Activation energies for diffusion processes in damaged materials via Monte Carlo

1. Activation energies for diffusion processes in damaged materials via Monte Carlo Mark Calleja Martin Dove

2. Outline • An admission… • Origin: cationic diffusion in twinned quartz • Simulation details (Mishin’s way) • O2--vacancy diffusion in twinned perovskite • Cationic diffusion in metamict ZrSiO4

3. Previously… • The effects of domain walls in quartz on cationic conductivity were studied using empirical-potential molecular dynamics. • To compensate for short timescales (~ ps regime), strong electric fields were applied to promote forced diffusion. • This works, but is a roundabout way of getting information on the activation energies present. • Problems with large activation energies.

4. A Monte Carlo approach (Mishin) The standard Metropolis MC method is used for sampling our ensemble’s equilibrium distribution, which for a proposed configuration change from state m→n in NVT is : δVnm ≤ 0 accept move δVnm > 0 accept move if exp(-β δVnm) > rand (0,1) In fact, we tend to work in NPT which changes this slightly, but the idea’s the same (NPT involves changes in cell volume). Our interactions are 2- and 3-body empirical potentials plus (long-ranged) coulombic terms (treat using Ewald sum).

5. Now allow for ionic migration… • Select an ion, and march towards desired location in ‘small’ hops (steps of ~0.1-0.2 Å). • At every such hop, allow the entire crystal to relax (requires many attempted moves per ion). • However, hopping ion can only relax normal to its last jump direction. • If hopping towards a vacancy, allow all ions to further relax at the end of the hopping sequence, in order to locate true preferred position.

6. Na+ hopping along c-axis in bulk α-quartz at 10K

7. Na+ hopping along c-axis in the wall of twinned α-quartz

8. Twin walls in orthorhombic CaTiO3 We consider the effects of 90°rotation twins on ionic motion. Set up manually, starting from the bulk crystal (picture on right shows view down the c-axis). Careful at the walls to avoid unphysical interatomic distances. Then simmer…

9. Unrelaxed 90° rotation wall

10. Relaxed 90° wall (energy ~ 0.33(2) Jm-2)

11. Activation energies for O-vacancy hopping • Relax crystal using MC. • Create oxygen vacancy. • Charge balance O2- vacancy by removing distant Ca2+ ion. • Relax crystal again. • Force an oxygen towards the vacancy, relaxing at each step.

12. Vacancy motion on domain wall Pink balls are Ti4+ Blue balls are Ca2+ Red balls are O2- White ball is migrating oxygen

13. Vacancy motion on domain wall Pink balls are Ti4+ Blue balls are Ca2+ Red balls are O2- White ball is migrating oxygen

14. Activation energy for intra-wall hop

15. Vacancy motion off the domain wall

16. Vacancy motion off the domain wall

17. Activation energy for vacancy moving off wall

18. Activation energy for vacancy in corresponding bulk site

19. Na+ migration in a small damaged region. Some initial zircon results

20. Na+ migration in a small damaged region. Some initial zircon results

21. Na+ ion migrating into damaged zircon region at 10K

22. Na+ ion migrating into damaged zircon region at 300K

23. Summary • Modified MC approach appears to give reasonable results for activation energies in diffusive processes. • Results for twinned quartz are in agreement with MD data. • Depending on system, can get good high temperature results. • Metropolis MC is not efficient at low temperatures (slow equilibration times).

24. Outlook (“To do” list) • Improve implementation! • Consider much larger damaged samples. • Embed non-empirical methods?