The Ideal Rubber

The Ideal Rubber.—The data available at present as summarized above show convincingly that for natural rubber (dE/dL)T,v is equal to zero within experimental error up to extensions where crystalhzation sets in (see Sec. le). The experiments of Meyer and van der Wyk on rubber in shear indicate that this coefficient does not exceed a few percent of the stress even at very small deformations. This implies not only that the energy of intermolecular interaction (van der Waals interaction) is affected negligibly by deformation at constant volume—which is hardly surprising inasmuch as the average intermolecular distance must remain unchanged—but also that con- 

Similarly, the condition dE/dV)T = 0 for an ideal gas demands direct proportionality between P and T, for according to Eq. (14) we then have 

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Poly-isobutylene (PIB) is a very useful rubber because of its very low gas permeability. Co-polymerised with small amounts of isoprene (to enable vulcanisation with sulphur) to butyl rubber (HR), it is the ideal rubber for inner tubes. If PIB would crystallise, it could not be used as a technical rubber The same holds for the rubbers BR and IR. 

The strain energy function (equation 7.39) follows directly from equation 7.111 assuming AH = 0 as in the ideal rubber case. The function for a swollen sample will be modified by the inclusion of the term. This term that is essentially a scaling factor relating the unswollen dimensions to the swollen dimensions for a stretched sample, is given by... 

We shall now attempt to establish an equation of state for the ideal rubber by analogy with an ideal gas. We start with equation (17). If we introduce kin W for s and eliminate W from equation (15) we obtain ... 

The state of the ideal rubber can be specified by the locations of all the junction points, ij, and by fce end-to-end vectors for all tire chains connecting the junction points,. The first postulate of the statistical theory of rubber elasticity is that, in the rest state with no external constraints, the distribution fimction for the set of chain end-to-end vectors is a Gaussian distribution witii a mean-squared end-to-end distance that is proportional to the molecular weight of the chains between jimcnons ... 

The second postulate of the ideal-rubber theory is that, after deformation, the distribution of chain end-to-end vectors is perturbed in exactly the ratio determined by the macroscopic deformation. This assumption is called the principle of affine deformation. The distribution of chain end-to-end vectors is now given by ... 

Fig. 13.28. Stress-extension curve for a sample of natural rubber compared with the ideal rubber prediction [Eq. (35)] (Reprinted from Paul J. Flory, Principles of Polymer...

Now, we consider the second class of experiments and check for the predictions of Lodge s model with regard to extensional flows. Using again an equation from the ideal rubbers we can directly write down the time dependent Finger tensor B(f,t ). It has the form... 

The ideal rubber should respond to an external stress only by uncoiling, that is, dE/dL = 0. Experiments in which f, L, and T are varied confirm that for many amorphous, cross-linked materials above T, dE/dL is very small (Figure 9.17). Below we have said that polymer segments cannot deform in the timescale in which was measured, so dS/dL = 0. In a glass we expect external stresses to be countered by bond bending and stretching. 

For small deformations, oc/n 1, the above equation reduces to the ideal rubber elasticity formula of Equation 9.70 when one keeps only the lowest term in Equation 9.77. For higher deformations, Equation 9.78 gives much better agreanent with experimental results as it has the additional parameter... 

Note the analogy to Equation 9.70 where the phenomenological coefficient 2Cj can be equated to RTN, the elastic modulus of the ideal rubber elasticity model. If we also keep the next term in the series in Equation 9.85, for i = 0 and j = 1, we get... 

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