1. Aluminum Nanoparticles: Energetics, Structure, and Chemical Imaging at 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. A necessary starting point is
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. Tested 43 functionals with MG3 basis: 6-311++G(3d2f,2df,2p)
DFT: All DFT is not the same — depends on functional and basis.
Tested 43 functionals with MG3 basis: G(3d2f,2df,2p)
GGA
hybrid
meta
hybrid meta
r, r
r, r, HFE
r, r, t
r, r, t, HFE
 = 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.
best lit.
MEC
ave. lit.
best lit.
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) Al
Al ,000
Al ,000,000
N CPU Time (hours) Al Al
Al ,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. Structural Preferences
Cohesive energy
(eV/atom)
Bulk
Al13 clusters
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 of Aln 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. Structural Preferences, 0 K (cont.)
Transition between icosahedral and FCC occurs around 1 nm.
Al55
Al55 is two geometric shells.
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. Bond lengths (FCC structures, 0 K)
 bulk
2.84 Å
diameter (nm)
0.9
1.5
1.9
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. 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
clusters
nano
20 – 177
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 2 3
4 – 19
number of atoms
Abandon this approach.

16. Literature Potentials for Aln
Popular approach: fit to bulk and extrapolate down
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
0.15
NP-B: modified embedded atom
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. Aln: Accurate Methods For Nanoparticle Simulation
Tight Binding
MCG3/3
PBE0/MG3
PBE0/MEC
Analytic
Accuracy (in eV/atom):
0.01
0.02
0.02
0.03
0.03–0.08
(PRB 2005, 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 N
-1/3
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. Sphericality Parameter (L) of liquid nanoparticles
1.0 L = 3 I
unique i S
1000K
1500K I - i
unique =
max ( )
0.8
2500K
Ii = moments of inertia † † †
Prolates: 3 ≥ L > 1
Spherical: L = 1
Oblates: 0 ≤ L < 1
0.6
400
800
1200
Other oblate spheroids:
Earth: L = 0.997
L definition from
Mingos, McGrady, Rohl (1992)
Hockey puck: L = 0.600

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

23. Nanoparticles, as we have heard —
have properties intermediate between clusters and the bulk
— tunable, changing size = number n of atoms
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.

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

27. Coordination number imaging of nanodroplets
Coordination Number: number of atoms bonded to a specific center
Solid (FCC):
Liquid 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. 3D Imaging of Ensemble-Averaged Coordination Number
Al55
Al400
Al1000
1000K 12 8 4
T = 1000, 1500, 2500K CN
2500K ! 5 10 6 12 18 6 12 18 24
r (Å)
r (Å)
r (Å) † 12 8 4 6%
mean
DCNrms/CN
fluctuation 4% CN 2% † † 0%
400
800
1200
400
800
1200 n n

29. 3D Imaging of Vacancy Formation Energy
Binding Energy: BE +
T = 1000, 1500, 2500K
BE (eV) 5 4 3 2
1000K
2500K 5 10 6 12 18 6 12 18 24
r (Å)
r (Å)
r (Å)
4.25 2%
mean
DBErms/BE
fluctuation
BE (eV)
3.75 1%
3.25 0%
400
800
1200
400
800
1200 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: K K K K K 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)            2791
DHvap,1100 (kcal/mol)

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