Steady-state elongational flow field

Using Eq. (2.13) in Eq. (2.14), we have the following expression for the rate-of-strain tensor d in uniaxial elongational flow 

For equal biaxial stretching, the rate-of-strain tensor d can be expressed as 

In Chapter 7 of Volume 2, we present the rheological response of polymeric liquids in steady-state biaxial elongational flow. 

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Figures 7a and 7b show the time evolution of the diagonal components Cxx, Cyy, and c z of the conformation tensor for the C24 and C78 melts, respectively. For both systems, the initial value of c x is significantly higher than 1, whereas those of Cyy and Czz are a little less than 1, indicative of the oriented conformations induced by the imposed steady-state elongational structure of flow field a x- As time evolves, c x decreases whereas Cyy and Czz increase continuously, approaching the steady-state, field-free value of 1, indicative of fully equilibrated, isotropic structures in the absence of any deforming or orienting field.

Under steady-state conditions, as in the Couette flow, the strain rate is constant over the reaction volume for a long period of time (several hours) and the system of Eq. (87) could be solved exactly with the matrix technique developed by Basedow et al. [153], Transient elongational flow, on the other hand, has two distinctive features, i.e. a short residence time (a few ps) and a non-uniform flow field, which must be incorporated into the kinetics equations. In transient elongational flow, each rate constant is a strongfunction of the strain-rate which varies with time in the Lagrangian frame moving with the center of mass of the macromolecule the local value of the strain rate for each spatial coordinate must be known before Eq. (87) can be solved. 

LDPE, and with polypropylene, PP, was studied In steady state shear, dynamic shear and uniaxial extenslonal fields. Interrelations between diverse rheological functions are discussed In terms of the linear viscoelastic behavior and Its modification by phase separation Into complex morphology. One of the more Important observations Is the difference In elongational flow behavior of LLDPE/PP blends from that of the other blends the strain hardening (Important for e.g. fllm blowing and wire coating) occurs In the latter ones but not In the former. 

Mechanical degradation of polymers has been studied for more than 70 years in several flow fields encompassing strong elongation components. In certain flow fields the streamlines are symmetric with a stagnation point. In the vicinity of the stagnation point, the dwell time of the fluid element is longer than the timescale for coU extension. Such flow is referred to as quasi-steady-state-flow (QSSF). hi most other cases the dwell time is shorter than the coil extension time and the flow is referred to as fast-transient-flow (FTF). 

For most blends, the morphology changes with the imposed strain. Thus, it is expected that the dynamic low strain data will not follow the pattern observed for the steady-state flow. One may formulate it more strongly in polymer blends the material morphology and the flow behavior depend on the deformation field, thus under different flow conditions different materials are being tested. Consequently, for immiscible blends the steady state relation may be quite different from the dynamic one in shear or elongation. 

In polymer blends, both the morphology and flow behavior depend on the deformation field. Under different flow conditions the blend may adopt different structures, hence behave as different materials. Note that in multiphase systems, the relationships between the steady state, dynamic and elongational viscosities (known for simple fluids) are not observed. Similarly, the time-temperature (t-D superposition principle that has been a cornerstone of viscoelastometry is not valid. 

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