Nanoscopic Materials

Size-Dependent Phenomena

By Emil Roduner

The Royal Society of Chemistry

Copyright © 2006 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-0-85404-857-1

Contents

Chapter 1 Introduction, 1,
Chapter 2 Bulk and Interface, 5,
Chapter 3 Geometric Structure, Magic Numbers, and Coordination Numbers of Small Clusters, 21,
Chapter 4 Electronic Structure, 41,
Chapter 5 Magnetic Properties, 81,
Chapter 6 Thermodynamics for Finite Size Systems, 119,
Chapter 7 Adsorption, Phase Behaviour and Dynamics of Surface Layers and in Pores, 163,
Chapter 8 Nucleation, Phase Transitions and Dynamics of Clusters, 209,
Chapter 9 Phase Transitions of Two-Dimensional Systems, 233,
Chapter 10 Catalysis by Metallic Nanoparticles, 239,
Chapter 11 Applications: Facts and Fictions, 263,
Subject Index, 281,

CHAPTER 1

Introduction

1 Clusters and Nanoparticles

Particles with a size between 1 and 100 nm are normally regarded as nanomaterials. Figure 1 shows the size of nanoparticles in comparison with other small particles. In general, nanomaterials may have globular, plate-like, rod-like or more complex geometries. Near-spherical particles which are smaller than 10 nm are typically called clusters. The number of atoms in a cluster increases greatly with its diameter, demonstrated in Figure 2 for sodium clusters. At 1 nm diameter there are 13 atoms in a cluster and at 100 nm diameter the cluster can accommodate more than 107 atoms. Clusters may have a symmetrical structure which is, however, often different in symmetry from that of the bulk. They may also have an irregular or amorphous shape. As the number of atoms in a cluster increases, there is a critical size above which a particular bond geometry that is characteristic of the extended (bulk) solid is energetically preferred so that the structure switches to that of the bulk.

It is below a dimension of 100 nm where properties such as melting point, colour (i.e. band gap and wavelength of optical transitions), ionisation potential, hardness, catalytic activity and selectivity, or magnetic properties such as coercivity, permeability and saturation magnetisation, which we are used to thinking of as constant, vary with size. We basically distinguish two types of variations as a function of size:

• Scalable effects: Surface atoms are different from bulk atoms. As the particle size increases, the surface-to-volume ratio decreases proportionally to the inverse particle size. Thus, all properties which depend on the surface-to-volume ratio change continuously and extrapolate slowly to bulk values.

• Quantum effects: When the molecular electronic wave function is delocalised over the entire particle then a small, molecule-like cluster has discrete energy levels so that it may be regarded like an atom (sometimes called a super atom). The simplest model for it is that of a particle in a box. Adding more atoms to the cluster changes the size of the box continuously so that the energy levels close up to some extent. More importantly, adding more atoms means adding more valence electrons to the system. Thus, whenever a shell of sometimes multiple degenerate energy levels is filled the next electron has to be accommodated in the next shell of higher energy. The situation is analogous to the evolution of properties with increasing atomic number in the periodic table. Filled shells represent a particularly stable configuration. Properties such as ionisation potential and electron affinity are well known to display a discontinuous behaviour as one moves along the periodic table. For clusters consisting of atoms with strongly overlapping atomic orbitals, i.e. for metals and semiconductors, the situation is analogous.

Quantum effects are more pronounced for small clusters and often superimposed on a smoothly varying background of a scalable effect. Clusters are interesting intermediates between single atoms and bulk matter and represent a natural laboratory to ‘see both ends from the middle’.

2 Feynman’s Vision

On 29 December 1959, at the annual meeting of the American Physical Society, Richard Feynman addressed the audience with his visionary and by now historical and legendary lecture under the title – There is Plenty of Room at the Bottom: Invitation to Enter a New Field of Physics. With this talk on the problem of manipulating things on a small scale, Feynman opened the field of nanotechnology. Today, more than four decades later, the field is finally seen to really take off. It is amazing how closely some of the key developments follow Feynman’s vision. Why cannot we write the entire 24 volumes of the Encyclopaedia Britannica on the head of a pin? he asked. We know how successful information technology has been in its work towards this goal and we should be aware of how much this has influenced our lives. Make the electron microscope hundred times better, Feynman said. The development of the atomic force microscope was one of the milestones on the way not only to observe but also to manipulate in atomic dimensions. Amazingly, Rohrer and Binnig achieved this goal with their cantilever-based instrument in a single step, rather than by a hierarchy of smaller and smaller robots – training an ant to train a mite – as Feynman suggested. In 1986, the two scientists were honoured for their achievement with the Nobel Prize in Physics. It is quite obvious today that the invention of the scanning tunnelling microscope finally triggered the boom in nanotechnology to which the direct observation of very small scale structures down to individual atoms is essential. Obviously, seeing things directly is more convincing for vision-based beings than just having measurements which are in agreement with a model.

Some of the inspiration came from biology. Feynman understood that information is stored on a molecular level in biology, that cells manufacture substances and operate on a small scale, that the human brain is a wonderful and efficient miniaturised computer. Consider the possibility that we too can make a thing very small which does what we want, an object that manoeuvres at that level, he suggested. While his talk was primarily technology oriented, he knew that physics, chemistry, biology and engineering are all relevant and must all be involved. He realised that making things smaller was not just a technological problem of scaling down. He saw that certain things changed principally. Magnetism, for example, is a cooperative phenomenon and involves domains which cannot be reduced down to an atomic size. Atoms differ from the bulk in their quantum nature. Most significantly, Feynman predicted that when we have some control of the arrangement of things on a small scale we will get an enormously greater range of possible properties that substances can have. It is exactly the arrangement of things on a small scale which is the foundation for all the excitement about nanomaterials and for the success of modern materials science.

CHAPTER 2

Bulk and Interface

1 Gradients Near Surfaces

A simplistic view distinguishes between gaseous, liquid, and solid phases of a single chemical component, and it assumes that there is a sharp phase boundary where properties change discontinuously between two homogeneous neighbouring phases. In nature there are of course no discontinuities, and when we focus our view onto the interface we first find that it is corrugated on a level of atomic or molecular dimensions. More importantly, although the chemical potential of an equilibrated system is by definition the same in the bulk of both phases and does not change through the phase boundary, most other properties depend on the phase, but they change continuously along the coordinate perpendicular to the interface, often with superimposed oscillations. An example is given in Figure 1.

Let us assume that we cleave a perfect single crystal in vacuum. After cleavage, the density drops to zero instantaneously at the new interface, and the atoms or molecules near the surface experience an asymmetric interaction because some of the partners on one side have disappeared. Thus, they have to adjust to find a new balance of interacting forces. This is achieved by relaxation of the lattice. As a consequence, the local lattice parameters change, and the density becomes a function of distance from the surface. It is plausible that this gradual change in density is paralleled by changes in most other properties. The wave function is modified, bond lengths, bond strengths, molecular orientation, atomic or ionic mobility, conductivity, and even the index of refraction change. Essentially, as we approach a phase boundary, there is a finite gradient of all properties which may be of interest to us. Note, however, that quite often interfaces are nonequilibrium metastable regions so that not even the chemical potential is constant across a phase boundary.

The length scale λ0 over which these gradients extend depends on the range of the interactions between the particles. For example, the cohesion of noble gas atoms is governed by van der Waals forces. They have an attractive contribution that decays approximately with r-6, thus it has a relatively short range of a few atomic diameters. The Coulomb interaction in an ion pair goes as 1/r, it is thus in principle of much longer range. However, in an ionic lattice the alternation of anions and cations leads to an effective Coulomb potential which decays as r-5.

As long as a system has dimensions which are large compared to λ0 its behaviour is determined by its bulk properties, but when its diameter reaches values as low as a few times λ0 then its properties become dominated by those of the boundary region. Such a system is not strictly periodic, but it is also not homogeneously amorphous, rather the surface is always different from the interior so that there is a radial gradient in all properties.

2 The Coordination Number Rules the Game

An important parameter for the description of size-dependent phenomena is the surface-to-volume ratio, A/V. For spherical particles of diameter d (radius R) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The d-1 dependence holds for simple geometries such cubes, long cylinders or thin plates, but for complicated structures the relation is less straightforward. Equilibrated matter often adopts simple shapes. Many properties therefore obey to a good approximation a linear dependence if plotted against d-1. For spherical or cubic particles the diameter scales with the inverse third power of the number of atoms or molecules, N1/3, therefore an equivalent straight-line plot is obtained as a function of N-1/3.

Another expression that is often encountered in this context is the dispersion F, which designates the fraction of atoms in the surface shell of a material. For cubic clusters with n atoms of radius r0 along the edge the total number of atoms is N = n3, the number of atoms at the surface is given by 6 n2 for the six faces, corrected for double counts of the 12 edges (12 n) and reinstalling the 8 corners, so that the dispersion becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

This behaviour is illustrated in Figure 2.

For large spherical clusters of radius R the dispersion is proportional to the volume of a shell of thickness 2 r0 divided by the total volume. Assuming the same packing density and recognising that N = R3/r03 the dispersion becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](3)

which is the same as from Equation (2). Accounting for the fact that the packing density at the surface is lower, other sources use F ≈ 4/N1/3. The above expression is applicable for N larger than about 100, for smaller clusters detailed structural information is required. On this basis we obtain F= 0.4 for N = 103, and F = 0.04 for N = 106.

The key parameter which permits a more precise interpretation on a molecular level of the effects described by the surface-to-volume ratio and the dispersion is the number of direct neighbours, the coordination number. Atoms or molecules at and near the surface, and even more at edges and corners, have fewer neighbours and are therefore less strongly bound than those in the bulk. This is the reason why the surface has a higher energy, why it often melts first, and why it affects many other properties of the particle. It will be come clear throughout this text that a large number of size-dependent phenomena can be explained on the basis of the coordination number. This is the case whenever a property scales with the inverse particle diameter, or equivalently, with the inverse cube root of the number of atoms in a system.

The term coordination number is often explained based on a simplified model crystal built up of cubic atoms (Figure 3). It is the number of nearest neighbours which are bound via a face of the cube. Each such neighbour contributes with a certain binding energy to the stabilisation of an atom. The most stable atom is one in the bulk (not shown) which has six such neighbours (12 for close packed hard spheres); the least stable is a surface adatom which has only one. The numbers in Figure 3 are at the same time the coordination numbers. They are a first measure of the energy of the atom at a certain site, which is of importance in crystal growth and in catalysis.

In a better approximation the neighbours which are bound via an edge and those in contact via a corner would also contribute to stabilisation, but to a lesser extent. This would also distinguish between the energy of a corner atom (3) which has three face neighbours and three edge neighbours, and that of a kink atom (3′) which has also three face neighbours but six edge neighbours.

3 Surface Science, a Source of Information for Nanoscience

Properties near surfaces of macroscopic single crystals and of other interfaces have been studied both theoretically and experimentally in a number of areas over the last century. The early decades of the 20th century provided us with the macroscopic concepts of surface chemistry, ranging from adsorption isotherms to the dissociation of diatomic molecules and their desorption from metal surfaces. Much of this progress is due to Langmuir, but chemists like Emmett, Polany, Freundlich, Bodenstein, and Rideal, to name a few, have made other major contributions. From the 1960s new techniques permitted studies of atoms and molecules on surfaces.

Over 65 techniques including photon, electron, molecule and ion scattering as well as scanning probe methods are available today for the investigation of composition, atomic and electronic structures, and the dynamics of their motion. Their sensitivity extends from below 1% of a monolayer up to coverages reached under high pressure conditions. This makes surface science one of the main disciplines of physical chemistry. Interfacial systems can thus serve as models and provide a source of plenty and very valuable information related to nanoscale materials.

The surface reconstruction of silicon, which has been studied by low energy electron diffraction (LEED), may serve as an example. The driving force for such effects is the attempt of the surface to lower its energy by saturating the dangling bonds which result from the missing nearest neighbours. At the outermost surface of silicon (100) this is achieved to some extent through the formation of dimers, as illustrated in Figure 4. The relaxation resulting from this perturbation extends three to four atomic layers into the bulk.

Many metal surfaces also reconstruct. As a result of the one-sided coordination, the interatomic distances of the atoms at the topmost layer of Ir, Pt, and Au (100) surfaces shrink by a few percent. It then becomes more favourable for the top layer to adopt a hexagonally close packed structure rather than to maintain the square lattice of the underlying layer.

Already quite some time ago Benson and co-workers calculated the relaxation near the (100) surface of sodium chloride. The displacements which are displayed in Figure 5 show a tendency toward NaCl ion pair formation. While the large negative chloride ions are displaced away from the bulk interior, the smaller sodium cations show an oscillating behaviour. They move in and out alternating between subsequent layers. The consequence of this is a surface enrichment with chloride ions and an oscillating charge density as a function of distance from the surface (compare Figure 1).

The free energy of formation of a surface is always positive since creation of a surface from the bulk requires energy. Particles consisting of more than a single component have, in addition to surface reconstruction, a different way of reducing their energy: atoms or molecules which lower the surface free energy accumulate at the surface. This phenomenon is called surface segregation. It makes the surface composition different from the composition in the bulk. Impurities such as carbon or sulfur accumulate on the surface by diffusion from the bulk. In alloys it is often the lower melting component which has the lower surface energy and which therefore accumulates at the surface. Thus, in a silver–gold alloy, silver will be found in considerable excess in the first layer of the surface.

(Continues…)Excerpted from Nanoscopic Materials by Emil Roduner. Copyright © 2006 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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