Polymer solutions in elongational flow: suspension of extensible rods model

A simple mechanistic model of a polymer solution in strong elongational Bow is studied. A molecule is modelled by uniform slender rod of an incompress ible rubber-like elastic material imbedded into a coaxial cylindrical cell filled with a viscous fluid (solvent). It is assumed that the polymer solut ion can be treated as a regular lattice of identical individual cells align ed along the extension direction. So only dynamics of an individual cell is studied. Under these assumptions the cell deforms affinely with bulk flow and the shear stress vanishes at the cell boundary. The rod-fluid interacti on within the cell is studied using lubrication-layer approximation. Then t he rod extension dynamics is described by a nonlinear parabolic equation fo r local stretch ratio with a source term responsible for the externally imp osed flow. This equation is essentially the same as that derived by Hinch [ E.J. Hinch, Phys. Fluids, 20 (1977) 22] for an individual molecule in exten sional flow. It should be solved subjected to boundary conditions correspon ding to vanishing elastic force in the rod at the rod ends. First, steady-s tate solutions for extension with constant strain rate are considered. Two steady-state solutions exist at small (subcritical) extension rate, one cor responding to a slightly extended molecule, and another to a highly extende d one. It is argued that the second steady state is unstable. The steady st ates cease to exist beyond a critical strain rate. A 'molecule' extends unb oundedly in supercritical flow. At constant strain rate the extension asymp totically (after an adjustment period) tends to exponential elongation of t he rod affinely with the bulk flow. Then relaxation of a highly extended ro d is considered. For an uniformly extended initial state relaxation proceed s non-uniformly, propagating from the rod ends to the center. As a result, the model predicts slow relaxation in the case of high initial extension. T he model predictions expressed in terms of the microscopic variables, namel y the elastic strain and force distribution along a 'molecule' can be relat ed to bulk variables, such as extension rate and effective tensile stress. In particular, the elastic stress proves to be proportional to the elastic force in the rod integrated over its length. More general extension regimes are studied using numerical modelling of the rod dynamics equation. Those include constant-strain-rate extension with subsequent relaxation, step tes t with switching from high to a lower extension rate, and extension followe d by periodic extension regime with zero mean extension rate. Among the eff ects observed are those predicted by the asymptotic analysis, such as exist ence of a critical value of the extension rate beyond which the rod extends unboundedly, exponential affine extension after some adjustment period, sl ow relaxation after strong extension, and hysteresis phenomena: unbounded e xtension at subcritical extension rate and asymmetric behavior of stress in oscillatory extension regime after preliminary strong extension. It is arg ued that the simple mechanistic model can be useful in explaining some expe rimental observations, in particular, unexpectedly successful application o f the Maxwell-Oldroyd model to predominantly elongational strong flows. (C) 1999 Elsevier Science B.V. All rights reserved.