Aluminum Nanoparticles: Energetics, Structure, and Chemical Imaging at 0 K and Finite Temperature
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Transcript of Aluminum Nanoparticles: Energetics, Structure, and Chemical Imaging at 0 K and Finite Temperature
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Aluminum Nanoparticles:Energetics, Structure, and Chemical Imaging
at 0 K and Finite TemperatureNov. 17, 2005, Aberdeen, MD
Nate SchultzAhren JasperPrzemek StaszewskiGrazyna StaszewskaDivesh BhattJ. Ilja SiepmannZhenhua LiMark Iron
and Don Truhlar
Dept. of Chemistry andSupercomputing Institute
University of Minnesota
Defense-University ResearchInitiative in NanoTechnology
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Aluminum nanoparticles are technologically important forenergetic fuels, and much can be learned from simulations.
A necessary starting point is • energetics
& structure
Let’s start there …
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Phase One: Validating PotentialsValidate Against Experiment?
Al2, Al3:bond energies,frequencies,ion data
Bulk data:cohesive energies,lattice constants,stress tensors, etc.
lack of nanoparticle data
Previous potentials for Al are fit to small clusters or bulk data.
Difficult to assess their accuracy for nanoparticles.
Use electronic structure theory and large-scale computing to generate accurate nanoparticle data.
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Multi-level DFT
(all-electron)
DFT Analytic Potentials
Tight Bindingmethods,
e.g., MCG3 (effective core potential)
Multiscale Scheme For Validating Potentials
affordability:
n ~ 7 n ~ 13 n ~ 100 n ~ 4,000 n >> 10,000
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Tested 43 functionals with MG3 basis: 6-311++G(3d2f,2df,2p)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
method
MUE (eV/atom)
PBE0 TPSS TPSSh TPSS1KCIS
= Alx
+ = AlxCyHz
= both
hybrid meta hybrid metaGGA
BPW91
HFE HFE
Key Result: PBE0/MG3 works well
DFT: All DFT is not the same — depends on functional and basis.
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Next step: Effective core potential
Errors relative to all-electron results:
0.130.13
0.06
0.01
0.034
0.018
0.006
MEC MECbest lit.ave. lit. ave. lit.best lit.
CEP-121G*
MU
E (
eV/a
tom
)
MU
E (
Å)
bond energies bond lengths
Allows smaller basis set — lowers cost
New: MN Effective CoreAverage over 7 from theliterature, only including ones with polarization functions
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Basis Sets
N CPU Time (hours)
Al13 96
Al55 30,000
Al177 33,000,000
6-311++G(3d2f,2df,2p) (all-electron basis)
est.
N CPU Time (hours)
Al13 0.2
Al55 16
Al177 8,000
MEC (MN effective core method)
Largest Calculation:
Al177 1D optimization with effective core potential
CPU time: 8,000 hours = 30 hours 256 processors
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Creation of Al Nanoparticle Database by DFT Calculations
1. Many SCF convergence issues for larger clusters
• near degeneracy (gap as size )
We found NWChem to performbest due to most stable integration grids
0
75
150
225
300
375
0 100 200Number of Atoms
SC
F C
ycle
s
0
5
10
15
0 100 200Number of Atoms
Mul
tipl
icit
y
2. Must find lowest-energy multiplicity
Special difficulties
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Structural Preferences
3.43
3.39
2.43
2.48
2.53
FCC
HCP
Icosahedral
(JT-distorted)
FCC
HCPBulk crystal structures are not
preferred in small clusters
BCC2.42
BCC 3.33
≈
Cohesive energy(eV/atom)Bulk Al13 clusters
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Structural Preferences of Aln Nanocrystals, 0 K
11 13 15 17 19 21 23 25 27
0.1
0.05
Structures of global minima are icosahedral-like for these nanocrystals.
cohe
sive
ene
rgy
(eV
/ato
m)
2.7
2.6
2.5
2.4= BCC = FCC = HCP
n
= global min.
0.9 nmOur potential gives correct ordering for bulk.
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Al55
1.5 nm
Icosahedral FCC2.82 eV/atom2.77 eV/atom
Transition between icosahedral and FCC occurs around 1 nm.
Structural Preferences, 0 K (cont.)
Al55 is two geometric shells.
Cohesive energy:
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Structural Preferences of Nanocrystals, 0 K
+ = FCC
= HCP
= BCC
10 30 50 70 90 110 130 150 170
FCC favored for large n
HCP & FCC oscillate for intermediate sizes
BCC, HCP, FCC energetically competitive for small n
cohe
sive
ene
rgy
(eV
/ato
m)
number of atoms (n)
3.0
2.8
2.6
2.42.10.9 1.5 1.9diameter (nm)
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Bond lengths (FCC structures, 0 K)B
ond
leng
th (
Å)
Bond lengths rapidly converge for small clusters < 1 nm
bulk 2.84 Å
2.70
2.72
2.74
2.76
2.78
2.80
2.82
0 50 100 150 200
number of atoms
2.10.9 1.5 1.9diameter (nm)
2.1 nm
Al177: 2.81 Å • 1% < bulk value
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MCG3/3 PBE0/MG3 PBE0/MEC Analytic Potentials
Tight Binding
Potentials for Multiple Scales
accuracy:
0.01 0.02 0.02
7
13
177
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∑∑>>>
+=γβαβα32 VVV
Accurate 2- & 3-body fits
• 402 Al3 geometries
• MUE = 0.03 eV/atom
Many-body expansion: 2-body, 3-body
Abandon this approach.
MU
E (
eV/a
tom
)
number of atoms
nano20 – 177
bulk∞2 3 4 – 19
clusters
= 2 body fit
= 3-body fit
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
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cluster2 – 19
nano20 – 177
bulk∞
Literature Potentials for Aln
• 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.
Popular approach: fit to bulk and extrapolate downPairwise
2 + 3 body
simple embedded atom 3 or 4 parameters
modified embedded atom5+ parametersM
UE
(eV
/ato
m)
n
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Fit to small clusters (n = 2 -13) and bulk
0.00
0.05
0.10
0.15
0.20
0.25
number of atoms
MU
E (
eV/a
tom
)
NP-B: modified embedded atom
NP-A: two-body + screening & coordination number
NP-A and NP-B show that this strategy works
— only slight improvement if fit to all data.
Fit 33 different potential forms containing various physical effects.
cluster2 – 19
nano20 – 177
bulk∞
Literature errors
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MCG3/3 PBE0/MG3 PBE0/MEC Analytic Tight
Binding
Accuracy (in eV/atom):
0.01 0.02 0.02 0.03–0.080.03(PRB 2005, 71, 45423)
Aln: Accurate Methods For Nanoparticle Simulation
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2.3
2.5
2.7
2.9
3.1
3.3
3.5
0.0 0.1 0.2 0.3 0.4 0.5N -1/3
Ene
rgy
per
atom
, eV
2.3
2.5
2.7
2.9
3.1
3.3
3.5
0.0 0.1 0.2 0.3 0.4 0.5N -1/3
Energy per atom, eV
Compare TB to analytic potentials: cohesive energy, 0 KFCC – redHCP – greenBCC – blue
Analytic (NP-A)Tight binding(Wolfsberg-Helmholtz)
bulk
Quasispherical clusters
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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
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0.6
0.7
0.8
0.9
1.0
0 400 800 1200
Sphericality Parameter (L) of liquid nanoparticles
†
L3Iunique
iIi
† †
IiIIuniquemax
Ii = moments of inertia
Other oblate spheroids:
Hockey puck: L = 0.600
Earth: L = 0.997
Prolates: 3 ≥ L > 1
Spherical: L = 1
Oblates: 0 ≤ L < 1
L definition from Mingos, McGrady, Rohl (1992)
QuickTime™ and aTIFF (LZW) decompressorare needed to see this picture.
1.0
0.6
0.8
400 800 12000
1000K
2500K
1500K
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36 †
r
Radial Distribution Function,
€
g(r) =1
4πr2Δr
n r +Δr
2
⎛
⎝ ⎜
⎞
⎠ ⎟− n r −
Δr
2
⎛
⎝ ⎜
⎞
⎠ ⎟
ρbulk,T
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
3
Al55 Al400 Al1000
r (Å) r (Å)
0 10 2010 200 0 15 3015 300r (Å)
0 5 10
6
3
g(r)
at g
iven
T
5 100
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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.
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Nanodroplet Densities at 1000 KComputed the nanoparticle density by averaging over the droplet volumes (computed with overlapping van der Waals spheres)
bulk density = 2.4 g/ml
2.1
2.2
2.3
2.4
0 500 1000 1500
number of atoms
dens
ity
(g/m
l)
55
400
1,000
96%
94%
89%
1.7 2.9 3.8diameter (nm)
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
Density as a Function of Position in Nanodroplet
r (Å)
dens
ity
(g/m
l)
Bulk liquid 55 400 1,000
inhomogenous
distribution
rr
Compute in shells as a function of distance from center of mass at 1,000 K
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0.0
0.7
1.4
2.1
2.8
0 2 4 6 8 10 0 5 10 15 200 5 10 15
0.0%
1.0%
2.0%
0 400 800 1200n
2.00
2.25
2.50
0 400 800 1200n
rm
s/
2500K
1000K
T =
100
0, 1
500,
250
0K
3D Imaging of Ensemble Averaged Densities2.8
2.1
1.4
0.7
0.02 6 10
r (Å)0 5 10 15 0 5 10 15 20
r (Å)r (Å)
2.50
2.25
2.00400 800 12000
2%
1%
0% 12008004000
mean fluctuation
1000 K
Al55 Al400 Al1000
2500 K
1500 K
---Bulk liquid
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0
2
4
6
8
10
12
0 5 10 15 20
Coordination number imaging of nanodroplets
r (Å)
coor
dina
tion
num
ber
55 400 1000
Interior:converging to 10.5
Surface:converging to ~4
• Coordination Number: number of atoms bonded to a specific center
Solid (FCC):
12
Liquid (exp. @ 1000 K): 10.2 ± 1 Black & Cundall 1965
or 10.6 Gamertsfelder 1941
2 nm
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0 6 12 18 0 6 12 18 240
4
8
12
0 5 10
3D Imaging of Ensemble-Averaged Coordination NumberC
N
T =
100
0, 1
500,
250
0K
4.0
8.0
12.0
0 400 800 1200n
0%
2%
4%
6%
0 400 800 1200
†
†
n
Al55 Al400 Al1000
2500K
!
1000K
CN
rms/C
N
mean fluctuation
12
8
4
0
12
8
4400 800 12000 400 800 12000
6%
4%
2%
0%
0 5 10r (Å)0 6 12 18
r (Å)0 6 12 18 24
r (Å)
CN
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2
3
4
5
0 5 10 0 6 12 18 240 6 12 18
3.25
3.75
4.25
0 400 800 12000.0%
1.0%
2.0%
0 400 800 1200
+
T =
100
0, 1
500,
250
0K
BE
(eV
)
1000K
2500K
BE
(eV
)
3D Imaging of Vacancy Formation Energy
mean fluctuationB
Erm
s/BE
5
4
3
20 5 10 0 6 12 18 0 6 12 18 24
r (Å)r (Å)r (Å)
400 800 12000
4.25
3.75
3.25n 400 800 12000
2%
1%
0%n
Binding Energy: BE
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QuickTime™ and aTIFF (LZW) decompressor
are needed to see this picture.
Critical properties of aluminum
For example, the critical temperature hasbeen measured only for Hg, Cs, Rb.
Various authors have tried to estimate
the Tc of Al in various ways, suchas approximate eqs. of state:
1962 8550 K1971 7151 K1984 5726 K1996 8860 K2003 12100 K2003 6400 K
The high-temperature properties of Al are given by the equation of state.High-temperature equations of state of metals are poorly known.
We will estimate Tc for Al
by Gibbs ensemble configurational-bias Monte Carlo calculations.
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Critical temperature of aluminumby Gibbs ensemble Monte Carlo calculations.
Tc
Tc = 6300 K for our nanoparticle potential
Tc = 3380 K forMei-Davenportembedded-atom potentialfit to bulk solid data
Vapor-liquid coexistence curvesExperimental liquid density
Checks on potential for liquid-vapor equilibria Embedded-atom Our potential Experiment
fit to solid + nanoparticles
Boiling point (K) 1802 2993 2791
Hvap,1100 (kcal/mol) 24 74.3 74.6
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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
Chemicalimaging
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Aluminum Nanoparticles:Energetics, Structure, and Chemical Imaging
at 0 K and Finite TemperatureNov. 17, 2005, Aberdeen, MD
Nate SchultzAhren JasperPrzemek StaszewskiGrazyna StaszewskaDivesh BhattJ. Ilja SiepmannZhenhua Li
and Don Truhlar
Dept. of Chemistry andSupercomputing Institute
University of Minnesota
Defense-University ResearchInitiative in NanoTechnology
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Bulk Limit
Results for NP-A (NP-B results are similar)
Correct ordering, but HCP crystal is overbound by 0.025 eV/atom
12 14 16 18 20
3.0
3.4
3.2
BCC
HCP
FCC
= accurate� = PEF
atomic volume (Å3)
cohe
sive
ene
rgy
(eV
/ato
m)