Simulating the Structure and Dynamics of Heterogenous

Nanoclusters

François G. Amar

Department of Chemistry

University of Maine

Department of Chemistry & Physics, UNE January 23, 2009

Acknowledgements

•Jinasena Hewage (now at University of Ruhuna)•James Smaby•TJ Preston

•Gérard Torchet & Marie-Françoise de Feraudy•Marcus Lundwall & Swedish group at Lund

Motivation•Interested in how material properties vary as system size grows from atom to bulk.•Novel or special properties of ultra-small chunks of matter: catalysis, transport, fabrication.•Varying the stoichiometry as well as size of small particles adds another “tunable” parameter.

Finite heterogeneous systems are also of intrinsic interest.

Ballistic deposition of Cu on Ag yields subsurface shells: ”onion”

(MD with embedded atom potentials)

At intermediate temperature fcc-A-core supports A-B-A growth while ico-A-core does not.

F. Balleto, C. Mottet, and R. Ferrando, PRL 90(13) 5504 (2003)

Molecular Beam Source

Ar P=1 bar T=300 K V=550 m/s

€

V =2CPT0

M=

5RT0

M

A Closer Look at the Beam

V=550 m/s

vs=13 m/s

In skimmed beam, T ~ 0.5 K

Mach # = V/ vs ~ 40

How do we know what clusters are predominant in the jet?

•Sizes•Structure•Stoichiometry

Perform experiments!

V=550 m/s

electron diffraction

10 millisecond time window!

h

UV-vis or IRspectrum

ionizinge- beam

mass spectrum

Notice maxima at rare gases

Why is mercury a metal? When does it become so?Recall, Hg: [Xe] (4f)14(5d)10(6s)2

Let’s look at that ionization energy again.

Hg Hg (l)

10.3 eV4.49 eV

Hgn

?

Rademan, Kaiser, Even, Hensel, PRL 59, 2319 (1987)

Around n=20, change in slope suggests transition from van der Waals to metallic behavior

1/R ~ 1/n1/3

What about the theory of these small objects?

Structure•ab initio quantum mechanics or DFT(for smaller clusters)•Semi-empirical quantum theory•Empirical force-fields or potential energy surfaces

van der Waals systems larger clustersPES from scattering experiments…

Dynamics

•Quantum dynamics (9 degrees of freedom)•Quantum MD (classical MD with forces from DFT)

aka Car-Parrinello•Classical dynamics on potential energy surfaces

Large clusters Long times: 10 nanoseconds! Solve:

€

rF i = mi

r a i for i =1,N (Newton's 2nd Law)

wherer F i = −∇V [ Fx i

= -∂

∂xij≠1

∑ V(rij) ]

to get the trajectory : {r r i(t)}

Thermodynamics•Path integral Monte Carlo; diffusion MC•Multiple histogram methods using classical dynamics on potential energy surfaces ; adiabatic switching

accurate classical densities of states (“heavy” atoms) larger clusters

Two recent projects from our group

Ar/N2 cluster structure and dynamics

Simulating the photoelectron spectra of Arn, Xen, ArnXem clusters

Two recent projects from our group

Ar/N2 cluster structure and dynamics

Simulating the photoelectron spectra of Arn, Xen, ArnXem clusters

V(r) for Ar2

Ar/N2 Potential Models

• Ar-Ar (Aziz-Chen4) Re=3.75 Å; De=99.4 cm-1

•N2-N2 (exp-6 + 3-charge quadrupole2)

Canted parallel: Re=3.98 Å; De=102.5 cm-1

T-shape: Re =4.15 Å; De=102.8 cm-1

• Ar-N2 (damped dispersion model fit to

ab initio3)

Re=3.64 Å; De=111.9 cm-1

Quench energies of (N 2)nAr13-n Clusters

-44

-43

-42

-41

-40

-39

-38

0 1 2 3 4 5 6 7 8 9 10 11 12 13

No. of N2 molecules

Ar at the centerN2 at the center

Ar7(N2)6

with Ar at center

Despite the stronger pair interaction, N2 appears to be less easily incorporated into the center of the cluster than Ar due to frustration effects.

Ar7(N2)6

with N2 at center

Ar centered

N2 centered

The Caloric Curve: Heat both the Ar-centered and N2-centered isomers

Inflection is a signature of “melting”

Caloric curve

RMS bond fluctuation parameter

Orientational order parameter

What does melting mean away from the thermodynamic limit (large N)?

€

dE

dT

Tm

E

Tm

N --> ∞

E

Tm

Tm

€

dE

dT

small N

0 10 20 30 40 50 60

T / K

Ar-centered clusters

Ar-centered clusters

N2-centered clusters

t = 255 ps

t = 0 ps

t = 5 ps t = 450 ps

t = 500 psN2 molecules mix throughoutcluster and migrate to surface

(N2)13Ar42 Dynamics

Initial structure: quenched cuboctahedron with N2 in centerT = 41 K (liquid-like)

Two recent projects from our group

Ar/N2 cluster structure and dynamics

Simulating the photoelectron spectra of Arn, Xen, ArnXem clusters

“Phase” diagram of A55B55:

=AB/AA =BB/ AA

A.S. Clarke, R. Kapral, and G.N. Patey, JCP 101, 2432 (1994)

What does the photoelectron experiment measure?

So… …calculate the final state polarization energy

(the signal electrons--at 50 eV--leave in 10-16 seconds)

Potentials

Dimer Re / Å De / K

Ar-Ara 3.771 142.331

Ar-Xeb 4.0668 188.63

Xe-Xea 4.420 263.417aSlavicek et al, JCP 119, 2102 (2003)bAziz et al, JCP 78, 2402 (1982)

HFD type potentials with accurate well depths and equilibrium bond lengths.

Cubic splines used for potential and force.

=0.72

=0.54

Making clusters

Start with perfect ordered structures such as icosahedra and then warm and anneal within a bounding sphere.

Xe300

ico 0pdp 5hcp 67fcc 52unknown 176

The induced dipoles

are iterated to self-consistency, taking about 6 to 8 iterations to achieve self-consistent energies to 1 part in 106.

The polarization energies are binned to construct a histogram and we typically average over an ensemble of 10 to 20 clusters.

[Xe=4.04Å3; Ar=1.64Å3]

Polarization energy calculation

€

Eion = −eμ j ⋅rij

rij3

j =1

n

∑i =1

n

∑ + μ i ⋅Tij ⋅i <j

∑ μ j +μ i ⋅μ i

2α ii =1

n

∑

Self-consistent polarization energy calculation in which each atom in a cluster takes the role of the ion

€

μ i = α iE i = α i E icoul − Tij ⋅μ j

j =1

n

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥ i = 1,n€

where Tij = I −3rijrij

rij2

⎡

⎣ ⎢

⎤

⎦ ⎥1

rij3

Pure Xenon “4d5/2”

Xe150

Xe1000

Xe500

Xe250

Pure Argon “2p3/2”

Ar150

Ar250

Ar500

Ar1000

Signal Attenuation

€

Assuming a spherical cluster and random cluster orientation relative to

the detector, we can do one non - trivial integration over θ to obtain :

P =1

πe

−1

λr cosθ + r2 cos2θ + R2−r2 ⎛

⎝ ⎜

⎞

⎠ ⎟

0

π∫ dθ

€

P = e−d / λ probability an electron will travel

a distance d in a uniform medium

λ is an effective mean free path

r

d

R

What is the mean free path?Dependent on material and electron kinetic energy.

Tchaplyguine et al, permit an estimate of:

=17 Å and 9 Å for Xe and Ar clusters,

respectively for 50 eV signal electrons.

Alternatively, the TPP-2M formula gives (IMFP)

Xe6.5 Å and Ar=10.9 Å

at the same energy.

We use Xe6.5 Å and Ar=9 Å in the following.

Broadening

Convolute screened histogram data with Voight profile of isolated atom signal provided by experimentalists

Atom h /eV KEel / eV FWHM / eVXe 120 ~ 50 0.138Ar 310 ~ 60 0.207

Simulated Xe 4d5/2

spectrumStructure in the raw histogram bulk peak reflects local environment but is no longer apparent after broadening.

For Xe309, screening tends to “reduce” the bulk peak.

Xe309 “raw”

Xe309 broadened

Xe309 screened broadened

Pure Ar cluster spectra at 50 eV (=9Å)

Exp: <N>≈300

Ar500

Ar250

Ar1000

Pure Xe spectra

Exp: <N>=900

Xe150

Xe250

=17Å

Xe500

Exp: <N>=900

Xe500

=6.5ÅXe250

Xe1000

The polarization shift calculation appears to give semi-quantitative shifts

1) Experimentalists report a Gaussian size distribution in their beam with a FWHM =<N>.

2) Point dipole model may be inadequate.

3) Thermal treatments may affect final spectrum

Mixed Clusters

Experimental data*

*Thanks to to M. Lundwall for sharing these results prior to publication.

Xe500Ar500

core/shell structure

Xe spectrum

Ar spectrum

1)

SubstituteAr atoms with single Xe atoms

(“pepper”)

Substitute Ar atoms with small clusters of Xe (“plum”)

Modified clusters

Start with Xe/Ar core-shell.

Xe1000

Xe396Ar527

“plum”

Ar250

Xe396Ar527

“plum”

Conclusions• Polarization energy shift model captures the

essential physics and is quantitative to within about 5%.

• The bulk/surface shift model for pure clusters of Tchaplyguine et al is well supported by our atomistic calculations.

• Our preliminary calculations of mixed clusters supports the layering model proposed by the Swedish group. As the Ar/Xe ratio in the beam increases it appears that the cluster will consist of a core/shell structure with trapped Xe atoms and/or small clusters in the outer Ar layer.

Continuing Work

TJ Preston is refining the pure Ar and Xe cluster simulations and will be tackling the mixed cluster problem for his thesis.

FIN

Xenon spectrum of mixed clusters (Xe=6.5 Å; Ar=9 Å)

Argon spectrum of mixed clusters (Xe=6.5 Å; Ar=9 Å)

Xe396Ar527

raw “plum”

Xe396Ar527

screened

Xe374Ar549 raw “pepper”

Xe374Ar549

screened

Xe396Ar527 raw “plum”

Xe396Ar527

screened

Xe374Ar549 raw “pepper”

Xe374Ar549

screened

Xenon spectrum Argon spectrum

Making clusters IStart with perfect ordered structures such as icosahedra and then warm and anneal.

Xe300 (initially a cuboctahedron):

ico 0 pdp 0 hcp 58 fcc 71 unknown 171

Making clusters II

Grow from a small seed by bombarding with monomers while annealing (velocity scaling) within a bounding sphere.

Xe300

ico 0pdp 5hcp 67fcc 52unknown 176

TPP-2M formulaRG IMFP/Å EE/ÅM/g-mol-1 r/g-cm-3

Ar 10.9 9 39.95 3.655 14.3 8Xe 6.5 17 131.29 1.65 9.28 8

Compare single cluster spectrum with spectrum of a size distribution

N %77 16

309 68

464 14

773 2

Xe309 and Xe<309>

ensemble spectra:

=6.5 Å

Polarization models (Böttcher)Consider an ion (1) and a neutral (2) a distance s apart:

€

Mutually polarizable : W12 = −α 2e1

2

2s4

1

A4

where A4 =1−4α 1α 2

s6

€

Neutral only polarizable : W12 = −α 2e1

2

2s4

€

Homogeneous polarizability :

W12 = −α 2e1

2

2s41+

2ε + 4

2ε + 3

a

s

⎛

⎝ ⎜

⎞

⎠ ⎟2

+3ε + 6

3ε + 4

a

s

⎛

⎝ ⎜

⎞

⎠ ⎟4

+L ⎡

⎣ ⎢

⎤

⎦ ⎥ where α =

ε -1

ε +2a3

s

a

Ratio of homogeneous and point polarization energies

0.5

0.75

1

1.25

1.5

0 2 4 6 8 10

s/a

Wh/Wp

s/a

Re

What does theory already say about mixed

rare gas clusters?

L. Perera and F. G. Amar, JCP 93, 4884 (1990).

Garzon et al studied A13B13 systems (1989).

Single guest/host systems:

Scoles, LeRoy, FGA, …

Xe in Ar (Scharf, et al)

What is the mean free path, ?Dependent on material and electron kinetic energy.

Tchaplyguine et al, permit an estimate of:

Xe= 17 Å and Ar = 9 Å

for 50 eV signal electrons.

We use these values in the calculations presented here.

• Swedish experiments/data

Pure Ar cluster spectra at 50 eV (=9Å)

<N>≈300

Ar150

Ar250

Ar500

Simulated Xe 4d5/2

spectrum

Structure in the raw histogram bulk peak reflects local environment but is no longer apparent after broadening.

For Xe309, screening tends to “equalize” the two peaks.

Experimental data on Mixed clusters

Experimentalists report a Gaussian size distribution in their beam with a FWHM =<N>.

N %77 16

309 68

464 14

773 2

Compare

Xe309 and Xe<309>

ensemble spectra:

=17Å

X50Ar200 --> Xe

=17 Å

=17 Å

L=6.5

L=6.5

Pepper

Plum

Xe396Ar527

raw “plum”

Xe396Ar527

screened

Xe374Ar549

raw “pepper”

Xe374Ar549

screened

Modified clusters

Start with Xe/Ar core-shell cluster (XeAr), then:

1) Substitute Ar atoms with small clusters of Xe (“plum”)

2) 2) SubstituteAr atoms with single Xe atoms (“pepper”)