Force-extension relation

In Ref. [76] we showed that the necklace conformations can exist also in the presence of counterions and that they exhibit a variety of conformational transitions as a function of density. The end-to-end distance was found to be a non-monotonic function of concentration and showed a strong minimum in the semi-dilute regime. Here we have found for short chains a collapse of each chain into a globular stable state which repel each other due to their remaining net charge. The focus of a more recent work was to analyze, by extensive computer simulations in detail, three possible experimental observables, namely the form factor, the structure factor and the force-extension relation, which can be probed by scattering and AFM techniques [77]. The details of the simulation techniques can be found in Refs. [76, 77]. 

Fig. 15 Force-extension relation for a transition from two to three pearls. Shown are the average over all conformations, over only two-pearl configurations and only three-pearl configurations...

FIGURE 1.8 Force-extension relation for simple extension. 

A more exact calculation involves the calculation of the energy stored in a rubber network because of network deformation. This stored energy is expressed in terms of the Helmholtz free energy (A) and is derived from entropy considerations. The force-extension relation can then be calculated by taking the derivative of A with respect to elongation, as described below and in Chapter 3.2. 

The force-extension relation derived previously from statistical considerations does not agree well with experimental data at small extensions. As an example, a plot of unixial force-elongation data for natural rubber falls below the curve calculated from the theoretical equation (equation 7.48) in the region between 1.1 to 2.0 elongation... 

Flow Stress, Flow Curve, Fig. 2 Force-extension relation in simple tension for a mild steel... 

FIGURE 8 Force-extension relation for simple extension. —, Linear relation obtaining at infinitesimal strains. (From Gent [37].)... 

Fig. 3.7. Non-Gaussian force-extension relation (eqn (3.43) fitted to experimental data, with NkT = 0-273 N mm, n = 75. (From Treloar, 1975.)...

Figure 3.3. The Complete Force-Extension Relation for a Random Chain Compared to the Gaussian Prediction...

Besides the assumption of ideality inherent in Equation 9.61, several approximations were made in obtaining the force-extension relation of Equation 9.70. These assumptions are as follows ... 

Figure 10.4 Force-extension relation for a freely jointed chain. (Reprinted from Treloar, L. R. G. The Physics of Ruhher Elasticity, 3rd ed.. Clarendon, Oxford, UK, 1975, hy permission of Oxford University Press.)...

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