Elongational shear-free flows

Three kinds of viscometric flows are used by rheologists to obtain rheological polymer melt functions and to study the rheological phenomena that are characteristic of these materials steady simple shear flows, dynamic (sinusoidally varying) simple shear flows, and extensional, elongational, or shear-free flows. 

Extensional, elongational or shear-free flows play a dominant role in the post die-forming step, such as stretching of melt strands in spinning, uniaxial stretching of molten films... 

The two functions and 2 depend on the parameter b as well as on the elongation rate e in steady shear-free flows. When b = 0, the function 2 is zero, and is replaced by the symbol tj, which is called the elongational viscosity . The elongational viscosity describes the resistance to elongational flow if 8 is positive and the resistance to biaxial stretching if e is negative (the terms extensional viscosity and Trouton viscosity have also been used for ). 

The material functions tii and rji depend on both t and 8q, and of course on the parameter b that specifies the type of shear-free flow. For elongational flow, with b = 0 and Sq positive, rj becomes rj, the elongational stress growth function. This quantity has been measured for a number of polymer melts. Further information on elongational properties can be found in several extensive... 

The constant elements of the tensor represent the possibility of a shear and/or a shear-free, elongation flow. The condition for incompressional or volume-preserving flow in the latter case is = 0. If at / = 0 our unit cell is rectangular,... 

The choice of the kinematics used for probing non-Newtonian behavior is arbitrary, and hence, the number of choices is infinite. However, the rheological community has settled on a small number of standard flows. These standard flows are simple and realizable by experimentalists. Classic flows are simple shear and simple shear-free elongation [2]. 

Simple shear-free elongational flows are described by... 

The morphologies discussed so far have all been formed from isotropie melts. In practice melts are often subjected to shear forces and this ean induce alignment in the chains, which aids crystalhzation in particular directions. Orientation of the melt causes an increase in the free energy and this itself constitutes an important factor in practical processing. Shish-kebabs ° are formed from solutions that are subjected to elongational flow, which induces orientation of the solute molecules (Figure 5.12). 

Free energy versus relative elongation (X = r/R) for a single poljnner coil in a longitudinal shear flow. 

The form of the stochastic equations (105) and (106) is motivated by the following considerations. The first term in (105), dF/da, is the thermodynamic force due to bending energy and volume constraints it is calculated from the free energy F a). The second term of (105) is the deformation force due to the shear flow. Since the hydrodynamic forces elongate the vesicle for 0 < 0 < r/2 but push to reduce the elongation for - r/2 < 0 < 0, the flow forces should be proportional to sin(20) to leading order. The amplitude A is assumed to be independent of the asphericity a. C,a and A can be estimated [205] from the results of a perturbation theory [199] in the quasi-spherical limit. Equation (106) is adapted from KeUer-Skalak theory. While B is a constant in KeUer-Skalak theory, it is now a function of the (time-dependent) asphericity a in (106). 

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