Transcript Introduction to Clusters - University of Birmingham

Atomic & Molecular Clusters
1. Introduction
Introduction
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What are clusters?
Types of cluster
Why study clusters?
Cluster size effects
Scaling Laws
Spherical cluster
approximation
• Cluster-surface analogy
• Nanoscience
• Importance of theory in
cluster science
What Are Clusters?
• Aggregates of 2–10n (n  6 or 7) particles
(atoms or molecules).
• Constituent particles may be identical or
they can be two or more different species.
• Clusters may be studied in the gas phase, in
a cluster “molecular” beam, adsorbed onto a
surface or trapped in an inert matrix.
The Abundance of Clusters
• Clusters are formed by most of the elements in the
periodic table – even the noble gases!
• Clusters of the coinage metals (copper, silver and
gold) are found in stained glass windows.
• Silver clusters are important in photography.
• Molecular clusters are present in the atmosphere.
• Carbon nanoclusters (e.g. C60 and related fullerenes)
may be present in soot and even in space.
Types of Cluster (1)
Metal Clusters
• s-block metals (e.g. alkali
and alkaline earth metals)
– bonding is “metallic”
(delocalised and nondirectional) involving
mainly the valence sorbitals.
• sp-metals (e.g. aluminium)
– bonding has some
covalent character.
• Transition metals – greater
degree of covalency and
directionality in the
bonding - involving the
valence d orbitals.
Types of Cluster (2)
Semiconductor Clusters
• Composed of elements
(e.g. C, Si & Ge) which
are semiconductors in the
solid state.
• Includes compound
semiconductor
(heteroatomic) clusters,
with polar covalent-bonds
(e.g. GaxAsy).
• Bonding is covalent –
bonds are directional and
strong.
Types of Cluster (3)
Ionic Clusters
• Heteroatomic clusters
composed from atoms
with large difference
in electronegativity.
• Bonding is
predominately ionic
(electrostatic).
• Examples: alkali metal
halides [NaxCly](xy)+
and magnesium oxides
[MgxOy]2(xy)+.
Types of Cluster (4)
Rare Gas Clusters
• Form at low
temperatures.
• Bound by weak van
der Waals dispersion
forces.
• Inter-atomic attraction
increases with
increasing atomic
mass (HeRn).
Types of Cluster (5)
Molecular Clusters
• Clusters of molecules.
• Types of bonding
include: van der
Waals, dipole-dipole
interactions, higherorder multipolar
interactions and
hydrogen bonding.
• Examples: (N2)n ;
(C6H6)n ; (HF)n ;
(H2O)n.
Types of Cluster (6)
Cluster Molecules
• Rich chemistry of inorganic and organometallic
clusters – developed over the second half of the 20th
century.
• Generally thermodynamically and/or kinetically
stable with respect to coalescence and can exist in
the solid, liquid and vapour phases.
• Examples: P4 ; [B12H12]2- ; Os6(CO)18.
Why Study Clusters? (1)
• Clusters are of fundamental interest:
due to their intrinsic properties
 because of their central position between molecular and
condensed matter science.

• Clusters span a wide range of particle size – from
molecular (well separated, quantized states) to microcrystalline (quasi-continuous states).
• Clusters constitute new materials (nano-particles)
which may have properties that are distinct from
those of discrete molecules or bulk matter.
Why Study Clusters? (2)
• Fundamental interest in the evolution of geometric and
electronic structures of clusters (and chemical and physical
properties) with cluster size.
• “How large must a cluster be before its properties resemble
those of the bulk element?”
• (Answer depends on the type of cluster and which property).
• The high ratio of surface to interior atoms in clusters means
that there are many common features between clusters and
bulk surfaces.
Cluster Size Effects (1)
• To what extent do cluster properties resemble those
of discrete molecules or infinite solids?
• Can the study of large finite clusters tell us anything
about the bonding or explain the properties of bulk
solids?
• How rapidly do cluster structures and other
properties converge towards those of the bulk as the
nuclearity (size) increases?
Cluster Size Effects (2)
• Can the evolution of band structure with increasing cluster
size be detected?
• For clusters of metallic elements, at what cluster size is
metallic conductivity first observed?
• Can phase transitions be observed and are they of the same
type found for bulk solids and surfaces?
• By studying the geometric structures of clusters, how their
structures change as a function of size, and cluster growth
patterns, can we gain an understanding of crystal growth at
the microscopic level?
Cluster Size Effects (3)
• Evolution of band structure (e.g. metal clusters):
E  kT
Atoms &
Molecules
E = 0
Ef
Clusters
Bulk
Cluster Size Effects (4)
• Size-induced structural phase transitions
Cluster Size Regimes (1)
Cluster Size
Small
Medium
Large
N
Diameter / nm
(NaN)
Fs
 102
 1.9
 0.86
102–104
1.9–8.6
0.86–0.19
> 104
> 8.6
 0.19
Cluster Size Regimes (2)
MACROSCOPIC
D
104
Å
N  1010
MESOSCOPIC
MICROSCOPIC
D ~ 102-104 Å
D ~ 10-102 Å
D  10 Å
N ~ 104-1010
N ~ 10-104
N  10
ATOMS &
MOLECULES
BULK
COLLOIDS
NANOCLUSTERS
Scaling Laws (1)
• In the large cluster regime, many cluster properties (e.g.
ionisation energy, electron affinity, melting temperature and
cohesive energy) show a smooth variation with cluster size.
• The following scaling laws apply for a general property (G)
GR   G  aR

GN  G  bN 
• Usually  = 1,  = 1/3.
Scaling Laws (2)
Large
Medium
Small
R, N
G(N,R)
1
G(1)
Liquid Drop
Behaviour
Quantum Size
and Surface
Effects
G()

R , Nb
Scaling Laws (3)
Examples
• Ionisation energies of potassium clusters (N100):
IP R  / eV  2.3  5.35R / Å 
1
IPN  / eV  2.3  2.04N

1
3
• Melting temperatures of gold clusters:
Tm R  / K  1336 .15  5543 .65R / Å 
1
Scaling Laws (4)
Deviations from Scaling Laws
• Large deviations (oscillations about the smooth
trend) are observed for many properties in the
medium and (especially) the small cluster size
regimes.
• Deviations arise due to Quantum Size Effects
(electronic shell closings) and Surface Effects
(geometric shell closings).
Spherical Cluster Approximation (1)
• N-atom cluster modelled by sphere.
• Cluster volume:
Vc = NVa
• Cluster radius:
Rc = N1/3Ra
• Cluster surface area:
Ac = N2/3Aa
• No. surface atoms:
Ns = 4N2/3
• Fraction of surface atoms:
Fs = Ns/N = 4N1/3
• Fs < 0.01 (1%) for N > 64,000,000
atoms.
Atom
Radius = R a
Cluster
Rc
Spherical Cluster Approximation (2)
• Plot of fraction of surface
atoms (Fs) against N1/3
for icosahedral shell
clusters.
1
N K   (10K 3  15K 2  11K  3)
3
• K – number of shells.
Cluster-Surface Analogy (1)
• Clusters have a high percentage of their atoms on the surface.
• There is a strong link between the chemistry and physics of
clusters and that of the surfaces of bulk matter.
• Cluster surface rearrangements, analogous to the
reconstructions observed for bulk surfaces, which lower the
cluster’s surface energy by forming additional surface bonds.
• Clusters may be stabilised by the coordination of ligands to
the surface.
Cluster-Surface Analogy (2)
• The reactivity of under-coordinated surface atoms makes
clusters of interest as models for heterogeneous catalysis on
bulk metal surfaces.
• Metal clusters (generally supported on an inert oxide
substrate) can themselves be used as very finely dispersed
metal for catalysis.
Nanoscience (1)
• The study of particles and structures with dimensions of the
order of a nanometre. Quantum dots etc.
•
Derives from an article by Richard Feynman (1960):
“There’s Plenty of Room at the Bottom”.
• Feynman challenged scientists to develop a new field of study
where devices and machines could be constructed from
components consisting of a small number of atoms.
• Clusters will play an increasingly important role as building
blocks in nanoscale devices.
Nanoscience (2)
• Example: Heath and coworkers (UCLA) have
observed a reversible,
pressure-induced
insulator-metal transition
in a langmuir film of thiolcoated silver clusters.
D
d
• Onset of metallicity
depends on ratio d/D.
• Could be used as a
nanoscale switching
device.
ligand shell
cluster core
The Importance of Theory
• Many cluster properties (e.g. cluster geometries, binding
energies and energy barriers) are not easily measured directly
from experiment.
• Theoretical models and computational methods have been
very useful in helping to interpret spectroscopic (e.g.
UV/visible and photoelectron) and mass spectrometric data.
• Clusters constitute an exacting testing ground for theoretical
methods – testing the range of validity of theoretical models
derived from the extremes of atomic/molecular and solid state
physics.
Atomic & Molecular Clusters
2. Experimental Methods
Generic Cluster Experiment
Generation (including Nucleation, Growth & Condensation)
Sources
Flow/Expansion
Interaction/Investigation
Detection
Ionization
Fragmentation
Spectroscopy
Reaction
Ions
Neutrals
Photons
Cluster Generation
The study, design and production of molecular beams is integrally related to
the production and study of clusters. There are many variations on the
molecular beam theme; however, they are in general based on one of the
two types of sources: Knudsen and supersonic free jets.
1. Knudsen Cell (Effusive Source)
• Thermal source (Maxwell-Boltzmann distribution of energies).
• Heat solid (or liquid) of low vapour pressure
– mean free path of particles in cell > daperture
– very few collisions before leaving cell
• daperture small enough: solid–gas eqm. not perturbed.
• Generates M1 , M2 , M3 .... MN in equilibrium distribution
– cluster intensity:
I(N) ≈ ae-bN
– broad energy and angular distribution
– continuous source
Advantages
simple (cheap)
I(N)  Eb (small clusters)
Disadvantages
low cluster flux
low nuclearity
limited to volatile elements
2. Supersonic (Free) Jet Sources
•Generally used for rare gas and molecular clusters and clusters of
low melting metals (e.g. Hg, alkali metals).
•Source can be continuous or pulsed.
•Slit nozzles can be used  2-d beams for optical measurements
requiring longer path lengths.
• Atomic vapour expanded from high pressure region (104–107 Pa) into
vacuum through a small nozzle (daperture ~ 0.03–1 mm).
• Mean free path « d
– many collisions during expansion
– adiabatic + isenthalpic expansion
– no collisions after expansion
• Mean velocity v increases (therefore KE rises)
• BUT random thermal motions reduced
– very low relative velocities
– low T in beam but high absolute velocity
• Can study pure clusters or mixtures
• Can pass beam through a pick-up chamber
– pick up atoms, ions, electrons, molecules
• Ionized clusters can be generated by introducing an electric discharge
on the high or low pressure side of the aperture
Mach Number: M = u/c
(u = stream velocity, c = local speed of sound)
Effusive beam (Knudsen cell) M ≈ 0 – broad v distribution
As M increases the velocity distribution narrows
Supersonic jets characterised by large values of M
Cluster Formation
• 3 Factors influence clustering:
– stagnation (or backing) pressure (p0)
– aperture cross section (daperture)
– initial gas temperature (T0)
• The cluster content (number of clusters) and average
cluster size N increase with increasing p0 and daperture, and
decrease with increasing T0.
• Low T0, high collision rate  cluster nucleation, growth
and coalescence.
Nucleation, Growth and Condensation
• Cluster Nucleation
M + M + M  M2 + M
– excess energy removed by third atom (KE)
– dimer acts as site of further growth and condensation
• Cluster Growth
MN + M  MN+1
• Cluster Condensation
MN + MP  MN+P (or MN+P-X + X M)
Cluster Temperature
• It is difficult to measure and define T for an individual cluster.
• If there is negligible clustering T is generally low.
• BUT cluster formation causes an increase in T:
– cluster formation is exothermic
– internal energy increases due to heat of condensation of added
atoms
– in heavily clustered beams (high ratio clusters/atoms) the clusters
are very hot (possibly molten).
Mechanisms for Cluster Cooling
1. Collisional Cooling
By collision (and energy transfer) with other atoms in the beam:
MN**(T1) + A  MN*(T2<T1) + A (inc. KE)
A = M or rare gas (usually in excess).
2. Evaporative Cooling
By evaporation (generally loss of one atom at a time for rare gas and
metal clusters) – energy released as KE of emitted atom:
MN**(T1)  MN-1*(T2<T1) + M (KE)
The dominant cooling mechanism once free flight is achieved.
3. Radiative Cooling
By emission of IR radiation:
MN**(T1)  MN*(T2<T1) + h (IR)
Inefficient and slow (compared with time scales of experiments).
Distribution of Cluster Sizes
Depends on initial conditions (T0, p0), presence of carrier gas, size and
shape of nozzle etc.
Photoionization mass spectra for sodium
in inert gas-seeded beams. A 0.3 mm
aperture cylindrical nozzle was used
throughout. The oven temperature was
800°C, corresponding to a Na vapour
pressure of 350 Torr. Inert gas backing
pressures were 1.3 atm in each case.
Larger Na cluster ions were detected for
heavier carrier gases. Note that expansion
of Na vapour through the same nozzle
without carrier gas results in
comparatively small cluster abundances.
Alternative Cluster Sources
1.
Laser Vaporisation-Flow Condensation Source
Ion detector
High pressure
Helium
Vaporization
Laser
Time-of-f light
mass spectrometer
Skimmer
Poppet
valve
Chemical reactor
Rod
Ionizing
laser
Ion
extraction
electrodes
•Combination of laser ablation (vaporization) and supersonic jet expansion.
•Related sources employ ion-beam sputtering (e.g. using heavy ions Xe+, KE = 30 keV) or
spark erosion to vaporize material.
•Applicable to high-melting point (refractory) elements (C, Si, transition metals etc.).
•Intense visible/UV pulse (> 107 W/cm2 i.e. ~10 mJ per mm2 in a 10 ns pulse) generates
10141015 atoms per mm2  plasma with T ~ 104 K.
•Rapid clustering induced after cooling by carrier gas (e.g. flowing He). Clusters subsequently
cool by evaporation.
2.
Gas Aggregation/Smoke Sources
•Evaporate (or sputter) material into slow (i.e. cold) flow of carrier gas (Ar or He; 50-500 Pa).
•Mild expansion out of stagnation chamber through a 1–5 mm aperture.
•Atoms slowed by collisions  aggregation
M + M + Ar M2 + Ar
•Further collisions lead to cooling and subsequent cluster aggregation.
•Analogous to cloud or smoke formation.
•Tends to generate larger clusters (more collisions and higher density of cooling gas compared
to Knudsen source).
•Has been used to make macroscopic quantities of C60.
Investigation of Clusters
1. Media in which Clusters can be Studied
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Molecular (cluster) beams – “free clusters”
•
Matrix isolated – “trapped clusters”
•
•
Matrix = liquid, glass or crystal (e.g. zeolites), condensed rare gases etc.
Surface adsorbed – “supported clusters”
2. Types of Investigation
•
•
Free Clusters
•
mostly using mass-selected cluster ions
•
detect by mass spectroscopy
•
photo-depletion spectroscopy
•
Laser-induced fluorescence (LIF) …
Trapped & Supported Clusters
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direct UV/vis., IR spectroscopy, ESR …
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microscopy – TEM, SEM, STM, AFM …
3. Types of Experiment
•
•
•
Electron Removal
•
electron ejection from neutrals or anions
•
collect and study cluster cations (measure Q/M) or electrons (measure KE)
Cluster Excitation
•
excite clusters (neutral or charged) by light irradiation or electron/atom
impact
•
study radiation or particles (atoms/cluster fragments) emitted
Cluster Reaction
•
reactions with small molecules (pick-up experiments)
•
cluster collisions (including collision-induced fragmentation)
4. Generation of Cluster Ions
• Cationic Clusters
• photoionization
• electron impact
(often leads to cluster fragmentation).
MN + e (KE)  [MN+]** + 2e  [MNP+]* + MP + 2e  …
• Anionic Clusters
• electron transfer from low IP atoms (e.g. alkali metals):
MN + Rb  MN + Rb+
• capture of low energy electrons
• Both cationic and anionic clusters can be generated in the plasmas generated by
laser ablation and electric discharge (spark) sources.
5. Size (Mass) Selection of Clusters
• Use magnetic or electric fields to select size of charged clusters by deflection
(depending on their Q/M ratio).
• Clusters can be studied as charged or neutral (after re-neutralization).
• (Mass selection of small neutral clusters can also be carried out by deflection
following collision with a beam of rare gas atoms/ions).
Factors Influencing Mass Spectral Intensities
IN = F (PN, SN, XN, DN)
•
PN – Cluster Production Efficiency (Source)
–
–
–
–
•
Type of source
Source conditions (p,T)
Type and % of carrier gas
Diameter of nozzle
SN – Cluster Stability (Lifetime)
– Stability of neutral cluster
» May depend on electron count or geometric structure
» Magic Numbers
•
XN – Ionization Efficiency
– Ionization probability (ionization cross section)
– Fragmentation probability during ionization
– Photoionization
• “Softer” than electron impact ionization
– Less likely to cause cluster fragmentation.
– Near Threshold Photoionization
• Most stable closed-shell clusters have high IPs and lower
ionization cross sections
– Dip in MS intensity for magic numbers.
•
DN – Detection Efficiency
• Usually varies smoothly with cluster size