Theory of the Elastic Properties of Rubber

Abstract

Previous discussions of the kinetic theory of rubber elasticity have dealt with individual long chain molecules, but the theory of the structure of bulk rubber has been almost entirely undeveloped. The present paper goes beyond earlier ones both in the more detailed treatment of the individual chains and in the development of a clear-cut model of the bulk material. From the consideration of familiar properties of rubber, it is concluded that in the lightly vulcanized state it consists of a coherent network of flexible molecular chains (this involving a considerable fraction of the total material) together with other molecules not actively involved in the network, but acting like a fluid mass through which the network extends and in which it moves with Brownian motion. The idea of an effective internal pressure is advanced and discussed. A simplified model for bulk rubber is proposed, consisting of a network of idealized flexible chains extending through the material and a fluid filling it, the bounding surfaces being in equilibrium under all the forces acting on them—internal pressure, pull of the molecular network, and any external forces. For the quantitative treatment of this model the principal problem is that of computing the forces exerted by the molecular network, because of its thermal agitation, on the bounding surfaces. Two models of the flexible molecular chains have been used—one with a linear stress-extension relation (Gaussian chain), the second with independent links of fixed length, having, like real molecular chains, a definite maximum extension. Methods for the statistical treatment of chains of independent links at all extensions are developed and applied to the computation of stress-strain relations for the second of the above models. It is shown that an irregular network of Gaussian chains is equivalent to a simple set of independent chains; the corresponding reduction in the case of non-Gaussian chains is only approximate. Making this reduction in all cases, the model is applied to the quantitative computation of stress-strain curves for unilateral stretch of rubber and for stretch in two directions. A prediction is made of the change in elastic properties of rubber in the swollen state. The thermoelastic properties of rubber are worked out, and the change in slope of the nearly linear stress-temperature curves from negative to positive values with increasing strain is explained. The location of the thermoelastic inversion point, at which this slope is zero, is shown to depend only upon the cubic expansion coefficient of unstretched rubber. The linear thermal expansion coefficient of stretched rubber in the direction of the stress is shown to be positive (and of the order of the same coefficient for unstretched rubber) below the thermoelastic inversion point. Above the thermoelastic inversion point this coefficient becomes negative and is—for appreciable extensions—of the order of the cubic expansion coefficient for a gas, independent of the composition of rubber. The linear coefficient of thermal expansion perpendicular to the direction of stress is always positive. Stress-strain curves for bulk rubber at constant temperature have been compared with the theory for extensions up to 400 percent. The agreement is particularly good as concerns that part of the total stress due to changing entropy. In some rubber compounds minor deviations, variable from sample to sample, are found for small extensions. These are attributed to van der Waals forces not taken into account by the theory. The characteristic knee in rubber stress-strain curves is shown to be due to change in internal pressure in the material. This varies inversely with the extended length, and in unstretched rubber is of the order of 5 kg/cm2. The upward curvature of the stress-strain curve for larger extensions, which appears even before the onset of crystallization, is explained as due to the approach of the molecular network to its maximum extension, which is of the order of 10 times its extension in the unstretched bulk rubber. Crystallization may in general enhance the S-shape of the stress-strain curve for natural cured rubber, Neoprene and butyl rubber. Synthetic rubber of the Buna type, while having an S-shaped stress-strain curve, does not exhibit any crystallization at all. It is estimated that, in the materials considered, roughly a fourth of the rubber chains are actively involved in the network. Internal pressure, maximum extensions, and the fraction of active material all change with the state of vulcanization. The general theory developed in this paper provides a basis for the treatment of many other physical properties of stretched rubber.

Thermal Fluctuations and Rubber Elasticity

“…This contrasts with the aforementioned result of James and Guth [8] for a phantom network, where such a constraint, and concomitantly the dependence on the imposed deformation, are absent. The structure of Z v Λ for the unstrained case of g = I is identical to that of a U (1) gauge field theory in the transverse gauge.…”

“…It is noteworthy that these elastic fluctuations were previously studied by James and Guth [8] in a phantom network, where interactions between neighboring polymer chains are ignored. They concluded that these fluctuations are independent of the imposed strain deformation and therefore have no effect on the elasticity.…”

Mechanics-based model for non-affine swelling in perfluorosulfonic acid (PFSA) membranes

“…However, the exact nature of the polymer matrix is irrelevant to our calculations, since we model it as elastic springs. This is analogous in some ways to the classical characterization of the deformation of polymer chains in elastically deformed polymers [64][65][66].…”

Energy functions for rubber from microscopic potentials

“…We have only, from the first equation, that C 0 100 and C 3 100 correspond to the term in sЈ · sЈ. A thermodynamic treatment that takes into account the finite length of the polymer chains has been given by James and Guth 14 and subsequent authors ͑see Treloar 3 ͒. It yields the force on a molecule which has n links of length l as a function of the separation r of its ends as This can be integrated for the energy, giving even powers only of sЈ · sЈ.…”