Rubber elasticity molecular theory

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. 

The effective molecular mass Mc of the network strands was determined experimentally from the moduli of the polymers at temperatures above the glass transition (Sect. 3) [11]. lVlc was derived from the theory of rubber elasticity. Mc and the calculated molecular mass MR (Eq. 2.1) of the polymers A to D are compared in Table 3.1. 

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]. 

Small deformations of the polymers will not cause undue stretching of the randomly coiled chains between crosslinks. Therefore, the established theory of rubber elasticity [8, 23, 24, 25] is applicable if the strands are freely fluctuating. At temperatures well above their glass transition, the molecular strands are usually quite mobile. Under these premises the Young s modulus of the rubberlike polymer in thermal equilibrium is given by ... 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

Anseth et al. [20] have reviewed the literature dealing with the mechanical properties of hydrogels and have considered in detail the effects of gel molecular structure, e.g., cross-linking, on bulk mechanical properties using theories of rubber elasticity and viscoelasticity. 

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free energies of the individual chains. In this section, the elasticity of the single chain is discussed first, followed by the elementary theory of elasticity of a network. Corrections to the theory coming from intermolecular correlations, which are not accounted for in the elementary theory, are discussed separately. 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

In our first paper, the molecular theory of rubber elasticity was briefly reviewed, especially the basic assumptions and topics still subject to discussion (21). we will now focus on the effects of the structure and the functionality f of the crosslinks and the relevant theory. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

To compare the predictions of the various molecular theories of rubber elasticity, three sets of high functionality networks were prepared and tested In this Investigation. The first set of networks tested were formed In bulk and attained a high extent of the endllnklng reaction, i.e., eX).9 where e Is the extent of reaction of the terminal vinyl groups. The second set of networks studied were formed In the presence of diluent and also achieved a high extent of reaction (e>0.9). The final group of experiments were performed on networks formed In bulk at low extents of reaction (0.4 [Pg.333]

It is clearly shown that chain entangling plays a major role in networks of 1,2-polybutadiene produced by cross-linking of long linear chains. The two-network method should provide critical tests for new molecular theories of rubber elasticity which take chain entangling into account. 

The change of chemical potential due to the elastic retractive forces of the polymer chains can be determined from the theory of rubber elasticity (Flory, 1953 Treloar, 1958). Upon equaling these two contributions an expression for determining the molecular weight between two adjacent crosslinks of a neutral hydrogel prepared in the absence of... 

Figure 1. Stress relaxation curves for three different extension ratios. Uncross-linked high-vinyl polybutadiene with a weight average molecular weight of 2 million and a reference temperature of 283 K. G is the apparent rubber elasticity modulus calculated from classical affine theory. (Solid line is data from Ref. 1).

Finally, it should be pointed out that no molecular theory of rubber elasticity is required and that no assumptions were made in order to reach above conclusions. 

Gluck-Uirsch, J., Kokini, J. L. (1997). Determination of the molecular weight between cross-links of waxy maize starches using the theory of rubber elasticity. J. RheoL, 41, 129-139. 

Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

It is important to examine the temperature dependence of bW/dlf for development of a more exact molecular theory of rubber elasticity. Figure 1744,4S illustrates this... 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

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. 

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. 

In the current statistical theory of rubber elasticity, it is suggested that the front-factor molecular forces. They have proposed a semiempirical equation of state taking into account the dependence... 

Thus, this consideration shows that the thermoelasticity of the majority of the new models is considerably more complex than that of the phantom networks. However, the new models contain temperature-dependent parameters which are difficult to relate to molecular characteristics of a real rubber-elastic body. It is necessary to note that recent analysis by Gottlieb and Gaylord 63> has demonstrated that only the Gaylord tube model and the Flory constrained junction fluctuation model agree well with the experimental data on the uniaxial stress-strain response. On the other hand, their analysis has shown that all of the existing molecular theories cannot satisfactorily describe swelling behaviour with a physically reasonable set of parameters. The thermoelastic behaviour of the new models has not yet been analysed. 

According to the theory of rubber elasticity, the elastic response of molecular networks is characterized by two mechanisms. The first one is connected with the deformation of the network, and the free energy change is determined by the conformational changes of the elastically active network chains. In the early theories, the free energy change on deformation of polymeric networks has been completely identified with the change of conformational entropy of chains. The molecular structure of the chains... 

The statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

Jerry and Monnerie l " have proposed a modified theory of rubber elasticity which includes anisotropic intermolecular interactions U12 (favoring the alignment of neighbouring chain segments) in the form U,2 = ZUL(r12) PL(0i) PL(02), where r,2 is the intermolecular distance, 0X and 02 are the angles between the molecular axes... 

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