Rubber elasticity concepts

Through the research of Guth and James (31-35), Treloar (36), Wall (37), and Flory (38), the quantitative relations between chain extension and entropy reduction were clarified. In brief, the number of conformations that a polymer chain can assume in space were calculated. As the chain is extended, the number of such conformations diminishes. (A fully extended chain, in the shape of a rod, has only one conformation, and its conformational entropy is zero.) 

The idea was developed, in accordance with the second law of thermodynamics, that the retractive stress of an elastomer arises through the reduction of entropy rather than through changes in enthalpy. Thus long-chain molecules, capable of reasonably free rotation about their backbone, and joined together in a continuous, monolithic network are required for rubber elasticity. 

In brief, the basic equation relating the retractive stress, a, of an elastomer in simple extension to its extension ratio, a, is given by 

The quantity Me in a polymer network (the present case) is the number-average molecular weight between cross-links. 

The quantity Me is the molecular weight between the entanglements. Wool (43) defines Me = (4/9)M/ for linear polymers but points out that experimentally Me = (1/2)M/. 

---

Although the basic concept of macromolecular networks and entropic elasticity [18] were expressed more then 50 years ago, work on the physics of rubber elasticity [8, 19, 20, 21] is still active. Moreover, the molecular theories of rubber elasticity are advancing to give increasingly realistic models for polymer networks [7, 22]. 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

Since Meyer ) introduced the concept of kinetic molecular chain into the physics of polymers in 1932, remarkable progress has been made in the molecular-theoretical interpretation of elastic behavior of rubber vulcanizates and polymer solids in general2- ), and one can appreciate the present status of knowledge on this subject by a number of review articles and reference books. On the other hand, the phenomeno-logic approach to rubber elasticity has not aroused much interest in the field of polymer research. This is understandable because polymer scientists are primarily concerned with affairs of the molecular world. 

Viscoelastic properties of molten polymers conditioning the major regularities of polymer extension are usually explained within the framework of the network concept according to which the interaction of polymer molecules is localized in individual, spaced rather far apart, engagement nodes. The early network theories were developed by Green and Tobolsky 49) and stemmed from successful network theories of rubber elasticity. These theories were elaborated more fully in works by Lodge50) and Yamamoto S1). The major elasticity. These theories is their simplicity. However, they have a serious drawback the absence of molecular weight in the theory. 

In Section I, the discussion dealt with the significant role of nonbonded interactions in the development of the full stress tensor, mean plus deviatoric, in rubber elasticity, in the important high reduced density regime p > 1. Here, we present some concepts and formulations that apply to this regime. 

The concept of a long chain molecule acting as an entropic spring plays a central role in most molecular theories of rubber elasticity. To what extent does this concept remain valid and useful in dense systems of interacting chains This question has been considered by MD simulation in Ref. [12]. 

CHAPTER I 1 Rubber Elasticity Basic Concepts and Behavior... 

The physics of rubber elasticity is characterized by a great variety of approaches, models and concepts. It is the aim of this review to summarise the situation and to formulate a comprehensive picture of this topic from the point of view of the authors. In the introduction, this situation will be illustrated by a few examples of current problems and developments. 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

We shall close this chapter by pointing out that Huggins in recent work, has developed a general theory of rubber elasticity, which does not make use of the concept of randomly coiled molecules. For this reason it falls outside the scope of this chapter. 

---