Elongational flows

In this flow, the diffusion equation can be solved rigorously, so that it is possible to check the accuracy of the decoupling approximation. Since k 

In Fig. 8.6, eqns (8.153) and (8.157) are compared. Though the decoupling approximation is not correct for small the difference becomes smaller as becomes larger. 

The visco ty calculated by the decoupling qiproximation is S/4 times larger than the rigorous value of eqn (8.13S). For large k, is dominated by the viscous stress 

Thus the intrinsic viscosity decreases in proportion to k. On the other hand, rigorous asymptotic analysis and the numerical calculation show that the intrinsic viscosity decreases as 

For these first experiments, a temperature relatively close to Tg, T=123°C, was chosen with the intention of minimizing the relaxation of stress and chain orientation during the quenching the weight-average relaxation time of sample SI at 123°C is calculated from that at 140°C and the thermal shift factor between 123°C and 140°C Xw(123°C) 380s. On the other hand the cooling time of the stretched specimens can be estimated to a few seconds [19], which is very small compared to the polymer relaxation time at the temperature of the experiments. 

For strain rates lower than 8x10 s, it was found that the rheological behaviour is nearly linear viscoelastic Fig. 5 shows the tensile stress-growth function CT (0,t) = EXT]E (E,t) at 123°C for three different strain rates in the linear range after about 1000s, the stress reaches a 

Comparison of cr (6,t)for different samples stretched at the same strain rate shows that the reproducibility of the tensile test is satisfactory. 

Further tests have been carried out on sample SI at two higher strain rates. The stress-growth coefficient corresponding to these experiments is represented in Fig. 8, where the 

It has to be mentioned that the radii of gyration of the high molecular weight chains (which are of the order of 300 A) could not be determined in the fiame of the present study. Reaching scattering vectors as low as 10 A , which would have been required, involves technical difSculties on the spectrometer and very long counting times. 

During its passage from r to r -I- dz, its cross-section changes from Ato A+ d A and its length from llol + d/. Since its volume remains constant it follows that. 

The change in logarithmic strain as the element moves from r to r + dr is 

according to Trouton s definition of the elongational viscosity A, a liquid instantaneously elongating at de,/df generates a tensile stress a ( = F/4), 

The determination of A involves experiments of this type, i.e. the measurement of F, u, and d/l/dz, or related experiments in which a cylinder of liquid is extended in a rapid tensile test. 

A filament of molten polymer is extruded at a volume flow rate 10 mVsand is hauled olT under a tension of 3 N. At a certain point downstream of the die the elongaiional viscosity is 5 x 10 N s/m and the velocity has reached 1 m/s. Calculate the velocity gradient at this point. 

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The elongation viscosity defined by Equation (1.19) represents a uni-axial extension. Elongational flows based on biaxial extensions can also be considered. In an equi-biaxial extension the rate of deformation tensor is defined as... 

Stevenson, J.F., 1972. Elongational flow of polymer melts. AIChE. J. 18, 540-547. 

In packed beds of particles possessing small pores, dilute aqueous solutions of hydroly2ed polyacrylamide will sometimes exhibit dilatant behavior iastead of the usual shear thinning behavior seen ia simple shear or Couette flow. In elongational flow, such as flow through porous sandstone, flow resistance can iacrease with flow rate due to iacreases ia elongational viscosity and normal stress differences. The iacrease ia normal stress differences with shear rate is typical of isotropic polymer solutions. Normal stress differences of anisotropic polymers, such as xanthan ia water, are shear rate iadependent (25,26). 

C. Characteristically, these nematic melts show the persistence of orientational order under the influence of elongational flow fields which result in low melt viscosities under typical fiber formation conditions even at high molecular weights. 

Chain Conformation in Non-Steady State Elongational Flow 99... 

The inclusion of internal viscosity raises considerably the free-energy storage capacity of a rapidly deforming macromolecule as compared to the idealized Hookean spring model and could play a decisive role in mechanochemical reactivity in transient elongational flow. 

Fig. 9. Stress distribution on a solid cylinder in elongational flow (the flow direction is x)...

Using different elongational flow geometries, the CS transition of a flexible polymer chain as well as the phenomenon of hysteresis for the reversed process of SC transition has been confirmed experimentally by the Bristol group [8, 9], the Cal-Tech group [10, 53] and the Paris group [11, 54]. 

Fig. 15. Simulation of conformational changes in elongational flow by preferred rotation of bonds perpendicular to flow direction (according to Ref. [70])...

The existence of kinks was recently explicitly taken into account by Larson (Fig. 14) as a possible model for chain unravelling in the flow [69]. At the same time, Kausch developed a similar model to explain degradation results measured in transient elongational flow (Fig. 15) [70]. With this difference from the Larson model, kinks in the latter model can support compressive stress chain elastic modulii range from 16 to 110 GPa, depending on the number of defects within the kinked region. 

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