definition of torsion angles

15. LCBOPII • Extension of the angular dependency via a middle range (2.2 - 4 Å) potential, to account for dangling bonds • Angular correlations flexibly coordination dependent • New definition of torsional angles • New scheme for multidimensional smoothing

16. LCBOPII: radial distribution functions in the liquid

17. In summary: • State-of-the-artbond order potential for carbon (LCBOPII !) • Carbon phase diagram by free-energy calculation • No first order liquid-liquid phase transition Perspectives Surface reconstructions Size dependent phase diagram Nano-materials

18. This work was done in collaboration with: Jan H. Los, Radboud Universiteit Nijmegen Annalisa Fasolino, Radboud Universiteit Nijmegen and Universiteit van Amsterdam Evert Jan Meijer, Universiteit van Amsterdam Daan Frenkel, AMOLF and Universiteit van Amsterdam. Support: FOM and NCF For more information: LCBOP: Los and Fasolino, Phys. Rev. B 68, 024107 (2003) Diamond-like liquid: Ghiringhelli et al., Phys. Rev. B 69, 100101(R) (2004) Phase diagram: Ghiringhelli et al., Phys. Rev. Lett. 94, 145601 (2005) LCBOPII (theory): Los, Ghiringhelli et al., accepted on Phys. Rev. B LCBOPII (application): Ghiringhelli, Los et al., accepted on Phys. Rev. B