Nonlinear Polymer Rheology
Macroscopic phenomenology and molecular foundation
Shi-Qing Wang
Department of Polymer Science
University of Akron
http://www3.uakron.edu/rheology/
updated Dec., 2014
PREFACE
0. Introduction
The objective of polymer rheology is two-fold, first as a characterization tool, usually
involving linear viscoelastic measurements, and second as a core science concerning
nonlinear mechanical responses to deformation and flow of polymeric materials. As a
characterization method, it relies heavily on molecular modeling. As the foundation for
fluid mechanics of polymeric liquids, nonlinear rheology needs to be developed adequately
to provide guidance for polymer processing. This book focuses on the latter.
PART ONE
LINEAR VISCOELASTICITY AND EXPERIMENTAL METHODS
1. Phenomenological linear viscoelasticity (LVE)
1.1 Basic modes of deformation
1.1.1 Startup deformation
1.1.2 Step strain
1.1.3 Dynamic or oscillatory shear
1.2 Linear responses
1.2.1 Elastic Hookean solids
1.2.2 Viscous Newtonian liquids
1.2.3Viscoelastic responses
i. Boltzmann superposition principle of linear response
ii. General materials functions in oscillatory shear
iii. Stress relaxation from step strain or steady-state shear
1.2.4 Maxwell model for viscoelastic liquids
i. Stress relaxation from step strain
ii. Startup deformation
iii. Oscillatory (dynamic) shear
1.2.5 General features of viscoelastic liquids
i. Generalized Maxwell model
ii. Lack of linear response in step strain: a real dilemma
1.2.6 Kelvin-Voigt model for viscoelastic solids
i. Creep experiment
ii. Strain recovery in stress-free state
1.2.7 Weissenberg number and voluntary yielding
1.3 Classical rubber elasticity theory
1.3.1 Chain conformational entropy and elastic force
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1.3.2 Network elasticity and stress-strain relation
1.3.3 Alternative expression in terms of retraction force and areal crosslinking density
This Chapter 1 provides a basic discussion of how mechanical properties of viscoelastic
materials can be characterized experimentally and phenomenologically and ends with an
account of rubber elasticity theory, which is being brought to such a prominent position
because it appears nonlinear rheology can trace its physical origin to chain elasticity.
2. Molecular characterization in LVE regime
2.1 Dilute limit
2.1.1 Viscosity of Einstein suspensions
2.1.2 Kirkwood-Riseman model
2.1.3 Zimm model
2.1.4 Rouse model
i. Stokes law of friction
ii. Brownian motion and Einstein-Stokes formula for solid particles
iii. Equations of motion and Rouse relaxation time R
iv. Rouse dynamics for unentangled melts
2.1.5 Relationship between diffusion and relaxation
2.2 Entangled state
2.2.1 Phenomenological evidence of chain entanglement
i. Elastic recovery phenomenon
ii. Rubbery plateau in creep compliance
iii. Stress relaxation
2.2.2 Transient network models
2.2.3 Models depicting onset of chain entanglement
i. Packing model
ii. Percolation model
2.3 Molecular-level descriptions of entanglement dynamics
2.3.1 Reptation idea of de Gennes
2.3.2 Tube model of Doi and Edwards
2.3.3 Polymer mode coupling theory of Schweizer
2.3.4 Self diffusion vs. zero-shear viscosity
2.3.5 Entangled Solutions
2.4 Temperature dependence
2.4.1 Time-temperature equivalence
2.4.2 Thermo-rheological complexity
2.4.3 Segmental friction and terminal relaxation dynamics
This Chapter 2 offers the important and elementary information about all classical
molecular theories of polymer dynamics that form the bedrock to describe anything
viscoelastic about polymeric liquids. Without this chapter nothing can be discussed
about our latest understanding of polymer rheology. Nevertheless, the treatment is
somewhat brief here because the content may be found in many books mentioned above,
with the except of 2.2.3 that has not been incorporated into any textbook to the best of my
knowledge although it should be regarded as elementary and standard materials for any
book that discusses chain entanglement. For completeness, a brief exposure to the
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temperature dependence of polymer dynamics is given at the end in 2.4, which is a
subject too important to omit.
3. Experimental Methods
3.1 Shear rheometry
3.1.1 Linear displacement
3.1.2 Rotational motion
i. Parallel disks
ii. Cone-plate
iii. Couette
3.1.3 Pressure-driven apparatus
i. Capillary die
ii. Channel slit
3.2 Extensional rheometry
3.2.1 Basic definitions of strain and stress
3.2.2 Three types of device
i. Instron type stretcher
ii. Extender at fixed length
iii. Filament stretching rheometer
3.3 Rheo-optical (in situ) methods
3.3.1 Flow birefringence
i. Stress optical rule (SOR)
ii. Breakdown of SOR
3.3.2 Scattering (X-ray, light, neutron)
3.3.3 Spectroscopy (NMR, fluorescence, IR, Raman, dielectric)
3.4 Advanced rheometric methods
3.4.1 Superposition of small amplitude oscillatory shear and small step strain during
steady continuous shear
i. SAOS in steady shear
ii. Small step strain in steady shear
3.4.2 Rate or stress switching multi-step platform
i. Rate jump during startup deformation and after cessation of deformation
ii. Elastic recovery at various stages of startup deformation and during relaxation
odani@kit.ac.jp
Chapter 3 covers experimental methods in terms of rheometric instruments, their
improvement or modification. We have equal coverage on extensional rheometry in
terms of its complementary role in the measurement science of rheometry. A second part
discussed versatile protocols to program rheometry to gain additional rheological
information.
4. Characterization of deformation field
4.1 Basic features in simple shear
4.1.1 Working principle for displacement-controlled rheometry: homogeneous shear
4.1.2 Stress-controlled shear
4.2 Yield stress in Bingham type (yield-stress) fluids
4.3 Cases of homogeneous shear
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4.4 Particle tracking velocimetry (PTV)
4.4.1 Simple shear
i. Motions in XZ plane
ii. Imaging in XY plane
4.4.2 Channel flow
4.4.3 Other geometries
4.5 Single molecule imaging velocimetry (SMIV)
4.5.1 Simple shear
4.5.2 Channel flow
4.6 Other methods
Chapter 4 begins by reviewing the basic principle that leads to the basic premise for
rheometry – homogeneous shear. It discusses when this premise is valid. We then select a
class of material behaviors known as yield stress phenomena to discuss critically whether
the traditional depiction is adequate. The discussion of shear homogeneity sets the stage for
an introduction of the particle-tracking velocimetric (PTV) method as well as the nextgeneration single-molecule imaging velocimetry.
5. Improved and other rheometric apparatuses
5.1 Linearly displaced co-cylinder for simple shear
5.2 Cone-partitioned plate for rotational shear
5.3 Other forms of high shear deformation
i. Deformation at die entry
ii. One-dimensional squeezing
iii. Planar extension
Rheometric measurements of nonlinear responses from entangled polymers are inherently
difficult due to a number of factors. It remains challenging to obtain reliable experimental
information. Chapter 5 first indicates how edge instabilities can be deferred and minimized
and then mentions the intrinsic problem of interfacial failure that can be due to true wall
slip, leading to the topic of the next chapter.
PART TWO YIELDING – PRIMARY NONLINEAR RESPONSES TO ONGOING
DEFORMATION
6. Wall slip – Interfacial yielding
6.1 Basic notion of wall slip in steady shear
6.1.1 Slip velocity Vs and Navier-de Gennes extrapolation length b
6.1.2 Correction of shear field due to wall slip
6.1.3 Origin of wall slip and condition to achieve complete slip
6.2 Stick-slip transition (in stress-controlled mode)
6.2.1 Stick-slip transition in capillary extrusion
i. Analytical description
ii. Experimental data
6.2.2 Stick-slip transition in simple shear (co-cylinder)
6.2.3 Maximum slip velocity Vs(max) for different polymer melts
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6.3 Wall slip during startup shear - Interfacial yielding
6.3.1 Theoretical discussions
6.3.2 Experimental data
6.4 Relationship between slip and bulk shear deformation
6.4.1 Transition from wall slip to bulk nonlinear response
6.4.2 Experimental evidence of stress plateau associated with wall slip
i. Case based on entangled DNA solution
ii. Entangled PB solution in small gap distance H ~ 50 m
6.4.3 Influence of shear thinning on slip
6.5 Molecular evidence of disentanglement during wall slip
6.6 Uncertainty in boundary condition
i. Oscillation between entanglement and disentanglement
ii. Polymer desorption on weak surfaces: permanent slip
iii. Two more flow oscillations
6.7 Theoretical accounts
6.7.1 Small surface coverage - Brochard-de Gennes theory
6.7.2 Saturated adsorption – disentanglement picture
6.7.3 Origin of stick-slip transition: interfacial cohesive breakdown
Wall slip is the leading form of inhomogeneous shear response and is the subject of Chapter
6. The topic occupies a central position in the book. Everything we now know about
nonlinear rheology of entangled polymers can be traced back to our understanding of wall
slip. In particular, it not only emphasize de Gennes’s pioneering contribution to our
understanding of wall slip but also shows that our understanding actually has extended
beyond de Gennes’s primitive analysis. Today, we call wall slip produced in startup shear
“interfacial yielding” because we have now been able to unify the depiction of shear
induced yielding.
We will provide the first treatment to account for the transformation from no slip to slip. All
previous studies on wall slip only concerns with wall slip in steady state. Although wall slip
is an essential subject in rheology, existing books have hardly treated the topic in any
adequate detail.
7. Yielding during startup deformation: from elastic deformation to flow
7.1 Voluntary yielding at Wi < 1 and steady shear thinning
7.1.1 Elastic deformation and yielding in terminal flow
7.1.2 Steady shear rheology: shear thinning
7.2 Stress overshoot in fast startup shear
7.2.1 Scaling characteristics of the overshoot
i. Viscoelastic regime
ii. Elastic deformation regime
7.2.2 Elastic recoil from startup shear: evidence of yielding
7.3 Nature of steady shear
7.3.1 Superposition of small-amplitude oscillatory shear onto steady state shear
7.3.2 Two other methods to probe steady shear
7.4 From terminal flow to fast flow under creep: entanglement-disentanglement transition
7.5 Yielding in startup uniaxial extension
7.5.1 Myth with Considère criterion
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7.5.2 Tensile force (engineering stress) vs. (true) stress
7.5.3 Tensile force maximum
i. Terminal flow
ii. Significance of engineering stress engr
iii. Yielding for Wi > 1
vi. Elastic recoil
This Chapter 7 is in some sense the soul of the book. Yielding is what must occur in
entangled polymer upon startup deformation in both simple and uniaxial extension. This is
a most difficult concept for some to accept because of a wide-spread confusion (thanks to
Barnes and Walters for their discussion of yield-stress concept) that yielding was only to be
used to describe mechanical behavior of solids.
8. Strain hardening in extension
8.1 Conceptual pictures
8.2 Origin of "strain hardening" in uniaxial extension
8.1.1 Geometric condensation of entanglement network
8.1.2 False strain hardening in LCB polymers such as LDPE
8.1.3 True strain hardening: finite extensibility limit and non-Gaussian stretching
8.2 Kinematic difference between extension and shear
Strain hardening has been misused and over-used to describe the transient responses of
entangled melts in extension. Chapter 8 shows that the real physics concerns about when
and how yielding via chain disentanglement takes place. We also show an example of how
true strain hardening is made possible in simple shear by LCB.
9. Shear banding in startup and oscillatory shear: PTV observations
9.1 Shear banding after overshoot in startup shear
9.1.1 Brief historical background
9.1.2. Influential factors
i. Sample requirements
ii. Controlling slip velocity
iii. Edge effects
iv. Criterion for shear strain localization
9.1.3 Shear banding in conventional rheometric device
i. Shear banding of entangled DNA solutions
ii. Shear banding of entangled 1,4-polybutadiene solutions
9.1.4 From wall slip to shear banding in small-gap distance
9.2 Overcoming slip during startup shear
9.3 Shear banding in LAOS
Shear banding is of course a key observation from applications of PTV and is extensively
discussed in this Chapter 9. It is a key chapter of the book because the strain localization
has fundamentally altered our previous understanding of polymer rheology. Serious
questions to ask here include whether shear banding is a steady state property or only a
metastable character.
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10. Pressure-driven deformation in capillary extrusion, squeezing, planar extension
10.1 Capillary rheometry in rate-controlled mode
10.1.1 Analogy with simple shear
10.1.2 Yielding at entrance
10.1.3 Collapse of continuum mechanical description
10.2 Particle-tracking velocimetric observations
10.2.1 Vortex formation
10.2.2 Stagnation at corners – internal slip
10.3 Squeezing deformation
10.2.1 Interfacial failure
10.2.2 Internal strain localization through EDT under constant pressure
10.4 Planar extension
Chapter 10 shows the some valuable fruit from the studies of simple shear and uniaxial
extension behavior. We extend our PTV method to observe the real phenomenon under real
processing conditions. The findings are eye opening to say the least. In particular, we truly
understand why polymers such as LDPE show vortex flow in the die entrance whereas
linear polymer melts show stagnant corners. This all has to do with whether the polymer is
capable of undergoing yielding for a limited amount of straining or whether there is a
sufficient inherent extrapolation length scale in the system that shares the same definition as
the slip length rediscovered by de Gennes since Navier introduced it in the 19th century.
11. Different modes of structural failure during startup uniaxial extension
11.1 Tensile-like failure at low rates
11.2 Shear yielding and necking-like strain localization
11.3 Rupture without crosslinking: where is disentanglement?
11.4 Strain localization vs. steady-flow: SER vs. FSR
Chapter 11 summarizes how the different ways to undergo strain localization during startup
extension can be understood in terms of how disentanglement may take place in extension
differently from shear.
PART THREE DECOHESION AND ELASTIC YIELDING AFTER LARGE
DEFORMATION
12. Elastic yielding in stepwise simple shear
12.1 Strain softening after large step strain
12.1.1 Phenomenology
12.1.2 Tube model interpretation
i. Normal DE behavior
ii. Type C as elastic instability
12.2 PTV revelation of non-quiescent relaxation: localized elastic yielding
12.2.1 Non-quiescent relaxation in polymer solutions
i. Elastic yielding in polybutadiene solutions
ii. Suppression of breakup by reduction in b
iii. Non-quiescent relaxation in polystyrene solutions
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iv. Strain localization in absence of edge instability
12.2.2 Non-quiescent relaxation in polymer melts
i. Induction time and molecular weight dependence
ii. Severe shear banding before shear cessation and immediate breakup
iii. Rate dependence of elastic breakup
iv. Unconventional "step strain" at WiR < 1
12.3 Quiescent elastic yielding
12.3.1 General comments
12.3.2 Condition for uniform yielding and quiescent stress relaxation
12.3.3 Homogeneous elastic yielding probed by sequential shearing
12.4 Arrested wall slip: elastic yielding at interfaces
12.4.1 Entangled solutions
12.4.2 Entangled melts
Elastic yielding is the most profound phenomenon discovered with PTV. In Chapter 12, we
describe elastic yielding after stepwise simple shear, where the first example is the arrested
wall slip, i.e., elastic yielding at polymer-wall interfaces. It is a “self destruction” process in
the sense that the decohesion occurs upon cessation of external deformation. Elastic yielding is
a unique example of how chain elasticity plays an essential role in polymer dynamics. This
chapter along with Chapter 7 forms the foundation of nonlinear polymer rheology.
13. Elastic breakup in stepwise uniaxial extension
13.1 Rupture-like failure during relaxation (WiR < 1)
13.2 Shear-yielding induced failure upon fast large stepwise extension (WiR > 1)
13.3 Nature of the elastic breakup probed by IR measurements
13.4 Primitive phenomenological explanations
In Chapter 13, we describe elastic yielding after stepwise uniaxial extension, analogous to the
behavior observed after step shear.
14. Finite cohesion and role of chain architecture
14.1 Cohesive strength of entanglement network
14.2 Enhancing cohesion barrier with long-chain branching to prevent structural breakup
This Chapter 14 emphasizes that long-chain-branching strongly affect nonlinear rheological
behavior by strengthening the entanglement network.
PART FOUR EMERGING CONCEPTUAL FRAMEWORK
15. Homogeneous entanglement
15.1 What is chain entanglement?
15.2 When, how and why disentanglement occurs
15.3 Criterion for homogeneous shear
15.4 Constitutive non-monotonicity
15.5 Metastable nature of shear banding
We begin with a definition of chain entanglement: a dynamic concept originating from
chain uncrossability (excluded volume and chain connectivity). The assumption of
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homogeneous state of entanglement is true only under special rheological conditions or
when the response is still largely elastic, unless the system is weakly entangled. Chapter 15
emphasizes that the central issue is whether non-affine deformation could occur without
strain localization.
16. Molecular network as a conceptual foundation
16.1 Introduction: the tube model and its predictions
16.1.1 Smooth-out of interchain interactions
16.1.2 Barrier-less chain retraction
16.1.3 Non-monotonicity inherent from the simplification
i. Stress overshoot upon startup shear
ii. Strain softening: damping function for stress relaxation
iii. Excessive shear thinning: proposal of shear banding base on the symptom of
shear stress maximum in constitutive curve
iv. Anticipation of necking based on Considère criterion
16.1.4 Inconsistencies with the tube model
16.2 Definition of three forces in entanglement network
16.2.1 Intra-chain elastic retraction force
16.2.2 Intermolecular gripping force (IGF)
16.2.3 Entanglement (cohesion) force arising from entropic barrier: finite cohesion
16.3 Overcoming finite cohesion after stepwise deformation: Quiescent or not
16.3.1 Non-quiescence from severe elastic yielding
i. With WiR >1
ii. With WiR < 1
16.3.2 Homogeneous elastic yielding: quiescent relaxation
16.4 Microscopic yielding during rate-controlled startup deformation
16.4.1 Chain disentanglement at WiR < 1
16.4.2 Molecular force imbalance and scaling at WiR > 1
16.4.3 Yielding as a universal response
16.5 Interfacial yielding by disentanglement
16.6 Decohesion in startup creep: entanglement-disentanglement transition
16.7 Emerging microscopic theory of Sussman and Schweizer
16.8 New tests to reveal the nature of polymer deformation
16.8.1 Molecular dynamics simulations
16.8.2 Small angle neutron scattering measurements
After becoming extensively informed by the emerging experimental observations, it is
natural to ask why what would be a coherent conceptual framework. After a detailed
review of the prevailing tube model, Chapter 16 describes alternative phenomenological
and conceptual foundation for nonlinear rheology of entangled polymers where the concept
of yielding and role of intrachain retractive force are emphasized.
17. "Anomalous phenomena"
17.1 Breakdown of time-temperature superposition during transient response: both shear
and extension
17.2 Strain hardening in simple shear of certain polymer solutions
17.3 Lack of universal nonlinear responses: solutions vs. melts
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17.4 Emergence of transient intersegmental elasticity and viscous stress
Chapter 17 discusses several frontier topics to demonstrate that the field is still vibrant and
active investigation although the core aspects of nonlinear rheology of entangled linear
polymeric liquids have become clear.
18. Strain localization and fluid mechanics of polymeric liquids
18.1 Relationship between wall slip and banding: a rheological-state diagram
18.2 Failure of conventional continuum (fluid) mechanics for entangled polymers
18.2.1 Spatial discontinuity arising from discontinuity in state of entanglement
18.2.2 Unconventional constitutive equations containing long-range correlations
18.3 Challenges in polymer processing
18.3.1 Extrudate distortions
i. Sharkskin melt fracture (due to exit boundary discontinuity)
ii. Gross (melt fracture) distortions (due to entry instability)
18.3.2 Optimal extrusion conditions
18.3.3 Melt strength
The ultimate objective of polymer rheology is to provide the basis for fluid mechanics of
polymeric liquids. Chapter 18 describes how and why strain localization is behavior
beyond the conventional scope of any rheology theory. Whether shear homogeneity and a
spatially uniform state of chain entanglement can be assured depends on the ratio of the
extrapolation length b to the minimal dimension of the system (i.e., the thickness H). Any
theoretical account of fluid mechanical behavior of polymers has to involve a dimensionless
parameter of y/b. Most processing instabilities are possible to understand in terms of chain
disentanglement and strain localization, which reflects spatial discontinuity in the state of
entanglement.
A second important component is the discussion on the relation between wall slip and shear
banding, which gives rise to a phase diagram, essentially offering a roadmap to guide us
about what to expect from startup for any linear entangled polymers.
19. Conclusions
19.1 Theoretical challenges
19.2 Experimental difficulties
It appears that the challenge to formulate a realistic theory for nonlinear rheology of
entangled polymer is upon us. The task to overcome the inherent finite-size effects in
rheometry is a daunting one. Experimental visualization of dynamical behavior at the
molecular level remains a dream to be realized. Computer simulations hold the key to our
further understanding in the near future.
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