Elasticity molecular theories

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 cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

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]

The strain is given by the relative length or elongation a = L/L where L and are the lengths of the sample in the deformed and undeformed states, respectively. Dividing the stress f/A by the strain function (a - a- ) indicated in the simplest molecular theories of rubberlike elasticity (38,39) then gives the elastic modulus or reduced stress (38-42)... 

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. 

All gas particles have some volume. All gas particles have some degree of interparticle attraction or repulsion. No collision of gas particles is perfectly elastic. But imperfection is no reason to remain unemployed or lonely. Neither is it a reason to abandon the kinetic molecular theory of ideal gases. In this chapter, you re introduced to a wide variety of applications of kinetic theory, which come in the form of the so-called gas laws. ... 

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. 

The molecular theory of elasticity of polymeric networks which leads to the equation of state, Eq. (28), rests on the following basic postulates Undeformed polymeric chains of elastic networks adopt random configurations or spatial arrangements in the bulk amorphous state. The stress resulting from the deformation of such networks originates within the elastically active chains and not from interactions between them. It means that the stress exhibited by a strained network is assumed to be entirely intramolecular in origin and intermolecular interactions play no role in deformations (at constant volume and composition). 

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. 

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. 

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]. 

The kinetic molecular theory explains the behavior of gases in terms of characteristics of their molecules. It postulates that gases are made up of molecules that are in constant, random motion and whose sizes are insignificant relative to the total volume of the gas. Forces of attraction between the molecules are negligible, and when the molecules collide, the collisions are perfectly elastic. The average kinetic energy of the gas molecules is directly proportional to the absolute temperature (Section 12.10). 

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. 

Because these networks have a known degree of cross-linking (as inversely measured by Mj, they can be used to test the molecular theories of rubberlike elasticity, particularly with regard to the possible effects of interchain entanglements (26, 38-42). Intentionally imperfect networks can also be prepared (Figure 6) (38). Such networks have known numbers and... 

When De 1 and molecular elasticity is negligible, the flow properties of polymeric nematics can, in principle, be described by the Leslie-Ericksen equations (see Section 10.2.3). However, at moderate and high De, the Leslie-Ericksen continuum theory fails, and a molecular theory is required to describe the effect of flow on the distribution of molecular orientations. 

Gas particles in constant motion collide with each other and with the walls of their container. The kinetic-molecular theory states that the pressure exerted by a gas is a result of collisions of the molecules against the walls of the container, as shown in Figure 7. The kinetic-molecular theory considers collisions of gas particles to be perfectly elastic that is, energy is completely transferred during collisions. The total energy of the system, however, remains constant. 

As alluded to in the introduction to this entry, the LE theory was conceived for small molecule LCs while molecular theories are intended for LCPs. LC molecules retain their equilibrium orientation distribution. LCPs are susceptible to disturbances to their distribution function T (m) its temporal relaxation gives rise to molecular viscoelasticity, while its spatial gradient produces distortional elasticity. A natural question is whether the molecular theories reduce properly to the continuum LE theory in the limiting case of an undisturbed orientation distribution. This situation arises in the weak flow limit where the flow is weak De < . 1) and spatial distortions are small ([V l [Pg.2962]

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