The Extensional Viscosity

If a filament or rod of a virgin (stress-free) material is stretched at a constant extension rate, the total stress component ajj, initially zero, will increase with time in a manner which depends on the nature of the polymer. It is then possible to define a tensile stress growth coefficient  

If aji tends to a constant value as stretching proceeds, will also approach a limiting value this is termed either the elongational, extensional, or tensile viscosity, The exten-sional viscosity is a true material property, and is independent both of the technique of measurement and of any assumptions concerning the constitutive behavior of the polymer. For homogeneous materials, its numerical value can be a function of the stretch rate and of the temperature at which the measurement is made. 

For a Newtonian liquid of shear viscosity t], the various components of the extra-stress tensor are related to the components of the rate of strain tensor y by the equation(5,s,9) 

for Newtonian liquids, the tensile stress growth coefficient is a constant quantity and the extensional viscosity is three times the shear viscosity. This result was verified experimentally in 1906 by TroutonOD and is known as Trouton s rule. In addition, the ratio of the extensional viscosity to the shear viscosity is known as Trouton s ratio. 

For polymeric liquids, Trouton s law holds only in the limit of vanishingly low deformation rates,(12) j e., 

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Extensional Viscosity. In addition to the shear viscosity Tj, two other rheological constants can be defined for fluids the bulk viscosity, iC, and the extensional or elongational viscosity, Tj (34,49,100—107). The bulk viscosity relates the hydrostatic pressure to the rate of deformation of volume, whereas the extensional viscosity relates the tensile stress to the rate of extensional deformation of the fluid. Extensional viscosity is important in a number of industrial processes and problems (34,100,108—110). Shear properties alone are insufficient for the characterization of many fluids, particularly polymer melts (101,107,111,112). 

Additional complications can occur if the mode of deformation of the material in the process differs from that of the measurement method. Most fluid rheology measurements are made under shear. If the material is extended, broken into droplets, or drawn into filaments, the extensional viscosity may be a more appropriate quantity for correlation with performance. This is the case in the parting nip of a roUer in which filamenting paint can cause roUer spatter if the extensional viscosity exceeds certain limits (109). In a number of cases shear stress is the key factor rather than shear rate, and controlled stress measurements are necessary. 

Compared to partially hydrolyzed polyacrylamide, xanthan gum is more expensive, more susceptible to bacterial degradation, and less stable at elevated temperatures (1). However, xanthan gum is more soluble in saline waters, particularly those containing divalent metal ions generally adsorbs less on rock surfaces and is substantially more resistant to shear degradation (1,34). The extensional viscosity of the semi-rigid xanthan molecule is less that that of the flexible polyacrylamide (263). 

Electrostatic repulsion of the anionic carboxylate groups elongates the polymer chain of partially hydrolyzed polyacrylamides increasing the hydrodynamic volume and solution viscosity. The extensional viscosity is responsible for increased resistance to flow at rapid flow rates in high permeability zones (313). The screen factor is primarily a measure of the extensional (elonga-tional) viscosity (314). The solution properties of polyacrylamides have been studied as a function of NaCl concentra-tion and the parameters of the Mark-Houwink-Sakaruda equation calculated... 

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

Spatiael et al. [77] studied the foaming behaviors of several TPV formulations containing various amounts of branched PP resin with water as the blowing agent, while the extensional viscosity of the materials with different formulations was measured and considered. The authors indicated that the replacement of a small amount of linear PP with branched PP improved the foam density and cellular structure. However, as the added content of branched PP was increased, a worse foamability was observed. They concluded that there exists an optimal amount of... 

Since the flow is axially symmetric, the stress is also symmetric and analogous to shear viscosity, the extensional viscosity can be defined as... 

The extensional viscosity of the two fluids was determined using a punch that splits the liquid thread with a total length of 1 mm to 6.5 mm (method see [15]). The resulting time-dependent thread diameter and the calculated extensional viscosities are shown in Fig. 3.22. 

Figure 3.22 Measurement of the extensional viscosities of the PEO solution and of silicone oil Baysilone M 1000.

The exponential decrease in the thread diameter and the highly time-dependent increase in extensional viscosity is clearly visible in the case of the PEO solution. However, the silicone oil displays a linear drop in the diameter of the thread (Newtonian fluid) and no increase of the extensional viscosity over time the extensional viscosity corresponds to approximately three times the shear viscosity. 

The extensional viscosity of semi-solid fat-based products such as butter, ice cream and some cheeses can be measured by lubricated squeezing flow rheometry (Campanella and Peleg, 2002 Gunasekaran and Ak, 2002). 

Because also the extensional viscosity is dependent on the strain rate, we have ... 

In these experiments, the tensile force is measured as a function of time, so that at a constant rate of deformation e it is possible to calculate the true tensile stress and the extensional viscosity r/c elastic properties of the deformation can be determined by measuring the elastic strain e. 

By way of example, the experimental results of Meissner (1971) on low-density polyethylene have been represented in Fig. 15.22, by plotting rie/))eo against t/x0 with as a parameter. For low values of all points lie on a single curve, which shows some correspondence to the curves of Fig. 15.16 for rj/rj0 against yxQ. If e > 1, however, the extensional viscosity increases considerably with increasing extension. 

This effect may be responsible for the popular belief that the extensional viscosity of polymer melts increases with increasing rate of deformation. Obviously, this statement is too simplistic, as one more parameter is needed to describe the relationship between extensional viscosity and rate of deformation. The situation is even more complicated. It is certain that the correlation of Fig. 15.22 has no universal validity, but depends on the nature of the polymer. Therefore, at this moment it is not possible to predict the extensional viscosity behaviour of an arbitrary polymer. 

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. 

For mobile liquids, the use of this kind of controllable instruments is practically impossible. For these liquids, the non-controllable measurement techniques are available only and in general an apparent transient viscosity will be obtained. Nevertheless these measurements are still of great value, because in many cases they approximate industrial process conditions. Mostly used is the spinning line rheometer, where an elastic liquid is pressed through a spinneret and the liquid is pulled from the die by winding the filament around a rotating drum or by sucking the tread into a capillary tube. This is schematically shown in Fig. 15.25. A serious problem is the translation of the obtained data to the extensional viscosity. Many other non-controllable devices are discussed by,... 

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

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 Sect. 15.4 it was shown how the shear thinning behaviour of the viscosity could be described empirically with the aid of many suggestions found in literature. It was not mentioned there that the first normal stress coefficient also shows shear thinning behaviour. In this Sect. 15.5 it became clear that also the extensional viscosity is not a constant, but depending on the strain rate upon increasing the strain rate qe the extensional viscosity depart from the Trouton behaviour and increases (called strain hardening) to a maximum value, followed by a decrease to values below the zero extensional viscosity. It has to be emphasised that results in literature may show different behaviour for the extensional behaviour, but in many cases this is due to the limited extensions used,... 

For concentrated polymer solutions, the behaviour in extensional deformation shows a great correspondence to that of polymer melts. At low rates of deformation the extensional viscosity has the theoretical value of three times the shear viscosity. At higher rates of deformation, the experimental results show different types of behaviour. In some cases, the extensional viscosity decreases with increasing rate of extension in the same way as the shear viscosity decreases with increasing shear rate. In other cases, however, a slight increase of the extensional viscosity with increasing rate of extension was observed. 

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