Rubber molecular theory

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 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. 

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. 

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. 

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. 

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... 

Attempts were made to remove the third assumption above, and it was shown that correct considerations of the limited extensibility of the chain adequately explain the S-shaped feature of stress-strain curves observed in uniaxial extension of vulcanized rubbers. However, the improved theory still gave zero for bW/bI2. Up to now there is no molecular theory available which predicts bW/bI2 that varies with//. 

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. 

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. 

It is worth noting at this point that the various scientific theories that quantitatively and mathematically formulate natural phenomena are in fact mathematical models of nature. Such, for example, are the kinetic theory of gases and rubber elasticity, Bohr s atomic model, molecular theories of polymer solutions, and even the equations of transport phenomena cited earlier in this chapter. Not unlike the engineering mathematical models, they contain simplifying assumptions. For example, the transport equations involve the assumption that matter can be viewed as a continuum and that even in fast, irreversible processes, local equilibrium can be achieved. The paramount difference between a mathematical model of a natural process and that of an engineering system is the required level of accuracy and, of course, the generality of the phenomena involved. 

The early molecular theories of rubber elasticity were based on models of networks of long chains in molecules, each acting as an entropic spring. That is, because the configurational entropy of a chain increased as the distance between the atoms decreased, an external force was necessary to prevent its collapse. It was understood that collapse of the network to zero volume in the absence of an externally applied stress was prevented by repulsive excluded volume (EV) interactions. The term nonbonded interactions was applied to those between atom pairs that were not neighboring atoms along a chain and interacting via a covalent bond. 

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]. 

You should notice that the first term has the same form as that given by simple theories of rubber elasticity. The equation fits extension data for deformations up to about 300% very well, but cannot fit compression data using the same values of the constants C, and C2. Attempts to obtain the second term (in this form) using a molecular theory have not, as yet, been very successful, so we ll say no more about it. 

The equation of state for rubber elasticity, embodied by any of equations (6-53) through (6-60), is important not only because it is historically the first quantitative treatment of molecular theories for elastomers but also because it laid a conceptual foundation for theories for the physical properties of polymers in general. Some of these have been discussed in detail in previous chapters. Perhaps the single most significant contribution is its recognition of the role of... 

The condition of swelling equilibrium can be calculated by means of two theoretical approaches. It is assumed that the chemical potential of mixing for a network is the same as the chemical potential of mixing an uncross-linked polymer of high molecular weight and of the same structure as the network polymer. The mixing term can be described by means of the Flory-Huggins (FH) equation. The calculation of the elastic deformation term is based on the rubber elasticity theory (RET). 

The average molecular weight between crosslinking points can be estimated, based on Ideal rubber elasticity theory In which... 

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