Effect of the inverse Langevin approximation on the solution of the Fokker–Planck equation of non-linear dilute polymer

Abstract

The Langevin function is defined by L(x)=coth(x)−1/x. Its inverse is useful for many applications and especially for polymer science. As the inverse exact expression has no analytic representation, many approximations have been established. The most famous approximation is the one traditionally used for the finitely extensible non-linear elastic (FENE) dumbbell model in which the inverse is approximated by L−1(y)=3y/(1−y2). Recently Martin Kröger has published a paper entitled ‘Simple, admissible and accurate approximations of the inverse Langevin and Brillouin functions, relevant for strong polymer deformation and flows’ (Kröger, 2015) in which he proposed approximations with very reduced error in relation to the numeric inverse of the Langevin function. The question we aim to analyze in this short communication is: when one uses the traditional approximation rather than the more accurate one proposed by Kröger is that really significant regarding the value of the probability distribution function (PDF) in the frame work of a kinetic theory simulation? If yes when we move to the upper scale by evaluating the value of the stress, can we observe a significant difference?By making some simple 1D simulations in homogeneous extensional flow it is demonstrated in this short communication that the PDF prediction within kinetic theory framework as well as the macroscopic stress value are both affected by the quality of the approximation.