Modulus rubber elastic plateau

The stress relaxation properties of a high molecular weight polybutadiene with a narrow molecular weight distribution are shown in Figure 1. The behavior is shown in terms of the apparent rubber elasticity stress relaxation modulus for three differrent extension ratios and the experiment is carried on until rupture in all three cases. A very wide rubber plateau extending over nearly 6 decades in time is observed for the smallest extension ratio. However, the plateau is observed to become narrower with increasing extension... 

For long chains, the modulus at about 106-107 dynes/cm2 passes into a domain of much slower relaxation (the plateau region). The width of the plateau depends strongly on molecular weight, while the characteristic modulus of the plateau region G is independent of molecular weight. The presence of the plateau confers rubber-like elastic properties over an intermediate range of frequencies or times. 

Chain entanglements are the cause of rubber-elastic properties in the liquid. Below the "critical" molecular mass (Mc) there are no indications of a rubbery plateau. The length of the latter is strongly dependent on the length of the molecular chains, i.e. on the molar mass of the polymer. From the shear modulus of the pseudo rubber plateau the molecular weight between entanglements may be calculated ... 

Mechanical oscillation measurements could only be carried out on nonionic PAAm-BisAAm gels. Because of their high degree of swelling, the saponified swelled gels could not be cut into suitable samples without their being destroyed. Via the theory of rubber elasticity developed by Flory (32), the value of the plateau modulus (Gp ) obtained by mechanical oscillation measurements is directly related to the number of elastically effective chains per unit volume (v J. 

We return now to the difference in behavior between the two types of PMMA with molecular weights of 1.5 x 10 and 3.6 x 10 . We note that the rubbery modulus of the type with the higher molecular weight reaches a plateau at /iR = 3.4 MPa. As we discuss in Chapter 6 on rubber elasticity, the entanglement molecular weight Me can be determined from this modulus through the statistical theory of rubber elasticity (Ferry, 1980) as... 

Classical rubber elasticity theory predicts that the plateau modulus,, for a diluted polymer, is proportional to the square of the volume fraction of the polymer. Scaling law interpretations for dilute solutions by P.G. 

The storage modulus of a 3wt% solution shows almost rubber elastic behaviour with a slope of approximately 0.035. The slope of the i] curve, i.e. 0.94, is in reasonable agreement with the slope of G. The rubber plateau decreases gradually (almost monotonically) with increasing temperature, although a weak transition is perceptible at about 40 °C, where the slope changes little. 

Estimation of the crosslink density of a thermoset network can be obtained from the storage modulus values in the rubbery plateau region. In principle, the crosslink density of a cured thermoset network could be calculated from the theory of rubber elasticity. The shear modulus G of a crosslinked rubbery network is given by [33] ... 

Vilgis and Erman that the constraint models and slip-link models have much in common, (iv) elucidating the effects of cross-link functionality and degree of cross linking, (v) exploring a variety of elastomeric polymers, particularly those having very different values of the plateau modulus, and (vi) generalizing rubber-elasticity models to include viscoelastic effects as well. 

In equation 76a the reptation time A,rep is the time for the chain to escape from the tube (orientation relaxation occurs from the end to the center of the chain). Gn is the entanglement plateau modulus (this value is slightly different from that implied from rubber elasticity of an entangled network) and f pit) is a normalized relaxation modulus for the reptation process. In this time regime, equation 76a implies that the modulus is separable into a time fimction and a modulus function. This becomes important in discussing the nonlinear response, which is done, in more detail, below. Some other viscoelastic functions from the DE tube model of reptation are... 

In the earlier discussion of the plateau modulus, it was remarked that a liquid of long chains acts, at intermediate times or frequencies, like a network. The theory of rubber elasticity predicts a relationship between the shear modulus and the concentration of network strands (Chapter 1). This relationship is used to evaluate Me, the equivalent molecular weight of a strand in the entanglement network, which is called the entanglement molecular weight [1] ... 

The so-called entanglement molar mass (Mg) is calculated from the plateau value of the shear modulus (Gg, Fig. 6.8) and, using classical rubber elasticity... 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

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