Extensional flow, start

The tensor B for uniaxial extension is given in Exercise 2.8.1a. If the uniaxial extensional flow starts at time r = 0, by analogy with Example 4.3.5 we have... 

Because of the difficulties involved in continuing extensional flow start-up experiments to steady state, few reliable extensional viscosity data have been published. While plastics processing operations never involve steady-state extensional flow, the behavior of %(f) is of considerable importance with regard to the relationship between molecular structure and rheological behavior. 

P. N. Dunlap and L. G. Leal, Dilute polystyrene solutions in extensional flow birefringence and flow modification, J. Non-Newt. Fluid Mech., 23, 5 (1987). E. Geffroy and L. G. Leal, Flow birefringence studies of a concentrated polystyrene solution in a two-roll mill. 1. Steady flow and start-up of steady flow, J. Polym. Sci., Polym. Phys., 30,1329 (1992). 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

The deformation of dispersed drops in immiscible polymer blends with the viscosity ratio X = 0.005-13 during extensional flow was studied by Delaby et al. [1994, 1995]. In the latter paper, the time-dependent drop deformation during a start-up flow at constant deformation rate was derived. The model is restricted to small drop deformations. 

The same approach may also be used to improve mixing in the extensional flow held. However, since the surface generation in extension is efficient to start with, the improvement is not as dramatic as that expected in shear. 

In the plastic response of semi-crystalline polymers the starting material has an initial spherulitic morphology and, in the process of simple extensional flow, either in tension or in plane-strain compression, ends up with a highly perfect... 

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)...

Starting from the above expression, one can classify steady extensional flows into three categories, as shown below. 

Light intensity due to birefringence versus time after the start-up of purely extensional flow for the polystyrene-tricresyl phosphate solution. From Fuller and Leal (1981). 

In principle, some types of nonspherical particles could be packed more tightly than spheres, although they would start to interact at lower concentrations. In reality, higher viscosities are normally found with nonspherical particles. The concentration law is approximately exponential at low to moderate concentrations, but equations similar to eq. 10.5.1 can still be used as well. The empirical value of 4>m can be much smaller than that for spherical particles (e.g., 0.44 for rough crystals with aspect ratios close to unity Kitano et al., 1981). If fibers are used, this value drops even further, down to 0.18 for an aspect ratio of 27 (see also Metzner, 1985). The decrease with aspect ratio seems to be roughly linear. Homogeneous suspensions of fibers with large aspect ratios are difficult to prepare and handle. As in dilute systems, the type of flow will determine the extent of the shape effect. Extensional flows are discussed below. 

When comparing extensional flow data with the LVE prediction, care must be taken in the calculation of the linear prediction. If the data used to establish the relaxation spectrum do not include very short-time (high-frequency) data, the initial portion of the curve will not be correct. It may thus be better to use data from start-up of steady simple shear to measure tfit) directly rather than inferring it from complex modulus data. 

Extensional flows yield information about rheological behavior that cannot be inferred from shear flow data. The test most widely used is start-up of steady, uniaxial extension. It is common practice to compare the transient tensile stress with the response predicted by the Boltzmann superposition principle using the linear relaxation spectrum a nonlinear response should approach this curve at short times or low strain rates. A transient response that rises significantly above this curve is said to reflect strain-hardening behavior, while a material whose stress falls... 

Eq. (15.98) strongly resembles Eq. (15.61), that we obtained for shear stress build up after starting a steady shear flow with shear rate q at time t = 0. Accordingly, the elongational or extensional viscosity, which is defined to be... 

The most commonly used experiment is start-up of steady simple extension, and the main problems that arise are the support of the sample and the marked decrease in sample cross section at Hencky strains sufficient to generate nonlinear information. The small cross-section gives rise to necking and rupture that prematurely terminate experiments. The only commercial extensional rheometers now available are devices designed for use as fixtures in standard rotational rheometers. These are easy to use and inexpensive, but the stability of the flow at large strains remains a limitation. 

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