In Chapter 6 it was pointed out that comparison of the mean unperturbed solution dimensions, i.e., the characteristic ratio, derived from experiment and from calculations based on conformational models could form an important testing ground for the structural models. It was concluded, however (Section 6.1.4), that in order for realistic comparisons to be made, the computation of characteristic ratio would have to allow for interactions between nearby bonds. The steric interference between a pair of adjacent gauche bonds of opposite sense in a three-state chain was given as an important example of such interactions between nearby bonds. It is possible to formulate a treatment of chains where adjacent bond pairs interact. This is accomplished by means of a representation of the statistical mechanical partition function of a one-dimensional chain using the methods of matrix algebra (Kramers and Wannier, 1941) and adapted to the polymer chain conformation problem (Birshtein and Ptitsyn, 1959, 1966; Birshtein, 1959; Nagai, 1959; Lifson, 1959). This development and some applications to real chains are described in this chapter.

The partition function and scalar averages for a one dimensional chain with interacting bonds

In Section 6.3.1 the classical configurational partition function was written in equation (6.44) as an integration over all chain configurations. In the present development, attention will be focused on a discrete number of local bond states, those judged as important in the rotational isomeric state context (see Chapter 5, especially Section 5.1.1).

A material that can be deformed quickly to several hundred per cent strain, recovers rapidly and completely upon removal of the stress and is capable of having the process repeated numerous times is described as rubbery or elastomeric. The possibility of this behavior is due to the flexible long chain nature of polymer molecules and presents a type of response to mechanical deformation that is fundamentally quite different from the response given by rigid materials such as metals, ceramics and glassy or semi-crystalline polymers. This behavior is often called ‘entropy elasticity’ in contrast to the ‘energy elasticity’ of more familiar materials. The resistance to deformation is due largely to an entropy decrease rather than an energy increase. Entropy elasticity demonstrates itself in easily observed thermodynamic behavior such as the contraction of a stretched rubber band with increasing temperature as described by Gough (1805) in the quotation above. In this chapter, the contrast of energy vs entropy elasticity in thermodynamic behavior is developed. Then the molecular theory of rubber elasticity is discussed and its applicability to real elastomers is considered. Rubber elasticity is intimately connected with the presence of a network.

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.