From intermediate anisotropic to isotropic friction at large strain rates to account for viscosity thickening in polymer solutions

Abstract

The steady-state extensional viscosity of dense polymeric liquids in elongational flows is known to be peculiar in the sense that for entangled polymer melts it monotonically decreases-whereas for concentrated polymer solutions it increases-with increasing strain rate beyond the inverse Rouse time. To shed light on this issue, we solve the kinetic theory model for concentrated polymer solutions and entangled melts proposed by Curtiss and Bird, also known as the tumbling-snake model, supplemented by a variable link tension coefficient that we relate to the uniaxial nematic order parameter of the polymer. As a result, the friction tensor is increasingly becoming isotropic at large strain rates as the polymer concentration decreases, and the model is seen to capture the experimentally observed behavior. Additional refinements may supplement the present model to capture very strong flows. We furthermore derive analytic expressions for small rates and the linear viscoelastic behavior. This work builds upon our earlier work on the use of the tumbling-snake model under shear and demonstrates its capacity to improve our microscopic understanding of the rheology of entangled polymer melts and concentrated polymer solutions.

Tumbling-Snake Model for Polymeric Liquids Subjected to Biaxial Elongational Flows with a Focus on Planar Elongation

“…This is in disaccord with the predictions for the uniaxial elongation viscosity of the tumbling-snake model, ηnormalE, which seems to be passing from a maximum when ε0′>0 (see inset of Figure 2a of Ref. [39]). However, both share the same power-law behavior at large elongation rates which is always equal to −1 irrespective of the value of ε0′, as for the Doi & Edwards model [18].…”

Assessment of the Tumbling-Snake Model against Linear and Nonlinear Rheological Data of Bidisperse Polymer Blends

“…Finally, to accurately compare with the predictions of the shear viscosity we need to employ a volume-fraction dependent ε0 of the form, ε0=0.18(1−2ϕL/3). We had reached the same conclusion recently when we needed to have the value of ε0 to increase as the polymer concentration, in polymer solutions, decreases to match the behavior of the stationary extensional viscosity of PS polymer solutions [29]. This is tantamount to having the friction tensor become more isotropic with decreasing polymer concentration.…”

“…As a result, similar undershoots are not expected to occur in elongational flows, due to their irrotational character, which is indeed predicted, without further modifications, by the tumbling-snake model [29,30]. The tumbling-snake model is also able of capturing the peculiar experimental extensional rheological data, according to which the extensional viscosity of polymer solutions is noted to exhibit thinning below the inverse Rouse time and thickening above, whereas the extensional viscosity of polymer melts is monotonically decreasing irrespective of the strain rate [39,40,41], by having the strength ε0 of the link tension coefficient to increase as the polymer concentration decreases [29].…”

“…In conclusion, the tumbling-snake model, which amends the problems of the original Curtiss-Bird model by employing a variable link-tension coefficient, has been shown able of providing a very adequate description of the available rheological measurements of monodisperse entangled polymer melts and concentrated polymer solutions when subjected to shear [24,27,28], uniaxial elongation [29], and planar elongation [30], and, in this work, for entangled bidisperse polymer blends when subjected to shear. Solving it necessitates the use of very simple Brownian dynamics simulations coupled with the analytical predictions at small rates.…”

“…We have recently solved the tumbling-snake model for ε′>0, via the use of Brownian Dynamics simulations, for both steady-state [24,27,28] and time-dependent shear flow [24,28], as well as to steady-state and time-dependent uniaxial [29] and planar elongation [30]. This solution scheme entails the numerical integration of two coupled Itô stochastic differential equations for the variables boldUt (segment unit vector realization at time t ) and σt∈[0,1] (relative contour position at time t ) (see Equation (5) of Ref.…”