A critical examination of force–extension relationship for freely jointed chain model

The pioneering paper by Kuhn and Grün (1942) provided a thorough understanding of the finite extensibility of the freely-jointed chain model in polymer science. In their work, the relative number of conformations are evaluated through the introduction of a non-Gaussian probability distribution for the free (subjected to zero force) chains. However, Flory (1969) pointed out that this distribution does not truly describe the probability for the end-to-end vector of the chain. To tackle this issue, a corrected probability distribution was proposed and compared with the original form. Despite this improvement, numerous research studies still rely upon the original formulation of Kuhn and Grün, without being aware of Flory’s correction. The present work attempts to clarify the fundamental difference between the two probabilities through recognition of the distinct underlying ensembles. Then, the influence of this correction is studied on the force–extension relationship of the individual chain, as well as on some well-known micromechanics-based network models. Finally, it is demonstrated that misuse of the probability distribution can lead to significant discrepancy in the force–extension relationship of a coil-rod chain, a model for the building element of some biopolymer gels.