Langevin function approximation

This expression may determine the relationship between ea as a function of Aa in intense fields, e.g., for x > 3, when the approximation L(x) A = 1 - l/xA applies. However, it assumes classical statistics, so it will require a quantum correction for x values in this range, assuming that the Langevin function is still applicable. For smaller values of xA, Eq. (48) may be solved graphically. A more transparent expression for small xA values would be useful. Equation (42) with m = 1/2 leads to a quartic equation in ea which may be simplified by expansion to a cubic expression. A solution is the use of Booth s approximation ea and e0 n2, with substitution ofEq. (47) into Eq. (42). This gives a simple and useful approximation ... 

Strain hardening at high deformations A can be explained by the non-Gaussian statistics of strongly deformed chains. Recall that the Gaussian approximation for a freely jointed chain model is valid for end-to-end distances much shorter than that for a fully stretched state R < / max = bN. In Section 2.6.2, the Langevin functional dependence of normalized end-to-end distance R/Nb on the normalized force Jb/ kT) for a freely jointed chain [Eq. (2.112)] was derived ... 

O.S.Thus, for small poling fields, when the linear approximation for the Langevin functions holds, the two independent susceptibility tensor elements given by Eqs. (66) and (67) give... 

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. 

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. 

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 ... 

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

Figure 2 Dependence of the Langevin function L(a) (solid line) together with the linear approximation (dashed line). The linear approximation (dashed line) L(a) = alZ holds for a< 1.

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

That the terminal acceleration should most likely vanish is true almost by definition of the steady state the system returns to equilibrium with a constant velocity that is proportional to the initial displacement, and hence the acceleration must be zero. It is stressed that this result only holds in the intermediate regime, for x not too large. Hence and in particular, this constant velocity (linear decrease in displacement with time) is not inconsistent with the exponential return to equilibrium that is conventionally predicted by the Langevin equation, since the present analysis cannot be extrapolated directly beyond the small time regime where the exponential can be approximated by a linear function. 

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