Vibrational spectroscopy

Molecular vibrations

Molecules consist of atoms bound together by what are usually called chemical bonds. The nature of these bonds will be discussed more fully in chapter 2 and it is only necessary to note here that the bonds and the angles between them are not rigid. To a first approximation the force required to make a small change in the length of a bond, or a small change in the angle between two bonds, is proportional to the change produced; similarly, the torque required to twist one part of a molecule through a small angle with respect to the rest about a bond is approximately proportional to the angle of twist. The molecule thus consists of a set of coupled harmonic oscillators and if it is disturbed from its equilibrium state it will vibrate in such a way that the motion can be considered to be a superposition of a number of simple harmonic vibrations. In each of these so-called normal modes every atom in the molecule vibrates with the same frequency, and in the simplest molecules all atoms pass through their respective positions of zero displacement simultaneously.

There are three principal methods by which the vibrations may be studied: infrared and Raman spectroscopies and inelastic neutron scattering. The first two methods are available in very many laboratories, since the equipment required is relatively small and cheap.

Introduction

This chapter is largely concerned with two aspects of the vibrational spectroscopy of polymers. The first of these is the interpretation of spectra, i.e. the assignment of the observed peaks in the infrared and Raman spectra at one or more of the levels listed in subsection 4.3.1, and the second is the application of a knowledge of such assignments to various aspects of the chemical analysis of polymers. Chapter 6 deals with applications aimed at understanding various aspects of the microstructure of the polymer which are generally classed as physical rather than chemical aspects, although the distinction is sometimes rather arbitrary. It should be emphasized that the division of spectral studies into separate phases of ‘interpretation’ and ‘application’ is somewhat artificial, since the two processes often take place side by side. Generally speaking, however, the aims of individual studies will be largely directed towards one or the other of these two aspects of vibrational spectroscopy.

The approach adopted for the interpretation of a particular vibrational spectrum depends upon the nature of the polymer under examination and the information being sought. If the purpose is simply the identification of the polymer, the spectrum is used on a ‘finger-print’ basis; strong peaks are assigned to various chemical groups and detailed assignments in terms of specific vibrational modes are unnecessary. If the objective is the quantitative analysis of a mixture, or of a copolymer, a rather less superficial approach is required.

Interatomic forces and molecular vibrations

Before considering the vibrations of polymers we shall consider those of isolated small molecules. The vibrational motion of such a molecule is independent of its overall translational motion and we shall neglect the effect of overall rotational motion of the molecule on its vibrational motion, since it can be shown that the two types of motion are independent to a good approximation. For polymers, particularly in the solid state, there is no significant rotational or translational motion of the molecules as a whole, but we shall see later, in chapter 3, that there are vibrational modes of crystalline polymers which are related to these types of motion.

When talking about the vibrations of a molecule it is convenient to consider the molecule to be a collection of atoms held together by chemical bonds. The atoms are imagined to be point masses and the bonds rather like springs, so that the whole assembly can vibrate as the masses move to and fro and the springs stretch and compress. This is the simple classical model that will be adopted in this chapter and it is worth a brief justification because at first sight it appears, when the true natures of the atoms and of the bonds are considered, to be rather far from the truth.

The distribution of copolymerized units in the polymer chain

General principles

If a mixture is made of two or more types of monomer which polymerize by similar mechanisms, e.g. the free radical process, they may be polymerized to form chains in which the structural units of the respective homopolymers are present to varying extents and in sequences whose lengths depend on the polymerization conditions. Such copolymers are important commercially and they are also the source of a considerable amount of information on polymerization processes. Their commercial importance is that they provide the means for modifying to advantage the physical properties of the parent homopolymers. They may provide a higher tensile strength, a greater impact strength, a superior stress crack resistance, an improved resistance to thermal and photochemical degradation or a modified crystallization behaviour. At the fundamental level, the molecular structure of the copolymer can often provide information on the polymerization mechanism, such as the factors which determine the reactivity of a particular radical. It is also possible to prepare polymers with known structural defects in order to study their effects. For example, the copolymerization of small amounts of ally1 chloride, CH2=CH—CH2CI, or acetylene, CH≡CH, with viny1 ch1oride may be used to introduce ch1oromethy1 branches, —CH2C1, or double bonds, respectively, into the viny1 chain in order to assess whether these types of defect are the sites of the initiation process for thermal degradation.

A number of crystal growth techniques have been developed for the production of specimens which have the form of a succession of layers. In molecular-beam epitaxy (MBE) beams of atomic or molecular species passing through ultra-high vacuum impinge on a single-crystal substrate and in the right conditions crystal growth occurs epitaxially, that is, with the crystal structures in register. In metallo-organic chemical vapour-phase deposition (MOCVD) growth occurs by deposition from a flowing vapour. Both MBE and MOCVD are used to produce semiconductor specimens in which, for example, single-crystal layers of GaAs and A1xGa1-xAs alternate. Metallic specimens, including those in which one or both constituents may be magnetic, are made by sputtering. A review of MBE is given by Joyce (1985). Parker (1985) and Chang and Ploog (1985) include detailed chapters on MBE as well as on the physics of the resulting specimens, while the latter also contains a chapter on MOCVD.

The growth techniques can be used to prepare specimens consisting of alternating layers of thickness d1 of constituent 1 and thickness d2 of constituent 2. Specimens can be prepared so that d1 and d2 have any value from two or three atomic spacings up to the order of 100 nm typically.

In dealing with magnetic surface modes in Chapters 3 and 4 we found it useful to classify modes, to some extent, by the range of the dominant restoring force. Thus Fig. 4.1 shows that for large enough wavevector |q∥| ≳ 108 m-1, the short-range exchange forces determine the nature of the surface modes. Over a wide range of intermediate wavevectors, approximately 104 m-1 < |q∥| < 107 m-1 the dipole—dipole force is dominant, and the important magnetostatic approximation holds. The properties of the magnetostatic surface modes are found by solving Maxwell's equations without retardation in the presence of the frequency-dependent tensorial magnetic susceptibility χ given by (4.15) to (4.17). Finally, for long-wavelength modes, |q∥| < 103 m-1, retardation cannot be neglected, and it is necessary to solve the full form of Maxwell's equations with susceptibility χ. This is the polariton, or electromagnetic, region, discussion of which has been deferred to this chapter.

Most of this chapter is concerned with electromagnetic, or polariton, modes arising from frequency-dispersion in the dielectric function rather than the magnetic susceptibility. The origin of dispersion was discussed in Chapter 5. Generally it results from the presence of a resonant mode carrying a dipole moment, like the optical phonon in a polar crystal.

In this book we are concerned with the ways in which surfaces or interfaces modify the properties of solids and liquids. Various different effects may be identified. First, there may be a modification to the equilibrium configuration in a medium close to a surface; this is known as surface reconstruction. For example, the atoms near to a surface may have a different crystallographic arrangement compared with those in the bulk, or they may be disordered. Another example is a ferromagnetic solid, in which the interactions between the magnetic moments at the surface may differ from those in the bulk, leading to a different value of the magnetisation. Clearly this type of effect may be temperature dependent, and it is particularly relevant when there is a phase transition (e.g. close to the Curie temperature in a ferromagnet). Second, the excitations within the system (such as the phonons in the lattice dynamics of a crystal or the magnons of a ferromagnet) are modified by a surface. In an infinite medium the bulk (or volume) excitations are characterised by an amplitude that varies in a wave-like fashion in three dimensions. When surfaces are present the bulk excitations are required to satisfy appropriate boundary conditions.

In earlier chapters of this book we have seen how in some circumstances a surface mode can arise at a frequency within a forbidden gap for the corresponding bulk mode. For example, the calculation of §2.1.2 for a semi-infinite diatomic lattice shows such a mode appearing in the frequency gap between the acoustic and optic branches. The first calculation of this kind for electronic structure was carried out by Tamm (1932), who considered a model of a simple 1D system comprising a semi-infinite lattice of square wells. For about twenty years after his calculation was published, various other calculations for model systems were performed. Attention then shifted to the question of performing realistic calculations for actual surfaces, the theoretical methods being developed from those used for calculation of bulk electronic band structure. A full treatment of this later work would involve an extensive account of band-structure theory, and is beyond the scope of this book. Instead, we restrict attention in §5.1 to a brief account of model calculations together with a description of some relevant experimental results. The reader interested in more recent work on electronic surface states may refer to Ehrenreich et al. (1980), and in particular the article by Kelly (1980) contains a list of earlier reviews.

The past twenty years have seen a great expansion in the study of surface properties. One part of this activity has been concerned with the various acoustic, magnetic and optic modes that propagate at the surface of a solid or liquid. These modes have a great deal in common, for example, they are often characterised by an amplitude that decays as an exponential (or sometimes a sum of exponentials) with distance from the surface. The generality of the concepts is well known to research groups working on surface modes, and most of them have made contributions across the board. However, although a number of excellent advanced monographs and review articles have appeared, there is no introduction to the field. The present work is designed to fill this gap.

Our intention is to provide an introductory text for someone starting research on surface modes or extending their range from one type of surface mode to another. It is hoped in addition that much of the material will be useful for advanced undergraduate teaching. In keeping with this pedagogical character, we have provided problems at the end of each chapter, and the lists of references are extensive, although we do not claim that they are comprehensive.