Introduction to polymer rheology

411 Pages·2012·21.43 MB·English

by  Montgomery T Shaw

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Introduction to Polymer Rheology
This page intentionally left blank
Introduction to Polymer Rheology
Montgomery T. Shaw

WILEY
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Preface
I am keenly aware of the quote in a book written by the late Arthur Lodge
saying roughly: "Who needs another rheology book"? Agreed. For teaching, I
personally am a fan of Dynamics of Polymeric Liquids, written by R. Byron
Bird, and Ole Hassager (Volume 1) and R. Byron Bird, Charles F. Curtiss,
Robert C. Armstrong and Ole Hassager (Volume 2). As far as I am concerned,
this book obviated forever the need for another rheology textbook.
Do I use it for teaching my graduate rheology course? No. First of all, at
600+ pages, it's expensive. Even Volume 1 retails for more than $200 on
amazon.com. At this price, the average graduate student will consider either
doing without, dropping the course, or buying an illegal copy. But my most

serious reservation with this and many other texts is the slant. It is directed at
engineers who have had basic transport phenomena, along with linear algebra,
differential equations and numerical analysis. Very few of my polymer
students are so privileged. Short-hand tensor notation, while convenient for the
expert, is baffling to them.
Why then do polymer students, mostly with organic chemistry backgrounds,
bother with a rheology course? It doesn't take them long to figure out that
training in polymers carries with it the necessity for an acquaintance with their
mechanical, viscoelastic and rheological properties. Sure, molecular
spectroscopy, thermal analysis and microscopy are the mainstays of polymer
analysis, but they hear the news from their friends in industry—learn about
rheology.
The long and short of this discussion is that a textbook aimed a bit lower
seemed like a valuable addition. The popular text Introduction to Polymer
Viscoelasticity was taken as the starting point. A reproduction of the relaxed
v
vi PREFACE
style of this text was attempted. Certainly one cannot learn about rheological
behavior without dealing in some fashion with three-dimensional mechanics,
but the basics are really enough. Tensors can be presented consistently in
matrix form instead of with shorthand. This is tough on the author, but helps
the student to sort out the important aspects of the various categories of
deformation and flow.
The knowledgeable rheologists will encounter in this volume some shortcuts
that they will find somewhat bothersome and certainly lacking in rigor. For
example, the continuity equation is barely mentioned and rarely used. No one
needs the continuity equation to figure out that the velocity in outward
axisymmetric radial flow is (1) positive and (2) falls as the reciprocal of the
radius. This is true with many other one- and two-dimensional flows as well.
Where possible, shell force balances are used as opposed to plowing through
the collection of confusing terms in the differential momentum balance.
One annoying aspect of a math-based physical science is nomenclature. In
rheology, it's not just about symbols, but also there is the sign issue and a factor
of two in the definition of the rate of deformation. This text sticks as closely as
possible to the conventions endorsed by the Society of Rheology. Thus
σ-rff - ρδ, for example. The symbol τ is introduced for the extra (dynamic,
deviatoric) stress tensor, as a convenience. The use of the "positive for
tension" sign convention is noted near each equation by the abbreviation (ssc),
which stands for "solids sign convention," because this convention was a
product of tensile testing of solid samples. However, the student is often

reminded that the "fluids sign convention" (fsc) is in wide use, and examples
are given. Another annoyance for the fastidious is the use of a few symbols for
two meanings. For example, the symbol r might appear in one place to mean
shear stress and another place as a time constant, even in the same equation.
Confusing? Not really, as the context readily differentiates the two, especially
after the student becomes familiar with the concept of a dimensionless term.
Also, in most cases, rfor stress carries two subscripts, e.g., r .
21
In addressing three-dimensional mechanics, the stress tensor is introduced
first (Chapter 2). Stress is very physical concept that sits well with most
students. The nuances of how a stress can be applied to a sample are explained
in perhaps more detail than in most texts. How does one turn the tensile force
in a string into a nice uniform uniaxial stress in a sample of finite size? To
solve this, a construct termed "ideal clamp" is introduced (although also used in
Introduction to Polymer Viscoelasticity, 3rd edition), which has properties that
can only be approached by real mechanical devices (some of which are
pictured). The complexities of "simple" shear and plane stress are worked over
at some length
PREFACE vii
After a discussion of stress, one expects a discussion of strain; however,
strain is left to the last possible moment (Chapter 8). There is no question that
the transition from linear viscoelasticity to finite strain of fluids is difficult to
learn and difficult to teach. Rather than jumping right into this treacherous
topic, an entire chapter (Chapter 3) is devoted to a discussion of the rate-of-
deformation tensor and the magnitude of the rate. Rate, in spite of being a
derivative, is much simpler to understand and use. Certainly, most applications
of nonlinear polymer rheology in industry will involve analysis using
Newtonian (Chapter 4) and generalized Newtonian fluid models (Chapter 5) in
steady or quasi-steady flows.
Moving to strain (Chapter 8) involves heavy use of the concept of
displacement relative to the present configuration and begins with the
infinitesimal strain tensor. In my experience, getting students to accept the
concept of the present position as the reference condition is certainly one of the
biggest hurdles of rheology instruction. This is especially true for those who
have had instruction in linear viscoelasticity, but they can see that it all works
out when the strain is small. However, the morphing of the Boltzmann
superposition principle into finite-strain integral models still produces
frustration and doubt, but this is reduced if time is spent explaining why the
initial condition no longer makes sense for finite strains of fluids. However, the
doubt always returns when it is explained that strains before the sample is
touched (/' < 0) must contribute to the stress if the present position is the
reference. Equally difficult is accepting the fact that the strain at f = t is zero

when clearly the sample has been deformed. Several examples are provided to
help with this admittedly complicated topic. Important also is pointing out,
with more examples, why using the undeformed configuration as the reference
leads to problems.
Most polymer science students are interested in using rheology as an
analytical tool. They usually are knowledgeable about techniques based on
linear viscoelasticity, but are much less familiar with steady-flow techniques
and usually unaware of the attractions and difficulties of extensional and
transient flows. Thus, Chapter 7 goes through how to find viscosity and normal
stresses using rotational geometries, and some of the issues faced with these
devices. Capillary viscometry is also addressed, along with its many problems.
The section on extensional flows starts out with a brief history of this
challenging measurement and moves to a description of the broad array of
techniques that have been introduced.
The connection of molecular structure to rheological response is an
important aspect of polymer rheology. In fact, it is so important that it is not
confined to one chapter (Chapter 9), but is spread throughout so the connection
viii PREFACE
with each rheological function is clear. While the basic ideas of molecular
motion are discussed in Chapter 9, there is no attempt to go into the details of,
for example, the exciting and rapidly growing field of molecular dynamics of
chain structures.
Oddly enough, very little space is devoted to the application of rheology to
polymer processing (Chapter 10). Polymer processing is an exceedingly
diverse and complicated subject requiring techniques that are far removed from
the interests and abilities of most polymer science students. Instead, the
chapter is devoted to explanation of the lubrication approximation, and its
application to the simple analysis of flows involved in common laboratory
processing methods. The goal of most laboratory processing is to make a
sample that can be analyzed or tested. Typical tests include infrared analysis,
contact angle, x-ray diffraction, microscopy, dielectric analysis and light
mechanical testing. Processing methods may be limited to solution casting,
spin casting and compression molding because the sample mass may often be
less than a gram. These methods are discussed and examined, with analyses
often confined to Newtonian and generalized Newtonian fluids. The goal here
is to provide an understanding of how rheology can help the student adjust their
sample preparation methods and conditions to avoid problems with the
fabricated object in subsequent characterization.
Most of the polymer students end up in an industrial position, and many call
about rheology problems they are experiencing on the job. Their "rheometers"
are sometimes rudimentary quality-control devices that they have never seen

before they joined the company. For this reason, Chapter 11 deals with typical
quality-control measurements such as melt-flow index, Mooney viscosity and
Rossi-Peakes flow. The goal is not only to define and describe these
measurements to the students, but to convince them that such methods have
very good reasons for existing.
Typically a semester rheology course runs out of time before the last chapter
which deals with the influence of polymer modification of rheological
properties. Again, the idea is to describe some of the key aspects of this very
broad area, which is covered thoroughly in texts such as The Structure and
Rheology of Complex Fluids (R. G. Larson, Oxford, 1998). Short sections on
fillers, crosslinking, liquid crystallinity, and physical intermolecular
interactions are included. The goal is to inform the student that rheological
properties are very sensitive to these structural variables.
Problems are a key feature of the book. Every chapter has at least ten and
some well over twenty problems at the end of the chapter, in addition to worked
sample problems with the text. As with Introduction to Polymer
Viscoelasticity, many of the problems have solutions in the final section of the