Equilibrium extensional viscosity

Both types of behavior are evident in confirming the trend from elastic (at low strain) to viscous response as the temperature is increased. The estimated extensional viscosity at 500 K from the simulations is of the order of 0.01 Pa s . Although this is much lower than the equilibrium extensional viscosity of polyethylene it is known that the viscosity does decrease significantly with increasing strain rate. At the extension rates used in our simulations, 10 —>10 s , the behavior is expected to be strongly non-Newtonian. 

As mentioned above, it is far more difficult to measure extensional viscosity than shear viscosity, in particular of mobile liquids. The problem is not only to achieve a constant stretch rate, but also to maintain it for a sufficient time. As shown before, in many cases Hencky strains, e = qet, of at least 7 are needed to reach the equilibrium values of the extensional viscosity and even that is questionable, because it seems that a stress overshoot is reached at those high Hencky strains. Moreover, if one realises that that for a Hencky strain of 7 the length of the original sample has increased 1100 times, whereas the diameter of the sample of 1 mm has decreased at the same time to 33 pm, then it will be clear that the forces involved with those high Hencky strains become extremely small during the experiment. 

Rheological measurements are performed so as to obtain a test fluid s material functions. Under viscometric flows we have seen that the shear viscosity and the primary and secondary normal stress differences suffice to rheologically characterize the fluid. If the flow field is extensional and the material is able to attain a state of dynamic equilibrium, then one measures the extensional viscosity otherwise, we measure the extensional viscosity growth or decay functions. In this section, we will examine steady and dynamic shear plus uniaxial extensional tests, since these make up the majority of routine rheological characterization. 

Another method for measuring uniaxial extensional viscosity is by bubble collapse. A small bubble is blown at the end of a Ciq>illary tube placed in the test fluid (see Figure 7.6.1). It comes to equilibrium with the surrounding pressure and surface tension. Then at time r = 0 the pressure inside the bubble is suddenly lowered or the surrounding pressure increased. The decrease in bubble radius with time is recorded. If the deformation is reversed (i.e., the pressure inside the bubble is suddenly increased), the growing bubble radius can be used to give the equibiaxial viscosity. This flow appears to be less stable and has not been studied as a rheometer. 

As discussed in Chapters 10 and 11, rheology can be very sensitive to the microstructure of liquids. For example, the viscosity of entangled polymer melts depends on molecular weight to the 3.4 power, T]o Ml . Equilibrium creep compliance is very sensitive to molecular weight distribution. The yield stress and low frequency G are good indicators of the flocculation state of colloids. Extensional viscosity can be an important indicator of bread dough quality (Padmanabhan, 1993). 

Tube models have been used to predict this material function for linear, monodisperse polymers, and a so-called standard molecular theory [159] gives the prediction shovm in Fig. 10.17. This theory takes into account reptation, chain-end fluctuations, and thermal constraint release, which contribute to linear viscoelasticity, as well as the three sources of nonlinearity, namely orientation, retraction after chain stretch and convective constraint release, which is not very important in extensional flows. At strain rates less than the reciprocal of the disengagement (or reptation) time, molecules have time to maintain their equilibrium state, and the Trouton ratio is one, i.e., % = 3 7o (zone I in Fig. 10.17). For rates larger than this, but smaller than the reciprocal of the Rouse time, the tubes reach their maximum orientation, but there is no stretch, and CCR has little effect, with the result that the stress is predicted to be constant so that the viscosity decreases with the inverse of the strain rate, as shown in zone II of Fig. 10.17. When the strain rate becomes comparable to the inverse of the Rouse time, chain stretch occurs, leading to an increase in the viscosity until maximum stretch is obtained, and the viscosity becomes constant again. Deviations from this prediction are described in Section 10.10.1, and possible reasons for them are presented in Chapter 11. 

Here was adopted for simplicity a = A and a 1 (the latter inequality is satisfied for bubbles with Rg > >1 mkm). Phase plot of this equation is presented in Figme 7.2.2. It is seen that for k = -1 (collapsing cavity) z->Zj ast- oo rfzo>Z2. The stationary point z = Zg is unstable. The rate of the cavity collapse z = Zj in the asymptotic regime satisfies inequality Zp < Zj < 0, where Zp = -RCp is equal to the collapse rate of the cavity in a pure viscous fluid with viscosity of polymeric solution q. It means that the cavity closure in viscoelastic solution of polymer at asymptotic stage is slower than in a viscous liquid with same equilibrium viscosity. On the contrary, the expansion under the same conditions is faster at k = 1, Zp < Zj < z, where Zp = RCp and z = Re = (1 - P) RCp is the asymptotic rate of the cavity expansion in a pure solvent with the viscosity (1 - P)q. This result is ejqrlained by different behavior of the stress tensor component controlling the fluid rheology effect on the cavity dynamics, in extensional and compressional flows, respec-... 

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