Transcript Document 7270797

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
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 …
Phase One: Validating Potentials
Validate Against Experiment?
Al2, Al3:
bond energies,
frequencies,
ion data
lack of nanoparticle data
Bulk data:
cohesive energies,
lattice constants,
stress tensors, etc.
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.
Multiscale Scheme For Validating Potentials
Multi-level
DFT
methods,
e.g., MCG3 (all-electron)
DFT
(effective core potential)
Tight
Binding
Analytic
Potentials
affordability:
n~7
n ~ 13
n ~ 100
n ~ 4,000
n >> 10,000
DFT: All DFT is not the same — depends on functional and basis.
Tested 43 functionals with MG3 basis: 6-311++G(3d2f,2df,2p)
0.45
0.40
GGA
r, r
hybrid
meta hybrid meta
r, r, HFE
r, r, t r, r, t, HFE
MUE (eV/atom)
0.35
 = Alx
0.30
+ = AlxCyHz
0.25
 = both
0.20
0.15
0.10
0.05
0.00
method
BPW91
PBE0 TPSS TPSSh TPSS1KCIS
Key Result: PBE0/MG3 works well
Next step: Effective core potential
Allows smaller basis set — lowers cost
Errors relative to all-electron results:
bond energies
bond lengths
0.04
0.13
0.13
0.034
0.03
0.10
MUE (Å)
MUE (eV/atom)
0.15
0.06
0.05
0.02
0.018
0.01
0.006
0.01
0.00
0.00
ave. lit. best lit.
MEC
Average over 7 from the
CEP-121G*
literature, only including
ones with polarization functions
ave. lit.
best lit.
MEC
New: MN Effective Core
Basis Sets
6-311++G(3d2f,2df,2p)
(all-electron basis)
MEC
(MN effective core method)
N CPU Time (hours)
N CPU Time (hours)
96
Al13
Al55
30,000
Al55
Al177
33,000,000
est.
Al13
Al177
0.2
16
8,000
Largest Calculation:
Al177 1D optimization with effective core potential
CPU time: 8,000 hours = 30 hours  256 processors
Creation of Al Nanoparticle
Database by DFT Calculations
Special difficulties
1. Many SCF convergence issues for larger clusters
2. Must find lowest-energy multiplicity
300
225
150
75
0
15
0
100
200
Multiplicity
SCF Cycles
• near degeneracy (gap as size )
375
Number of Atoms
We found NWChem to perform
best due to most stable integration grids
10
5
0
0
100
Number of Atoms
200
Structural Preferences
Cohesive energy
(eV/atom) Al13 clusters
Bulk
≈
BCC
3.33
HCP
3.39
FCC
3.43
2.42
BCC
2.43
FCC
2.48
HCP
2.53
Icosahedral
(JT-distorted)
Bulk crystal structures are not
preferred in small clusters
Structural Preferences of Aln Nanocrystals, 0 K
0.9 nm
cohesive energy (eV/atom)
2.4
2.5
Our potential gives correct ordering for bulk.
= BCC
 = FCC
 = HCP
0.1
2.6
 = global min.
0.05
2.7
11
13
15
17
19
21
23
25
n
27
Structures of global minima are icosahedral-like for these nanocrystals.
Structural Preferences, 0 K (cont.)
Transition between icosahedral and FCC occurs around 1 nm.
Al55
Al55 is two
geometric shells.
Cohesive energy:
1.5 nm
Icosahedral
2.77 eV/atom
FCC
2.82 eV/atom
Structural Preferences of Nanocrystals, 0 K
cohesive energy (eV/atom)
0.9
diameter (nm)
1.5
1.9
2.1
2.4
BCC, HCP, FCC energetically competitive for small n



HCP & FCC oscillate for intermediate sizes
2.6


FCC favored for large n
2.8

+ = FCC
= HCP
 = BCC
3.0
10
30
50
70
90
110 130
number of atoms (n)
150
170
Bond lengths (FCC structures, 0 K)
0.9
Bond length (Å)
2.82
1.5
diameter (nm)
1.9
2.1
bulk
2.84 Å
2.80
2.78
2.76
Al177: 2.81 Å
2.74
• 1% < bulk value
2.72
2.1 nm
2.70
0
50
100
number of atoms
Bond lengths rapidly converge
for small clusters < 1 nm
150
200
Potentials for Multiple Scales
MCG3/3
PBE0/MG3
PBE0/MEC
7
13
177
accuracy:
0.01
0.02
0.02
Tight
Binding
Analytic
Potentials
V
V2  V3
 
   
Accurate 2- & 3-body fits
• 402 Al3 geometries
• MUE = 0.03 eV/atom
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
MUE (eV/atom)
Many-body expansion: 2-body, 3-body
50
45
40
35
30
25
20
15
10
5
0
 = 3-body fit
 = 2 body fit
2 3
nano bulk
clusters
20 – 177 ∞
4 – 19
number of atoms
Abandon this approach.
Literature Potentials for Aln
MUE (eV/atom)
Popular approach: fit to bulk and extrapolate down
Pairwise
1.4
1.2
2 + 3 body
deSPH
CoxJM
SutC
PapCEP
BetH
Gol
StrM
MeiD
MisFMP
1.0
0.8
0.6
simple embedded atom
3 or 4 parameters
modified embedded atom
5+ parameters
0.4
0.2
0.0
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.
Fit to small clusters (n = 2 -13) and bulk
Fit 33 different potential forms containing various physical effects.
0.25
MUE (eV/atom)
Literature errors
0.20
0.15
NP-B: modified embedded atom
0.10
NP-A: two-body + screening &
coordination
number
0.05
0.00
cluster
2 – 19
nano
bulk
20 – 177
∞
number of atoms
NP-A and NP-B show that
this strategy works
— only slight improvement
if fit to all data.
Aln: Accurate Methods For Nanoparticle Simulation
MCG3/3
PBE0/MG3
PBE0/MEC
Tight
Binding
Analytic
Accuracy (in eV/atom):
0.01
0.02
0.02
0.03
(PRB 2005, 71, 45423)
0.03–0.08
Compare TB to analytic potentials: cohesive energy, 0 K
Quasispherical clusters
FCC – red
HCP – green
BCC – blue
3.5
Tight binding
(Wolfsberg-Helmholtz)
3.3
3.1
2.9
2.7
2.5
Energy per atom, eV
Energy per atom, eV
3.5
3.3
Analytic (NP-A)
3.1
2.9
2.7
2.5
2.3
2.3
0.0 0.1 0.2 0.3 0.4 0.5
N -1/3
bulk
0.0 0.1 0.2 0.3 0.4 0.5
N -1/3
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
Sphericality Parameter (L) of liquid nanoparticles
L
1.0
1.0
3Iunique
S i Ii
1500K
0.9
Iunique  max( Ii - I )
Ii = moments of inertia

1000K
0.8
0.8
2500K

0.7

Prolates: 3 ≥ L > 1
Spherical: L = 1
Oblates: 0 ≤ L < 1
0.6
0.6
00
400
400
1200
1200
Other oblate spheroids:
Earth:
L definition from
Mingos, McGrady, Rohl (1992)
800
800
QuickTime™ and a
TIFF (LZ W) decompressor
are needed to see t his picture.
L = 0.997
Hockey puck: L = 0.600
Radial Distribution Function,
Al400
Al1000

6
6
Al55
  r   r  
n r   - n r -  

1
2 
2 
 
g(r) 
rbulk,T

4 r 2r 




3
3
3
g(r) at given T

r
00
55
r (Å)
1010 00
10
10
r (Å)
2020 00
15
15
r (Å)
3030
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.
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
density (g/ml)
2.4
96%
2.3
94%


1,000
400
2.2
89%

2.1
0
55
500
1000
number of atoms
1500
Density as a Function of Position in Nanodroplet
d
r r
density (g/ml)
55
400
Compute in shells as a
function of distance from
center of mass at 1,000 K
1,000
Bulk liquid
3.0
2.5
2.0
1.5
inhomogenous
1.0
r distribution
0.5
0.0
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
r (Å)
T = 1000, 1500, 2500K
r
3D Imaging of Ensemble Averaged Densities
2.8
2.8
1000K
2.1
2.1
---Bulk liquid
1.4
1.4
0.7
0.7
2500K
Al55
0.0
0.0
0
22
4
r (Å)
Al400
66
8
10
10
00
55
r (Å)
2.50
2.50
1500 K
rrms/r
r
2.25
2.25
15
15
00
2.0%
2%
mean
1000 K
10
10
Al1000
10 15
15 20
20
55 10
r (Å)
fluctuation
1.0%
1%
2500 K
2.00
2.00
00
400
400
n
800
800
1200
1200
0.0%
0%
00
400
400
n
800
800
1200
1200
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
12
coordination number
or 10.6 Gamertsfelder 1941
12
Interior:
10
converging to 10.5
8
6
55
4
Surface:
converging to ~4
1000
400
2
0
0
5
10
r (Å)
15
20
2 nm
T = 1000, 1500, 2500K
CN
3D Imaging of Ensemble-Averaged Coordination Number
Al55
12
12
Al400
88
2500K
44
00
!
00
r (Å)
12
10
10 00
55
66
12
12
8

0
400
400

n
800 1200
800
1200
CNrms/CN
CN
mean
0
66
12 18
18
24
24
12
r (Å)
6%
6%
8.0
4
00
18
18
r (Å)
12.0
4.0
Al1000
1000K
fluctuation
4%
4%
2%
2%
0%
0%
00
400
400
800
1200
n 800 1200
3D Imaging of Vacancy Formation Energy
T = 1000, 1500, 2500K
+
55
1000K
44
33
2500K
00
r (Å)
4.25
4.25
55
10
10
00
r (Å)
12
12
18
18
2%
00
66
12
18
24
18
24
12
r (Å)
2.0%
mean
3.75
3.75
3.25
3.25
00
66
BErms/BE
22
BE (eV)
BE (eV)
Binding Energy: BE
400 n 800 1200
400
800
1200
fluctuation
1%
1.0%
0%
00
0.0%
400
800
1200
400
n 800 1200
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
1971
1984
1996
2003
2003
8550 K
7151 K
5726 K
8860 K
12100 K
6400 K
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
We will estimate Tc for Al
by Gibbs ensemble
configurational-bias
Monte Carlo calculations.
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
Boiling point (K)
Hvap,1100 (kcal/mol)
Our potential Experiment
fit to solid
+ nanoparticles
1802
24
2993
74.3
2791
74.6
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
Chemical
imaging
– In progress
• Dynamics: association and dissociation rate constants
• Heteronuclear systems: potentials for Al + hydrocarbon fragments
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
cohesive energy (eV/atom)
Bulk Limit
Results for NP-A (NP-B results are similar)
3.0
•= accurate
 = PEF
BCC
HCP
3.2
3.4
12
FCC
14
16
atomic volume (Å3)
18
20
Correct ordering, but HCP crystal is overbound by 0.025 eV/atom