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