Kinetic theory of rubber elasticity

Kawabata44 has panted out on the basis of a simple network model that of the two derivatives, bW/blt and bW/bI2, the former should be related primarily to intramolecular forces such as the entropy force which plays a major role in the kinetic theory of rubber elasticity, while the latter should be a manifestation of intramolecular interactions. He predicted the possibility that bW/bI2 assumes negative values in the region of small defamation. In fact, the prediction was confirmed experimentally by Becker and also by the present authos. 

In 1944, Flory (3) noted that the moduli of cross-linked butyl rubbers generally differ somewhat from values calculated from the crosslink density according to the kinetic theory of rubber elasticity. In many cases, the modulus also depends on the primary (uncross-linked) molecular weight distribution of the polymer. He attributed both observations to three kinds of network defects chain ends, loops, and chain entanglements. The latter are latent in the system prior to cross-linking and become permanent features of the network when cross-links are added. 

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

Each of the viscoelastic parameters G°, rj0, and Je° has associated with it a characteristic molecular weight which either measures an equivalent spacing of entanglement couples along the chain (Me, deduced from G with the kinetic theory of rubber elasticity), or marks the onset of behavior attributed to the presence of entanglements (Mc and AT, deduced from r/0 and Je° as functions of molecular weight). Table 5.2 lists Me, Mc, and M c for several polymers. Aside from certain difficulties in their evaluation, each is a rather direct and independent reflection of experimental fact. 

Analysis of networks in terms of molecular structure relies heavily on the kinetic theory of rubber elasticity. Although the theory is very well established in broad outline, there remain some troublesome questions that plague its use in quantitative applications of the kind required here. The following section reviews these problems as they relate to the subject of entanglement. 

The kinetic theory of rubber elasticity is so well known and exhaustively discussed (17, 27, 256-257, 267) that the remarks here will be confined to questions which relate only to its application in determining the concentration of elastically effective strands. In principle, both network swelling properties and elasticity measurements can provide information on network characteristics. However, swelling measurements require the evaluation of an additional parameter, the polymer-solvent interaction coefficient. They also involve examining the network in two states, one of which differs from its as-formed state. This raises some theoretical difficulties which will be discussed later. Questions on local non-uniformity in swelling (17) also complicate the interpretation. The results described here will therefore concern elasticity measurements alone. 

At temperatures well below Tg, when entropic motions are frozen and only elastic bond deformations are possible, polymers exhibit a relatively high modulus, called the glassy modulus (Eg) which is on the order of 3 Gpa. As the temperature is increased through Tg the stiffness drops dramatically, by perhaps two orders of magnitude, to a value called rubbery modulus Er. In elastomers that have been permanently crosslinked by sulphur vulcanization or other means, the values of Er, is determined primarily by the crosslink density the kinetics theory of rubber elasticity gives the relation as... 

The kinetic theory of rubber elasticity was developed by Kuhn (1936-1942), Guth, James and Mark (1946), Flory (1944-1946), Gee (1946) and Treloar (1958). It leads, for Young s modulus at low strains, to the following equation ... 

Guth E, James HM and Mark IT, "The Kinetic Theory of Rubber Elasticity", in Mark H and Whitby GS (Eds), "Scientific Progress in the Field of Rubber and Synthetic Elastomers" Interscience Publishers, New York, Vol. II, pp 253-299,1946. 

Unfortunately the materials do not have a sufficiently well-developed rubbery modulus for use in calculations. One therefore resorts to the equivalent ultimate Maxwell element from which the maximiun relaxation time was computed, and utilizes the modulus corresponding to that ultimate element for subsequent computations. Now if La" " " ions act as crosslinks, then the values should be directly proportional to their concentration, c, since both and c are inversely proportional to the molecular weight between crosslinks. Mg. The former relationship is due to the kinetic theory of rubber elasticity (E = 03qRTIMc where 0 is the front factor, q is the density, and R the gas constant), and the latter to simple stoichiometry (c = g/2Mj) for tetrafunctional crosslinks. A plot of vs. c was shown in Fig. 9, both for La" " " " and for Ca++ indicating that both ions act as crosslinks, at least at low concentrations and only for the ultimate Maxwell element. 

For a crossllnked rubber sample, one simple parameter which can be used to roughly characterize the material is the crosslink density (v) or the average molecular weight between crosslinks (Mg a 1/v). It should be clear that this single parameter cannot completely represent a network in general. Nevertheless, it is well known that the viscoelastic behavior of a polymer network will vary with crosslink density as schematically depicted in Figure 1 for the creep behavior of a polymer at two crosslink densities < Vq. Here the kinetic theory of rubber elasticity... 

From the kinetic theory of rubber elasticity, the stress, f(t), at time t is related to the initial stress, f(0), by where N(t) is... 

It remains to define the individual Els here one normally relies on the kinetic theory of rubber elasticity as applied to the submolecules. Once again,... 

In analogy to the kinetic theory of ideal gases, the statistical theory of rubber elasticity is often called the kinetic theory of rubber elasticity. Reflect upon the similarities and differences between the basic philosophies of these two theories. 

Among the major developments are a new approach to long-range relaxational motions known as the theory of reptation, and the further elucidation of the kinetic theory of rubber elasticity. In this second edition, we have attempted to take account of some of these developments on a level consistent with the introductory nature of the text. We have also added an entirely new chapter on dielectric relaxation, a technique now widely used to investigate molecular motions in polar polymers. Finally, we have tried to strengthen and clarify several other sections as well as eliminate errors or inconsistencies in the first edition that have been pointed out to us by colleagues and students. 

With the basic structure of polymers of macromolecules clarified, scientists now searched for a quantitative understanding of the various polymerization processes, the action of specific catalysts, and initiation and inhibitors. In addition, they strived to develop methods to study the microstructure of long-chain compounds and to establish preliminary relations between these structures and the resulting properties. In this period also falls the origin of the kinetic theory of rubber elasticity and the origin of the thermodynamics and hydrodynamics of polymer solutions. Industrially polystyrene, poly(vinyl chloride), synthetic rubber, and nylon appeared on the scene as products of immense value and utility. One particularly gratifying, unexpected event was the polymerization of ethylene at very high pressures. 

When the degree of cross-linking is low (H, is large), the glass transition temperature increases slightly as cross-linking proceeds. As approaches values in the hundreds, T becomes a sensitive measure of cure. Equation 10 can be derived in its essential features from the kinetic theory of rubber elasticity (62). giving... 

The number of crosslinked polymer chains per unit volume of NC gel, N, can be estimated by Eq. (1), based on the kinetic theory of rubber elasticity [59] ... 

The firs t systematic study of the reversible networks was the transient network theory developed by Green and Tobolsky [ 11 ], in which stress relaxation in rubber-Uke polymer networks was treated by the kinetic theory of rubber elasticity suitably extended so as to allow the creation and annihilation of junctions during the network deformation. 

In Figure 7, we compare the d)mamic moduli at 0.1 cps of a series of copolymers containing 2-4 mole percent of the comonomer and in Figure 8 for the copolymers containing 7-10 mole percent of the comonomer. In these plots, the moduli are corrected to 80 C by multiplying by the factor T (K)/353, which assmes that the kinetic theory of rubber elasticity is obeyed. 

Inspection of Figure 5 shows a very broad glass-to-rubber transition range which extends from below -100°C to above 0°C for the polyurethane adhesive. The relaxation modulus E(t) - 400 Kg/cm which occurs at the rubbery inflection temperature - 40°C - 313 K describes an effective molecular weight M as defined by kinetic theory of rubber elasticity ... 

The molecular foundations of the kinetic theory of rubber elasticity were conceived by Kuhn [l] and Meyer [2] and are well established. Subsequent work by James... 

For undiluted polymers and concentrated solutions, there are two types of theories (a) the singlechain theories or reptation theories , in which one focuses on the motions of one polymer molecule in the fluid as it moves in some kind of mean field provided by the surrounding polymer molecules and (b) the network theories , in which one visualizes the fluid as a loosely joined network in which the network junctions have a distribution of lifetimes. The chain theories are similar in structure to the dilute solution theories, and one has to make some kinds of assumptions about how the surrounding molecules affect the hydrodynamic drag and the Brownian motion. The network theories are similar in structure to the kinetic theory of rubber elasticity, and one has to make some kinds of assumptions about the junction kinetics. 

It has been observed in a variety of polymers that the velocity of sound becomes essentially independent of temperature, corresponding to a plateau modulus condition. Analysis of the data using the kinetic theory of rubber elasticity yields the relation ... 

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