The Inverse Langevin Approximation

An early development of the simple molecular network theory is the so-called inverse Langevin approximation for the probability distribution. The Gaussian approximation is 

For the inverse Langevin approximation, we have from Equation (4.36) 

The final part of the exercise is to reconsider the stress-strain relations using the inverse Langevin distribution function. This was done by James and Guth using an analogous development to that for the Gaussian distribution function. 

The analogue of the three-chain Equation (4.41) for uniaxial stretching becomes 

Arruda and Boyce [22] showed that the eight-chain model performed much better than three- and four-chain models in predicting the behaviour of vulcanised mbber in uniaxial and biaxial tension and shear, and also performed excellently in modelling uniaxial and plane strain compression of gum and neoprene mbber. 

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Other network models based on the inverse Langevin function are the tetrahedral model of Flory and Rehner (1943) subsequently modified by Treloar (1946) and the inverse Langevin approximation (Treloar, 1954). The relative merits of these approaches, which yield similar results have been discussed by Treloar (1975) who points out the overwhelming advantages of the three-chain model in ease of computation. 

It can be seen from Figure 4.12 that there is broad agreement between the form of the force-extension curve predicted by the inverse Langevin approximation and that observed in practice. 

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 ... 

It has been shown that the inverse Langevin function 1(x), with 0closed-form expression called a Pade approximant [140] ... 

With a Fade approximation of the inverse Langevin function [15],... 

Using Fade approximation of the inverse Langevin function (4.13) the elastic free energy of a non-Gaussian chain with the chain extension h/Na expresses by the following closed formula. 

The distribution functions for 25- and 100-link random chains obtained from the Gaussian and inverse Langevin approximations respectively are compared in Figure 4.11. 

Figure 4.11 Distribution functions for 25- and 100-link random chains (a) Gaussian approximation and (b) inverse Langevin approximation. (Reproduced from Treloar, L.R.C. (1975) The Physics of Rubber Elasticity, 3rd edn, Oxford University Press, Oxford. Copyright (1975) Oxford University Press.)...

Cohen, A. A Pad6 Approximant to the inverse langevin function. RheoL Acta (1991) 30, pp. 270-273... 

Transient effects in the kinetics of oriented nucleation are considered for melt processing in a wide range of deformation rates using a theory of non-linear chain statistics with transient effects. Inverse Langevin elastic free energy of a polymer chain in a Pade approximation, averaged with transient distribution of the chain end-to-end vectors, as well as Peterlin s approximation for the modulus of nonlinear elasticity are used. The effects of transient orientation distribution of the chain segments is also considered. 

To describe the soft phase contribution, one needs to develop a hyperelastic model taking into account (i) rubber elasticity behavior (ii) strain amplification due to the trapped hard phase inclusions (iii) strain hardening as the chains approach their maximum extensibility. Typically, one could approximate these effects using an inverse Langevin function or its Fade approximation (see ref. [39], Chapter 11), and using a strain multiplication factor. Here, we use a somewhat simplified expression that retains most of the required features ... 

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