Extensional thickening

For melts the nonlinearity in the extensional viscosity is usually more sensitive to the molecular architecture (e.g., the presence 

Uniaxial extensional viscosity T) versus time for different extension rates e for four different melts. The solid lines are the fits to the data of a special form of the integral equation, eq4.4.15. The curves at the lowest extension rates correspond to linear viscoelastic response. Note that curves have been shifted vertically by the multiplier indicated along the ordinate. Replotted from Laun (1984). 

Uniaxial extensional viscosity and shear viscosity 77+ as functions of time after inception of steady straining for lUPAC A low density polyethylene. The open symbols are elongational viscosities the solid and half-open symbols are shear viscosities. Adapted from Meissner (1972). 

In addition to the nonlinear phenomena we have discussed in this section, namely normal stresses, shear thinning, and ex- 

Stressing viscosity (i, for uniaxial, biaxial, and planar extension, stressing viscosity for planar extension, and shear viscosity as functions of time ter inception of steady straining for polyisobutylene. The solid line is the low shear rate limit of. Extension and shear rates are 0.08s except the biaxial which is 0.02s-. From Retting and Lawn, 1991. 

With an increase in flow time or strain at low yxx, tJe increases to its asymptotic steady-state value given by Eq. (32). 

---

The extensional thickening of polymer solutions is one form of viscoelastic behavior. This ability to support a tensile stress can also be demonstrated in a tubeless syphon with dilute aqueous solutions of polsrmers such as polyacrylamide or polyethylene oxide. If you suck up solution with a medicine dropper attached to a water aspirator and then lift the dropper out of the solution, the solution will still be sucked up. In shear, viscoelastic fluids develop normal stresses, which causes rod climbing on a rotating shaft, as opposed to the vortex and depressed surfaces that form with Newtonian liquids. Polsrmer solutions and semiliquid poljnners exhibit other viscoelastic behaviors, where, on short time scales, they behave as elastic solids. Silly putty, a childrens toy, can be formed into a ball and will slowly turn into a puddle if left on a flat surface. But if dropped to the floor it boimces. 

The extensional viscosity functions of emulsions were characterized using a Capillary Breakup Extensional Rheometry (CaBER). The apparent extensional viscosities ) of SE-la emulsions as a function of strain rate e are given in Fig. 23.5 for the different mean emulsion drop sizes of 2,4, and 10 pm. From this, an extensional viscosity characteristics for the emulsion with a drop size of 10 pm was derived close to constant and rather low (ca. 0.2 Pas, Newtonian-like). For emulsion drop sizes of 2 and 4 pm, pronounced extensional thinning was observed in the low elongation rate domain, whereas some pronounced extensional thickening behavior was monitored in the higher elongation rate domain. 

Typically nsUho) is a shear thinning function of the Cross or Carreau form and tjuilho) is an extensionally thickening function of the same form (but n > 1). The ratio Wni/S = 1 in simple shear and 0 in pure extension. 

The approaches above can describe only steady state, time-independent viscosities. In Chapter 4 we will show that for time-dependent viscoelastic models, like Maxwell s, extensional thickening arises naturally. 

In fact we have lost something over Chapter 1. The Finger tensor B is able to give us normal stresses and extensional thickening. We will have to wait until Chapter 4 to get these factors back into our models. But in the next chapter we will see how to bring in the phenomenon of time dependence, which is so important for polymeric systems. We should note that for concentrated suspensions, especially flocculated systems, there is little elastic recovery, and time dependence is often either very short or extremely long. Thus the viscous models of this chapter are often quite adequate (recall Figures 2.5.3 and 2.5.4 and look ahead to Chapter 10). 

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... 

Suppose that for viscoelastic fluid X we measure the ratio as -0.2 at low shear rates. In a slow steady uniaxial extensional flow, should we expect fluid X to show extensional thickening or extensional thinning ... 

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. 

For 77, — 3jj, the Trouton limit, we see by comparing Figures 10.3.5 and 10.3.7 that [ij], = [77]. In fact, at low Pe, the intrinsic viscosity is the same in extension as in shear. This is as expected because at low Pe there is no preferred orientation and extension does not differ from shear. However, at large positive Pe these cigar-shaped particles line up with the flow and increase the drag, thus producing extensional thickening. 

---