Inverse Langevin function

Here, I n is the reduced Bessel function, and its argument L (m) is the inverse Langevin function (see Callen and Callen 1963, 1965). Using the values given above for... 

Equations (28) and (29) are derived from the statistical theory based on the Gaussian statistics which describes the network behaviour if the network is not deformed beyond the limit of the applicability of the Gaussian approximation33). For long chains, this limit is close to 30 % of the maximum chain extension. For values of r, which are comparable with rmax, the force-strain dependence is usually expressed using the inverse Langevin function 33,34)... 

For a network of Gaussian chains having the same number n of links, uniaxially stretched by an amount L/Lo = A., the assumptions of affine displacement of jimction points and initial Gaussiein distribution of end-to-end vectors allows one to calculate the optical anisotropy of the network by integrating Eq.lO over the distribution of end-to-end vectors in the stretched state. By taking Treloar s expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of Unks per chain ... 

Figure 3.18 Elastic spring force versus molecular extension for the Warner spring, for the freely jointed chain (which is described by the inverse Langevin function), and for the linear spring. (From Tanner, copyright 1985 by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc.)...

Demonstrate that the probability distribution function of the end-to-end distance i of a freely jointed chain can be expressed in terms of the inverse Langevin function C (x ) of the ratio x = RjR y of the end-to-end distance R to its maximum value i max =... 

The force / required to stretch a single chain to an end-to-end distance R can be expressed through the inverse Langevin function ... 

A more general theoretical model [29], which results in Equation 11.38, provides a better description of an elastomeric stress-strain curve all the way up to fracture, by including the finite extensibilities of the chains. In this equation, 1 is the inverse Langevin function. " is a transcendental function which is defined by Equation 11.39. Coth is the hyperbolic cotangent function. The superscript of -1 represents the functional inversion (and not merely simple reciprocal) of the function in square brackets. 

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

If one expands the inverse Langevin function (equation 2.124) in a series... 

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. 

In Eq. 3, n denotes the number of statistical chain segments of length / in the strand. Also, is the inverse Langevin function and cr is a proportionality constant. 

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