1. Multiscale Modeling of Quasicrystals Outline: Introduction to quasicrystals Noncrystallographic symmetry, Penrose tiling model Interatomic pair potentials Derived from first principles, Force & energy matching Quasi-lattice gas model Identify the essential degrees of freedom Hierarchy of tiling models Tile Hamiltonians Current state of the art in quasicrystal modeling Back to atomistics, Thermodynamics

2. Co-workers: Ibrahim Al-Lehyani (CMU/KAU) Eric Cockayne (NIST) Chris Henley (Cornell) Marek Mihalkovic (Slovakia/CMU) Nassrin Moghadam (ORNL/Florida) John Moriarty (LLNL) Siddartha Naidu (CMU) Yang Wang (Pittsburgh Supercomputer Center) Funding: NSF

3. What’s wrong with 5-fold crystalline symmetry? Rotation Translation (shortest) Combination: New translation (shorter) Translationally periodic structure cannot have 5x axis  = (1+√5)/2 = 1.61803… is the Golden Mean, the “most irrational number”.

4. Flux-grown Quasicrystals (Ian Fisher, et al.) i-AlGaPdMn i-ZnMgHo d-AlCoNi

5. Penrose Tiling Model (Levine and Steinhardt) 36° and 72° rhombus tiling Open questions: a) Where are the atoms? b) Where are the tiles? “phason flip”

6. Total energy of a structure Perturbation expansion around uniform electron gas (Hafner, Moriarty) Pair potential E vol is structure-independent volume energy, v 3 is three-body term

7. Pair Potentials GPT (Moriarty&Widom, 1997) Fitted (Mihalkovic, 2006)

8. Primary tiling structure Atoms at vertices of rhombi rhombus edge length a=2.45 Å Tiling layer separation c/2=2.04 Å Important separations: Al-Al 2.88 Å Al-TM 2.45 Å TM-TM 2.54 and 4.5 Å Secondary tiling structure Atoms decorate Hexagon-Boat-Star (HBS) tiles Tile edge length a=2.45 Å Tiling layer separation c=4.08 Å Tile flip degeneracy Al rotations inside star

9. Tertiary tiling structure Tile flip degeneracy, Al flip inside hexagon & boat, Al rotations inside star

10. Phason flip of tertiary tiling Typical energy cost ~ 0.1 eV. Atomic displacements: 2 swaps of (NiNi) → (AlCo) 4 Al relaxations

11. Further abstraction: Tile Hamiltonians Effective tile energies and interactions favor edge zig-zagsfavor stars

12. R || Cut and project method Atomic surfaces RR

13. AlCoNi Cut & project model Aluminum Cobalt Nickel Atomic Surface #1 Atomic Surface #2

15. Current modeling state of the art: Ab-initio total energy calculations Experimental Al-rich phase diagram Ab-initio low temperature phase diagram Quasicrystal unstable by 20- 50 meV/atom. Improve structure model or include entropy?

16. Conclusions First-principles derived pair potentials capture important structural principles. Structural motifs follow Friedel oscillations of pair potential. Quasi-lattice gas Monte Carlo method effective in discovering these motifs. Forcing local motifs allows efficient simulation at higher length scales. Explaining quasicrystal stability remains a mystery and requires both improved models and incorporation of thermal excitations.