Extensional rate

In simple shear flow where vorticity and extensional rate are equal in magnitude (cf. Eq. (79), Sect. 4), the molecular coil rotates in the transverse velocity gradient and interacts successively for a limited time with the elongational and the compressional flow component during each turn. Because of the finite relaxation time (xz) of the chain, it is believed that the macromolecule can no more follow these alternative deformations and remains in a steady deformed state above some critical shear rate (y ) given by [193] (Fig. 65) ... 

Turning to Fig. 3.2, Case 1, we see that the tensile force F needed to sustain the applied constant extensional rate e, either levels off to a constant F (s) or exhibits strain hardening increasing with time, occasionally in an unbounded fashion the force is then represented as F (i . t). For this uniform extensional flow... 

Finally, it is worth discussing briefly the flow singularity at the exit comer of pressure-flow dies used for forming fibers and film, which are consequently stretched to orient and structure them. At that location we have to reconcile the fact that the wall melt flow layer must, in nearly zero distance, accelerate from a zero to a finite velocity. Irrespective of the details of this high acceleration, the surface layer undergoes high extensional rate flows,... 

The extensional melt behavior was assessed with the new SER Universal Testing Platform from X-pansion Instruments, described by Sentmanat (53,54) and shown in Fig. 12. 25. The obtained tensile stress of the two resins at 170°C and extensional rate of 20 s-1 are shown on Fig. 12.26. It is evident that Resin E has a higher modulus and higher tensile stress values, at a given strain below yield, than Resin C. From this, and the experimental data discussed previously, we see that the values of the critical shear rate and shear stress for the onset of sharkskin fracture are inversely proportional to the magnitude of the tensile stress of the resins. This suggests that the rapid increase... 

We now turn to the gross melt fracture behavior. Estimates of the prevailing extensional rates at the capillary entrance indicate that their value corresponding to the critical gross melt fracture conditions is larger that the 20 s 1 used in this work. Nevertheless, since... 

Extensional flow (also called elongational flow) is defined as a flow where the velocity changes in the direction of the flow dvi/ dxy in contradistinction with shear flow where the velocity changes normal to the direction of flow (dv1/dx2). In uniaxial flow in the x1 direction the extensional rate of strain is defined as ... 

For constant extensional rate of strain, qe, the length of a line element increases exponentially in the redirection ... 

For correct measurements, the extensional rate of strain is in general controlled and the extensional stress is measured. 

FIG. 15.23 Extensional rheometer, designed by Miinstedt (1979). A servo control system is used to maintain a specified extensional rate of strain or a specified tensile stress. Courtesy Society of Rheology. For a modern version, see Miinstedt et al., 1998. 

The result is that the extensional viscosity increases with increasing extensional rate of strain. This is also in agreement with practice. 

As soon as the flow becomes so fast that qe > 1 the first term in Eq. (15.97) increases without limit with time. However, as long as t is still small, Ni(t)/qe is still independent of qe. The deviation from this linear behaviour occurs earlier, i.e. at shorter times, when the extensional rate of strain becomes higher (approximately at qet = 0.5). 

In Fig. 15.26 an example is given of Eq. (15.97) for a Maxwell element with G = 1000 N/m2 and t = 1 s. For small extensional rates of strain, the extensional viscosity is constant and equal to 3000 N s/m2. For higher extensional rates of strain, the viscosity increases. At the extensional rate of strain of 0.5 s-1 there is a transition to infinite extensional viscosities. The dotted line is the transient shear viscosity r)+(t) at low shear rates and equal to 14 of the transient extensional viscosity r)+ (f) at low extensional rates of strain. 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

FIG. 15.27 Transient extensional viscosity, r/+ (f)of low-density polyethylene LDPEIUPAC A at 150 °C, vs. time, for various extensional rates of strain. From Munstedt and Laun (1979). Courtesy Springer Verlag. 

FIG. 15.30 Transient extensional viscosity of LDPE Melt I for extensional rates of strain varying from 0.001 to 1 s-1, at 150 °C. From Wagner and Meissner (1980). Courtesy John Wiley 8t Sons, Inc. 

We will finish this Section with some interesting experimental results on dilute polymer solutions. In Fig. 16.23 extensional viscosity data are shown, obtained from a spin-line rheometer on a solution of polybutadiene in decalin (Hudson and Ferguson, 1976). Here rie not only shows normal behaviour, starting with an increase, followed by a substantial decrease, but also at very high extensional rates a substantial increase. It is not clear yet whether this ultimate tension thickening also occurs for stiff polymeric systems. Concentration dependent extensional viscosities are presented in Fig. 16.24 for 0.0125-0.75%... 

For an unvulcanized polydimethylsiloxane, the biaxial viscosity was approximately six times the shear viscosity over the biaxial extensional rates from 0.003 to 1.0 s (Chatraei et al., 1981), a result expected for Newtonian fluids, that is, the relationship between the limiting value of biaxial extensional viscosity (j, ) at zero strain rate and the steady zero-shear viscosity (i o) of a non-Newtonian food is ... 

Hypochlorite hard surface and drain cleaner compositions exhibiting enhanced extensional viscosity are mentioned in U.S. Patents 5,728,665 and 5,916,859. The viscoelastic compositions are intended for use with trigger sprayers and the hexa-decyl amineoxide/organic counterion compositions provide low bleach odor and reduced spray misting. The patent contains extensional viscosity data in support of the claims. Viscosity as a function of shear rate at various Cm diphenyloxide disulfonate concentrations is shown in Figure 4.5. Examples of steady shear and extensional viscosity as a function of shear rate and extensional rate are shown in Figure 4.6 and Figure 4.7. 

Doughs with different protein contents (13.2, 16.0, and 18.8% based on 14%MB) showed different biaxial extensional viscosities (Huang and Kokini, 1993). Figure 31 shows that the biaxial extensional viscosity approached 6tj at an extensional rate of 7.3 x 10 sec . Strain thinning behavior was observed in dough during biaxial extension. 

The importance of extensional flow in the mixing process has been pointed out by Gotten (16) and thoroughly analyzed by Nakajima (8) in the case of carbon black-filled compounds. A steady elongational flow can be developed only if the extensional rate increases exponentially (versus time) (17). Nakajima demonstrated that this type of deformation induces an anisotropy of the material, enhanced in highly filled compounds or containing oriented fibers. Therefore, the steady state is nearly impossible and, with polymers, the elongational flow is not a pure deformation and necessarily involves a shear component. 

Transforming this deviatoric shear-rate expression, eq. (10.7), into a uniaxial extensional rate expression gives... 

Fig. 7.12 Illustration of extensional viscosity versus the extensional rate curve predicted by the molecular theory based on the standard tube model for the stable extensional flow of linear polymers. Starting from the low extensional rate, the viscosity first keeps in 3%, then decays, after deformation begins to increase, till to saturation (Marrucci and lannirubertok 2004) (Readapted with permission)...

It is assumed that with respect to the stress form the polymers are activated in extensional fields. As the extensional rates are not directly measurable/ we characterize the flow fields by means of the wall shear stress. The onset wall shear stress is presented in fig. 6 as a function of the mean molecular weight of the PAAm sample. Apparently there is a lower limit for below which no drag reduction can be attained for a realistic value of T. On the other hand a certain value ap-... 

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... 

The PTT model predicts shear thinning and first normal stress difference for both steady and transient shearing. It also predicts a non-zero (negative) second normal stress difference in simple shear flow N2 — — Nil2. The main advantage of the PPT model is that it predicts reasonable extensional flow behavior at all extensional rates. 

The uniaxial extensional rate may be constant or vary in the Xi direction of flow. When s is constant, i.e. when the axial velocity is proportional to Xj, the resulting flow is steady uniaxial extensional flow. In such a flow situation, a cylindrical rod of length I is stretched along its axis according to the following equation ... 

The rotational viscometers and the capillary rheometers described in sections 3.1 and 3.2 are those applicable for shear flows. However, there are processing operations that involve extensional flows. These flows have to be treated differently for making mecisurements of extensional viscosity. The extensional viscosity of a material is a measure of its resistance to flow when stress is applied to extend it. In general, measurement of steady-state extensional viscosity has proven to be extremely difficult. Steady extensional rate would be achieved by pulling Ihe ends of the sample apart such that I = Zq exp(ef) or in other words, at a rate that increases exponentially with time. Steady-state is reached when the force is constant. However, often d e sample breaks before steady-state is achieved or the limits of the equipment are exceeded or at the other extreme, die forces become too small for the transducer to differentiate between noise etnd response signal. Nevertheless, there have been various methods attempted for the measurement of extensional viscosity. 

There are other variations of the filament stretching technique. For example, filaments are clamped at one end and taken up on a rotating roll [86,87]. This reduces the amoimt of filament stretching to a more uniform level and produces a more constant extensional rate. In fact, when the following filament is taken up on a cold roll [87] a better constancy in the extensional rate is obtained. 

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