RECENT IDEAS IN EXTENSIONAL RHEOLOGY

Extended abstract for presentation at the PPS Europe/Africa Region 1997 meeting in Göteborg, Sweden, August 1997.
Introduction
This review starts from two main questions: "What is the present state of knowledge of the extensional flow behaviour of polymer melts?" and "What contributions does theoretical rheology make to our understanding of this behaviour?" Extensional rheometry and the choice of a constitutive equation are two obvious topics for our attention. There are a number of other topics which we shall touch on briefly as being areas where a theorist and a polymer processor should have some common interests. In conclusion we comment on Kurtz's list of key challenges [1].

Extensional rheometry
Nearly half of the review of polymer melt rheometry by Meissner [2] is devoted to extensional flows. The picture is clearer for melts than in the more recent review for polymer solutions by James and Walters [3]. Much recent work in extensional rheometry and in the exploration of constitutive equations has had to do with polymer solutions. The reasons for the chaos [see 3, Figure 2.1] are becoming clearer [4] - largely the variety of methods adopted for measurement of extensional viscosity. This arose from the problem encountered with mobile liquids that the "simple" tensile test experiments [2] were not thought feasible. This part of polymer solution history is not directly relevant to polymer melts. However any understanding of non-ideal flows such as fibre spinning and converging or contraction flows will be useful for all polymeric liquids.

Methods of measuring the true extensional viscosities require great care [2] and one problem is to relate the results of easier experiments to the fundamental material property. It is also important to be able to use the results in real processing flows, and that may seem to offer a dilemma. On the one hand, a true extensional viscosity is only obtained (reliably) from a steady uniform flow and for viscoelastic fluids the effect of unsteadiness or non-uniformity may be highly significant. On the other hand, few processing flows even approximate to steady uniform flow and tests involving more relevant flows are tempting.

As far as the "ideal" flows are concerned, there are data on a number of polymer melts obtained mainly by Meissner and his collaborators (past and present) using the well known, but ever-improving, uniaxial extensional rheometer. Meissner's rheometer for multiaxial extension offers a variety of geometrical modes of deformation and programmes with changes in rate or even direction of extension. In all cases the measurement of recoverable strain is always a valuable addition and should not be an optional extra. It may also be worth considering stress relaxation after extension, a measurement which is attracting attention in current work on polymer solutions. One very important distinction is between a genuine extensional viscosity and a stress growth function (or "transient extensional viscosity") obtained in a spatially uniform flow which does not reach a steady state.

Constitutive equations
One important test of a constitutive equation, if it is to be used for a complex flow, is whether it will adequately describe behaviour in different geometrical situations. There is also the practical test of whether the equation is simple enough to use in, for example, a theoretical or computational analysis of a particular process. The compromises that this necessitates constitute the art of mathematical modelling of industrial processes. For polymer melts, the Wagner equation and some related Kaye-BKZ equations offer the best compromise between the ability to fit rheometrical data from different experiments and the feasibility of use in computational modelling of complex flows. There are a number of variants of this model and the choice depends on what data are available. It is sometimes thought easier to avoid use of an integral equation, in which case the Phan-Thien- Tanner and Acierno-Marrucci equations may be useful.

Converging flow
This is probably one of the topics which is most in need of clarification. There have been a number of recent investigations, both experimental [5], theoretical [6] and with an engineering view [7]. One aspect of conventional ways of analysing these flows, which has been noted [4], is that they do not give the correct result for the extensional viscosity of a Newtonian liquid.

Flow stability
The most important rheological issue as far as flow stability is concerned is whether we can predict the onset of observed flow instability and then, from a theoretical understanding, suggest measures for avoiding instability or reducing its effect. Mention of instability in polymer processing brings melt fracture and sharkskin to mind and extensional flow is not necessarily irrelevant here. However the obvious areas of relevance for extensional flows are fibre spinning, film blowing, flat film casting and perhaps coating flows. The distinction between different phenomena is important here too [8] and care is needed in reading some of the literature.

Rupture
The rupture behaviour of polymer melts is often neglected in rheological studies of materials and this may explain a lack of explanation of the conditions for rupture during extensional flow [9]. The theories that have been advanced do not explain adequately such data as are available.

Film blowing
This is process which is geometrically complex and where there is a strong desire for simple ideas. There is no guarantee that such simplicity can be found, and part of the theorist's task here is the rather negative one of curbing the enthusiasm of would-be simplifiers. There are, however, many questions to be resolved after valid approximations to the dynamical equations are obtained [10] which explains continuing interest in the process [11-13].

Conclusion
Kurtz's list [1] of five key challenges in polymer processing technology included two relating directly to extensional flow stability, namely "Blown film bubble stability" and "Draw resonance". The other three were "Screw wear", "Sharkskin melt fracture" and "Scale up problems" and clearly this last is important in processes involving extensional flow. I would add "Converging flow" and "Rupture" to the list of challenges and claim that extensional flow is the area of polymer rheology presenting the most interesting and important challenges to theorists, experimenters, computer packages and production managers alike.

References

  1. Kurtz, S.J. "Some key challenges in polymer processing technology", in "Recent Advances in Non-Newtonian Flows", ASME, AMD-Vol 153/PED-Vol 141, 1-13 (1992).
  2. Meissner, J. "Rheometry of polymer melts", Ann. Revs. Fluid Mech., 17, 45-64 (1985).
  3. James, D.F. and Walters, K. "A critical appraisal of available methods for the measurement of extensional properties of mobile systems", Chap. 2 in "Techniques of Rheological Measurement", Ed. A.A. Collyer, Elsevier, New York, 33-53 (1994).
  4. Petrie, C.J.S. "Extensional flow - a mathematical perspective", Rheol. Acta, 34, 12-26 (1995).
  5. Groves, D.J., Martyn, M.T. and Coates, P.D. "Off line and in process converging flow measurements for polyethylene melts", Plastics, Rubber and Composites Proc. & Appl., 26, 13-22 (1997).
  6. Davies, A.R., Farah, I.M., Rides, M. and Thomas, K. "Numerical evaluation of a method for determining extensional viscosity of fluids using contraction flow analysis", Makromol. Chem., Macromol. Symp., 68, 25-39 (1993).
  7. Mackay, M.E. and Astarita, G. "Analysis of entry flow to determine elongation flow properties revisited", J. Non-Newtonian Fluid Mech., 70, 219-235 (1997).
  8. Petrie, C.J.S. "Some remarks on the stability of extensional flows", Prog. Trends Rheol., II, 9-14 (1988).
  9. Malkin, A.Ya. and Petrie, C.J.S. "Some conditions for rupture of polymer liquids in extension", J. Rheol., 41, 1-25 (1997).
  10. Petrie, C.J.S. "Film blowing, blow moulding and thermoforming" Chap. 7 in "Computational Analysis of Polymer Processing", Eds. J.R.A. Pearson and S.M. Richardson, Applied Science Publishers, London, 217-241 (1983).
  11. Tas, P. "Film blowing: from polymer to product", Doctoral thesis, Technical University of Eindhoven, (1994).
  12. Kurtz, S.J. "Relationship of stresses in blown-film processes", Int. Polym. Proc., 10, 148-154 (1995).
  13. Lui, C.-C., Bogue, D.C. and Spruiell, J.E. "Tubular film blowing. Part 2. Theoretical modelling", Int. Polym. Proc., 10, 230-236 (1995).


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