Aluminum Nanoparticles: Energetics, Structure, and Chemical Imaging at 0 K and Finite Temperature

1. Aluminum Nanoparticles:Energetics, Structure, and Chemical Imagingat 0 K and Finite Temperature Nov. 17, 2005, Aberdeen, MD Nate Schultz Ahren Jasper Przemek Staszewski Grazyna Staszewska Divesh Bhatt J. Ilja Siepmann Zhenhua Li Mark Iron and Don Truhlar Dept. of Chemistry and Supercomputing Institute University of Minnesota Defense-University Research Initiative in NanoTechnology

2. Aluminum nanoparticles are technologically important for energetic fuels, and much can be learned from simulations. A necessary starting point is •energetics & structure Let’s start there …

3. Phase One: Validating Potentials Validate Against Experiment? Al2, Al3: bond energies, frequencies, ion data Bulk data: cohesive energies, lattice constants, stress tensors, etc. lack of nanoparticle data Use electronic structure theory and large-scale computing to generate accurate nanoparticle data. Previous potentials for Al are fit to small clusters or bulk data. Difficult to assess their accuracy for nanoparticles.

4. Multiscale Scheme For Validating Potentials Multi-level DFT DFT Tight Binding Analytic Potentials methods,e.g., MCG3 (all-electron) (effective core potential) affordability: n ~ 7 n ~ 13 n ~ 100 n ~ 4,000 n >> 10,000

5. DFT: All DFT is not the same — depends on functional and basis. Tested 43 functionals with MG3 basis: 6-311++G(3d2f,2df,2p) GGA hybrid meta hybrid meta r, r, HFE r, r, t, HFE r, r r, r, t  = Alx + = AlxCyHz  = both BPW91 PBE0 TPSS TPSSh TPSS1KCIS Key Result: PBE0/MG3 works well

6. Next step: Effective core potential Allows smaller basis set — lowers cost Errors relative to all-electron results: bond energies bond lengths 0.13 0.13 0.034 MUE (eV/atom) MUE (Å) 0.06 0.018 0.006 0.01 ave.lit. bestlit. MEC ave.lit. bestlit. MEC Average over 7 from the literature, only including ones with polarization functions CEP-121G* New: MN Effective Core

7. Basis Sets       6-311++G(3d2f,2df,2p)     (all-electron basis)                   MEC (MN effective core method) N CPU Time (hours) Al13 0.2 Al55 16 Al177 8,000 N CPU Time (hours) Al13 96 Al55 30,000 Al177 33,000,000 est. Largest Calculation: Al177 1D optimization with effective core potential CPU time: 8,000 hours = 30 hours  256 processors

8. Creation of Al Nanoparticle Database by DFT Calculations Special difficulties • 1.Many SCF convergence issues for larger clusters • near degeneracy (gap as size ) • We found NWChem to perform • best due to most stable integration grids 2. Must find lowest-energy multiplicity SCF Cycles Multiplicity Number of Atoms Number of Atoms

9. Cohesive energy (eV/atom) Bulk Al13 clusters Structural Preferences 2.42 BCC 2.43 FCC 2.48 HCP 2.53 Icosahedral (JT-distorted) ≈ BCC 3.33 HCP 3.39 Bulk crystal structures are not preferred in small clusters FCC 3.43

10. Structural Preferences ofAln Nanocrystals, 0 K Our potential gives correct ordering for bulk. 0.9 nm 2.4 • = BCC  = FCC  = HCP 0.1 2.5 cohesive energy (eV/atom) 2.6  = global min. 0.05 2.7 n  Structures of global minima are icosahedral-like for these nanocrystals.

11. Al55 is two geometric shells. Structural Preferences, 0 K (cont.) Transition between icosahedral and FCC occurs around 1 nm. Al55 1.5 nm Icosahedral FCC Cohesive energy: 2.77 eV/atom 2.82 eV/atom

12. Structural Preferences of Nanocrystals, 0 K diameter (nm) 0.9 1.5 1.9 2.1 2.4 BCC, HCP, FCC energetically competitive for small n    HCP & FCC oscillate for intermediate sizes 2.6   cohesive energy (eV/atom) FCC favored for large n 2.8  + = FCC • = HCP  = BCC 3.0 number of atoms (n)

13.  bulk 2.84 Å        Bond lengths (FCC structures, 0 K) diameter (nm) 1.9 0.9 1.5 2.1 Bond length (Å) • Al177: 2.81 Å • 1% < bulk value 2.1 nm number of atoms Bond lengths rapidly converge  for small clusters < 1 nm

14. Potentials for Multiple Scales Tight Binding Analytic Potentials MCG3/3 PBE0/MG3 PBE0/MEC 7 13 accuracy: 177 0.01 0.02 0.02

15. Abandon this approach. Many-body expansion: 2-body, 3-body  = 3-body fit MUE (eV/atom) • Accurate 2- & 3-body fits • 402 Al3 geometries • MUE = 0.03 eV/atom  = 2 body fit nano 20 – 177 clusters bulk ∞ 808 energies for Al2 – Al177 divided into 11 groups: Natom = 2, 3, 4, 7, 9-13, 14-19, 20-43, 50-55, 56-79, 80-88, and 89-177 2 3 4 – 19 number of atoms

16. Popular approach: fit to bulk and extrapolate down Literature Potentials for Aln Pairwise 2 + 3 body simple embedded atom 3 or 4 parameters MUE (eV/atom) modified embedded atom 5+ parameters cluster 2 – 19 nano 20 – 177 bulk ∞ n • Error is a function of n, will cause systematic errors in nucleation     or any size-dependent property. • Errors of literature methods  0.18 eV/atom for some n.

17. Fit to small clusters (n = 2 -13) and bulk Fit 33 different potential forms containing various physical effects. 0.25 Literature errors 0.20 NP-B: modified embedded atom 0.15 MUE (eV/atom) NP-A: two-body + screening &            coordination number 0.10 0.05 0.00 NP-A and NP-B show that this strategy works — only slight improvement if fit to all data. cluster 2 – 19 nano 20 – 177 bulk ∞ number of atoms

18. Tight Binding MCG3/3 Aln: Accurate Methods For Nanoparticle Simulation PBE0/MG3 PBE0/MEC Analytic Accuracy (in eV/atom): 0.01 0.02 0.02 0.03 0.03–0.08 (PRB2005, 71, 45423)

19. Compare TB to analytic potentials: cohesive energy, 0 K FCC – red HCP – green BCC – blue Quasispherical clusters 3.5 Tight binding (Wolfsberg-Helmholtz) Analytic (NP-A) 3.3 3.1 Energy per atom, eV 2.9 2.7 2.5 2.3 0.0 0.1 0.2 0.3 0.4 0.5 -1/3 N bulk

20. Simulation: Nanodroplets • • Monte Carlo Simulations at 1,000 K with NP-B Potential • can also use molecular dynamics with thermostat • Melting point of bulk Al is 933 K; cluster m.p. is lower • 3 cluster sizes in this talk: Al55, Al400, and Al1000 • Physical properties of the clusters: • shapes, densities, coordination numbers

21. 3 I unique L = ( ) I max I I = - unique i I S i i 0 800 1200 400 Sphericality Parameter (L) of liquid nanoparticles 1.0 1000K 1500K 0.8 2500K Ii = moments of inertia † † † Prolates: 3 ≥ L > 1 Spherical: L = 1 Oblates: 0 ≤ L < 1 0.6 Other oblate spheroids: Earth: L = 0.997 L definition from Mingos, McGrady, Rohl (1992) Hockey puck: L = 0.600

22. Al55 Al400 Al1000 6 3 g(r) at given T 5 10 0 10 15 0 20 0 30 r (Å) r (Å) r (Å) Radial Distribution Function, 6 † 3 3 r

23. Less often mentioned — nanoparticles properties show large fluctuations, even for a given n. Even less often mentioned — nanoparticles properties, even a given n, are inhomogeneous within a given particle. Nanoparticles, as we have heard — have properties intermediate between clusters and the bulk — tunable, changing size = number n of atoms

24. Nanodroplet Densities at 1000 K Computed the nanoparticle density by averaging over the droplet volumes (computed with overlapping van der Waals spheres) diameter (nm) 1.7 2.9 3.8 bulk density = 2.4 g/ml 96%  94% 1,000 density(g/ml)  400 89%  55 number of atoms

25. dr r Density as a Function of Position in Nanodroplet Compute in shells as a function of distance from center of mass at 1,000 K 55 400 1,000 Bulk liquid density (g/ml) inhomogenous r distribution r (Å)

26. 2.8 2.1 Al55 Al400 Al1000 1.4 T = 1000, 1500, 2500K r 0.7 0.0 10 15 20 0 5 0 5 10 15 r 3D Imaging of Ensemble Averaged Densities 1000K ---Bulk liquid 2500K 2 6 10 r (Å) r (Å) r (Å) 2.50 2% mean fluctuation 1000 K 2.25 Drrms/r 1% 1500 K 2500 K 2.00 0% 0 400 800 1200 1200 400 800 0 n n

27. Coordination number imaging of nanodroplets • Coordination Number: number of atoms bonded to a specific center Solid (FCC): Liquid (exp. @ 1000 K): 10.2 ± 1 Black & Cundall 1965 or 10.6 Gamertsfelder 1941 12 Interior: converging to 10.5 Surface: converging to ~4 coordination number 55 1000 400 r (Å) 2 nm

28. 12 8 4 0 5 10 0 0 6 12 18 18 24 0 6 12 r (Å) 12 CN 8 DCNrms/CN 4 0 400 800 1200 0 400 800 1200 3D Imaging of Ensemble-Averaged Coordination Number Al55 Al400 Al1000 1000K CN T = 1000, 1500, 2500K 2500K ! r (Å) r (Å) † 6% mean fluctuation 4% 2% † 0% † n n

29. 5 4 T = 1000, 1500, 2500K BE (eV) 3 5 10 0 0 6 12 18 18 24 0 6 12 r (Å) 2 DBErms/BE 0 0 400 400 800 800 1200 1200 3D Imaging of Vacancy Formation Energy Binding Energy: BE + 1000K 2500K r (Å) r (Å) 4.25 2% mean fluctuation BE (eV) 3.75 1% 3.25 0% n n

30. Critical properties of aluminum The high-temperature properties of Al are given by the equation of state. High-temperature equations of state of metals are poorly known. For example, the critical temperature has been measured only for Hg, Cs, Rb. Various authors have tried to estimate the Tc of Al in various ways, such as approximate eqs. of state: 1962 8550 K 1971 7151 K 1984 5726 K 1996 8860 K 2003 12100 K 2003 6400 K We will estimate Tc for Al by Gibbs ensemble configurational-bias Monte Carlo calculations.

31. Critical temperature of aluminum by Gibbs ensemble Monte Carlo calculations. Tc = 6300 K for our nanoparticle potential Tc Tc = 3380 K for Mei-Davenport embedded-atom potential fit to bulk solid data Experimental liquid density Vapor-liquid coexistence curves Checks on potential for liquid-vapor equilibria Embedded-atom Our potential Experiment fit to solid + nanoparticles Boiling point (K) 1802 2993           2791 DHvap,1100 (kcal/mol) 24 74.3 74.6

32. Summary • Development of accurate potentials for Al2 – Al∞ • validated PBE0 DFT method • developed improved effective core potentials • large and diverse database  new potentials • Structural characterizations of nanocrystals and nanodroplets • 0 K structural preferences and properties • High-T properties • Shapes • Oblate spheroids tending to spherical particles • Coordination numbers • bulk coordination for interior of Al400 and Al1,000 • Densities • bulk density for interior of Al400 and Al1,000 • In progress • Dynamics: association and dissociation rate constants • Heteronuclear systems: potentials for Al + hydrocarbon fragments Chemical imaging

33. Aluminum Nanoparticles:Energetics, Structure, and Chemical Imagingat 0 K and Finite Temperature Nov. 17, 2005, Aberdeen, MD Nate Schultz Ahren Jasper Przemek Staszewski Grazyna Staszewska Divesh Bhatt J. Ilja Siepmann Zhenhua Li and Don Truhlar Dept. of Chemistry and Supercomputing Institute University of Minnesota Defense-University Research Initiative in NanoTechnology

34. Bulk Limit Results for NP-A(NP-B results are similar) 3.0 BCC  = accurate  = PEF HCP 3.2 cohesive energy (eV/atom) 3.4 FCC atomic volume (Å3) Correct ordering, but HCP crystal is overbound by 0.025 eV/atom