The ability of polymers to dissolve in various media has great practical importance and can be of either positive or negative benefit. Processing of polymers is often aided by forming solutions but formed polymers in use would most often benefit from being impervious to the environmental effects of potential solvents. Solutions also form an important arena for the characterization of polymers. For example, the various means for molecular weight determination rely on solution measurements. Thus there is good reason to understand the factors governing solubility and to understand the molecular organization of solutions.

Solutions in general, not just polymer solutions, are obviously of high importance and a great deal of attention has been fixed on understanding them. In any such endeavour it is very useful to have a simple theory that conceptually encompasses many of the phenomena observed even if it is not necessarily quantitatively accurate. That role for solutions of simple organic molecules has been filled by the ‘regular solution’ model. In the case of polymer molecules their long chain connectivity requires significant modification of the regular solution model. Thus an appropriate first task here is briefly to review the theory of regular solutions and then to introduce the Flory-Huggins modification for polymer solutions.

Regular solutions of simple non-electrolytes

The regular solution model is based on assuming the spatial disposition of two kinds of molecules about each other in a two-component mixture is random and separately evaluating the energy and entropy of mixing on this basis.

Polymers are large molecules made up of many atoms linked together by covalent bonds. They usually contain carbon and often other atoms such as hydrogen, oxygen, nitrogen, halogens and so forth. Thus they are typically molecules considered to be in the province of organic chemistry. Implicit in the definition of a polymer is the presumption that it was synthesized by linking together in some systematic way groups of simpler building block molecules or monomers. Although the final molecular topology need not be entirely linear, it is usually the case that the linking process results in linear segments or imparts a chain-like character to the polymer molecule.

Most of the synthetic methods for linking together the building block molecules can be placed into one of two general classifications. The first of these results when the starting monomers react in such a way that groups of them that have already joined can react with other already joined groups. The linked groups have almost the same reactivity towards further reaction and linking together as the original monomers. This general class of reactions is called step polymerization. In the other general method, an especially reactive center is created and that center can react only with the original monomer molecules. Upon reaction and incorporating a monomer, the reactive center is maintained and can keep reacting with monomers, linking them together, until some other process interferes.

There are a number of methods for determining experimentally the molecular weights of polymers. These include both the measurement of number-average and weight-average molecular weights. It will be seen that solution viscosity offers a very convenient method but it is not an absolute method and does not give one of the simple averages. The resolution of molecular lengths into fractions and therefore measurement of molecular weight distribution is also possible experimentally.

End-group analysis

In linear polymers each molecule has two ends so it is clear that a measurement of total numbers of end-groups in a sample of known weight can result in a determination of number-average molecular weight (= sample weight/moles of chains). There is no general method for accomplishing this and essentially the task embraces organic functional group identification in analytical chemistry. An obvious complication is that the method must be very sensitive since the end-groups are present at very low concentrations in high molecular weight polymers. The available methods can perhaps be classified as chemical or physical. Chemical methods would include acid-base titration of acidic or basic end-groups (—CO2H, for example), reaction of end-groups with determinable amounts of specific reagents, and chemical degradation to identifiable products from end-groups. The most prominent physical method and the most useful method in general is probably infrared vibrational spectroscopy.

It is not possible to keep track of the details of the configurations of polymer molecules when they have become disordered or coiled through populating various local bond conformations. An elementary calculation is instructive. Consider a chain with three conformational states for each skeletal bond, a trans and two gauche states for example. Then a chain with N bonds capable of internal rotation will have 3N total possible conformational states. For N = 1000, a modest chain length, there are 10477 states possible! Obviously statistical descriptions are called for. This can take the form of directly finding the average value of a desired property or, in more detail, finding a distribution function for the property. For example, in the consideration of the relation between the solution viscosity and molecular weight (Section 3.3.3) it was apparent that a measure of average dimensions or size was needed. Under appropriate conditions, in a ‘theta’ solvent where phantom chain behavior obtains, the mean-square end-to-end distance can be directly calculated. Under these conditions, and where the chain length is long, it is also possible to calculate a distribution function for the probability of a chain having an arbitrary end-to-end extension. In this chapter these particular questions will be taken up, the calculation, under phantom conditions, of mean-square dimensions and the distribution function for end-to-end distance. The effects of non-self-intersection in good solvents will also be considered.

The science and technology connected with polymeric materials has grown into an immense subject. It is not possible in any single work to cover the field in useful detail. Thus there is a daunting task confronting the person, who, wishing to become acquainted with such materials, attempts to master some of the areas of his or her special interest. It is our belief that polymeric materials are best understood from a molecular basis and that there is a common core of knowledge and principles concerning polymer molecules that can be set out in a single introductory work.

We have taken the viewpoint that an introduction or textbook should undertake to explain and develop the principles selected and not just present results. That means, for most of the subjects, we have proceeded from a very elementary starting point and presented in fair detail the steps. The goal has been to arrive at a point where the reader or student can understand the principles and profitably read the literature connected with that subject.

A number of subjects have been selected based on answering the questions: ‘how are polymers made?,’ ‘what do they look like?’ and ‘how do they behave?’ With respect to the third question we have deliberately stayed away from properties associated directly with the aggregation of polymer molecules in bulk materials. It is of course the interest in bulk materials that is the basic motivation of many, if not most, of the readers and students we hope to reach.

The subject of lattice dynamics is taught in most undergraduate courses in solid state physics, usually to a very simple level. The theory of lattice dynamics is also central to many aspects of research into the behaviour of solids. In writing this book I have tried to include among the readership both undergraduate and graduate students, and established research workers who find themselves needing to get to grips with the subject.

A large part of the book (Chapters 1–9) is based on lectures I have given to second and third year undergraduates at Cambridge, and is therefore designed to be suitable for teaching lattice dynamics as part of an undergraduate degree course in solid state physics or chemistry. Where I have attempted to make the book more useful for teaching lattice dynamics than many conventional solid state physics textbooks is in using real examples of applications of the theory to materials more complex than simple metals.

I perceive that among research workers there will be two main groups of readers. The first contains those who use lattice dynamics for what I might call modelling studies. Calculations of vibrational frequencies provide useful tests of any proposed model interatomic interaction. Given a working microscopic model, lattice dynamics calculations enable the calculation of macroscopic thermodynamic properties. The systems that are tackled are usually more complex than the simple examples used in elementary texts, yet the theoretical methods do not need the sophistication found in more advanced texts. Therefore this book aims to be a half-way house, attempting to keep the theory at a sufficiently low level, but developed in such a way that its application to complex systems is readily understood.