Extensional flow time-dependent

To reach steady state, the residence time of the fluid in a constant stretch rate needs to be sufficiently long. For some polymer melts, this has been attained however, for polymer solutions this has proved to be a real challenge. It was not until the results of a world wide round robin test using the same polymer solution, code named Ml, became available that the difficulties in attaining steady state in most extensional rheometers became clearer. The fluid Ml consisted of a 0.244% polyisobutylene in a mixed solvent consisting of 7% kerosene in polybutene. The viscosity varied over a couple of decades on a logarithmic scale depending on the instrument used. The data analysis showed the cause to be different residence times in the extensional flow field... 

Winter et al. [119, 120] studied phase changes in the system PS/PVME under planar extensional as well as shear flow. They developed a lubrieated stagnation flow by the impingement of two rectangular jets in a specially built die having hyperbolic walls. Change of the turbidity of the blend was monitored at constant temperature. It has been found that flow-induced miscibility occurred after a duration of the order of seconds or minutes [119]. Miscibility was observed not only in planar extensional flow, but also near the die walls where the blend was subjected to shear flow. Moreover, the period of time required to induce miscibility was found to decrease with increasing flow rate. The LCST of PS/PVME was elevated in extensional flow as much as 12 K [120]. The shift depends on the extension rate, the strain and the blend composition. Flow-induced miscibility has been also found under shear flow between parallel plates when the samples were sheared near the equilibrium coexistence temperature. However, the effect of shear on polymer miscibility turned out to be less dramatic than the effect of extensional flow. The cloud point increased by 6 K at a shear rate of 2.9 s. ... 

Figure 1-14 compares the time-dependent extensional viscosity to the time-dependent shear viscosity, after onset of flow, at various shear and extension rates, for the same molten polyethylene described in Figs. 1-9 to 9-11 (Meissner 1972). This figure shows that the behavior of the extensional viscosity can be very different from that of the shear viscosity the former increases while the latter decreases with increasing strain rate at a fixed time after inception of steady flow. Thus, while the shear viscosity is shear thinning, the extensional viscosity is extension thickening. 

The temporary network model predicts many qualitative features of viscoelastic stresses, including a positive first normal stress difference in shear, gradual stress relaxation after cessation of flow, and elastic recovery of strain after removal of stress. It predicts that the time-dependent extensional viscosity rj rises steeply whenever the elongation rate, s, exceeds 1/2ti, where x is the longest relaxation time. This prediction is accurate for some melts, namely ones with multiple long side branches (see Fig. 3-10). (For melts composed of unbranched molecules, the rise in rj is much less dramatic, as shown in Fig. 3-39.) However, even for branched melts, the temporary network model is unrealistic in that it predicts that rj rises to infinity, whereas the data must level eventually off. A hint of this leveling off can be seen in the data of Fig. 3-10. A more realistic version of the temporary network model... 

The deformation of dispersed drops in immiscible polymer blends with the viscosity ratio X = 0.005-13 during extensional flow was studied by Delaby et al. [1994, 1995]. In the latter paper, the time-dependent drop deformation during a start-up flow at constant deformation rate was derived. The model is restricted to small drop deformations. 

Chauveteau and co-workers 24, 48) examined the flow of PEO and HPAA through the extensional flow produced in severe constrictions. They concluded that a coil-stretch transition was responsible for the dilatant behavior observed, and that the critical shear rate required was of the order of 10 times the reciprocal of the Rouse relaxation time. Perhaps the most extensive studies have been those of Haas and co-workers (25, 26, 49). They have explored the critical dilatant behavior on flow through porous media and pursued the hypothesis that the phenomenon is primarily due to a coil-stretch transition beyond a critical deformation rate. They attempted a semiquantitative description based upon the dependence of the lowest order relaxation time of the random coil upon polymer type, molecular weight, solvent quality, and ionic environment. 

Durst et al. [10], Cll] considered the extensional flow between turbulent eddies and derived initially a time scale, which was dependent on the local position, and from which they tried to obtain information about the local alteration of the velocity profiles caused by the polymer addition. After introducing a mean extensional rate, they put forward a formula for t, which is identical with equation (4). They chose the relaxation time derived by Bird et al. C123... 

For K > 2 the drops deform into stable filaments, which only upon reduction of k disintegrate by the capillarity forces into mini-droplets. The deformation and breakup processes require time - in shear flows the reduced time to break is tb > 100- When values of the capillarity number and the reduced time are within the region of drop breakup, the mechanism of breakup depends on the viscosity ratio, A, - in shear flow, when X > 3.8, the drops may deform, but they cannot break. Dispersing in extensional flow field is not subjected to this limitation. Furthermore, for this deformation mode Kcr (being proportional to drop diameter) is significantly smaller than that in shear (Grace 1982). 

Interest in the study of the extensional flow properties of polymer melts also stems from the fact that several industrially important, polymer processing operations such as fiber spinning, blow molding, and film blowing involve a predominantly extensional mode of deformation. In each of these processes a material experiences large and time-dependent stresses the theoretical prediction of these stresses is a necessary prerequisite for the rational design of any useful control strategy. 

Now, we consider the second class of experiments and check for the predictions of Lodge s model with regard to extensional flows. Using again an equation from the ideal rubbers we can directly write down the time dependent Finger tensor B(f,t ). It has the form... 

Figure 7.19 shows the time dependent viscosities derived from Eqs. (7.143) and (7.154) for both simple shear and extensional flow. For simplicity a single exponential relaxation with a relaxation time r is assumed for The... 

Fig. 7.19. Time dependent viscosities for shear and extensional flow, and as predicted by Lodge s equation of state. Calculations are performed for different Hencky strain rates en, assuming a single exponential relaxation modulus G(t) exp — t/r...

Although in a shear flow the viscosity of a polymeric fluid usually decreases with increasing deformation rate, in an extensional flow the viscosity frequently increases with increasing extension rate that is, the fluid is extensional thickening (recall Figure 2.1.3). Figure 4.2.5 shows the time-dependent uniaxial extensional viscosity... 

If shear thinning is the main phenomenon to be described, the simplest model is the general viscous fluid. Section 2.4. It has no time dependence, nor can it predict any normal stresses or extensional thickening (however, recaU eq. 2.4.24). Nevertheless, it should generally be the next step after a Newtonian solution to a complex process flow. The power law. Cross or Carreau-type models are available on all large-scale fluid mechanics computation codes. As discussed in Section 2.7, they accurately predict pressure drops in flow through channels, forces on rollers and blades, and torques on mixing blades. 

Figure 9.21 shows the time-dependent viscosities derived from Eqs. (9.178) and (9.189) for both simple shear and extensional flow. For simplicity a single exponential relaxation with a relaxation time t is assumed for G t"). The dotted line represents the time-dependent viscosity for simple shear, which is independent of 7. A qualitatively different result is found for the extensional flow. As we can see, the time-dependent extensional viscosity ff t) increases with ch and for en > 0.5t a strain hardening arises. 

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