Elastomer deformation statistical theory

Stress-strain properties for unfilled and filled silicon rubbers are studied in the temperature range 150-473 K. In this range, the increase of the modulus with temperature is significantly lower than predicted by the simple statistical theory of rubber elasticity. A moderate increase of the modulus with increasing temperature can be explained by the decrease of the number of adsorption junctions in the elastomer matrix as well as by the decrease of the ability of filler particles to share deformation caused by a weakening of PDMS-Aerosil interactions at higher temperatures. 

The presence of filler in the rubber as well as the increase of the surface ability of the Aerosil surface causes an increase in the modulus. The temperature dependence of the modulus is often used to analyze the network density in cured elastomers. According to the simple statistical theory of rubber elasticity, the modulus should increase twice for the double increase of the absolute temperature [35]. This behavior is observed for a cured xmfilled sample as shown in Fig. 15. However, for rubber filled with hydrophilic and hydrophobic Aerosil, the modulus increases by a factor of 1.3 and 1.6, respectively, as a function of temperature in the range of 225-450 K. It appears that less mobile chain units in the adsorption layer do not contribute directly to the rubber modulus, since the fraction of this layer is only a few percent [7, 8, 12, 21]. Since the influence of the secondary structure of fillers and filler-filler interaction is of importance only at moderate strain [43, 47], it is assumed that the change of the modulus with temperature is mainly caused by the properties of the elastomer matrix and the adsorption layer which cause the filler particles to share deformation. Therefore, the moderate decrease of the rubber modulus with increasing temperature, as compared to the value expected from the statistical theory, can be explained by the following reasons a decrease of the density of adsorption junctions as well as their strength, and a decrease of the ability of filler particles to share deformation due to a decrease of elastomer-filler interactions. 

In contrast to the filled samples, the deformation energy for the unfilled ones increases proportionally to the increase in the absolute temperature according to the prediction of the simple statistical theory of rubber elasticity. Thus, it appears that the change of the modulus and the deformation energy with increasing temperature reveals a decrease of the density of adsorption junctions in the elastomer matrix, as well as a decrease of the ability of filler particles to share deformation, resulting from a weakening of elastomer-filler interactions. 

The phenomenological theory, as its name implies, concerns itself only with the observed behavior of elastomers. It is not based on considerations of the molecular structure of the polymer. The central problem here is to find an expression for the elastic energy stored in the system, analogous to the free energy expression in the statistical theory [equation (6-72)]. Consider again the deformation of our unit cube in Figure 6-3. In order to arrive at the state of strain, a certain amount of work must be done which is stored in the body as strain energy ... 

The statistical theory is remarkable in that it enables the macroscopic deformation behaviour of an elastomer to be predicted from considerations of how the molecular structure responds to an applied strain. However, it is important to realize that it is only an approximation to the actual behaviour and has significant limitations. Perhaps the most obvious problem is with the assumption that end-to-end distances of the chains can be described by the Gaussian distribution. This problem has been highlighted earlier in connection with solution properties (Section 3.3) where it was shown that the distribution cannot be applied when the chains become extended. It can be overcome to a certain extent with the use of more sophisticated distribution functions, but the use of such functions is beyond the scope of this present discussion. Another problem concerns the value of N. This will be governed by the number of junction points in the polymer network which can be either chemical (crosslinks) or physical (entanglements) in nature. The structure of the chain network in an elastomer has been discussed earlier (Section 4.5). There will be chain ends and loops which do not contribute to the strength of the network, but if their presence is ignored it follows that if all network chains are anchored at two crosslinks then the density, p, of the polymer can be expressed as... 

The statistical theory allows the stress-strain behaviour of an elastomer to be predicted. The calculation is greatly simplified when the observation that elastomers tend to deform at constant volume is taken into account. This means that the product of the extension ratios must be unity... 

It is apparent from considerations of the structure in Section 4.2 that semi-crystalline polymers are essentially two-phase materials and that the increase in modulus is due to the presence of the crystals. Traditional ideas of the stiffening effect due to the presence of crystals were based upon the statistical theory of elastomer deformation (Section 5.3.2). It was thought that the crystals in the amorphous rubber behaved like crosslinks and produced the stiffening through an increase in crosslink density rather than through their own inherent stiffness. Although this mechanism may be relevant at very low degrees of crystallinity it is clear that most semi-... 

The isothermal reversible work of deformation w per unit volume of an elastomer is given by the statistical theory of elastomer deformation as... 

In our statistical treatment of an ideal elastomer, we have assumed that the elastic force is entirely attributable to the conformational entropy of deformation, energy effects being neglected. That the theory reproduces the essential features of the elasticity of real elastomers attests to the basic soundness of this assumption. On the other hand, we know that in real elastomers such energy effects cannot be entirely absent, and deviations from the ideal elastomer model may be expected to occur. Let us now examine in greater detail the extent to which the neglect of energy effects is justified. We can rewrite equation (6-28) ... 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

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