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

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 …

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.

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

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.

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

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

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

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

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.

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:

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)

       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

MCG3/3 PBE0/MG3 PBE0/MEC Analytic Potentials

Tight Binding

Potentials for Multiple Scales

accuracy:

0.01 0.02 0.02

7

13

177

∑∑>>>

+=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

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

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

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

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

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

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

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

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 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)

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

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

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

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

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

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.

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

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

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

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)