Extensional viscosity steady state

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... 

These results highlight the genuine need for carrying out steady state experiments for accurate measurement of dough extensional viscosity. At the present time, there is no other instrument available that can provide stress-growth profiles for doughs. 

For correlating extensional viscosity data, it is obvious to attempt the same method as was used for non-steady state shear viscosity. Thus, the ratio r)Jrjeo is presumed to be determined by two dimensionless groups i0 and f/i0. As e is constant (i.e. qe), the ratio of these groups is equal to the tensile deformation e. Therefore, t/t0 will likewise be a function of t/xa and . 

Often, it is not possible to reach a steady state in extension and it is convenient to define a transient extensional viscosity, tje, that is a function of time, t, and the extensional strain rate, e, (Barnes et al., 1989). 

The instantaneous extensional stress W(e, t) is the force F(e, t) along the cylinder axis required to pull the cylinder ends apart, divided by the instantaneous cross-sectional area A s, t) of the cylinder thus a(e, r) = F e, t)/A(e, t). The time-dependent extensional viscosity, rj(e, t), is then ct( , t)/e. If this viscosity reaches a time-independent value within the duration of the experiment, that value is called the steady-state extensional viscosity, r ( ). 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

Figure 3.19 The polymer contribution to the steady-state uniaxial extensional viscosity r divided by the polymer contribution to the zero-shear viscosity rjp = r/o — fjj for the dumbbell model with a nonlinear FENE spring and various values of B = ipL. (From Bird et al. Dynamics of Polymeric Liquids, Vol. 2, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

Strands that terminate with a branch point at both of its ends can neither reptate nor completely retract. Relaxation of such strands presumably occurs by more complex, hierarchical processes discussed by McLeish (1988b). Here we simply note that the presence of branch points at both ends of a strand leads to much more strain hardening in extensional flows (Bishko et al. 1997 McLeish and Larson 1998). Low-density polyethylenes (LDPEs), which are highly branched, are well known for their extreme strain hardening behavior in extensional flows (Meissner 1972 Laun 1984) (see Fig. 3-39). The steady-state shear viscosity, as a function of shear rate, seems to be little affected by long-chain branching, however. 

The steady-state uniaxial extensional viscosity rj is given by ([Pg.181]

While the fiber contribution to the steady-state stress tensor at steady-state is modest for shearing flow, its contribution to the stress in extensional flow is large at steady state. In a uniaxial extensional flow, the fibers orient in such a way that the viscous dissipation is maximized. Large values of the extensional viscosity are the result from Batchelor s (1971) theory the uniaxial extensional viscosity is... 

Since the steady-state shear viscosity is only slightly affected by the rods, the ratio of the extensional to the shear viscosity is of order 50 or so. 

In the extensional, irrotational field, under steady state conditions, the particles remain oriented in the direction of stress. In uniaxial flow, they align with the main axis in the flow direction, while in biaxial they lie on the stretch plane [Batchelor, 1970, 1971]. For dilute spherical suspensions in Newtonian liquid the extensional viscosity follows the Trouton rule, i.e., = 3q. 

Instead of imposing a constant stretch rate on a sample and measuring the steady-state stress, one may impose a constant stress and determine the resulting extensional strain. This is a creep experiment, and if the strain, initially zero, begins to increase linearly with time, a constant stretch rate is achieved. The extensional viscosity is again obtained as the ratio of the imposed stress to the resulting constant stretch rate. 

A cylindrical rod composed of a very viscous polymer can be stretched uniformly, meaning that the diameter changes with time but not with position on the rod. This may be done in several ways (1) at constant stretch rate, (2) at constant stress, (3) at constant velocity, or (4) at constant force. Although all four test conditions are encountered in the literature, two of the conditions have serious disadvantages. Thus, the stretch rate continually decreases in a constant-velocity experiment and the stress continually increases in a constant-force experiment. Consequently, a steady state cannot be obtained and the extensional viscosity cannot be determined if stretching is done either at constant force or at constant velocity. 

The major advantage of a constant-stress rheometer over a constant-strain-rate rheometer is that, for a given polymer, a steady state is frequently achieved in the former mode but not in the latter one.(28) Even when a steady state is obtained with the use of both instruments, the total strain needed to achieve a steady state is lower for the constant stress viscometer. This extends the range of strain rates at which the extensional viscosity can be determined for an apparatus of a given size. Finally, it has been observed(4 20) that the stress tends to decrease slightly in a constant stretch rate experiment even after a plateau appears to have been reached. The physical significance of this last observation is not entirely clear. (29)... 

The uniaxial extensiometers described so far are suitable for use with viscous materials only. They cannot, for example, be used to measure the steady extensional viscosity of such commercially important polymers as nylons and polyesters used in the textile industry, and which may have shear viscosities as low as 100 Pa sec at processing temperatures. As a consequence, other techniques are needed but these invariably involve nonuniform stretching. Here one cannot require that the stress or the stretch rate be constant. Also, the material is usually not in a virgin (stress-free) state to begin with. One can therefore not obtain the extensional viscosity directly from these measurements. Nonetheless, data from properly designed non-uniform stretching experiments can be profitably analyzed with the help of rheological constitutive equations. In addition, such data provide a simple measure of resistance that polymeric fluids offer to extensional deformation. 

The rheological responses measured at low values of strain better reflect the effects of the blend structure. For multiphase systems, there are serious disagreements between the predictions of continuum-based theories and experiments, that is, between the small and large deformation behavior. For example, the identity of zero-deformation rate dynamic and steady state viscosity is seldom found, and so is the Trouton rule. Similarly, the derived by Cogswell, relationship between the extensional viscosity and the capillary entrance pressure drop, and derived by Tanner equation for calculating the first normal stress difference from the extrudate swell, are rarely valid. 

The linear metallocene polyethylene (mPE) reference, polymer Clb P5, shows stress-independent steady-state viscosity throughout the stress range measured, whereas the LDPE shows strain hardening behavior typical for that range. At low stress, the response is equal to three times the LVE shear viscosity. Increasing the tensile stress leads to strain hardening up to a maximum stress, after which the response becomes extension thinning [121]. In contrast to the LDPE, the steady-state extensional viscosity ( /e) of branched mPE polymer C4 P1 appears... 

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. 

Figure 9.1 Variation of steady state extensional viscosity with extensional rate for filled polystyrene melts at 30vol% of various types of fillers as indicated. (Reprinted from Ref. 29 with kind permission from American Chemical Society, vyashington DC. USA)...

One of the effects of increasing filler concentration is that constant extensional viscosities, namely, steady-state conditions are reached more easily and earlier in the filled systems than in unfilled systems and the values decrease with increasing extensional rate. This point has been brought out in the work of Lobe and White [19] who studied the influence of carbon black on the rheological properties of a polyst3n"ene melt. 

Figure 9.11 shows the effect of surface treatment on extensional viscosity for 30% calcium carbonate filled polystyrene [27]. The data are presented in two forms, namely steady state extensional viscosity vs. extensional rate in Figure 9.11(a) and steady state extensional viscosity vs. tensile stress in Figure 9.11(b). Irrespective of the type of data representation, it is seen that surface treated calcium carbonate reduces the level of extensional viscosity and brings it closer to that of the unfilled polymer. The yield stress value is reduced considerably though the values of the ratio of yield stress in extension to that of shear is still maintained nearer to the von Mises value of 1.73 as can be seen from Table 9.1. Surface treatment tends to modify the forces of particle-particle interaction and hence show reduced yield stress values due to lowering of the interaction forces [2,27]. 

Most concentrated structured liquids shown strong viscoelastic effects at small deformations, and their measurement is very useful as a physical probe of the microstructure. However at large deformations such as steady-state flow, the manifestation of viscoelastic effects—even from those systems that show a large linear effects—can be quite different. Polymer melts show strong non-linear viscoelastic effects (see chap. 14), as do concentrated polymer solutions of linear coils, but other liquids ranging from a highly branched polymer such as Carbopol, through to flocculated suspensions, show no overt elastic effects such as normal forces, extrudate swell or an increase in extensional viscosity with extension rate [1]. 

Steady state shear conditions are relatively easy to establish with appropriate instruments in laboratory conditions but steady state extensional conditions require that an initial reference length (of the material) is growing exponentially with time. There are thus inherent complexities in assessing extensional viscosity, whose detailed discussion is outside the scope of this chapter but is found in many textbooks [6, 7]. Practically extensional rheometry is limited to very delicate and tedious laboratory practices, on a limited number of polymer systems. 

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