Langevin chain statistics, inverse

For fast flow deformations of polymer fluids, a non-linear theory of chain deformation and orientation is considered. To account for non-hnear effects and finite chain extensibility, inverse Langevin chain statistics is assumed. Time evolution of chain distribution function in the systems with inverse Langevin chain statistics has been discussed in earlier papers [12,13] providing physically sensible stress-orientation behaviour in the entire range of the deformation rates and chain deformations. 

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. 

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)... 

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. 

The use of the inverse Langevin function, as for example to derive eqn (3.20) overcomes the objection that the Gaussian analysis does not take into account the finite extensibility of the network chains. However in order to make the Langevin statistics reasonably mathematically tractable assumptions have to be introduced which strictly make the statistics invalid under non-Gaussian conditions. Their justification must lie in the fact that the resultant expressions give far better fits to the experimental data than the Gaussian statistics. 

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