Elasticity of a network

5 (a) A specimen of initial dimensions X Yj. and Z is deformed so that the dimensions become X A X, Y=AyYi, and Z=A Zj. (b) Erid A of the representative chain is at x y,. z, before deformation after deformation it is at x=A,Xj. y=Ayyj, z=A,rj. The end-to-end vector changes in exactly the same way as the specimen dimensions. Scale drawing for A,=j. 

It follows from this that the product of the three extension ratios is 

Because the deformation is affine, this vector in the deformed state becomes 

The change in the end-to-end vector produces a change in the end-to-end force (eqns 3.11 and 3.16) from /, before deformation, 

The stress-strain properties are calculated by equating the work done the internal forces with the work done by the external forces which generate the deformation. Consider first the work done by /, the z-component of f, as z changes from z-, to A Zg. Let this work for the representative chain be tv/. 

Consider the specimen to be of initial dimensions X-, Zj, and to be deformed by forces along the principal directions x, y, and z to the new dimensions X, F, Z (see Fig. 3.5(a)). When rubber is deformed it is always observed that there is essentially no change in volume. We may write, therefore. 

We assume that the deformation is afflne, which means that every part of the specimen deforms as does the whole. This will include the end to end vectors of all V chains including the representative chain OA in Fig. 3.5(b). It will be convenient to write the end-to-end vector in the undeformed state 

In reality a polymer mass will only be an effective rubber if the individual chains are joined into a network structure. This virtually eliminates the unlimited slippage of one chain past another causing viscous flow or creep. Subject to a number of assumptions it is possible to derive expressions relating stress-strain relationships in such a network structure. 

The rubber is incompressible. This is substantially true for solid rubbers under the deformational stresses usually encountered. It will not be true under high hydrostatic pressures nor in the case of swollen networks which may be important in some theoretical studies. 

The individual chains may be represented by a rjc model—the justification for which has already been considered. 

All chains have the same molecular weight. Whilst this may appear to be a severe restriction it will be seen that the expression to be derived is independent of molecular weight and the assumption is therefore of no consequence. 

No energy is stored in stretched, distorted or broken bonds. This assumption implies that elasticity is a result of configurational probabilities, i.e. that it is entirely entropic in origin. The term entropy spring has been used to describe a material showing high rubbery elasticity. This is now known to be an incorrect assumption. 

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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. 

Intermolecular crosslinking between pendant vinyl groups and radical centers located on different macromolecules produce crosslinks that are responsible for the aggregation of macromolecules, which leads to the formation of a macrogel. It must be remembered that both normal and multiple crosslinks may contribute to the rubber elasticity of a network, whereas small cycles are wasted links. 

L. R.G. Treloar (1946). The elasticity of a network oflong-chain molecules. III. Trans. Faraday Soc., 42, 83-94. 

We have explored what happens when an individual polymer chain is stretched. This was not just an exercise. We have shown that the elasticity of a network is built up from the elasticities of all the subchains (Figure 7.2), so we can make use of what we have found. There is one tricky question though. Let s imagine a highly elastic solid body, say, a rubber ball. The macromolecules are rather closely packed in it and interact strongly with each other. So can we really treat each subchain as an ideal polymer, with no volume interactions at all ... 

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... 

Rubinstein M, Colby RH (2003) Polymer physics. Oxford University Press, New York Sperling LH (2006) Introduction to physical polymer science, 4th edn. Wiley, Hoboken Treloar LRG (1943) The elasticity of a network of long chain molecules (I). Trans Faraday Soc 39 36-64... 

Theories have been developed to describe the mechanical properties of amorphous networks and their swelling behavior in terms of an average (1-3). Over the years there have been several modifications in the theories to account for the fluctuations of the junction points, the role of network defects such as dangling chains and loops and the role of trapped entanglements in determining the equilibrium elasticity of a network (4). 

Treloar, L.R.G. (1943) The elasticity of a network of long-chain molecules. II. Trans. 

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