Equivalent random link

To explain the difference between the experimental results and theory, Doherty et al. (4J have given an empirical and a theoretical hypothesis. The theoretical hypothesis concerns the question of the meaning to be attached to the concept of the "equivalent random link" in the statistical theory of the randomly-jointed chain. According to Doherty et al., the assumption that the optical properties of the chain are describable by a randomly jointed model, using the same value of n, as for the description of stress has no strictly logical foundation. 

In the derivation of eqn. (7) it was assumed that n (number of equivalent random links) is the same for all chains. For our samples (B2 system, Mw/Mn=1.45), this assumption is definitely not correct. Therefore, it is desirable to obtain birefringence results on networks prepared from monodisperse polymer (in that n is constant), before the validity of n itself is questioned. 

Calculate the root-mean-square length of a polyethylene chain of M = 250000 g moP assuming that the equivalent random link corresponds to 18.5 C—C bonds. 

When polymer chains are cross-linked to form a non-flowing rubber, a molecular network is obtained. It is shown in section 3.3.4 that the freely jointed random-link model of polymer ehains is appKeable to rubbers provided that the equivalent random link is eorreetly ehosen. In considering the network the following simplifying assumptions will therefore be made, leading to the simplest form of the theory. 

In the first two of these equations the subscript on the refractive index indicates that it refers to the mean refractive index and also distinguishes it from n in the third equation, which denotes the number of random links per chain. If the equivalent random link is taken as the structural unit. No is now the number of random links per unit volume and Q o and Aa are the mean value and the anisotropy of the polarisability of a random link. 

An estimate of the number of monomer units per equivalent random link can be obtained by dividing the value of Aa calculated from the stress-optical coefficient by the anisotropy of the polarisability of the monomer unit calculated from bond polarisabilities. This number can more interestingly be expressed in terms of the number of single bonds in the equivalent random link and is found to be about 5 for natural rubber, about 10 for gutta percha and about 18 for polyethylene. (For the last two the values are extrapolated from measurements at elevated temperature.) The number for polyethylene is considerably higher than the value of 3 suggested by the assumption of totally free rotation around the backbone bonds (see section 3.3.3 and problem 3.7). 

Let n and I be the num-ber and length of the equivalent random links and let the fully extended length of both chains be L. Then... 

In principle the value of a, the equivalent random link, may readily be compared with the length of a single bond or a monomer repeat unit. Values of the equivalent random link have been estimated by a number of methods (see Treloar, 1975) of which the optical anisotropy method is an example (Morgan and Treloar, 1972). This method gives a value for a of 1 73 isoprene units in the case of natural rubber (c/5-l,4-polyisoprene) and 3 39 for gutta percha (trafis-l,4-poly-isoprene) a result particularly interesting in view of the known differences in physical properties between the two polymers (see Chapter 4). 

Before concluding the discussion on the equivalent random link it may be mentioned that other model systems, such as the freely rotating tetrahedral chain model which was used to give eqn (3.6) may be corresponded to an rjc model. It can be shown that ... 

In spite of this, the failure envelopes are normal. Thus Figure 1 shows the envelopes for several Solithane 113-300 compositions (lO), These envelopes can be fitted by the inverse Langevin approximation (ll) to the stress-strain curve, and from the curve fit both the number of effective chains per cm and the niimber of equivalent random links N can be determined (l2). The fit for two compositions is shown in Figure l8 and the results of such an analysis (13) are given in Table II. It can be seen that the chain concentration is almost constant but N increases, i.e, the chains effectively become stiffer as the concentration of prepolymer is increased. 05iis is -ttie only elastomer system we are aware of in which such a change can be effected at constant Ye ... 

Broad changes in the composition of the basic elastomer family do not influence the crosslink density or effective chain concentration but rather change the nature of the crosslink site so that the number of the equivalent random links per chain is changed. 

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