Journal of Rheology -- May 2001 -- Volume 45, Issue 3 pp.
Deficiencies of FENE dumbbell models in describing the rapid stretching of dilute polymer solutions
Indranil Ghosh, Gareth H. McKinley, Robert A. Brown, and Robert C. Armstrong
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
(Received 6 September 2000; revised 17 January 2001)
A wide variety of bead-spring kinetic theory models have been proposed to explain the stress growth and hysteretic behavior of dilute polymer solutions in uniaxial extension. We analyze the Kramers chain, a fine-scale model for polymer dynamics, in order to assess the validity of the coarser-grained bead-spring models in these deformations. Whereas the spring force is a simple function of the dumbbell length for the FENE spring, we find that the relationship between the ensemble-averaged end-to-end force and the extension for a Kramers chain depends on the kinematic history to which it has been subjected. In a quiescent fluid, the Kramers chain force–extension relationship is identical to the FENE force law. However, during start up of elongational flow, the ensemble-averaged end-to-end force for a given (end-to-end) length of the molecule increases with strain until steady state is reached. If the extensional flow is suddenly stopped, the Kramers chain force–extension relationship relaxes back to the FENE force–extension function. For all positive strains, the FENE dumbbell force law underpredicts the ensemble-averaged end-to-end force in the Kramers chain. For a Weissenberg number of 11.4, the end-to-end forces in the two models can differ by three to four orders of magnitude, indicating the unsuitability of the FENE dumbbell for modeling polymers in strong transient extensional flows. This paper also presents a detailed analysis of the mechanisms causing stress– birefringence hysteresis. We find that it is essential for a dumbbell model to have an end-to-end force that depends upon the deformation history in order to capture configurational hysteresis.